# Degenerate Whittaker functions for $$\mathrm{Sp}_n(\mathbb{R})$$

Degenerate Whittaker functions for $$\mathrm{Sp}_n(\mathbb{R})$$ Abstract In this paper, we construct Whittaker functions with exponential growth for the degenerate principal series of the symplectic group of genus $$n$$ induced from the Siegel parabolic subgroup. This is achieved by explicitly constructing a certain Goodman–Wallach operator which yields an intertwining map from the degenerate principal series to the space of Whittaker functions, and by evaluating it on weight-$$\ell$$ standard sections. We define a differential operator on such Whittaker functions which can be viewed as generalization of the $$\xi$$-operator on harmonic Maass forms for $$\mathrm{SL}_2(\mathbb{R})$$. 1 Introduction The standard theory of automorphic forms focuses on the spectral decomposition of the space $$L^2(\Gamma\backslash G)$$ where $$G$$ is a connected real semi-simple Lie group and $$\Gamma$$ is a discrete subgroup of finite co-volume. This analysis involves functions of rapid decay, for example, cusp forms, or of moderate growth, for example, Eisenstein series. Classically, in the case of $$G=\mathrm{SL}_2(\mathbb R)$$ and $$\Gamma$$ commensurable to $$\mathrm{SL}_2({\mathbb Z})$$, the Fourier expansions of such functions involve the solution of the Whittaker ordinary differential equation that decays exponentially at infinity; this solution is uniquely characterized by this decay. In general, the uniqueness of the Whittaker model and the associated smooth Whittaker functional, Jacquet’s functional, plays a fundamental role and is the subject of a vast literature. Other Whittaker functionals and the associated “bad” Whittaker functions, which grow exponentially at infinity, have played a less prominent role in the theory of automorphic forms. An important exception to this is the work of Miatello and Wallach [21] where, for $$G$$ semi-simple of split rank $$1$$, a theory of Poincaré series constructed from such Whittaker functions is developed. More recently, in the case of $$G=\mathrm{SL}_2(\mathbb R)$$ and $$\Gamma$$ a congruence subgroup of $$\mathrm{SL}_2({\mathbb Z})$$, the space of (vector-valued) weak Maass forms—Maass forms that are allowed to grow exponentially at the cusps—and its subspace of harmonic weak Maass forms, analytic functions annihilated by the weight $$k$$ Laplacian, have been shown to have interesting and important arithmetic applications [7, 8, 10]. For example, the weakly holomorphic modular forms are the input for Borcherds celebrated construction of meromorphic modular forms with product formulas [2] and, more generally, harmonic Maass forms can be used to construct Arakelov-type Green functions for special divisors on orthogonal and unitary Shimura varieties [6, 8]. The harmonic weak Maass forms of negative (or low) weight play a role in the theory of mock modular forms and their relatives [4, 28, 29]. In particular, they are linked to holomorphic modular forms of a complementary positive weight by means of the $$\xi$$-operator introduced in [5, 6]. There are serious obstacles to extending these results to groups of higher rank. For example, the Koecher principle asserts that, for $$\Gamma$$ irreducible in $$G$$ of Hermitian type of reduced rank $${>}1$$, any holomorphic modular form on the associated bounded symmetric domain “extends holomorphically” to the cusps, that is, the notions of holomorphic modular form and weakly holomorphic modular form coincide. More generally, Miatello and Wallach [21, Section 5] conjecture that the same phenomenon occurs for general automorphic forms. Specifically, they conjecture that, for $$\Gamma$$ irreducible in $$G$$ of real reduced rank $${>}1$$, a smooth function $$f$$ on $$\Gamma\backslash G$$ that is $$K$$-finite and an eigenfunction of the center of the universal enveloping algebra of $$\mathfrak g =\mathrm{Lie}(G)_{\mathbb C}$$ is automatically of moderate growth. They prove this conjecture in the case of $$SO(n,1)$$ over a totally real field. However, in [5], the first author has shown that for $$G=\mathrm{SL}_2(\mathbb R)^d$$ and $$\Gamma$$ an arithmetic subgroup $$\Gamma$$ of $$\mathrm{SL}_2(O_{k})$$, where $$O_{k}$$ is the ring of integers in a totally real field $${k}$$ with $$|{k}:{\mathbb Q}|=d>1$$, it is possible to replace the non-existent space of harmonic weak Maass forms with a certain space of Whittaker functions. These Whittaker functions are invariant only under the unipotent subgroup $$\Gamma_\infty^u$$ of $$\Gamma_\infty$$ and the associated Poincaré series do not converge. Nevertheless, it is shown in [5] that they are linked to holomorphic Hilbert modular cusps forms via a $$\xi$$-operator and provide an adequate input for a Borcherds type construction. This suggests that it would be fruitful to consider analogous Whittaker functions for more general groups. The goal of this paper is to construct “bad” Whittaker functions for the degenerate principal series $$I(s)$$ induced from a character of the Levi factor $$M=\mathrm{GL}_n(\mathbb R)$$ of the Siegel parabolic $$P=MN$$ of $$G = \mathrm {Sp}_n(\mathbb R)$$. For $$s\in{\mathbb C}$$, let $$I(s) = I(s,\chi)$$ be the space of smooth of $$K$$-finite functions $$\phi$$ on $$G$$ such that   ϕ(n(b)m(a)g)=χ(deta)|det(a)|s+ρϕ(g),m(a)=(ata−1), n(b)=(1b1), (1.1)$$a\in \mathrm{GL}_n(\mathbb R)$$, $$b\in S:=\mathrm{Sym}_n(\mathbb R)$$, $$\chi(t) = \mathrm{sgn}(t)^\nu$$, $$\nu=0,$$$$1$$, $$\rho = \frac12(n+1)$$. Then $$I(s)$$ is a $$(\mathfrak g,K)$$-module, where $$\mathfrak g = \mathrm{Lie}(G)_{\mathbb C}$$ and $$K\simeq U(n)$$, $$k\mapsto\mathbf k$$. For $$T\in \mathrm{Sym}_n(\mathbb R)$$ with $$\det(T)\ne0$$, an algebraic Whittaker functional of type $$T$$ is an element $$\boldsymbol{\omega}^{\mathrm{T}}\in I(s)^* = \mathrm{Hom}_{\mathbb C}(I(s), {\mathbb C})$$ such that   ωT(n(X)ϕ)=2πitr(TX)ωT(ϕ),n(X)=(X0)∈g,X∈SC=Symn(C). (1.2) Such a functional determines a $$(\mathfrak g,K)$$-intertwining map   ωT:I(s)⟶WT(G),ωT(ϕ)(g)=ωT(π(g)ϕ), (1.3) where $$\mathcal W^{\mathrm{T}}(G)$$ is the space of smooth functions $$f$$ on $$G$$ such that   f(n(b)g)=e(tr(Tb))f(g). (1.4) Here $${e}(t) = \mathrm{e}^{2\pi i t}$$. The resulting generalized Whittaker functions are right $$K$$-finite and real analytic on $$G$$. Conversely, such an intertwining map (1.3) gives rise to a Whittaker functional $$\boldsymbol{\omega}^{\mathrm{T}}(\phi) = \omega^{\mathrm{T}}(\phi)(e)$$. One intertwining map is given by the integral   WT(g,s;ϕ)=∫Sϕ(w_n(b)g)e(−tr(Tb))db,w_=(1−1), which converges absolutely for $$\mathrm{Re}(s)>\rho$$ and has a meromorphic analytic continuation in $$s$$. For example, for $$\phi = \phi_{s,\ell}$$, the (unique) function in $$I(s)$$ such that $$\phi_{s,\ell}(k) = \det(\mathbf k)^\ell$$,   WT(n(b)m(a)k,s;ϕs,ℓ)=χ(det(a))|det(a)|s+ρe(tr(Tb))det(k)ℓξ(v,T;α′,β′), (1.5) where   ξ(v,T;α′,β′)=∫Sdet(b+iv)−α′det(b−iv)−β′e(−tr(Tb))db, (1.6) with $$v=a{}^ta$$, $$\alpha'= \frac12(s+\rho+\ell)$$ and $$\beta' = \frac12(s+\rho-\ell)$$, is the confluent hypergeometric function of matrix argument studied by Shimura [23]. If $$\epsilon\,T>0$$ with $$\epsilon=\pm1$$, then (1.6) can be written as   i−nℓ2−n(ρ−1)(2π)n(s+ρ)Γn(α)−1Γn(β)−1|det(T)|s ×e−2πϵtr(Tv)∫t>0e−2πtr(cvtct)det(t)α−ρdet(t+1)β−ρdt, (1.7) where $$\alpha=\frac12(s+\rho-\epsilon\ell)$$ and $$\beta=\frac12(s+\rho+\epsilon\ell)$$ and $$\epsilon\,T = c\,{}^tc$$. For notation not explained here see Section 2.1 and the Appendix. The Whittaker function (1.5), which decays exponentially as the trace of $$v$$ goes to infinity, plays a key role in many applications. The corresponding Whittaker functional on $$I(s)$$ is characterized, among all algebraic Whittaker functionals, by the fact that it extends to a continuous functional on the space $$I^{\mathrm{ sm}}(s)$$ of smooth functions on $$G$$ satisfying (1.1). In the general theory, such Jacquet functionals and the resulting good Whittaker functions have been studied very extensively, cf. [24] and the literature discussed there. To construct other Whittaker functionals, we apply Goodman–Wallach operators to conical vectors in $$I(s)^*$$. Such conical vectors correspond to embeddings into induced representations. The two relevant ones in our situation are given by   c1(ϕ)=ϕ(e), (1.8) and, for $$\mathrm{Re}(s)>\rho$$,   cw_(ϕ)=(A(s,w_)ϕ)(e)=∫Sϕ(w_n(b))db, (1.9) where $$A(s,\underline{w}):I(s) \longrightarrow I(-s)$$ is the intertwining operator defined by (3.3), with corresponding embeddings the identity map and $$A(s,\underline{w})$$, respectively. Matumoto’s generalization [20] of the results of [12] applies in our situation. Let   N¯={n−(x)∣x∈S},n−(x)=(1x1), (1.10) and let $$\bar{\mathfrak n} = \mathrm{Lie}(\bar N)_{\mathbb C}$$. Since $$\bar{\mathfrak n}$$ is abelian, $$U(\bar{\mathfrak n}) = S(\bar{\mathfrak n})$$, and the completion   S(n¯)[n¯]=lim⟵rS(n¯)/n¯rS(n¯) (1.11) is the ring of formal power series in elements of $$\bar{\mathfrak n}$$. The action of $$U(\bar{\mathfrak n})$$ on $$I(s)^*$$ extends to an action of $$S(\bar{\mathfrak n})_{[\bar{\mathfrak n}]}$$. (Here it is essential that we have taken the dual $$I (s)^*$$ of the $$K$$-finite vectors $$I (s)$$.) Then, by the results of [20], there are elements   gwsT,gw−sT∈S(n¯)[n¯] such that   ω1T:=gwsT⋅c1,ωw_T:=gw−sT⋅cw_ are Whittaker functionals of type $$T$$. (Here it is essential that we have taken the dual $$I(s)^*$$ of the $$K$$-finite vectors $$I(s)$$.) Following [12] one realizes the representation $$I(s)$$ on a space of functions on $$\bar N$$, where the Goodman–Wallach operator is given by a differential operator of infinite order. Taking advantage of the fact that $$\bar N$$ is abelian and passing to the Fourier transform, this operator is realized as multiplication by an analytic function. In the case of $$\mathrm{SL}_2(\mathbb R)$$, this function is given explicitly in the introduction to [12], where the corresponding formal power series in $$S(\bar{\mathfrak n})_{[\bar{\mathfrak n}]}$$ is determined by a simple recursion relation. For $$n\ge2$$, it does not seem feasible to apply this method, and, as far as we could see, explicit formulas for these kernel functions do not exist in the literature. Our first main result is that, in analogy to the case of $$\mathrm{SL}_2(\mathbb R)$$ given in [12], the kernel function for the Goodman–Wallach operator $$\mathrm{gw}_s^{\mathrm{T}}$$ for any $$n$$ is given explicitly by a Bessel function, now of a matrix argument. We now describe the result in more detail. For $$\phi\in I(s)$$, define a function $$\Psi(\phi)$$ on $$\bar N$$ by   Ψ(x;ϕ)=ϕ(n−(x)), (1.12) and let   Ψ^(y;ϕ)=∫Se(tr(xy))Ψ(x;ϕ)dx (1.13) be its Fourier transform. In this model of $$I(s)$$, the conical functional $$c_1$$ is given by   c1(ϕ)=Ψ(0;ϕ)=∫SΨ^(y;ϕ)dy. (1.14) To define the relevant hypergeometric function of matrix argument, we use the notation and results of [11] and [22]. For $$z$$ and $$w\in S_{\mathbb C}=\mathrm{Sym}_n({\mathbb C})$$ and for $$\mathbf m= (m_1,\ldots,m_n)$$, with integers $$m_j$$ with $$m_1\ge m_2\ge \ldots \ge m_n$$, let $$\Phi_\mathbf{m}(z)$$ be the spherical polynomial, and let $$\Phi_\mathbf{m}(z,w)$$ be its “bi-variant” version. For further explanation and notation, see the Appendix. Following [22], define the hypergeometric function   GWs(z,w):=∑m≥0(−1)|m|dm(s+ρ)m(ρ)mΦm(z,w). (1.15) This is a Bessel function of matrix argument, [11, 13]. Theorem A. The kernel for the Goodman–Wallach operator is given by $$\mathrm{GW}_s(\cdot, 2\pi T)$$. More precisely, for $$\phi\in I(s)$$,   ω1T(ϕ)=∫SGWs(2πy,2πT)Ψ^(y;ϕ)dy. □ Thus, the corresponding Whittaker function is   ω1T(g;ϕ)=ω1T(π(g)ϕ)=∫SGWs(2πy,2πT)Ψ^(y;π(g)ϕ)dy. Our second main result is an evaluation of the Whittaker function for $$\phi= \phi_{s,\ell}$$. Theorem B. For $$\epsilon=\pm1$$, suppose that $$\epsilon\,T\in \mathrm{Sym}_n(\mathbb R)_{>0}$$ and let $$f^{\mathrm{T}}_{s,\ell}(g) = \omega^{\mathrm{T}}_1(\pi(g)\phi_{s,\ell})$$ be the weight $$\ell$$ degenerate Whittaker function. Then, for $$g=n(b)m(a)k$$,   fs,ℓT(g)=χ(det(a))|det(a)|s+ρe(tr(Tb))det(k)ℓ ×2n(s−ρ+1)exp⁡(−2πϵtr(Tv))1F1(α,α+β;4πcvtc), where $$v=a{}^ta$$, $$\alpha= \frac12(s+\rho-\epsilon \ell)$$, $$\beta = \frac12(s+\rho+\epsilon \ell)$$, and $$\epsilon\,T= {}^tc c$$. Here   1F1(α,α+β;z) =∑m≥0(α)m(α+β)mdm(ρ)mΦm(z) =Γn(α+β)Γn(α)Γn(β)∫t>01−t>0etr(zt)det(t)α−ρdet(1−t)β−ρdt is the matrix argument hypergeometric function.□ It is instructive to compare the formula for $$f^{\mathrm{T}}_{s,\ell}$$ and the integral occurring here with the expressions in (1.5) and (1.7) defining the good Whittaker function. Note, for example, that in the case $$n=1$$, the functions   M(a,b;z)=Γ(b)Γ(a)Γ(b)∫01eztta−1(1−t)b−a−1dt and   U(a,b;z)=1Γ(a)∫0∞e−ztta−1(t+1)b−a−1dt, occurring in Theorem A and (1.7), respectively, are a standard basis for the space of solutions for the second-order Kummer equation [1, Chapter 13]. The proof of Theorem B depends on an elaborate calculation which makes essential use of the fact that, up to diagonalization, the orthogonal group of $$T$$ is $$O(n)$$. Thus, at present, we do not have a corresponding evaluation for $$T$$ of arbitrary signature. It is easy to check, cf. Lemma A.2, that for $$a$$, $$b$$, and $$z>0$$ real, with $$a>\rho$$, $$b>\rho$$, and for any $$\eta$$ with $$0<\eta<1$$,   |1F1(a,a+b;z)|≥Cηe(1−η)tr(z). Thus, for $$s$$ real and $$\alpha$$ and $$\beta >\rho$$,   |fs,ℓT(g)|≥Cη′det(v)12(s+ρ)e2π(1−2η)tr(ϵTv). This shows the exponential growth of $$f^{\mathrm{T}}_{s,\ell}$$ as the trace of $$v$$ goes to infinity. Due to its construction via a Whittaker functional, the function $$f^{\mathrm{T}}_{s,\ell}$$ on $$G$$ is an eigenfunction for the center of the universal enveloping algebra with eigencharacter given by the infinitesimal character of the degenerate principal series $$I(s)$$. Of course, for $$n=1$$, the results of Theorems A and B agree with the expressions given in the introduction of [12] in the case of $$\mathrm{SL}_2(\mathbb R)$$. Also, up to an elementary factor, the Whittaker function   fs,ℓT(g)=χ(a)|a|s+ρe(Tb)det(k)ℓ2sexp⁡(−2πϵTv)M(α,α+β;4πϵTv) in the $$n=1$$ case is precisely (the $$m_1$$-component of) the function utilized in the construction of [5], (4.13). Finally, we define an analogue of the $$\xi$$-operator introduced in [5, 6]. Since this operator is a variant of the $$\bar\partial$$-operator, it is best expressed in terms of vector bundles. Write $$\mathfrak H_n$$ for the Siegel upper half plane of genus $$n$$. For a discrete, torsion free subgroup $$\Gamma_0\subset G$$, let $$X=\Gamma_0\backslash \mathfrak H_n$$, and let $$\mathcal E^{a,b}$$ be the bundle of smooth differential forms of type $$(a,b)$$ on $$X$$. For a Hermitian vector bundle $$E$$ on $$X$$, let   ∗¯E:Ea,b⊗E⟶EN−a,N−b⊗E∗ be the Hodge $$*$$-operator [27, Chapter V, Section 2]. Here $$N=n\rho=\dim \mathfrak H_n$$. We include the cases where $$\Gamma_0$$ is $$\Gamma_\infty^u$$ or trivial. Definition. For a Hermitian vector bundle $$E$$ on $$X$$, the $$\xi$$-operator is defined as   ξ=ξE=∗¯E∘∂¯:Γ(X,Ea,b⊗E) ⟶ Γ(X,EN−a,N−b−1⊗E∗). (1.16) For a finite-dimensional representation $$(\sigma,\mathcal V_\sigma)$$ of $$\mathrm{GL}_n({\mathbb C})$$ with an admissible Hermitian norm, there is an associated homogeneous Hermitian vector bundle $$\mathcal L_\sigma$$ on $$X$$. For example, for an integer $$r$$, $$\mathcal L_r:= \mathcal L_{(\det)^{-r}}$$ is the line bundle whose sections correspond to functions on $$\mathfrak H_n$$ that transform like Siegel modular forms of weight $$r$$ with respect to $$\Gamma_0$$. In particular, $$\mathcal E^{N,0} \simeq \mathcal L_{n+1}$$. If $$\mathcal F_\nu$$ is a flat Hermitian bundle associated with a unitary representation $$(\nu, \mathcal F_{\nu})$$ of $$\Gamma_0$$, and $$\kappa$$ is an integer, then sections of $$\mathcal F_\nu\otimes \mathcal E^{0,N-1}\otimes\mathcal L_{n+1-\kappa}$$ can be viewed as $$\mathcal F_\nu$$-valued $$(0,N-1)$$-forms of weight $$n+1-\kappa$$, and $$\xi$$ carries such sections to sections of   Fν∨⊗EN,0⊗Lκ−n−1≃Fν∨⊗Lκ. For $$n=1$$ and $$\Gamma_0$$ a subgroup of finite index in $$\mathrm{SL}_2({\mathbb Z})$$, this reduces to the $$\xi$$-operator defined in [6] where $$(\nu, \mathcal F_\nu)$$ is a finite Weil representation. For simplicity, we now omit the bundle $$\mathcal F_\nu$$. Motivated by the construction of [5], we consider the $$\xi$$-operator applied to a space of Whittaker forms   ξ: W−T(E0,N−1⊗Ln+1−κ)⟶WT(Lκ). Here we take $$T\in \mathrm{Sym}_n({\mathbb Z})_{>0}^\vee$$ and $$\Gamma_0 = \Gamma^u_\infty=\mathrm {Sp}_n({\mathbb Z})\cap N$$. Lifted to $$G$$, this amounts to   ξ: [W−T(G)⊗∧N−1(p−∗)⊗C(κ−n−1) ]K⟶[WT(G)⊗C(−κ) ]K. A family of Whittaker forms in the space on the left here can be constructed by means of our Whittaker functional. Note that   σ∨=∧N−1(p−∗)⊗C(κ−n−1) is an irreducible representation of $$K$$. Since the $$K$$-types of $$I(s)$$ occur with multiplicity 1, we see that the space   [ I(s)⊗∧N−1(p−∗)⊗C(κ−n−1) ]K has dimension $$1$$. Let $$\phi_{s,\sigma}$$ be a basis vector. We then obtain a diagram   $$\$$ (1.17) and define the Whittaker form   fs,σ −T:=ω1−T(ϕs,σ). We finally determine the behavior of this family of Whittaker forms under the $$\xi$$-operator. Theorem C. The Whittaker form $$\boldsymbol{f}_{s,\sigma}^{\ -T}$$ has the following properties. (i) The form $$\boldsymbol{f}_{s,\sigma}^{\ -T}$$ is an eigenfunction of the center of the universal enveloping algebra of $$\mathfrak g$$ with eigencharacter the infinitesimal character of $$I(s)$$. In particular, for the Casimir operator $$C$$,   C⋅fs,σ −T=18(s+ρ)(s−ρ)fs,σ −T. (ii) For the $$\xi$$-operator,   ξ(fs,σ −T)=n(s¯−ρ+κ)fs,−κ−T¯, where   fs,−κ−T=ω1−T(ϕs,−κ). (iii) At $$s=s_0=\kappa-\rho$$,   ξ(fs0,σ −T)(g)=c(n,s0)WκT(g), where   WκT(n(b)m(a)k)=det(k)κdet(a)κe(tr(Tτ))=j(g,i)−κqT, (1.18) and $$c(n,s_0) = 2^{n(\kappa-2\rho+1)}$$. Here $$\tau = b+ \mathrm{i} a{}^ta$$ and $$q^{\mathrm{T}} = e(\mathrm{tr}(T\tau))$$.□ Thus, the $$\xi$$-operator carries $$\boldsymbol{f}^{\ -T}_{s_0,\sigma}$$ to the standard holomorphic Whittaker function of weight $$\kappa$$. Here note that, for $$\Gamma = \mathrm {Sp}_n({\mathbb Z})$$ and for $$\kappa>2n$$, the Poincaré series defined by   PΓ(WκT)(g)=∑γ∈Γ∞u∖ΓWκT(γg) is termwise absolutely convergent and defines a cusp form of weight $$\kappa$$. We define the global $$\xi$$-operator   ξΓ=PΓ∘ξ :Hn+1−κ(G)⟶Sκ(Γ), where $$\mathbb H_{n+1-\kappa}$$ is the subspace of   [ C∞(Γ∞u∖G)⊗∧N−1(p−∗) ]K spanned by the $$\boldsymbol{f}_{s_0,\sigma}^{\ -T}$$ for $$T\in \mathrm{Sym}_n({\mathbb Z})^\vee_{>0}$$. Since the Poincaré series spans $$S_\kappa(\Gamma)$$, the conjugate linear map $$\xi_\Gamma$$ is surjective and we obtain a kind of “resolution”   ker⁡(ξΓ)⟶Hn+1−κ⟶Sκ(Γ) (1.19) of the space of cusp forms which, for $$n>1$$, might be viewed as a kind of replacement for harmonic weak Maass forms in the higher genus case. It is our hope that the “resolution” of the space of cusp forms resulting from (1.19) will have interesting arithmetic applications. In particular, we will consider the Borcherds lift/regularized theta lift of such forms in a sequel to this paper. We remark that there are two points where our results could be extended. First, we have only determined the Whittaker function $$\omega_1^{\mathrm{T}}(\phi_{s,\ell})$$ for definite $$T$$. As mentioned above and explained in Section 4, our calculation depends on this assumption in an essential way, although it may be that some variant could be used for $$T$$ of arbitrary signature. This case distinction also occurs in [23] where the case of arbitrary signature requires a more elaborate argument. Second, we have not treated the Whittaker functions $$\boldsymbol{\omega}_{\underline{w}}^{\mathrm{T}}(f_{s,\ell})$$ arising from the other conical vector $$c_{\underline{w}}$$. There are two reasons for this. On the one hand, we do not need them for the applications we have in mind, and, on the other hand, already in the case $$n=1$$, some additional complications arise which we did not see how to handle for general $$n$$. We now briefly describe the contents of the various sections. In Section 2, we review background material about the degenerate principal series representation $$I(s)$$. In Section 3, we begin with a sketch of the theory of Goodman–Wallach operators relevant to our situation, intended to summarize some of the basic ideas from [12, 20] for nonspecialists (like the authors). We then state and prove our first main result, Theorem 3.1 (Theorem A). Its proof depends on some basic facts about matrix argument Bessel functions and Bessel operators from [22]. Everything up to this point could just as well have been formulated in terms of analysis on symmetric cones associated with Euclidean Jordan algebras, as in [11, 22, 23], and it should be possible to prove the analogue of Theorem A in this generality. We plan to do this in a sequel. In Section 4, we compute the “bad” Whittaker function with scalar $$K$$-type explicitly via an elaborate exercise with special functions of matrix argument. As the final answer is quite simple, we wonder if there is not a more direct derivation of it but did not succeed in finding one. In Section 5, we begin by defining the $$\xi$$-operator in some generality. We then show that its action on Whittaker forms can be determined from that of the corresponding operator on a complex associated with the degenerate principal series, cf. (5.16). We then construct certain Whittaker forms whose images under the $$\xi$$-operator interpolate, in the variable $$s$$, the standard Whittaker function $$W^{\mathrm{T}}_\kappa$$ occurring in the Fourier expansion of holomorphic Siegel cusp forms of weight $$\kappa$$, as explained in Theorem C. In Section 6, we briefly discuss the global $$\xi$$-operator and in the Appendix, we review some notation from [11], the inversion formula used in the proof of Theorem A, and an estimate for the growth of $${}_1F_1$$. 2 Background 2.1 Notation Let $$W$$, $$\langle \,{},{}\,\rangle$$ be a symplectic vector space of dimension $$2n$$ over $${\mathbb Q}$$ with standard basis $$e_1, \ldots, e_n, f_1,\ldots, f_n$$ with $$\langle \,{e_i},{f_j}\,\rangle = \delta_{ij}$$ and $$\langle \,{e_i},{e_j}\,\rangle= \langle \,{f_i},{f_j}\,\rangle=0$$. Let $$G= \mathrm {Sp}(W) \simeq \mathrm {Sp}_n/{\mathbb Q}$$. Following the tradition of [25, 26], we view $$W$$ as a space of row vectors with $$G$$ acting on the right. The Siegel parabolic $$P$$ is the stabilizer of the subspace spanned by the $$f_j$$’s, and we write $$P= M N$$ with Levi subgroup   M={m(a)=(ata−1)∣a∈GLn}, (2.1) and unipotent radical   N={n(b)=(1b1)∣b∈Symn} (2.2) and its opposite   N¯={n−(b)=(1b1)∣b∈Symn}. (2.3) For $$X\in \mathrm{Sym}_n$$, we write   n(X)=(X), and n−(X)=(X), for elements of $$\mathrm{Lie}(N)$$ and $$\mathrm{Lie}(\bar N)$$, respectively. We frequently write $$S = \mathrm{Sym}_n(\mathbb R)$$ and $$S_{\mathbb C}= \mathrm{Sym}_n({\mathbb C})$$. The stabilizer in $$G(\mathbb R)= \mathrm {Sp}_n(\mathbb R)$$ of the point $$i\cdot 1_n\in \mathfrak H_n$$, the Siegel space of genus $$n$$, is the maximal compact subgroup   K={k=(AB−BA)∣k=A+iB∈U(n)}. (2.4) If $$g = n m(a) k$$, then $$g = n m(a\mathbf k_0) k_0^{-1}k$$, where $$\mathbf k_0\in O(n)$$. In particular, in such a decomposition, we can always assume that $$\det(a)>0$$. Let   w_=(1n−1n), (2.5) so that $$\underline{w}$$ corresponds to $$i\,1_n\in U(n)$$ and lies in the center of $$K$$. For $$\tau\in \mathfrak H_n$$ and $$g\in G$$, $g = \begin{pmatrix} a&b\\c&d\end{pmatrix}$, we let $$j(g,\tau) = \det(c\tau+d)$$ be the standard scalar automorphy factor. Note that $$j(gk,i) = j(g,i)\,\det(\mathbf k)^{-1}$$. 2.2 Weil representations Let $$V$$, $$(\ ,\ )$$ be a non-degenerate inner product space over $${\mathbb Q}$$ of signature $$(p,q)$$. If $$\dim V = m = p+q$$ is even, $$\mathrm {Sp}_n(\mathbb R)\times\mathrm{ O}(V(\mathbb R))$$ acts on the space of Schwartz functions $$\mathcal S(V(\mathbb R)^n)$$ via the Weil representation:   ω(m(a))φ(x) =χV(det(a))|det(a)|m2φ(xa)ω(n(b))φ(x) =e(tr(Q(x)b))φ(x)ω(w_)φ(x) =γ(V)∫V(R)ne(tr(x,y))φ(y)dy, and $$\omega(h)\varphi(x) = \varphi(h^{-1}x)$$ for $$h\in O(V)(\mathbb R)$$. Here $$\chi_V(t) = (\mathrm{sgn}(t))^{\frac12(p-q)}$$ and $$\gamma(V) = e(\frac18(p-q))$$. Let $$D(V)$$ be the space of oriented negative $$q$$-planes in $$V(\mathbb R)$$. For $$z\in D$$, let $$(\,,\,)_z$$ be the majorant of $$(\,,\,)$$ defined by   (x,x)z=(x,x)−2(prz(x),prz(x)), and let $$\varphi_0(\cdot,z)\in \mathcal S(V(\mathbb R)^n)$$, given by   φ0(x,z)=exp⁡(−πtr((x,x)z)), be the associated Gaussian. It is an eigenfunction for $$K$$ with   ω(k)φ0(⋅,z)=det(k)p−q2φ0(⋅,z). 2.3 The degenerate principal series For $$G=\mathrm {Sp}_n(\mathbb R)$$ and the Siegel parabolic $$P=NM$$, with notation as in Section 2.1, let $$I^{\mathrm{ sm}}(s,\chi)$$ be the degenerate principal series representation given by right multiplication on the space of smooth functions $$\phi$$ on $$G$$ with   ϕ(n(b)m(a)g)=χ(det(a))|det(a)|s+ρϕ(g), (2.6) where $$\rho = \rho_n = \frac12(n+1)$$. In the case of interest to us, $$\chi(t) = \mathrm{sgn}(t)^{\nu}$$ for $$\nu=0$$, $$1$$. We let $$I(s)=I(s,\chi)$$ be the space of $$K$$-finite functions; it is the $$(\mathfrak g,K)$$-module associated with $$I^{\mathrm{ sm}}(s)$$. The structure of $$I(s)$$ is known [16–18]. We review the facts that we need and refer the reader to these papers for more information. 2.4 The infinitesimal character Let $$\mathfrak h\subset \mathfrak g= \mathrm{Lie}(G)_{\mathbb C}$$ be a Cartan subalgebra of $$\mathfrak k = \mathrm{Lie}(K)_{\mathbb C}$$ and hence also of $$\mathfrak g$$. Let $$\mathfrak z(\mathfrak g)$$ be the center of the universal enveloping algebra $$U(\mathfrak g)$$ and let   γ:z(g) ⟶∼ S(h)W be the Harish–Chandra isomorphism [9]. For $$\lambda\in \mathfrak h^*$$, let $$\chi_\lambda$$ be the character of $$\mathfrak z(\mathfrak g)$$ given by $$\chi_\lambda(Z) = \gamma(Z)(\lambda)$$. Following the notation of [16], for $$x=(x_1,\ldots, x_n)\in {\mathbb C}^n$$, we write   d(x)=diag(x1,…,xn),h(x)=(−id(x)id(x)), and take $$\mathfrak h = \{ h(x)\mid x\in {\mathbb C}^n\}$$. Then $$H_j = h(e_j)$$ is a basis for $$\mathfrak h$$ with dual basis $$\epsilon_j\in \mathfrak h^*$$, $$1\le j\le n$$. Then the infinitesimal character of $$I(s)$$ is $$\chi_{\lambda(s)+\rho_G}$$, where   λ(s)=(s−ρ)∑jϵj (2.7) and $$\rho_G= \sum_j (n-j+1)\epsilon_j$$, cf. [9, Theorem 4, p. 76], for example. Let $$C$$ be the Casimir operator of $$\mathfrak g$$. Then $$C$$ acts in $$I(s)$$ by the scalar   χλ(s)+ρG(C)=⟨λ(s)+ρG,λ(s)+ρG⟩−⟨ρG,ρG⟩=18(s+ρ)(s−ρ). (2.8) This is consistent with the fact that the trivial representation of $$G$$ is a constituent of $$I(s)$$ at the points $$s=\pm\rho$$. The Killing form on $$\mathfrak g\subset M_{2n}({\mathbb C})$$ is given by   ⟨X,Y⟩g=4ntr(XY), so that, since $$\mathrm{tr}(p_+(x)p_-(y)) = \mathrm{tr}(xy)$$,   C=Ck+14n∑αp+(eα)p−(eα∨)+p−(eα∨)p+(eα), (2.9) where $$C_{\mathfrak k}$$ is the $$\mathfrak k$$ component of $$C$$. 2.5 $$K$$-types For further details, cf. [16]. As a representation of $$K$$, we have   I(s)≃IndM∩KK(χ)≃IndO(n)U(n)sgn(det)ν. Thus the $$K$$-types of $$I(s)$$ have multiplicity 1 and an irreducible representation $$(\sigma, \mathcal V_\sigma)$$ of $$K$$ occurs precisely when its highest weight has the form   (ℓ1,…,ℓn),ℓ1≥…≥ℓn,ℓj∈ν+2Z, or, equivalently, precisely when its restriction to $$O(n) \simeq M\cap K$$ contains the representation $$(\det)^\nu$$. For such $$\sigma$$,   dim⁡HomK(σ,I(s))=dim⁡[I(s)⊗σ∨]K=1. (2.10) Suppose that $$v_0\in \sigma^\vee$$ is an eigenvector for $$O(n)$$, so that $$\sigma^\vee(k) v_0 = \det(k)^\nu v_0$$ for all $$k\in O(n)$$. The vector $$v_0$$ is unique up to a non-zero scalar. A standard basis element for $$[I(s)\otimes \sigma^\vee]^K$$ is then given by   ϕs,σ(nm(a)k)=χ(det(a))|det(a)|s+ρσ∨(k−1)v0. For example, for an integer $$\ell$$, with $$\ell\equiv \nu\mod 2$$, the unique function $$\phi_{s,\ell}\in I(s)$$ with scalar $$K$$-type $$\det(k)^\ell$$ is given by   ϕs,ℓ(n(b)m(a)k)=χ(det(a))|det(a)|s+ρdet(k)ℓ. (2.11) 2.6 Submodules For $$s\notin \nu+2{\mathbb Z}$$, the $$(\mathfrak g,K)$$-module $$I(s)$$ is irreducible. At points $$s\in \nu+2{\mathbb Z}$$, nontrivial submodules arise via the coinvariants for the Weil representation. For a quadratic space $$V$$ over $$\mathbb R$$ of signature $$(p,q)$$, $$p+q$$ even, with associated Weil representation $$(\omega, S(V^n))$$, for the additive character $$x \mapsto e(x)$$, there is an equivariant map   λV:S(Vn)⟶I(s0),φ↦(ω(g)φ)(0), where $$s_0 = \frac12(p+q) - \rho$$ and $$\nu \equiv \frac12(p-q) \mod 2$$. The image, $$R(p,q)$$, is the $$(\mathfrak g,K)$$-submodule of $$I(s_0)$$ generated by the scalar $$K$$-type $$(\det)^\ell$$ with $$\ell = \frac12(p-q)$$. Moreover, the vector $$\phi_{s_0,\frac12(p-q)}$$ is the image of the Gaussian $$\varphi^0_V \in S(V^n)$$, where   ωV(k)φV0=det(k)12(p−q)φV0. For example, for signature $$(m+2,0)$$ with $$m+2>2n+2$$, $$R(m+2,0)\subset I(s_0)$$ is a holomorphic discrete series representation with scalar $$K$$-type $$(\det)^{\kappa}$$, $$\kappa = \frac{m}2+1$$. In particular, the vector $$\phi_{s_0, \kappa}$$ for $$s_0= \kappa -\rho$$ is killed by $$\mathfrak p_-$$. On the other hand, for signature $$(m,2)$$, the vector   φKM∈[S(Vn)⊗∧(n,n)(pH∗)]KH, satisfies   ω(k)φKM=det(k)κφKM. The image of this vector under the map $$\lambda_{m,2}$$ is again $$\phi_{s_0,\kappa}$$, so that we have submodules   R(m+2,0)⊂R(m,2)⊂I(s0),s0=κ−ρ. Note that, by [16], $$R(p,q)$$ is the largest quotient of $$S(V^n)$$ on which the orthogonal group $$O(V)= O(p,q)$$ acts trivially, that is, the space of $$O(V)$$-coinvariants. 3 Goodman–Wallach Operators In our discussion of both conical and Whittaker vectors, we will only consider the degenerate principal series representation $$I(s)$$ and the Siegel parabolic $$P=NM$$. In this case, the simple example for $$\mathrm{SL}_2(\mathbb R)$$ worked out in the introduction of Goodman–Wallach [12] provides an adequate template. An essential feature is that the various classical special functions occurring there are replaced by their matrix argument generalizations. These results seem to be new; at least we could not find such explicit formulas in the literature. Since we work with intertwining operators which express our functions via integral representations, we derive, as a consequence, the behavior of our functions under the differential operators coming from the center of the enveloping algebra $$\mathfrak z(\mathfrak g)$$, whereas, in the case of $$\mathrm{SL}_2(\mathbb R)$$ one can work with classical solutions of the second-order ode satisfied by the radial part. For discussion of the more general theory including general results about the existence of Goodman–Wallach operators, cf. [20]. 3.1 Conical and Whittaker vectors Suppose that $$(\pi,\mathcal V)$$ is a continuous representation of $$G$$ on a Banach space $$\mathcal V$$ and that   Fcon:V⟶Ian(s) is a $$G$$-equivariant linear map. Here $$I^{\mathrm{ an}}(s)$$ is the space of real analytic functions on $$G$$ satisfying (2.6). The linear functional $$\mu_{\mathrm{ con}}\in \mathcal V^*$$ defined by $$\mu_{\mathrm{ con}}(v) = F_{\mathrm{ con}}(v)(e)$$ satisfies   μcon(π(nm(a))v)=χ(det(a))|det(a)|s+ρμcon(v). We refer to such a vector as a conical vector in $$\mathcal V^*$$ of type $$(P,s+\rho)$$. Conical vectors in the dual $$I^{\text{an}}(s)^*$$ are given by   c1(ϕ)=ϕ(e), (3.1) and   cw_(ϕ)=(A(s,w_)ϕ)(e)=∫Nϕ(w_n(b))db, (3.2) where, for $$\mathrm{Re}(s)>\rho$$, the intertwining operator $$A(s,\underline{w}):I(s) \longrightarrow I(-s)$$ is defined by the integral   (A(s,w_)ϕ)(g)=∫Nϕ(w_ng)dn, (3.3) for $$\underline{w}$$ given by (2.5). It has a meromorphic analytic continuation. According to our terminology, these are of type $$(P,s+\rho)$$ and $$(P,-s+\rho)$$, respectively. ({Here, we are introducing a compressed version of the standard terminology [12, 20] convenient for our special case.}) Similarly, for $$T\in \mathrm{Sym}_n({\mathbb C})$$, suppose that   Fwh:V⟶WT(G)an is a $$G$$-equivariant linear map, where $$\mathcal W^{\mathrm{T}}(G)^{\mathrm{ an}}$$ is the space of real analytic functions on $$G$$ satisfying (1.4). The linear functional $$\mu_{\mathrm{ wh}} \in \mathcal V^*$$ defined by $$\mu_{\mathrm{ wh}}(v) = F_{\mathrm{ wh}}(v)(e)$$ satisfies   μwh(π(n(b))v)=e(tr(Tb))μwh(v). We refer to such a vector in $$\mathcal V^*$$ as a Whittaker vector of type $$(N,T)$$. We can make analogous definitions for $$\mathcal V$$ an irreducible $$U(\mathfrak g)$$-module, and now explain an essential idea of [12] relating conical and Whittaker vectors in $$\mathcal V^*$$. Our goal is to motivate the explicit construction given below; for a more careful treatment cf. [12, 20]. Let $$\mathfrak n=\mathrm{Lie}(N)_{\mathbb C}$$, $$\mathfrak m = \mathrm{Lie}(M)_{\mathbb C}$$, and let $$\bar{\mathfrak n} = \mathrm{Lie}(\bar N)_{\mathbb C}$$ where $$\bar N$$ is the unipotent radical of the opposite maximal parabolic $$\bar P = M \bar N$$. Let $${\mathbb C}(s+\rho)$$ be the one-dimensional representation of $$\mathfrak m$$ determined by the representation $$|\det \,|^{s+\rho}$$ of $$M$$ and extend it to a representation of $$\mathfrak m+\mathfrak n$$, trivial on $$\mathfrak n$$. Define the generalized Verma modules   V(P,s+ρ)=U(g)⊗U(m+n)C(s+ρ) and   V¯(P¯,−s−ρ)=C(−s−ρ)⊗U(m+n¯)U(g). By the Poincaré–Birkoff–Witt theorem,   V(P,s+ρ)=U(n¯)⊗CC(s+ρ),andV¯(P¯,−s−ρ)=C(−s−ρ)⊗CU(n). There is a natural pairing   ⟨⟨ , ⟩⟩:V¯(P¯,−s−ρ)⊗CV(P,s+ρ)⟶V¯(P¯,−s−ρ)⊗U(g)V(P,s+ρ) ⟶∼ C, and, for $$Z\in U(\mathfrak g)$$, $$\bar u\in \bar V(\bar P,-s-\rho)$$, and $$u\in V(P,s+\rho)$$,   ⟨⟨u∗Z,u⟩⟩=⟨⟨u∗,Zu⟩⟩. This pairing is non-degenerate precisely when $$V(P,s+\rho)$$ is irreducible, [20, Section 3.1]. For the rest of our discussion, we suppose that this is the case. Since $$\mathfrak n$$ and $$\bar{\mathfrak n}$$ are abelian   U(n¯)=S(n¯)=⨁∂≥0S(n¯)d (3.4) is the symmetric algebra on $$\bar{\mathfrak n}$$, graded by degree, and the completion   S(n¯)[n¯]=U(n¯)[n¯]=lim⟵U(n¯)/n¯kU(n¯)≃∏∂≥0S(n¯)d, (3.5) is the ring of formal power series in such elements. Let   V^(P,s+ρ)=V(P,s+ρ)[n¯]=U(n¯)[n]¯⊗CC(s+ρ) be the $$\bar{\mathfrak n}$$-completion of $$V(P,s+\rho)$$. The pairing $$\langle\langle{\,},{}\rangle\rangle$$ extends to this space and induces an isomorphism   V^(P,s+ρ) ⟶∼ V¯(P¯,−s−ρ)∗. (3.6) Then, following the notation of [20], for an admissible character $$\psi$$ of $$\mathfrak n$$, the spaces   Whn,ψ∗(V¯(P¯,−s−ρ))={w∈V¯(P¯,−s−ρ)∗∣X⋅w=ψ(X)w, ∀X∈n}, (3.7) and   Whn,ψ(V^(P,s+ρ))={w∈V^(P,s+ρ)∣X⋅w=ψ(X)w, ∀X∈n} (3.8) are isomorphic via (3.6). ({In our situation, for $$X\in \mathfrak n\simeq \mathrm{Sym}_n({\mathbb C})$$, and $$\psi(X) = \mathrm{tr}(TX)$$, admissibility simply means that $$\det(T)\ne0$$.}) If we extend $$\psi$$ to an algebra homomorphism $$\psi: U(\mathfrak n)\longrightarrow {\mathbb C}$$, then there is an obvious basis for the space (3.7) given by the functional $$1\otimes\psi$$ on $$\bar V(\bar P,-s-\rho)\simeq {\mathbb C}(-s-\rho)\otimes_{\mathbb C} U(\mathfrak n)$$. We write $$\mathrm{gw}_s^\psi$$ for the corresponding element of $$\hat V(P,s+\rho)$$, viewed as a formal power series in the elements of $$\bar{\mathfrak n}$$, and we refer to it as the Goodman–Wallach element. It is characterized by   ⟨⟨A,gwsψ⟩⟩=ψ(A),for all A∈U(n). Returning to the irreducible $$U(\mathfrak g)$$-module $$(\pi, \mathcal V)$$, we let $$U(\mathfrak g)$$ act on $$\mathcal V^*$$ on the left by $$Z\cdot \mu = \mu\cdot {}^tZ$$, where $${}^t:U(\mathfrak g) \longrightarrow U(\mathfrak g)$$ is the involution restricting to $$X\mapsto -X$$ on $$\mathfrak g$$. We assume that $$\mathcal V$$ is finitely generated as a $$U(\mathfrak n)$$-module. For a conical vector $$\mu\in \mathcal V^*$$ of type $$(P,s+\rho)$$, we have a homomorphism   V^(P,−s−ρ)⟶V∗,Z↦Z⋅μ. Then, for the Goodman–Wallach element $$\mathrm{gw}_s^\psi\in \hat V(P, -s-\rho)$$, and for $$X\in \mathfrak n$$, we have   (gwsψ⋅μ)⋅X=(μ⋅tgwsψ)⋅X=μ⋅t(−Xgwsψ)=−ψ(X)gwsψ⋅μ. Thus, $$\mathrm{gw}_s^{\psi}\cdot \mu$$ is a Whittaker vector of type $$(N,-\psi)$$. Of course, we have only given a rough sketch of the idea here, including the %unrealistic restriction to the case where $$V(P,s+\rho)$$ is irreducible. After removing this restriction, a main point of the Goodman–Wallach theory is to give estimates on the growth of the components of the power series $$\mathrm{gw}_s^\psi$$ so that it can be used to define $$G$$-intertwining operators—the Goodman–Wallach operators—from principal series representations to spaces of Whittaker functions preserving Gevrey classes, that is, certain function spaces between real analytic and $$C^\infty$$, cf. the remark on p. 228 of [20] for a more precise statement. 3.2 An explicit formula for the Goodman–Wallach operator for $$\mathrm {Sp}_n(\mathbb R)$$ We now describe the matrix argument Bessel function giving the Goodman–Wallach operator. For $$\phi\in I(s)$$, the function $$\Psi(\phi)$$ defined by (1.12) and its Fourier transform (1.13) satisfy   Ψ(x;π(m(a))ϕ)=det(a)s+ρΨ(taxa;ϕ), (3.9) and   Ψ^(y;π(m(a))ϕ)=det(a)s−ρΨ^(a−1yta−1;ϕ). (3.10) The conical vectors $$c_1$$ and $$c_{\underline{w}}$$ defined by (3.1) and (3.2) can be written as   ⟨c1,ϕ⟩=∫SΨ^(y;ϕ)dy=Ψ(0;ϕ), (3.11) and   ⟨cw_,ϕ⟩=∫SΨ(y;π(w_)ϕ)dy=Ψ^(0;π(w_)ϕ). (3.12) Here note that   ∫Sϕ(w_n(b))db=∫Sϕ(n−(−b)w_)db=∫Sϕ(n−(b)w_)db. We next define the relevant Bessel type function on $$S$$, specializing some of the notation of [11, 22] to the present case. This notation is summarized in the Appendix, which the reader should consult for things not explained here. For $$z$$ and $$w\in S_{\mathbb C}$$, the function $$\mathrm{GW}_s(z,w)$$ defined by (1.15) coincides with the $$J$$-Bessel function   GWs(z,w)=Js+ρ(z,w), (3.13) in the notation of [22, p. 818 and p. 823]. Note that $$\Phi_{\mathbf m}(z,w)$$ is the function on $$S_{\mathbb C}\times S_{\mathbb C}$$ described in [22, Lemma 1.11]. In particular, this function is holomorphic in $$z$$, antiholomorphic in $$w$$, and satisfies $$\Phi_{\mathbf m}(z,1_n) = \Phi_{\mathbf m}(z)$$, where $$\Phi_{\mathbf m}(z)$$ is the spherical polynomial in [11, Chapter XI, Section 3]. Also recall that $$\Phi_{\mathbf m}(z)$$ is homogeneous of degree $$|\mathbf m| = \sum_i m_i$$. Moreover, for $$a\in \mathrm{GL}_n({\mathbb C})$$,   Φm(a⋅z,w)=Φm(z,ta¯⋅w), (3.14) and $$\overline{\Phi_{\mathbf m}(z,w)} = \Phi_{\mathbf m}(w,z)$$. The invariance (3.14) is inherited by $$\mathrm{GW}_s(z,w)$$. Theorem 3.1. For $$\phi\in I(s)$$, let   ω1(ϕ)=∫SGWs(2πy,2πw)Ψ^(y;ϕ)dy. (3.15) Recall that for $$X\in \mathrm{Sym}_n(\mathbb R)$$,   n(X)=(0X0). Then,   ω1(π(n(X))ϕ)=2πitr(Xw¯)ω1(ϕ). (3.16)□ Corollary 3.2. For $$Y\in \mathrm{Sym}_n({\mathbb C}) \simeq \bar{\mathfrak n}$$, view $$\mathrm{GW}_s(2\pi Y,2\pi w)$$ as a power series in $$S(\bar{\mathfrak n})_{[\bar{\mathfrak n}]}$$. Then this power series defines the Goodman–Wallach–Matumoto operator in $$\hat V(P,-s-\rho)$$ for the Siegel parabolic $$P$$ of $$G=\mathrm {Sp}_n(\mathbb R)$$ and the character $${\frak n}(X)\mapsto 2\pi \mathrm{i}\,\mathrm{tr}(X\bar w)$$ of $$\mathfrak n$$. (Concretely, write $$Y = \sum_{\alpha} Y_\alpha\, e_\alpha^\vee$$ and view $$\mathrm{GW}_s(2\pi Y,w)$$ as a power series in the $$Y_\alpha$$’s.) □ Definition 3.3. For $$T\in \mathrm{Sym}_n(\mathbb R)$$ with $$\det(T)\ne0$$, let $$\omega^{\mathrm{T}}_1$$ be the Whittaker functional constructed from the conical vector $$c_1$$, so that   ω1T(ϕ)=∫SGWs(2πy,2πT)Ψ^(y;ϕ)dy. (3.17) and   ω1T(ϕ)(n(b)g)=e(tr(Tb))ω1T(ϕ)(g). □ 3.3 Proof of the Goodman-Wallach identity We want to prove (3.16) and so we consider   ω1(n(X)ϕ)=∫SGWs(2πy,2πw)Ψ^(y;n(X)ϕ)dy. (3.18) We adopt some of the notation and setup from [11, 22]. Recall that $$S$$ is a simple Euclidean Jordan algebra with product $$x\cdot y = \frac12(xy+yx)$$. Endomorphisms $$P(x)$$ and $$P(x,y)$$ of $$S$$ are defined by, [22, p.794],   P(x)z=xzx,P(x,y)z=xzy+yzx. Let $$e_\alpha$$ be a basis for $$S$$ and let $$e_\alpha^\vee$$ be the dual basis with respect to the trace form, cf. Appendix. In particular, we write $$x = \sum_\alpha x_\alpha e_\alpha$$. Define vector-valued differential operators as follows. The gradient operator   ∂∂x=∑α∂∂xαeα∨ is characterized by   ∂a=tr(a∂∂x), where, for $$a\in S$$, $$\partial_a$$ is the directional derivative associated with the constant vector field $$a$$. For a complex scalar $$\lambda$$, the Bessel operator $$\mathcal B_\lambda$$ is defined by   Bλ=P(∂∂x)x+λ∂∂x=∂∂xx∂∂x+λ∂∂x. Thus, if $$f$$ is a $$C^2$$ function on $$S$$, then $$\mathcal B_\lambda f$$ is an $$S$$-valued function on $$S$$. The key fact is the following, see Proposition 3.3 in [14] and also [15],: Proposition 3.4. For $$X$$ and $$Y\in \mathrm{Sym}_n(\mathbb R)$$, (i)   Ψ(x;π(n(X))ϕ)=−tr(X(sx+x∂∂xx)Ψ(x;ϕ)), (ii)   Ψ(x;π(n−(Y))ϕ)=tr(Y∂∂x)Ψ(x;ϕ), (iii)   Ψ^(y;π(n(X))ϕ)=12πitr(XB−sΨ^(y;ϕ)), (iv)   Ψ^(y;π(n−(Y))ϕ)=−2πitr(Yy)Ψ^(y;ϕ). □ Proof First we note that $$\exp({\frak n}(X)) = n(X)$$, and that   (1x1)(1X1)=(1Xx1+xX)=(1∗1)(tA−1A)(1A−1x1), where $$A = 1+xX$$ and $$* = X(1+xX)^{-1}$$. Here, we are going to take $$tX$$ in place of $$X$$ so that $$1+xX$$ will be invertible for $$t$$ sufficiently small. Thus, in this range,   ϕ(n−(x)n(X))=det(1+xX)−s−ρϕ(n−((1+xX)−1x)). We now replace $$X$$ by $$tX$$ and take $$\mathrm{d}/\mathrm{d}t\vert_{t=0}$$. First we have   ddtdet(1+txX)−s−ρ|t=0=−(s+ρ)tr(xX), where we note that   det(1+txX)=1+tr(txX)+O(t2). Next, we let $$z = (1+xtX)^{-1}x$$ and compute   0=ddt((1+xtX)z)|t=0=xXx+dzdt|t=0, so that, writing $$z = \sum_\alpha z_\alpha e_\alpha$$, we have   dzdt|t=0=∑αdzαdteα=−∑α(xXx)αeα. Therefore   ddtϕ(z)|t=0=∑α∂ϕ∂zαdzαdt|t=0=−tr(xXx∂∂x)ϕ. Thus we have proved that   Ψ(x;π(n(X))ϕ)=−((s+ρ)tr(Xx)+tr(xXx∂∂x))Ψ(x;ϕ). But we have   tr(xXx∂∂x)=tr(Xx∂∂xx)−ρtr(xX). Indeed, writing $$\bullet$$ for the “evaluation” product, we have   (∑α∂∂xαeα∨)∙(∑αxαeα)=∑αeα∨eα=12(n+1)∑ieii. This gives (i). Now we consider the Fourier transform   −∫Se(tr(xy))(sx+x∂∂xx)Ψ(x;ϕ)dx. If we write $$y = \sum_\alpha y_\alpha e_\alpha^\vee$$, then $$(\partial/\partial y) = \sum_\alpha (\partial/\partial y_\alpha)\,e_\alpha$$, and we have   ∂∂ye(tr(xy))=(∑α∂∂yαeα)(e(∑αxαyα))=2πi∑αxαeα=2πix. Then the factor $$s\, x$$ (resp. $$x^2$$) can be obtained by applying   s2πi∂∂y,(resp. (2πi)−2(∂∂y)2) outside the integral. Also   ∫Se(tr(xy))∂∂xΨ(x;ϕ)dx=−2πiy∫Se(tr(xy))Ψ(x;ϕ)dx. We obtain   − ∫Se(tr(xy))(sx+x∂∂xx)Ψ(x;ϕ)dx =−12πi(s∂∂y−∂∂yy∂∂y)∫Se(tr(xy))Ψ(x;ϕ)dx =12πiB−sΨ^(y;ϕ). This proves (iii) of Proposition 3.4. The proofs of (ii) and (iv) are easy and omitted. ■ Now we return to (3.18) and, using (ii) of the previous proposition, obtain   ω1(π(n(X))ϕ) =∫SGWs(2πy,2πw)Ψ^(y;π(n(X))ϕ)dy =12πitr(X∫SGWs(2πy,2πw)B−sΨ^(y;ϕ)dy) =12πitr(X∫SB−s∗GWs(2πy,2πw)Ψ^(y;ϕ)dy ) =12πitr(X∫SBsGWs(2πy,2πw)Ψ^(y;ϕ)dy). Here we use the fact that the adjoint of $$\mathcal B_{-s}$$ is $$\mathcal B_{s}$$. But now by (3.13) and Proposition 3.6 of [22], we have   BsGWs(2πy,2πw)=−(2π)2w¯⋅GWs(2πy,2πw) so that we obtain   ω1(n(X)ϕ)=2πitr(Xw¯)ω1(ϕ), as required. This proves Theorem 3.1. 3.4 Proof of Corollary 3.2 To show that $$\mathrm{GW}_s(2\pi Y,2\pi w)$$ is indeed the Goodman–Wallach element, as claimed, we proceed as follows. The pairing   ⟨⟨,⟩⟩:V¯(P¯,s+ρ)⊗CV^(P,−s−ρ)⟶C is characterized by   ⟨⟨A⋅Z,B⟩⟩=⟨⟨A,Z⋅B⟩⟩, for $$A\in \bar V(\bar P, s+\rho)$$, $$B\in \hat V(P,-s-\rho)$$, and $$Z\in U(\mathfrak g)$$, and   ⟨⟨1⊗n(X),n−(Y)⊗1⟩⟩=(s+ρ)tr(XY). For any function $$\phi\in I(s)$$, and for $$A\in S(\mathfrak n)$$ and $$B\in U(\mathfrak g)$$, we define   ⟨⟨A,B⟩⟩ϕ=−∫SΨ^(y;t(A⋅B)⋅ϕ)dy=−Ψ(0;t(A⋅B)⋅ϕ). As a function of $$B$$ this map factors through $$V(P,-s-\rho)$$ and $$\langle\langle{A},{B}\rangle\rangle_\phi = \langle\langle{1},{A\cdot B}\rangle\rangle_\phi$$. Recall here that, as in 3.1, $$A\mapsto {}^tA$$ is the involution of $$U(\mathfrak g)$$ which is $$-1$$ on $$\mathfrak g$$. Moreover, we have   ⟨⟨n(X),n−(Y)⟩⟩ϕ =−Ψ(0;n−(−Y)n(−X)⋅ϕ) =−Ψ(0;([n−(−Y),n(−X)]+n(−X)n−(−Y))⋅ϕ) =(s+ρ)tr(XY)Ψ(0;ϕ). Taking $$\phi$$ with $$\Psi(0;\phi)=\phi(e)=1$$, we have $$\langle\langle{\,},{}\rangle\rangle_\phi = \langle\langle{\,},{}\rangle\rangle$$ on $$U(\mathfrak n)\times V(P,-s-\rho)$$ . Now by (iv) of Proposition 3.4, we take a power series $$\mathrm{ gw}_s \in S(\bar{\mathfrak n})_{[\bar{\mathfrak n}]}$$ so that, for all $$\phi$$,   Ψ^(y;tgws⋅ϕ)=GWs(2πy,2πw)Ψ^(y;ϕ). Then   ⟨⟨A,gws⟩⟩ =−∫SΨ^(y;t(A⋅gws)⋅ϕ)dy =−∫SGWs(2πy,2πw)Ψ^(y;tAϕ)dy =−ω1(tAϕ) =−ψ−2πiw¯(A)ω1(ϕ) =ψ−2πiw¯(A)⟨⟨1,gws⟩⟩, where $$\psi_{-2\pi \mathrm{i}\bar w}: S(\mathfrak n)\longrightarrow {\mathbb C}$$ is the character determined by $${\frak n}(X)\mapsto -2\pi \mathrm{i}\,\mathrm{tr}(X\bar w)$$. This in fact shows that the Goodman–Wallach element is actually given by   gws♮=⟨⟨1,gws⟩⟩−1gws. 4 Calculation of the “bad” Whittaker Function: the Scalar Case In this section, we determine the Whittaker function $$\omega_1^{ T}(\phi_{s,\ell})$$ for $$\phi_{s,\ell}\in I(s)$$ with a scalar $$K$$-type and for $$\epsilon\,T\in \mathrm{Sym}_n(\mathbb R)_{>0}$$ and $$\epsilon=\pm1$$. Whittaker functions for other $$K$$-types can then be obtained by applying differential operators. In the case of $$\mathrm{SL}_2(\mathbb R)$$, that is, for $$n=1$$, the radial part of the Whittaker function we obtain is essentially the classical confluent hypergeometric function $$M(a,b;z)$$ in the notation of [1], for example. On the other hand, again for $$n=1$$, the Whittaker functional obtained by applying the Goodman–Wallach operator to the conical vector $$c_w$$ yields a Whittaker function whose radial part is the classical function $$U(a,b;z)$$. Thus, this traditional basis for the solution space to the Whittaker ode arises in a natural way from the pair of conical vectors $$c_1$$ and $$c_w$$. The calculation of this section constructs the analogue of the $$M$$-Whittaker function for $$\mathrm {Sp}_n(\mathbb R)$$. As noted earlier, we do not have a corresponding evaluation for $$T$$ of arbitrary signature, and we will indicate in the course of the calculation where the assumption that $$T$$ is definite is used. For the standard vector $$\phi_{s,\ell}$$ with scalar $$K$$-type defined in (2.11), the Whittaker function   fs,ℓT(g):=ω1T(π(g)ϕs,ℓ)=∫SGWs(2πy,2πT)Ψ^(y;π(g)ϕs,ℓ)dy (4.1) satisfies   fs,ℓT(n(b)m(a)k)=e(tr(Tb))fs,ℓT(m(a))det(k)ℓ, (4.2) and hence is determined by its restriction to $$\mathrm{GL}_n(\mathbb R)$$. We will frequently omitted the $$T$$ as a superscript to lighten the notation. By analogy with the one variable case, for $$z\in S_{\mathbb C}$$ and the $$a$$ and $$b\in {\mathbb C}$$ with $$\mathrm{Re}(a)>\rho-1$$, $$\mathrm{Re}(b)>\rho-1$$, we let   Mn(a,b;z)=Γn(b)Γn(a)Γn(b−a)∫t>01−t>0etr(zt)det(t)a−ρdet(1−t)b−a−ρdt. (4.3) This is the standard matrix argument hypergeometric function   Mn(a,b;z)=1F1(a;b;z) (4.4) as in [11, 13], etc. Our first main result is the following. Theorem 4.1. Let $$v=a{}^ta$$, $$\alpha= \frac12(s+\rho- \epsilon \ell)$$ and $$\beta = \frac12(s+\rho+\epsilon \ell)$$. Then, writing $$\epsilon\,T= {}^tc c$$,   fs,ℓ(m(a))=2n(s−ρ+1)det(v)12(s+ρ)exp⁡(−2πtr(ϵTv))Mn(α,α+β;4πcvtc).□ For future reference, we give the full formula   fs,ℓ(n(b)m(a)k)=c(n,s)det(v)12(s+ρ)e(tr(Tb))exp⁡(−2πtr(ϵTv))Mn(α,α+β;4πcvtc)det(k)ℓ, (4.5) where   c(n,s)=2n(s−ρ+1). (4.6) Remark 4.2. The hypergeometric function $$M_n(\alpha,\alpha+\beta; z)$$ is given by a power series which is everywhere convergent in $$z$$ provided the values of $$s$$ for which some factor $$(\alpha+\beta)_\mathbf{m} = (s+\rho)_{\mathbf{m}}$$ in the denominator vanishes are excluded, [11]. The excluded values are thus $${\mathbb Z}_{<0}$$ for $$n=1$$ and   Z<0∪(−12+Z<0) for $$n\ge 2$$. □ By construction, the Whittaker function $$f_{s,\ell}$$ is an eigenfunction for the center $$\mathfrak z(\mathfrak g)$$ of the universal enveloping algebra $$U(\mathfrak g)$$ with the same eigencharacter as $$I(s)$$. Corollary 4.3. For all $$Z\in \mathfrak z(\mathfrak g)$$,   Z⋅fs,ℓ=χλ(s)+ρG(Z)fs,ℓ, where $$\chi_{\lambda(s)+\rho_G}$$ is the character of $$\mathfrak z(\mathfrak g)$$ given by (2.7). In particular, for the Casimir operator $$C$$, by (2.8),   C⋅fs,ℓ=18(s+ρ)(s−ρ)fs,ℓ.□ Proof of Theorem 4.1. This amounts to a long calculation. We first simplify by eliminating $$T$$. Writing $$2\pi\,\epsilon\, T = c^2$$ for $$c={}^tc >0$$, we have, by (3.14),   GWs(2πy,2πT)=GWs(2πϵtcyc,1n). Then, using (3.10), and setting $$\mathrm{GW}_s(z) = \mathrm{GW}_s(z,1_n)$$, we have   ∫SGWs(2πy,2πT)Ψ^(y;π(m(a))ϕ)dy=|det(2πT)|−12(s+ρ)∫SGWs(2πϵy)Ψ^(y;π(m(ca))ϕ)dy. Remark 4.4. It is at this point that we use the fact that $$T$$ is definite in an essential way. □ Therefore, it suffices to compute   ω1ϵ(π(m(a))ϕ):=∫SGWs(2πϵy)Ψ^(y;π(m(a))ϕ)dy. (4.7) We write this as   ω1ϵ(π(m(a))ϕ)=∑p+q=nω1ϵ(π(m(a))ϕ)p,q, where   ω1ϵ(π(m(a))ϕ)p,q=∫Sp,qGWs(2πϵy)Ψ^(y;π(m(a))ϕ)dy (4.8) for $$S_{p,q}$$ the subset of invertible matrices in $$S$$ of signature $$(p,q)$$. Specializing to the case $$\phi= \phi_{s,\ell}$$, the first step is the following. Proposition 4.5. For $$y\in S_{p,q}$$,   Ψ^(y;π(m(a))ϕℓ,s)=(2π)nρ2−n(ρ−1)Γn(α)Γn(β)det(v)−12(s+ρ)|det(y)|s(2π)ns2ns ×∫x+ϵp>0x+ϵq′>0e−2πtr(tcv−1c(1+2x))det(x+ϵp)α−ρdet(x+ϵq′)β−ρdx, (4.9) where $$y = c \epsilon_{p,q} {}^tc$$ and $$v= a{}^ta$$. Here, as in [23], $$\alpha = \frac12(s+\rho+\ell)$$, $$\beta= \frac12(s+\rho-\ell)$$, $$\epsilon_{p} = \mathrm{diag}(1_p,0)$$, and $$\epsilon'_q = \mathrm{diag}(0,1_q)$$. □ Proof If we write $$n_-(x) = n m k$$ for $$k=k(n_-(x))\in K$$, then   Ψ(x;ϕ)=det(1+x2)−12(s+ρ)ϕ(k(n−(x))) and   Ψ(x;ϕℓ,s)=det(1+ix)−αdet(1−ix)−β, where we are now using Shimura’s convention where $$\alpha=\frac12(s+\rho+\ell)$$ and $$\beta=\frac12(s+\rho-\ell)$$, so that, for example, $$\alpha+\beta = s+\rho$$. Then   Ψ^(y;ϕℓ,s)=∫Se(tr(xy))det(1+ix)−αdet(1−ix)−βdx, and, following the standard manipulations on pp. 274–5 of [23], we have   Ψ^(y;π(m(a))ϕℓ,s) =det(a)s−ρ∫Se(tr(xa−1yta−1))det(1+ix)−αdet(1−ix)−βdx =det(a)s−ρ∫Se(−tr(xa−1yta−1))det(1−ix)−αdet(1+ix)−βdx =in(α−β)det(a)s−ρξ(1,h;α,β) =det(a)s−ρ(2π)nρ2−n(ρ−1)Γn(α)−1Γn(β)−1 ×∫u>0,u>2πhe2tr(πh−u)det(u)α−ρdet(u−2πh)β−ρdu =det(a)s−ρ(2π)nρ2−n(ρ−1)Γn(α)−1Γn(β)−1η(2,πh;α,β), where $$h= a^{-1}y{}^ta^{-1}$$, cf. the top of p. 275 of [23]. Here $$\eta$$ is the function defined in (1.26) of loc.cit. Recalling (3.1) of loc. cit., for any $$a'\in \mathrm{GL}_n(\mathbb R)^+$$,   η(g,a′hta′;α,β)=det(a′)2sη(ta′ga′,h;α,β), and writing $$\pi y = c \epsilon_{p,q}{}^tc$$ so that $$\pi h = a^{-1} c \epsilon_{p,q}{}^tc{}^ta^{-1}$$, we have   η(2,πa−1yta−1;α,β)=det(a)−2s|det(πy)|sη(2tcta−1a−1c,ϵp,q;α,β). Next recall that Shimura writes, p. 288,   η(g,ϵp,q;α,β)=2n(α+β−ρ)ζp,q(2g;α,β), where   ζp,q(g;α,β)=e−12tr(g)∫x+ϵp>0x+ϵq′>0e−tr(gx)det(x+ϵp)α−ρdet(x+ϵq′)β−ρdx. Altogether this gives the claimed expression. ■ Now we return to the integral (4.8), using the expression just given for $$\hat\Psi(y;\pi(m(a))\phi_{\ell,s})$$. If we substitute the series expansion for $$\mathrm{GW}_s(2\pi\epsilon y)$$ and switch the order of integration, we obtain the expression   ω1ϵ(π(m(a))ϕℓ,s)p,q =(2π)nρ2−n(ρ−1)Γn(α)Γn(β)det(v)−12(s+ρ)(2π)ns2ns ×∫x+ϵp>0x+ϵq′>0det(x+ϵp)α−ρdet(x+ϵq′)β−ρ ×(∑m≥0dm(−2πϵ)|m|(ρ)m(s+ρ)m ×∫Sp,qe−2πtr(tcv−1c(1+2x))|det(y)|sΦm(y)dy)dx. (4.10) Before proceeding, we observe that, in the expression $$y = c \epsilon_{p,q} {}^tc$$, there is an ambiguity in the choice of $$c$$, that is, only the coset $$c\, O(p,q)$$ is well defined. More precisely, we have the following basic structural observations where, in particular, (a) implies that the ambiguity in the choice of $$c$$ has no effect on the double integral. The value of the inner integral does depend on the choice of $$c$$, however! Lemma 4.6. (a) There is a bijection   Xp,q={x∈S∣x+ϵp>0, x+ϵq′>0}↓Zp,q={z∈S∣z+ϵp,q>0, z−ϵp,q>0} given by $$x\mapsto 2x+1=z$$. The action of $$G= \mathrm{GL}_n(\mathbb R)$$ on $$S$$ induces an action of the group $$O(p,q)$$ on $$Z_{p,q}$$. Since $$z+ \epsilon_{p,q} = 2(x+\epsilon_p)$$ and $$z-\epsilon_{p,q}= 2(x+\epsilon'_q)$$, the quantities $$\det(x+\epsilon_p)$$ and $$\det(x+e'_q)$$ are constant on the $$O(p,q)$$-orbits in $$Z_{p,q}$$. (b) Let   Wp,q={w∈Sp,q∣1−w>0,w+1>0}=Sp,q∩S(−1,1), where   S(−1,1)={x∈S∣1−x>0and1+x>0}. The action of $$G= \mathrm{GL}_n(\mathbb R)$$ on $$S$$ induces an action of the group $$O(n)$$ on $$W_{p,q}$$. Moreover, there is a bijection on orbits   O(p,q)∖Zp,q⟷O(n)∖Wp,q defined as follows. For $$z \in Z_{p,q}$$, write $$z = \zeta{}^t\zeta$$ and let $$w=\zeta^{-1}\epsilon_{p,q}{}^t\zeta^{-1}$$. Then $$w\in W_{p,q}$$, $$w$$ depends only on the $$O(p,q)$$-orbit of $$z$$, and the $$O(n)$$-orbit of $$w$$ is independent of the choice of $$\zeta$$. Conversely, for $$w\in W_{p,q}$$, write $$w = \eta \epsilon_{p,q}{}^t\eta$$ and let $$z = \eta^{-1}{}^t\eta^{-1}$$. Then $$z\in Z_{p,q}$$, $$z$$ depends only on the $$O(n)$$-orbit of $$w$$, and the $$O(p,q)$$-orbit of $$z$$ is independent of the choice of $$\eta$$. □ Proof First note that   x+ϵp+x+ϵq′=2x+1, so that $$2x+1$$ is automatically positive definite. This also follows from the given conditions on $$z$$, viz.   z+ϵp,q+z−ϵp,q=2z>0. Now, given $$x\in X_{p,q}$$, we have   2x+1+ϵp,q=2(x+ϵp)>0,2x+1−ϵp,q=2(x+ϵq′)>0, so that $$2x+1$$ lies in $$Z_{p,q}$$. Conversely, if $$z \in Z_{p,q}$$, then   12(z−1)+ϵp=12(z+ϵp,q)>0,12(z−1)+ϵq′=12(z−ϵp,q)>0, so that $$\frac12(z-1)$$ lies in $$X_{p,q}$$. This proves (a). To prove (b), we first note that $$w \in S_{p,q}$$, by construction. We have   1±w=1±ζ−1ϵp,qζ−1=ζ−1(ζtζ±ϵp,q)tζ−1=ζ−1(z±ϵp,q)tζ−1>0, so that $$w\in S_{p,q}\cap S_{(-1,1)}$$, as claimed. The other direction is analogous.■ Remark 4.7. It will be useful to note that, under the bijection of part (b),   2ndet(x+ϵp) =det(z+ϵp,q),2ndet(x+ϵq′) =det(z−ϵp,q), and   det(z±ϵp,q) =|det(w)|−1det(1±w). □ We can make one simplification in the inner integral in (4.10) as follows. Writing $$v = a{}^ta=a k {}^tk{}^ta$$ with $$k\in O(n)$$ and setting $$z=1+2x$$, as in (a) of the previous lemma, we have    ∫Sp,qe−2πtr(tcta−1a−1cz)|det(y)|sΦm(y)dy =det(a)2(s+ρ)∫Sp,qe−2πtr(tccz)|det(y)|sΦm(ak⋅y)dy. (4.11) Since the whole expression is independent of $$k$$, integrating over $$O(n)$$ has no effect, but bringing the $$O(n)$$ integration inside the $$S_{p,q}$$-integral, we have   ∫O(n)Φm(ak⋅y)dk=Φm(a⋅1n)Φm(y)=Φm(v)Φm(y), by Corollary XI.3.2 in [11]. Thus (4.11) is equal to   det(v)s+ρΦm(v)∫Sp,qe−2πtr(tccz)|det(y)|sΦm(y)dy. Noting that $$\mathrm{d}z = 2^{n\rho}\,\mathrm{d}x$$, for (4.10) we have   ω1ϵ(π(m(a))ϕℓ,s)p,q =(2π)nρ2−n(ρ−1)Γn(α)Γn(β)det(v)12(s+ρ)(2π)ns ×∫Zp.qdet(z+ϵp,q)α−ρdet(z−ϵp,q)β−ρ (×∑m≥0dm(−2πϵ)|m|(ρ)m(s+ρ)mΦm(v) ×∫Sp,qe−2πtr(tccz)|det(y)|sΦm(y)dy)dz. (4.12) We need some additional structural information. Let   Rp,q+={δ=diag(δ1,…,δn)∣δ1>⋯>δp>0,δn>⋯>δp+1>0}. There is then a map   O(n)×Rp,q+⟶Sp,q,(u,δ)↦u⋅δϵp,q=uδϵp,qtu=y (4.13) with open dense image, and, by TheoremVI.2.3 of [11],   dy=Ξp,q(δ)dδdu, where   Ξp,q(δ)=c00∏1≤i<j≤porp<i<j≤n|δi−δj|∏i≤p<j(δi+δj), and   dδ=dδ1…dδn. Here $$c_{00}$$ is a certain positive constant depending only on $$n$$. The map (4.13) is $$2^n$$ to $$1$$, due to the fact that the stabilizer in $$O(n)$$ of an element $$\delta\epsilon_{p,q}$$ is the diagonal subgroup, isomorphic to $$(\mu_2)^n$$. Let   Ap,q+={δ12∣δ∈Rp,q+}. Then we have a map   O(n)×Ap,q+×O(p,q)⟶G,(u,a,h)↦uah=g with open dense image, and a left invariant measure $$\mathrm{d}g$$ on $$G$$ has pullback   dg=det(g)−2ρΞp,q(δ)dδdudh, (4.14) where $$a^2=\delta$$ and $$\mathrm{d}h$$ is a Haar measure on $$H=O(p,q)$$. Let   Rp,q+0={δ∈Rp,q+∣δj∈(0,1) for all j }. Then (4.13) restricts to a map   O(n)×Rp,q+0⟶Wp,q with open dense image and we have an injection $$R^{+0}_{p,q} \hookrightarrow O(n)\backslash W_{p,q}$$. Similarly, we have a map   O(p,q)×Rp,q+0⟶Zp,q,(h,δ)↦h⋅δ−1=hδ−1th, with open dense image and an injection $$R^{+0}_{p,q}\hookrightarrow O(p,q)\backslash Z_{p,q}$$. We now return to (4.12) and write $$z = h\cdot \delta_w^{-1}$$ with $$h\in O(p,q)$$ so that   dz=Ξp,q(δw−1)d(δw−1)dh=det(δw)−2ρΞp,q(δw)dδwdh. The double integral becomes    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)−2ρΞp,q(δw) ×∫O(p,q)∫Sp,qe−2πtr(ch⋅δw−1)|det(y)|sΦm(c⋅ϵp,q,1n)dydδwdh. Writing $$g = ch$$, we have $$\mathrm{d}y\,\mathrm{d}h = (\det g)^{2\rho}\,\mathrm{d}g$$ and this becomes    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)−2ρΞp,q(δw) ×∫Ge−2πtr(g⋅δw−1)|det(g)|2sΦm(g⋅ϵp,q,1n)(detg)2ρdgdδw. Now we put $$g\delta_w^{\frac12}$$ for $$g$$ and have    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)s−ρΞp,q(δw) ×∫Ge−2πtr(gtg)|det(g)|2sΦm(g⋅δwϵp,q,1n)(detg)2ρdgdδw. This is    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)s−ρΞp,q(δw) ×∫Ge−2πtr(tgg)|det(g)|2sΦm(δwϵp,q,tgg)(detg)2ρdgdδw, and so, setting $$y^\vee = {}^tgg$$, we arrive at the expression    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)s−ρΞp,q(δw) ×∫Sn,0e−2πtr(y∨)det(y∨)sΦm(δwϵp,q,y∨)dy∨dδw. By the inversion formula, cf. Lemma XI.2.3 of [11], we have    ∫Sn,0e−2πtr(y∨)det(y∨)sΦm(δwϵp,q,y∨)dy∨ =Γm(s+ρ)(2π)−n(s+ρ)(2π)−|m|Φm(δwϵp,q,1n). Returning to the inner sum in (4.12) and canceling the gamma factor, we have   Γn(s+ρ)(2π)−n(s+ρ)∑m≥0(−ϵ)|m|dm(ρ)mΦm(v)Φm(δwϵp,q). (4.15) Now, for $$v=a{}^ta$$, we write   Φm(v)Φm(δwϵp,q)=∫O(n)Φm(ak⋅δwϵp,q)dk, so that (4.15) becomes the integral over $$O(n)$$ of   Γn(s+ρ)(2π)−n(s+ρ)∑m≥0(−ϵ)|m|dm(ρ)mΦm(ak⋅δwϵp,q) =Γn(s+ρ)(2π)−n(s+ρ)exp⁡(−ϵtr(ak⋅δwϵp,q)), via the standard expansion of $$\exp(\mathrm{tr}(z))$$, Proposition XII.1.3 (i) of [11]. But now the last three lines of (4.12) amount to   Γn(s+ρ)(2π)−n(s+ρ)∫O(n)∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρexp⁡(−ϵtr(ak⋅δwϵp,q)) ×det(δw)s−ρΞp,q(δw)dkdδw=Γn(s+ρ)(2π)−n(s+ρ)∫Wp,q|det(w)|−(α+β−2ρ)det(1+w)α−ρdet(1−w)β−ρexp⁡(−ϵtr(a⋅w)) ×|det(w)|s−ρdw. Here the exponent of $$|\det(w)|$$ is   2ρ−α−β+s−ρ=0.(!!!) Taking $$a$$ with $$a={}^ta$$, we get simply   Γn(s+ρ)(2π)−n(s+ρ)∫Wp,qdet(1+w)α−ρdet(1−w)β−ρexp⁡(−ϵtr(vw))dw, and, altogether:   ω1ϵ(π(m(a))ϕℓ,s)p,q =Bn(α,β)−12−n(ρ−1)det(v)12(s+ρ) ×∫Wp,qdet(1+w)α−ρdet(1−w)β−ρexp⁡(−ϵtr(vw))dw. (4.16) Summing over the signatures, we have   ω1ϵ(π(m(a))ϕs,ℓ) =Bn(α,β)−12−n(ρ−1)det(v)12(s+ρ) ×∫1±w>0det(1+w)α−ρdet(1−w)β−ρexp⁡(−ϵtr(vw))dw. In the integral here we put $$2r = 1+w$$ and obtain   2nsexp⁡(ϵtr(v))∫r>01−r>0exp⁡(−2ϵtr(vr))det(r)α−ρdet(1−r)β−ρdr. For $$\epsilon=-1$$, this gives   ω1ϵ(π(m(a))ϕs,ℓ)=2n(s−ρ+1)det(v)12(s+ρ)exp⁡(−tr(v))Mn(α,α+β;2v) and for $$\epsilon=+1$$, this is   2nsexp⁡(−tr(v))∫r>01−r>0exp⁡(tr(2v(1−r))det(r)α−ρdet(1−r)β−ρdr, and hence   ω1ϵ(π(m(a))ϕs,ℓ)=2n(s−ρ+1)det(v)12(s+ρ)exp⁡(−tr(v))Mn(β,α+β;2v). Finally, taking into account the scaling transformation used to eliminate $$T$$, we obtain the claimed expression. This completes the proof of Theorem 4.1. ■ 5 The $$\xi$$-Operator In this section, we construct the $$\xi$$-operator, analogous to that defined in [5, 6], in our present situation. This operator is a slight modification of the $$\bar\partial$$-operator and is best expressed in terms of differential forms and the Hodge $$*$$-operator for homogeneous vector bundles on the Siegel space $$\mathfrak H_n$$. Here, as before, we write $$\tau = u+iv$$, $$v>0$$, for an element of $$\mathfrak H_n$$. 5.1 Homogeneous bundles and differential forms For a representation $$(\mu,\mathcal V_\mu)$$ of $$K$$, let $$\mathcal L_\mu$$ be the homogeneous vector bundle   Lμ=(G×Vμ)/K⟶G/K=Hn. Here $$K$$ acts by $$(g,v)\cdot k = (gk, \mu(k)^{-1}v)$$, and the $$C^\infty$$-sections are given by   Γ(Hn,Lμ)≃[C∞(G)⊗Vμ]K,μ(k)ϕ(gk)=ϕ(g). (5.1) If $$\Gamma$$ is a discrete subgroup $$Sp_n(R)$$—the main cases of interest will be $$\Gamma\subset \mathrm {Sp}_n({\mathbb Z})$$ an arithmetic subgroup, the intersection of such a subgroup with $$N$$, or the trivial subgroup — we use the same notation for the quotient bundle on $$X=\Gamma\backslash \mathfrak H_n$$. ({Here we should use orbifolds/stacks.}) We write $$\mathcal L_r = \mathcal L_{\det^{-r}}$$ with $$\mathcal V_{\det^{-r}}={\mathbb C}(-r)$$, so that sections of $$\mathcal L_r$$ satisfy $$\phi(gk) = \det(\mathbf k)^r\,\phi(g)$$. The function $$j(g,i)^{-r}$$ defines a smooth section of $$\mathcal L_r$$ on $$\mathfrak H_n$$. If $$\phi$$ is any section, we can write   ϕ(g)=j(g,i)−rf(τ), where $$f(g(i)) = j(g,i)^{r}\,\phi(g)$$, and invariance of $$\phi$$ under left multiplication by an element $$\gamma\in \Gamma$$ is equivalent to the invariance of $$f$$ under the corresponding weight $$r$$ slash operator for $$\gamma$$. The Petersson metric on $$\mathcal L_r$$ is given by $$|\phi(g)|^2 = |f(\tau)|^2\,\det(v)^r$$. More generally, suppose that the representation $$\mu$$ of $$K$$ on $$\mathcal V_\mu$$ extends to a representation of $$K_{\mathbb C} \simeq \mathrm{GL}_n({\mathbb C})$$. Let $$J(g,\tau) = c\tau+d$$ be the canonical automorphy factor $$J: G\times \mathfrak H_n \longrightarrow K_{\mathbb C}$$. Note that   J(k,i)=A−iB=k¯=tk−1. A general smooth section of $$\mathcal L_\mu$$ on $$\mathfrak H_n$$ can be written as   ϕ(g)=μ(tJ(g,i))f(τ), where $$f$$ is a smooth $$\mathcal V_\mu$$-valued function of $$\mathfrak H_n$$. The left invariance of the section $$\phi$$ under $$\gamma\in G$$ is equivalent to the invariance   f(γτ)=μ(tJ(γ,τ)−1)f(τ). (5.2) Now suppose, moreover, that $$\langle \,{},{}\,\rangle_\mu$$ is a Hermitian inner product on $$\mathcal V_\mu$$ such that $$\mu(a)^* = \mu({}^t\bar a)$$ for all $$a\in \mathrm{GL}_n({\mathbb C})$$. Such an inner product is “admissible” in the terminology of [3, p. 47]. Then we can define the Petersson metric on $$\mathcal L_\mu$$ by   ||ϕ(g)||μ2=⟨ϕ(g),ϕ(g)⟩μ=⟨f(τ),μ(v−1)f(τ)⟩μ,v=ℑ(τ)=ℑ(g(i)). (5.3) Since   v(γ(τ))=t(cτ+d)−1v(cτ¯+d)−1, the right side of (5.3) is $$\gamma$$-invariant if $$f$$ satisfies (5.2). If $$(\lambda, F_\lambda)$$ is a finite-dimensional unitary representation of $$\Gamma$$, there is an associated flat bundle $$\mathcal F_\lambda$$ on $$X$$ defined by   Fλ=Γ∖(Hn×Fλ), with Hermitian metric given by the norm on $$F_\lambda$$. The bundle $$\Omega^{N}$$ of top-degree holomorphic differential forms on $$\mathfrak H_n$$ is $$\mathcal L_{n+1}$$. Here $$N=\frac12n(n+1)$$. Writing $$\tau = \sum_\alpha \tau_\alpha \, e_\alpha$$, we let   dμ(τ)=∧αdτα and note that   dμ(g(τ))=j(g,τ)−2ρdμ(τ),ρ=12(n+1). If $$\phi$$ is a section of $$\mathcal L_{n+1}$$, the corresponding section of $$\Omega^N$$ is   ϕ(g)j(g,i)2ρdμ(τ)=f(τ)dμ(τ). Let $$\mathcal E^{a,b}$$ be the bundle of differential forms of type $$(a,b)$$ on $$\mathfrak H_n$$. We use the same notation for the corresponding bundle on $$X$$. Let   g=k+p++p− be the Harish-Chandra decomposition of $${\mathfrak g} = \mathrm{Lie}(G)\otimes_\mathbb R{\mathbb C}$$. Then we have an isomorphism   Γ(Hn,Ea,b)=A(a,b)(Hn) ⟶∼ [C∞(G)⊗∧a(p+∗)⊗∧b(p−∗) ]K. (5.4) More explicit coordinates can be given as follows. Let $$S= \mathrm{Sym}_n(\mathbb R)$$ with basis $$e_\alpha$$ and dual basis $$e_\alpha^\vee$$ with respect to the trace form. ({Recall that we take $$e_{jj}$$, $$1\le j \le n$$ and $$e_{ij}+e_{ji}$$, $$1\le i<j\le n$$, as basis for $$S$$ with dual basis is $$e_{jj}$$ and $$\frac12(e_{ij}+e_{ji})$$.}) There are isomorphisms   p±:SC ⟶∼ p±,p±(X)=12(X±iX±iX−X). (5.5) Then we have a basis $$L_\alpha = p_-(e_\alpha^\vee)$$ for $${\frak p}_-$$, and we write $$\eta'_\alpha\in {\frak p}_-^*$$ for the dual basis. The operator on the right side of (5.4) corresponding to $$\bar\partial$$ is then   ∂¯=∑αp−(eα∨)⊗ηα′, (5.6) where $$\eta'_\alpha$$ acts on $$\wedge^\bullet({\frak p}^*)$$ by exterior multiplication. Suppose that $$E$$ is any Hermitian vector bundle on $$X$$, and let $$\nu: E \ {\overset{\sim}{\longrightarrow}}\ E^*$$ be the conjugate linear isomorphism determined by the Hermitian inner product. Recall that the Hodge $$*$$-operator gives a conjugate linear operator [27, Chapter V, Section 2],   ∗¯E:Ea,b⊗E⟶EN−a,N−b⊗E∗,α⊗h↦(∗α¯)⊗ν(h). Definition. For a Hermitian vector bundle $$E$$ on $$X$$, the $$\xi$$-operator is defined as   ξ=ξE=∗¯E∂¯:Γ(X,Ea,b⊗E) ⟶ Γ(X,EN−a,N−b−1⊗E∗). (5.7) If $$E=\mathcal F_\lambda\otimes \mathcal L_\mu$$ for a unitary flat bundle $$\mathcal F_\lambda$$ and for $$\mathcal L_\mu$$ with the Petersson metric defined by (5.3), then $$E^* \simeq \mathcal F_{\lambda^\vee} \otimes \mathcal L_{\mu^\vee}$$ where $$\lambda^\vee$$ and $$\mu^\vee$$ are the contragradients of $$\lambda$$ and $$\mu$$. For example, for an integer $$\kappa$$, a $$C^\infty$$-section $$f$$ of the bundle   Fλ⊗E0,N−1⊗Ln+1−κ (5.8) can be viewed as an $$\mathcal F_\lambda$$-valued $$(0,N-1)$$-form on $$\mathfrak H_n$$ of weight $$n+1-\kappa$$. Then $$\xi(f)$$ is a section of   Fλ∨⊗EN,0⊗Lκ−n−1 ≃ Fλ∨⊗Lκ. For $$n=1$$, this coincides with the $$\xi$$-operator defined in [6]. From now on, to simplify things slightly, we will omit the flat bundle $$\mathcal F_\lambda$$. It is useful to note that we have the diagram   $$\$$ and that the two maps on the right are given explicitly by (5.6) and   ∗¯:ϕ⊗x⊗ω⟼ϕ¯⊗ν(x)⊗∗ω¯, (5.9) where $$\nu:\mathcal V_\mu \ {\overset{\sim}{\longrightarrow}}\ \mathcal V_{\mu^\vee}$$ is the conjugate linear isomorphism determined by $$\langle \,{},{}\,\rangle_\mu$$. 5.2 Whittaker forms As explained earlier, we consider a version of these operators involving Whittaker forms. For $$T\in\mathrm{Sym}_n(\mathbb R)$$, recall that $$\mathcal W^{\mathrm{T}}(G)$$ is the space of smooth functions $$\phi$$ on $$G$$ such that $$\phi(n(b)g) = e(\mathrm{tr}(Tb))\,\phi(g)$$. Define the space of Whittaker forms valued in $$\mathcal L_\mu$$ as   W−T(Ea,b⊗Lμ) ⟶∼ [W−T(G)⊗Vμ⊗∧a(p+∗)⊗∧b(p−∗) ]K. (5.10) There is a corresponding $$\xi$$-operator described by the diagram   $$\$$ where the maps in the right column are given by (5.6) and (5.9). Our Whittaker functionals (3.17) provide a supply of elements in these spaces via the diagram   $$\$$ Here the maps in the left column are again given by (5.6) and (5.9). We can utilize the fact that the $$K$$-spectrum of $$I(s)$$ is multiplicity free to produce various examples. 5.3 Some particular vectors In the construction of Whittaker forms, we will be interested in the following functions in $$I(s)$$. The isomorphism (5.5) satisfies   Ad(k)p+(X)=p+(k⋅X),k⋅X=kXtk. (5.11) Similarly, $$p_-(X) = \overline{p_+(X)}$$ and   Ad(k)p−(X)=p−(k¯⋅X)=p−(tk−1⋅X). The trace pairing   ⟨p+(X),p−(Y)⟩=tr(XY) is then invariant under the adjoint action of $$K$$, so that $$\mathfrak p_{\pm}^* \simeq \mathfrak p_{\mp}$$ as $$K$$-modules. Note that   ∧N(p+) ⟶∼ C(2ρ) as $$K$$-modules, where $$N= n\rho = \dim \mathfrak p_{\pm}$$. For the fixed integer $$\kappa$$, let $$r= \kappa - n-1$$, and consider the space   [ I(s)⊗∧N−1(p−∗)⊗C(r) ]K. (5.12) Fixing a basis vector $$\bar\omega$$ for $$\wedge^N(\mathfrak p_-^*)$$, we have a pairing   ∧N−1(p−∗)⊗p−∗⟶∧N(p−∗) ⟶∼ C(2ρ), and hence an isomorphism   ∧N−1(p−∗) ⟶∼ p−⊗C(2ρ). (5.13) Thus   σ∨:=∧N−1(p−∗)⊗C(r)≃Symn(C)⊗C(κ). The vector $$v_0= 1_n\in \mathrm{Sym}_n({\mathbb C})$$ is $$O(n)$$-invariant, and so, via (5.13), we have the standard function   ϕs,σ(nm(a)k)=χ(det(a))|det(a)|s+ρdet(k)−κtkk (5.14) in (5.12). By (2.10), it is characterized by the invariance property   π(k)ϕs,σ=det(k)−κtk⋅ϕs,σ, together with the normalization $$\phi_{s,\sigma}(e) = 1_n$$. For generic $$s$$, the function $$\phi=\phi_{s,\sigma}$$ can also be obtained by applying a certain differential operator to $$\phi_{s,-\kappa}$$. Here we use the conventions described in more detail in Section 3.3. Let $$e_\alpha$$ be a basis for $$S= \mathrm{Sym}_n(\mathbb R)$$ and let $$e_\alpha^\vee$$ be the dual basis with respect to the trace form. Let   D=∑αp+(eα)⊗eα∨ ∈ p+⊗S⊂ U(g)⊗S. (5.15) This operator has the following invariance property. Lemma 5.1. For $$k\in K$$,   Ad(k)D=tk⋅D=tkDk.□ Proof We compute using (5.11)   Ad(k)D =∑αp+(k⋅eα)⊗eα∨=∑α∑βtr((k⋅eα)eβ∨)p+(eβ)⊗eα∨ =∑βp+(eβ)⊗∑αtr(eα(tk⋅eβ∨))eα∨=∑βp+(eβ)⊗tk⋅eβ∨=tk⋅D.■ Now consider the function $$\pi(D)\phi_{s,-\kappa} \in I(s)\otimes \mathrm{Sym}_n({\mathbb C})$$. For $$k\in K$$, we have   π(k)π(D)ϕs,−κ =π(Ad(k)D)π(k)ϕs,−κ =det(k)−κtk⋅π(D)ϕs,−κ. Thus $$\pi(D)\phi_{s,-\kappa}$$ is a multiple of $$\phi_{s,\sigma}$$, and it remains to calculate the constant of proportionality. Lemma 5.2.   π(D)ϕs,−κ=12(s+ρ−κ)ϕs,σ=α(s)ϕs,σ.□ Proof An easy calculation shows that   p+(X)ϕs,ℓ(e)=12(s+ρ+ℓ)tr(X). Therefore   Dϕs,−κ=12(s+ρ−κ)1n.■ In particular, for $$s_0 = \kappa-\rho$$, we have $$p_+(X)\,\phi_{s_0,-\kappa} =0$$, so that $$\phi_{s_0,-\kappa}$$ is a highest weight vector. But this was already clear since this vector is the generator of $$R(0,m+2)$$, that is, the image in $$I(s_0)$$ of the Gaussian for the negative definite space of dimension $$m+2$$, cf. Section 2.6. Thus, we have basis vectors $$\phi_{s,-\kappa}$$ and $$\phi_{s,\sigma}$$ in the one-dimensional spaces on the upper left side of the diagram   $$\$$ (5.16) 5.4 Calculation of $$\bar\partial$$ and $$\xi$$ We now compute the image of $$\phi_{s,\sigma}$$ under the operator $$\xi$$ on the left side of (5.16). Proposition 5.3.   ∂¯ϕs,σ=n(s−ρ+κ)ϕs,−κ⋅dμ(τ¯), and   ξ(ϕs,σ)=n(s¯−ρ+κ)ϕs¯,κ.□ Here, we are slightly abusing notation and writing $$\mathrm{d}\mu(\bar\tau)$$ for the basis element of $$\wedge^{N}(\mathfrak p_-^*)$$ arising as the restriction of this global form to the tangent space at $$i$$. Proof First we apply $$\bar\partial$$:   [ I(s)⊗∧N−1(p−∗)⊗C(n+1−κ) ]K⟶∂¯[ I(s)⊗∧N(p−∗)⊗C(n+1−κ) ]K, noting that both spaces are one dimensional. Using (5.6), and (5.15),   ∂¯ϕs,σ=α(s)−1∂¯⋅Dϕs,−κ=α(s)−1∑αp−(eα∨)p+(eα)ϕs,−κ⋅dμ(τ¯). The second-order operator occurring here has the following expression in terms of the Casimir operator (2.9). We have   ∑αp+(eα)p−(eα∨)+p−(eα∨)p+(eα) =2∑αp−(eα∨)p+(eα)+[p+(eα),p−(eα∨)] =ρH+2∑αp−(eα∨)p+(eα), where $$H = \sum_j H_j$$, for $$H_j$$ as in Section 2.4. On the other hand, a short calculation shows that $$C_{\mathfrak k}$$ acts by $$\frac18\,\kappa^2$$, whereas $$H$$ acts by $$-n\kappa$$. Thus we have   2∑αp−(eα∨)p+(eα)ϕs,−κ =(4n(C−Ck)−ρH)ϕs,−κ =12n(s2−ρ2+2ρκ−κ2)ϕs,−κ =12n(s−ρ+κ)(s+ρ−κ)ϕs,−κ. This gives the first identity. Then   ∗¯∂¯ϕs,σ=n(s¯−ρ+κ)ϕs¯,κ⋅dμ(τ), so the second identity is immediate. ■ 5.5 Vector-valued Whittaker functions Now we can apply the Whittaker functionals to obtain Whittaker forms on the right side of (5.16). Recall that $$f_{s,-\kappa}(g) = \omega_1^{-T}(\pi(m(g))\phi_{s,-\kappa})$$ and let   fs,σ(g)=ω1−T(π(m(g))ϕs,σ). (5.17) Then by Lemma 5.2,   α(s)fs,σ(g)=Dfs,−κ(g)=∑αp+(eα)fs,−κ(g)eα∨, (5.18) where $$\alpha(s) = \frac12(s-s_0) = \frac12(s+\rho-\kappa)$$. Thus $$\boldsymbol{f}_{s,\sigma}$$ is $$S$$-valued and, by Lemma 5.1, satisfies   fs,σ(gk)=det(k)−κtkfs,σ(g)k. (5.19) The scaling relation   fs,σ −T(m(c)g)=det(c)s+ρfs,σ −cTtc(g),c∈GLn(R)+, (5.20) holds for $$\boldsymbol{f}_{s,\sigma}=\boldsymbol{f}_{s,\sigma}^{\ -T}$$, where we include the normally omitted superscript. As a consequence of our constructions, we have obtained the following. Theorem 5.4. The Whittaker forms $$\boldsymbol{f}_{s,\sigma}$$ defined by (5.18) lie in the space   [W−T(G)⊗∧N−1(p−∗)⊗C(r) ]K. (i) The infinitesimal character of $$\boldsymbol{f}_{s,\sigma}$$ is $$\chi_{\lambda(s)+\rho_G}$$. In particular, for the Casimir operator $$C$$,   C⋅fs,σ=18(s+ρ)(s−ρ)fs,σ. (ii) For the $$\xi$$-operator,   ξ(fs,σ)=n(s¯−ρ+κ)fs,−κ¯. (iii) At $$s=s_0=\kappa-\rho$$,   ξ(fs0,σ)(g)=c(n,s0)WκT(g), where   WκT(n(b)m(a)k)=det(k)κdet(a)κe(tr(Tτ))=j(g,i)−κqT, (5.21)$$v= a{}^ta$$, $$\tau = b+{\rm i}v$$, and   c(n,s0)=2n(κ−2ρ+1). (Here note that, although $$\alpha=\alpha(s_0)=0$$, the calculations of Section 5.6 show that the right side of (5.18) is divisible by $$\alpha$$ and that $$\boldsymbol{f}_{s_0,\sigma}$$ is given by a nice convergent power series in $$v$$.) □ Proof The first two statements follow from the commutativity of (5.16) and the corresponding results for the vectors on the left side. In particular,   ξ(fs,σ)=n(s¯−ρ+κ)fs,−κ¯. By (4.5), we have   fs,−κ¯(n(b)m(a)k)=c(n,s¯)qTdet(v)12(s¯+ρ)det(k)κMn(α,α+β;4πcvtc)¯. But, at $$s=s_0=\kappa-\rho$$, $$\alpha=0$$ and $$M_n(0,s_0;4\pi cv{}^tc)=1$$, so we get the claimed expression. ■ 5.6 Some explicit formulas In this section, we calculate the function $$\boldsymbol{f}_{s,\sigma}$$ more explicitly. In view of the scaling relation (5.20), it suffices to consider the case $$T=1_n$$. Recall that   Mn(α,α+β;z)=∑m≥0(α)m(s+ρ)mdm(ρ)mΦm(z), where $$\mathbf{m} = (m_1,\ldots,m_n)$$ with $$m_1\ge m_2\ge \cdots\ge m_m\ge0$$. Since   (α)m=(α)m1(α−12)m2…(α−12(n−1))mn, we have $$(\alpha)_0 =1$$ and $$(0)_{\mathbf{m}}=0$$ for $$\mathbf{m} \ne0$$. In particular, $$M_n(0,\beta;z) = 1$$. For $$\mathbf{m}\ne 0$$, let   [0]m=(α−1(α)m))|α=0. (5.22) Then   [0]m={(1)m1−1(−12)m2(−12)r(r+1)!,if m=(m1,m2,1,…,1,0,…,0),0,otherwise.  Here, in the first case, $$\mathbf{m}>0$$ with $$m_3\le 1$$ and with a string of $$r$$$$1$$’s following $$m_2$$. With this notation, we can state our result. Proposition 5.5. (i) For $$a\in \mathrm{GL}_n(\mathbb R)^+$$,   fs,σ(m(a)) =c(n,s)e−2πtr(v)det(v)12(s+ρ)Bn(α,β)−1 ×∫t>01−t>0(1n+α(s)−14πtata)e4πtr(tv)det(t)α−ρdet(1−t)β−ρdt. (5.23) Recall that $$c(n,s) = 2^{n(s-\rho+1)}$$. (ii) For $$n=1$$,   fs0,σ(m(a))=2κ−14πe2πvv12κ+1∫01e−4πtvtκ−1dt. (iii) For $$n\ge 2$$,   fs0,σ(m(a))=2n(κ−n)e−2πtr(v)det(v)12κ( 1n+∑m>0[0]m(κ)mdm(ρ)mta∂∂v{ Φm(4πv) }a). Here $$v = a{}^ta$$, as usual. □ Remark 5.6. In the case $$n=1$$, we recover the basic formula from [5]. Notice that, in the case $$n\ge 2$$, we do not yet have a complete evaluation of $$\boldsymbol{f}_{s_0,\sigma}(m(a))$$; it would be interesting to have a nicer closed formula. □ Proof Recalling (5.18), we begin by computing some derivatives. Lemma 5.7. For $$X\in S$$,   p+(X)fs,ℓ(m(a))=12(DX+ℓtr(X)+4πtr(aXta))⋅fs,ℓ(m(a)). Here, for a function $$h$$ on $$\mathrm{GL}_n(\mathbb R)$$,   DXh(a)=ddt(h(aexp⁡(tX)) )|t=0. In particular, for a function of $$v = a{}^ta$$,   DX=2tr(aXta∂∂v).□ This follows from a simple direct calculation. Then, recalling the expression of the function $$f_{s,\ell}$$ given in Theorem 4.1, we have the nice expression for our Whittaker form   α(s)fs,σ(m(a)) =∑αp+(eα)fs,−κ(m(a))eα∨ =c(n,s)e−2πtr(v)det(v)12(s+ρ)Bn(α,β)−1 ×∫t>01−t>0(α(s)1n+4πtata)e4πtr(tv)det(t)α−ρdet(1−t)β−ρdt. (5.24) This gives statement (i). When $$n=1$$, (5.24) amounts to   fs,σ(m(a)) =c(1,s)e−2πvv12(s+1)α(s)−1B(α,β)−1 ×∫01(α(s)+4πtv)e4πtvtα−1(1−t)β−1dt. (5.25) Lemma 5.8.   α−1B(α,β)−1∫01(α(s)+4πtv)e4πtvtα−1(1−t)β−1dt=M(α+1,s+1;4πv).□ Proof Initially, we have   α−1B(α,β)−1∫01(α(s)+4πtv)e4πtvtα−1(1−t)β−1dt =M(α,s+1;4πv)+α−1v∂∂v{M(α,s+1;4πv)}. But by 13.4.10 of [1],   4πvM′(α,s+1;4πv)=−αM(α,s+1;4πv)+αM(α+1,s+1;4πv).■ Thus   fs,σ(m(a))=2se−2πvv12(s+1)M(α+1,s+1;4πv). This agrees with the Whittaker form for $$m=1$$ in [5], up to a simple factor. To evaluate at $$s=s_0$$ we return to the original expression (5.25). Since   α−1B(α,β)−1=Γ(s+1)Γ(α+1)Γ(β), we can plug in $$s_0= \kappa-1$$ so that $$\alpha_0=0$$ and $$\beta_0= \kappa$$, and obtain the following expression   fs0,σ(g) =c0e−2πvv12κ∫01(4πtv)e4πtvt−1(1−t)κ−1dt =c04πe−2πvv12κ+1∫01e4πtv(1−t)κ−1dt =c04πe2πvv12κ+1∫01e−4πtvtκ−1dt =c04πe2πvv12κ+1[ (4πv)−κΓ(κ)−∫1∞e−4πtvtκ−1dt ], where $$c_0 = c(1,s_0) = 2^{\kappa-1}$$. The last expression here is in some ways more enlightening, although we do not record it in the statement (ii) of the proposition. Finally, for general $$n\ge 2$$, we want to evaluate (5.23) at $$s_0= \kappa-\rho$$. The term associated with $$1_n$$ is given by the scalar matrix   c(n,s)e−2πtr(v)det(v)12(s+ρ)Bn(α,β)−1 ×∫t>01−t>0e4πtr(tv)det(t)α−ρdet(1−t)β−ρdt⋅1n =c(n,s)e−2πtr(v)det(v)12(s+ρ)Mn(α,α+β;4πv)⋅1n. Since $$M_n(0,\kappa;z)=1$$, we obtain a contribution   2n(κ−n)e−2πtr(v)det(v)12κ⋅1n. The other contribution is the value of   c(n,s)e−2πtr(v)det(v)12(s+ρ) ×ta( α−1Bn(α,β)−1∫t>01−t>04πte4πtr(tv)det(t)α−ρdet(1−t)β−ρdt )a at $$s_0$$. The inner integral here is just   α−1∂∂v{ Mn(α,α+β;4πv) }=∑m>0α−1(α)m(s+ρ)mdm(ρ)m∂∂v{ Φm(4πv) }, where we can omit the term $$\mathbf{m}=0$$ since $$\Phi_{0}(z)=1$$ is killed by $$\partial/\partial v$$. For $$\mathbf{m}\ne 0$$, $$\alpha^{-1}(\alpha)_{\mathbf{m}}$$ is finite at $$\alpha=0$$ and vanishes if $$m_3>1$$. With the notation explained in (5.22), we arrive at the expression given in (iii). ■ 6 A Global Construction In this section, we define the space of Whittaker forms and discuss the “global” $$\xi$$-operator. For simplicity, we restrict to the case of $$\Gamma= \mathrm {Sp}_n({\mathbb Z})$$ and a positive even integer $$\kappa$$. For $$T\in \mathrm{Sym}_n({\mathbb Z})^\vee_{>0}$$, we consider the basic Whittaker form   fs,σ −T∈[W−T(G)⊗∧N−1(p−∗)⊗C(κ−n−1) ]K. (6.1) These forms are the analogues of those considered in Section 4 of [5] in the case of $$\mathrm{SL}_2(\mathbb R)$$, that is, $$n=1$$. By construction, they are invariant under the translation subgroup $$\Gamma_\infty^u$$ of $$\mathrm {Sp}_n({\mathbb Z})$$. Setting $$s=s_0$$, we have Whittaker forms $$\boldsymbol{f}_{s_0,\sigma}^{\ -T}$$ satisfying   C⋅fs0,σ −T=18κ(κ−n−1)fs0,σ −T, where $$C$$ is the Casimir operator for $$G$$. Let   Hn+1−κ(G) ⊂ [C∞(G)⊗∧N−1(p−∗)⊗C(κ−n−1) ]K be the subspace spanned by these forms as $$T$$ varies over $$\mathrm{Sym}_n({\mathbb Z})^\vee_{>0}$$. Clearly, the basic forms $$\boldsymbol{f}^{\ -T}_{s_0,\sigma}$$ are linearly independent. Let $$\mathbb M_\kappa(G)$$ be the subspace of $$C^\infty(G)$$ spanned by the functions $$W_T^{\kappa}$$ given by (5.21) as $$T$$ varies over $$\mathrm{Sym}_n({\mathbb Z})^\vee_{>0}$$. By (iii) of Theorem 5.4, the $$\xi$$-operator induces an isomorphism   ξ:Hn+1−κ(G) ⟶∼ Mκ(G). On the other hand, if $$\kappa > 2n$$, the classical theory of Poincaré series [19] implies that the series   PκT(g)=∑Γ∞u∖ΓWκT(γg) converges absolutely and uniformly on compact subsets of $$G$$. The resulting functions are “holomorphic” cusp forms and span the space $$S_\kappa(\Gamma)$$ of such forms. Thus we have constructed a diagram   $$\$$ (6.2) For example, assume that $$S_\kappa(\Gamma)$$ is one-dimensional and let $$\chi$$ be a generator (this is, e.g., the case for $$n=2$$ and $$\kappa=10$$ or $$12$$, where $$S_\kappa(\Gamma)$$ is spanned by the Igusa cusp form $$\chi_{10}$$ or $$\chi_{12}$$). Then, using the fundamental formula (7) of [19], it is easily seen that   ξΓ(fs0,σ −T)=A⋅(detT)n+12−κε(T)aT(χ)¯⋅χ‖χ‖2, where $$A$$ is a non-zero constant independent of $$T$$, and $$\varepsilon(T)$$ is the order of the stabilizer of $$T$$ in $$\mathrm{GL}_n({\mathbb Z})$$. Moreover, $$a_T(\chi)$$ denotes the $$T$$th Fourier coefficient of $$\chi$$, and $$\|\chi\|$$ its Petersson norm. Funding This work was supported by an Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant and Deutsche Forschungsgemeinshaft (DFG) grant BR-2163/4-1 within the research unit Symmetry, Geometry, and Arithmetic. Acknowledgements This study is part of an ongoing joint project begun during a visit by the third author to Darmstadt in December of 2011. The third author would like to acknowledge the support of an Oberwolfach Simons Visiting Professorship for a two-week visit to Darmstadt in connection with the Oberwolfach meeting on Modular Forms in April of 2014 as well as a Simons Visiting Professorship at MSRI, August–December 2014 for the Program on New Geometric Methods in Number Theory and Automorphic Forms. He also benefited from several additional productive visits to Darmstadt in 2012–2013. All authors benefited from stays at the ESI in Vienna during the program on Arithmetic Geometry and Automorphic Representations in April–May 2015, and much appreciate the excellent and congenial working conditions at all of these institutions. Finally, the authors would like to thank the referee for useful comments which clarified the exposition. Appendix A.1 Notation We summarize the slight variation of the notation from [11, 22] used in this paper. In particular, we specialize to the case of the formally real Jordan algebra $$S = \mathrm{Sym}_n(\mathbb R)$$. Here is a list of notation:   G =Spn(R),S =Symn(R). For $$g\in \mathrm{GL}_n({\mathbb C})$$ and $$x\in S_{\mathbb C}= \mathrm{Sym}_n({\mathbb C})$$, $$g\cdot x = g x \,{}^tg$$.   ρ =12(n+1)P(SC) =polynomial functions,ℓ(g)f(z)=f(g−1⋅z),Δj(z) =principal j×j-minor of z∈SC, Δn(z)=det(z) ,m =(m1≥m2≥⋯≥mn),mj∈Z,  Δm(z) =Δ1(z)m1−m2…Δj(z)mn−1−mnΔn(z)mn,Pm(SC) =subspace generated by GLn(C)-translates of Δm, m≥0,dm =dimC⁡Pm(SC),Φm(z) =∫O(n)Δm(k⋅z)dk,Γn(s) =(2π)14n(n−1)Γ(s)Γ(s−12)…Γ(s−12(n−1)),Bn(α,β) =Γn(α)Γn(β)Γn(α+β),Γm(λ) =Γn(λ+m)=(2π)14n(n−1)∏i=1nΓ(λ+mi−12(i−1)),Γm(λ) =(λ)mΓn(λ),(λ)m =∏i=1n(λ−12(i−1))mi,(s)m =s(s+1)…(s+m−1)=Γ(s+m)Γ(s). The notation $$\Gamma_{\mathbf m}(\lambda)$$ does not seem to be standard, but it is frequently convenient. The function $$\Phi_{\mathbf m}(z,w)$$ on $$S_{\mathbb C}\times S_{\mathbb C}$$ defined in [22, Lemma 1.11] is characterized by the following two properties: (i) $$\Phi_{\mathbf m}(z,1_n) = \Phi_{\mathbf m}(z).$$ (ii) For $$a\in \mathrm{GL}_n({\mathbb C})$$,   Φm(a⋅z,w)=Φm(z,ta¯⋅w). For the trace form $$\langle \,{x},{y}\,\rangle = \mathrm{tr}(xy)$$ on $$S$$ and the standard basis   {eα}={eii,eij+eji∣1≤i≤n,i<j}, the dual basis is   {eα∨}={eii,12(eij+eji)∣1≤i≤n,i<j}. At several points in the calculations, we need the following inversion formula [11, Chapter XI, Lemma XI.2.3]. For $$p\in \mathcal P_{\mathbf m}(S_{\mathbb C})$$, and $$\mathrm{Re}(\lambda)>\rho-1$$,   ∫x>0e−tr(xy)p(x)det(x)λ−ρdx=Γn(λ+m)det(y)−λp(y−1). We also recall that   etr(zw¯)=∑m≥0dm(ρ)mΦm(z,w), and that, for $$\lambda\in{\mathbb C}$$ with $$(\lambda)_{\mathbf m}\ne 0$$ for all $$\mathbf m\ge 0$$, $$z$$, $$w\in S_{\mathbb C}$$ the J-Bessel function is defined by [22, p.818],   Jλ(z,w)=∑m≥0(−1)|m|dm(λ)m(ρ)mΦm(z,w). A.2 An estimate The following classical estimate will be useful. Lemma A.2. (i) For $$\mathrm{Re}(\alpha)>\rho$$ and $$\mathrm{Re}(\beta-\alpha)>\rho$$,   |1F1(α,β;v)|≤etr(v)det(v)Re(α−β)Γn(Re(β−α))|Bn(α,β−α)|−1. (ii) Suppose that $$\alpha$$ and $$\beta$$ are real with $$\alpha>\rho$$ and $$\beta-\alpha>\rho$$. Then for any $$\epsilon$$ with $$0<\epsilon<1$$,   |1F1(α,β;v)|≥Cϵe(1−ϵ)tr(v), where $$C_\epsilon>0$$ depends on $$\epsilon$$, $$\alpha$$, and $$\beta$$ and can be taken uniformly for $$\alpha$$ and $$\beta$$ in a compact set.□ Proof Suppose that $$v = a^2$$ where $$a={}^ta$$, and, using (4.3), consider   1F1(α,β;v) =Bn(α,β−α)−1∫t>01−t>0etr(vt)det(t)α−ρdet(1−t)β−α−ρdt =Bn(α,β−α)−1etr(v)∫t>01−t>0e−tr(v(1−t))det(t)α−ρdet(1−t)β−α−ρdt =Bn(α,β−α)−1etr(v)∫r>01−r>0e−tr(vr)det(1−r)α−ρdet(r)β−α−ρdr =Bn(α,β−α)−1etr(v)det(v)α−β∫r>0v−r>0e−tr(r)det(1−a−1ra−1)α−ρdet(r)β−α−ρdr. Now $$1\ge a^{-1}ra^{-1} >0$$ so that, for $$\mathrm{Re}(\alpha)>\rho$$, the factor $$\det(1-a^{-1}ra^{-1})^{\alpha-\rho}$$ lies in $$(0,1)$$ and we have   |1F1(α,β;v)|≤|Bn(α,β−α)|−1etr(v)det(v)Re(α−β)∫r>0v−r>0e−tr(r)det(r)Re(β−α)−ρdr. The integral here is bounded by   ∫r>0e−tr(r)det(r)Re(β−α)−ρdr=Γn(Re(β−α)). For the lower bound, we have   |1F1(α,β;v)| ≥|Bn(α,β−α)|−1etr(v)∫r>12ϵ1nϵ1n−r>0e−tr(vr)det(1−r)α−ρdet(r)β−α−ρdr ≥|Bn(α,β−α)|−1e(1−ϵ)tr(v)∫r>12ϵ1nϵ1n−r>0det(1−r)α−ρdet(r)β−α−ρdr.■ References [1] Abramowitz M. and Stegun I. A. Handbook of Mathematical Functions, Applied Mathematics Series 55, 10th printing,  National Bureau of Standards, Washington, D.C., U.S. Government Printing Office, 1972. http://people.math.sfu.ca/~cbm/aands/. Google Scholar CrossRef Search ADS   [2] Borcherds R. “Automorphic forms with singularities on Grassmanians.” Inventiones Mathematicae  132 ( 1998): 491– 562. Google Scholar CrossRef Search ADS   [3] Borel A. and Wallach N. Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Annals of Mathematics Studies, 94.  Princeton NJ, Princeton University Press, 1980. Google Scholar CrossRef Search ADS   [4] Bringmann K. and Ono K. “Dyson’s ranks and Maass forms.” Annals of Mathematics  171, no. 2 ( 2010): 419– 49. 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# Degenerate Whittaker functions for $$\mathrm{Sp}_n(\mathbb{R})$$

, Volume 2018 (1) – Jan 1, 2018
56 pages

/lp/ou_press/degenerate-whittaker-functions-for-mathrm-sp-n-mathbb-r-aO0Xqi5Rqv
Publisher
Oxford University Press
ISSN
1073-7928
eISSN
1687-0247
D.O.I.
10.1093/imrn/rnw218
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### Abstract

Abstract In this paper, we construct Whittaker functions with exponential growth for the degenerate principal series of the symplectic group of genus $$n$$ induced from the Siegel parabolic subgroup. This is achieved by explicitly constructing a certain Goodman–Wallach operator which yields an intertwining map from the degenerate principal series to the space of Whittaker functions, and by evaluating it on weight-$$\ell$$ standard sections. We define a differential operator on such Whittaker functions which can be viewed as generalization of the $$\xi$$-operator on harmonic Maass forms for $$\mathrm{SL}_2(\mathbb{R})$$. 1 Introduction The standard theory of automorphic forms focuses on the spectral decomposition of the space $$L^2(\Gamma\backslash G)$$ where $$G$$ is a connected real semi-simple Lie group and $$\Gamma$$ is a discrete subgroup of finite co-volume. This analysis involves functions of rapid decay, for example, cusp forms, or of moderate growth, for example, Eisenstein series. Classically, in the case of $$G=\mathrm{SL}_2(\mathbb R)$$ and $$\Gamma$$ commensurable to $$\mathrm{SL}_2({\mathbb Z})$$, the Fourier expansions of such functions involve the solution of the Whittaker ordinary differential equation that decays exponentially at infinity; this solution is uniquely characterized by this decay. In general, the uniqueness of the Whittaker model and the associated smooth Whittaker functional, Jacquet’s functional, plays a fundamental role and is the subject of a vast literature. Other Whittaker functionals and the associated “bad” Whittaker functions, which grow exponentially at infinity, have played a less prominent role in the theory of automorphic forms. An important exception to this is the work of Miatello and Wallach [21] where, for $$G$$ semi-simple of split rank $$1$$, a theory of Poincaré series constructed from such Whittaker functions is developed. More recently, in the case of $$G=\mathrm{SL}_2(\mathbb R)$$ and $$\Gamma$$ a congruence subgroup of $$\mathrm{SL}_2({\mathbb Z})$$, the space of (vector-valued) weak Maass forms—Maass forms that are allowed to grow exponentially at the cusps—and its subspace of harmonic weak Maass forms, analytic functions annihilated by the weight $$k$$ Laplacian, have been shown to have interesting and important arithmetic applications [7, 8, 10]. For example, the weakly holomorphic modular forms are the input for Borcherds celebrated construction of meromorphic modular forms with product formulas [2] and, more generally, harmonic Maass forms can be used to construct Arakelov-type Green functions for special divisors on orthogonal and unitary Shimura varieties [6, 8]. The harmonic weak Maass forms of negative (or low) weight play a role in the theory of mock modular forms and their relatives [4, 28, 29]. In particular, they are linked to holomorphic modular forms of a complementary positive weight by means of the $$\xi$$-operator introduced in [5, 6]. There are serious obstacles to extending these results to groups of higher rank. For example, the Koecher principle asserts that, for $$\Gamma$$ irreducible in $$G$$ of Hermitian type of reduced rank $${>}1$$, any holomorphic modular form on the associated bounded symmetric domain “extends holomorphically” to the cusps, that is, the notions of holomorphic modular form and weakly holomorphic modular form coincide. More generally, Miatello and Wallach [21, Section 5] conjecture that the same phenomenon occurs for general automorphic forms. Specifically, they conjecture that, for $$\Gamma$$ irreducible in $$G$$ of real reduced rank $${>}1$$, a smooth function $$f$$ on $$\Gamma\backslash G$$ that is $$K$$-finite and an eigenfunction of the center of the universal enveloping algebra of $$\mathfrak g =\mathrm{Lie}(G)_{\mathbb C}$$ is automatically of moderate growth. They prove this conjecture in the case of $$SO(n,1)$$ over a totally real field. However, in [5], the first author has shown that for $$G=\mathrm{SL}_2(\mathbb R)^d$$ and $$\Gamma$$ an arithmetic subgroup $$\Gamma$$ of $$\mathrm{SL}_2(O_{k})$$, where $$O_{k}$$ is the ring of integers in a totally real field $${k}$$ with $$|{k}:{\mathbb Q}|=d>1$$, it is possible to replace the non-existent space of harmonic weak Maass forms with a certain space of Whittaker functions. These Whittaker functions are invariant only under the unipotent subgroup $$\Gamma_\infty^u$$ of $$\Gamma_\infty$$ and the associated Poincaré series do not converge. Nevertheless, it is shown in [5] that they are linked to holomorphic Hilbert modular cusps forms via a $$\xi$$-operator and provide an adequate input for a Borcherds type construction. This suggests that it would be fruitful to consider analogous Whittaker functions for more general groups. The goal of this paper is to construct “bad” Whittaker functions for the degenerate principal series $$I(s)$$ induced from a character of the Levi factor $$M=\mathrm{GL}_n(\mathbb R)$$ of the Siegel parabolic $$P=MN$$ of $$G = \mathrm {Sp}_n(\mathbb R)$$. For $$s\in{\mathbb C}$$, let $$I(s) = I(s,\chi)$$ be the space of smooth of $$K$$-finite functions $$\phi$$ on $$G$$ such that   ϕ(n(b)m(a)g)=χ(deta)|det(a)|s+ρϕ(g),m(a)=(ata−1), n(b)=(1b1), (1.1)$$a\in \mathrm{GL}_n(\mathbb R)$$, $$b\in S:=\mathrm{Sym}_n(\mathbb R)$$, $$\chi(t) = \mathrm{sgn}(t)^\nu$$, $$\nu=0,$$$$1$$, $$\rho = \frac12(n+1)$$. Then $$I(s)$$ is a $$(\mathfrak g,K)$$-module, where $$\mathfrak g = \mathrm{Lie}(G)_{\mathbb C}$$ and $$K\simeq U(n)$$, $$k\mapsto\mathbf k$$. For $$T\in \mathrm{Sym}_n(\mathbb R)$$ with $$\det(T)\ne0$$, an algebraic Whittaker functional of type $$T$$ is an element $$\boldsymbol{\omega}^{\mathrm{T}}\in I(s)^* = \mathrm{Hom}_{\mathbb C}(I(s), {\mathbb C})$$ such that   ωT(n(X)ϕ)=2πitr(TX)ωT(ϕ),n(X)=(X0)∈g,X∈SC=Symn(C). (1.2) Such a functional determines a $$(\mathfrak g,K)$$-intertwining map   ωT:I(s)⟶WT(G),ωT(ϕ)(g)=ωT(π(g)ϕ), (1.3) where $$\mathcal W^{\mathrm{T}}(G)$$ is the space of smooth functions $$f$$ on $$G$$ such that   f(n(b)g)=e(tr(Tb))f(g). (1.4) Here $${e}(t) = \mathrm{e}^{2\pi i t}$$. The resulting generalized Whittaker functions are right $$K$$-finite and real analytic on $$G$$. Conversely, such an intertwining map (1.3) gives rise to a Whittaker functional $$\boldsymbol{\omega}^{\mathrm{T}}(\phi) = \omega^{\mathrm{T}}(\phi)(e)$$. One intertwining map is given by the integral   WT(g,s;ϕ)=∫Sϕ(w_n(b)g)e(−tr(Tb))db,w_=(1−1), which converges absolutely for $$\mathrm{Re}(s)>\rho$$ and has a meromorphic analytic continuation in $$s$$. For example, for $$\phi = \phi_{s,\ell}$$, the (unique) function in $$I(s)$$ such that $$\phi_{s,\ell}(k) = \det(\mathbf k)^\ell$$,   WT(n(b)m(a)k,s;ϕs,ℓ)=χ(det(a))|det(a)|s+ρe(tr(Tb))det(k)ℓξ(v,T;α′,β′), (1.5) where   ξ(v,T;α′,β′)=∫Sdet(b+iv)−α′det(b−iv)−β′e(−tr(Tb))db, (1.6) with $$v=a{}^ta$$, $$\alpha'= \frac12(s+\rho+\ell)$$ and $$\beta' = \frac12(s+\rho-\ell)$$, is the confluent hypergeometric function of matrix argument studied by Shimura [23]. If $$\epsilon\,T>0$$ with $$\epsilon=\pm1$$, then (1.6) can be written as   i−nℓ2−n(ρ−1)(2π)n(s+ρ)Γn(α)−1Γn(β)−1|det(T)|s ×e−2πϵtr(Tv)∫t>0e−2πtr(cvtct)det(t)α−ρdet(t+1)β−ρdt, (1.7) where $$\alpha=\frac12(s+\rho-\epsilon\ell)$$ and $$\beta=\frac12(s+\rho+\epsilon\ell)$$ and $$\epsilon\,T = c\,{}^tc$$. For notation not explained here see Section 2.1 and the Appendix. The Whittaker function (1.5), which decays exponentially as the trace of $$v$$ goes to infinity, plays a key role in many applications. The corresponding Whittaker functional on $$I(s)$$ is characterized, among all algebraic Whittaker functionals, by the fact that it extends to a continuous functional on the space $$I^{\mathrm{ sm}}(s)$$ of smooth functions on $$G$$ satisfying (1.1). In the general theory, such Jacquet functionals and the resulting good Whittaker functions have been studied very extensively, cf. [24] and the literature discussed there. To construct other Whittaker functionals, we apply Goodman–Wallach operators to conical vectors in $$I(s)^*$$. Such conical vectors correspond to embeddings into induced representations. The two relevant ones in our situation are given by   c1(ϕ)=ϕ(e), (1.8) and, for $$\mathrm{Re}(s)>\rho$$,   cw_(ϕ)=(A(s,w_)ϕ)(e)=∫Sϕ(w_n(b))db, (1.9) where $$A(s,\underline{w}):I(s) \longrightarrow I(-s)$$ is the intertwining operator defined by (3.3), with corresponding embeddings the identity map and $$A(s,\underline{w})$$, respectively. Matumoto’s generalization [20] of the results of [12] applies in our situation. Let   N¯={n−(x)∣x∈S},n−(x)=(1x1), (1.10) and let $$\bar{\mathfrak n} = \mathrm{Lie}(\bar N)_{\mathbb C}$$. Since $$\bar{\mathfrak n}$$ is abelian, $$U(\bar{\mathfrak n}) = S(\bar{\mathfrak n})$$, and the completion   S(n¯)[n¯]=lim⟵rS(n¯)/n¯rS(n¯) (1.11) is the ring of formal power series in elements of $$\bar{\mathfrak n}$$. The action of $$U(\bar{\mathfrak n})$$ on $$I(s)^*$$ extends to an action of $$S(\bar{\mathfrak n})_{[\bar{\mathfrak n}]}$$. (Here it is essential that we have taken the dual $$I (s)^*$$ of the $$K$$-finite vectors $$I (s)$$.) Then, by the results of [20], there are elements   gwsT,gw−sT∈S(n¯)[n¯] such that   ω1T:=gwsT⋅c1,ωw_T:=gw−sT⋅cw_ are Whittaker functionals of type $$T$$. (Here it is essential that we have taken the dual $$I(s)^*$$ of the $$K$$-finite vectors $$I(s)$$.) Following [12] one realizes the representation $$I(s)$$ on a space of functions on $$\bar N$$, where the Goodman–Wallach operator is given by a differential operator of infinite order. Taking advantage of the fact that $$\bar N$$ is abelian and passing to the Fourier transform, this operator is realized as multiplication by an analytic function. In the case of $$\mathrm{SL}_2(\mathbb R)$$, this function is given explicitly in the introduction to [12], where the corresponding formal power series in $$S(\bar{\mathfrak n})_{[\bar{\mathfrak n}]}$$ is determined by a simple recursion relation. For $$n\ge2$$, it does not seem feasible to apply this method, and, as far as we could see, explicit formulas for these kernel functions do not exist in the literature. Our first main result is that, in analogy to the case of $$\mathrm{SL}_2(\mathbb R)$$ given in [12], the kernel function for the Goodman–Wallach operator $$\mathrm{gw}_s^{\mathrm{T}}$$ for any $$n$$ is given explicitly by a Bessel function, now of a matrix argument. We now describe the result in more detail. For $$\phi\in I(s)$$, define a function $$\Psi(\phi)$$ on $$\bar N$$ by   Ψ(x;ϕ)=ϕ(n−(x)), (1.12) and let   Ψ^(y;ϕ)=∫Se(tr(xy))Ψ(x;ϕ)dx (1.13) be its Fourier transform. In this model of $$I(s)$$, the conical functional $$c_1$$ is given by   c1(ϕ)=Ψ(0;ϕ)=∫SΨ^(y;ϕ)dy. (1.14) To define the relevant hypergeometric function of matrix argument, we use the notation and results of [11] and [22]. For $$z$$ and $$w\in S_{\mathbb C}=\mathrm{Sym}_n({\mathbb C})$$ and for $$\mathbf m= (m_1,\ldots,m_n)$$, with integers $$m_j$$ with $$m_1\ge m_2\ge \ldots \ge m_n$$, let $$\Phi_\mathbf{m}(z)$$ be the spherical polynomial, and let $$\Phi_\mathbf{m}(z,w)$$ be its “bi-variant” version. For further explanation and notation, see the Appendix. Following [22], define the hypergeometric function   GWs(z,w):=∑m≥0(−1)|m|dm(s+ρ)m(ρ)mΦm(z,w). (1.15) This is a Bessel function of matrix argument, [11, 13]. Theorem A. The kernel for the Goodman–Wallach operator is given by $$\mathrm{GW}_s(\cdot, 2\pi T)$$. More precisely, for $$\phi\in I(s)$$,   ω1T(ϕ)=∫SGWs(2πy,2πT)Ψ^(y;ϕ)dy. □ Thus, the corresponding Whittaker function is   ω1T(g;ϕ)=ω1T(π(g)ϕ)=∫SGWs(2πy,2πT)Ψ^(y;π(g)ϕ)dy. Our second main result is an evaluation of the Whittaker function for $$\phi= \phi_{s,\ell}$$. Theorem B. For $$\epsilon=\pm1$$, suppose that $$\epsilon\,T\in \mathrm{Sym}_n(\mathbb R)_{>0}$$ and let $$f^{\mathrm{T}}_{s,\ell}(g) = \omega^{\mathrm{T}}_1(\pi(g)\phi_{s,\ell})$$ be the weight $$\ell$$ degenerate Whittaker function. Then, for $$g=n(b)m(a)k$$,   fs,ℓT(g)=χ(det(a))|det(a)|s+ρe(tr(Tb))det(k)ℓ ×2n(s−ρ+1)exp⁡(−2πϵtr(Tv))1F1(α,α+β;4πcvtc), where $$v=a{}^ta$$, $$\alpha= \frac12(s+\rho-\epsilon \ell)$$, $$\beta = \frac12(s+\rho+\epsilon \ell)$$, and $$\epsilon\,T= {}^tc c$$. Here   1F1(α,α+β;z) =∑m≥0(α)m(α+β)mdm(ρ)mΦm(z) =Γn(α+β)Γn(α)Γn(β)∫t>01−t>0etr(zt)det(t)α−ρdet(1−t)β−ρdt is the matrix argument hypergeometric function.□ It is instructive to compare the formula for $$f^{\mathrm{T}}_{s,\ell}$$ and the integral occurring here with the expressions in (1.5) and (1.7) defining the good Whittaker function. Note, for example, that in the case $$n=1$$, the functions   M(a,b;z)=Γ(b)Γ(a)Γ(b)∫01eztta−1(1−t)b−a−1dt and   U(a,b;z)=1Γ(a)∫0∞e−ztta−1(t+1)b−a−1dt, occurring in Theorem A and (1.7), respectively, are a standard basis for the space of solutions for the second-order Kummer equation [1, Chapter 13]. The proof of Theorem B depends on an elaborate calculation which makes essential use of the fact that, up to diagonalization, the orthogonal group of $$T$$ is $$O(n)$$. Thus, at present, we do not have a corresponding evaluation for $$T$$ of arbitrary signature. It is easy to check, cf. Lemma A.2, that for $$a$$, $$b$$, and $$z>0$$ real, with $$a>\rho$$, $$b>\rho$$, and for any $$\eta$$ with $$0<\eta<1$$,   |1F1(a,a+b;z)|≥Cηe(1−η)tr(z). Thus, for $$s$$ real and $$\alpha$$ and $$\beta >\rho$$,   |fs,ℓT(g)|≥Cη′det(v)12(s+ρ)e2π(1−2η)tr(ϵTv). This shows the exponential growth of $$f^{\mathrm{T}}_{s,\ell}$$ as the trace of $$v$$ goes to infinity. Due to its construction via a Whittaker functional, the function $$f^{\mathrm{T}}_{s,\ell}$$ on $$G$$ is an eigenfunction for the center of the universal enveloping algebra with eigencharacter given by the infinitesimal character of the degenerate principal series $$I(s)$$. Of course, for $$n=1$$, the results of Theorems A and B agree with the expressions given in the introduction of [12] in the case of $$\mathrm{SL}_2(\mathbb R)$$. Also, up to an elementary factor, the Whittaker function   fs,ℓT(g)=χ(a)|a|s+ρe(Tb)det(k)ℓ2sexp⁡(−2πϵTv)M(α,α+β;4πϵTv) in the $$n=1$$ case is precisely (the $$m_1$$-component of) the function utilized in the construction of [5], (4.13). Finally, we define an analogue of the $$\xi$$-operator introduced in [5, 6]. Since this operator is a variant of the $$\bar\partial$$-operator, it is best expressed in terms of vector bundles. Write $$\mathfrak H_n$$ for the Siegel upper half plane of genus $$n$$. For a discrete, torsion free subgroup $$\Gamma_0\subset G$$, let $$X=\Gamma_0\backslash \mathfrak H_n$$, and let $$\mathcal E^{a,b}$$ be the bundle of smooth differential forms of type $$(a,b)$$ on $$X$$. For a Hermitian vector bundle $$E$$ on $$X$$, let   ∗¯E:Ea,b⊗E⟶EN−a,N−b⊗E∗ be the Hodge $$*$$-operator [27, Chapter V, Section 2]. Here $$N=n\rho=\dim \mathfrak H_n$$. We include the cases where $$\Gamma_0$$ is $$\Gamma_\infty^u$$ or trivial. Definition. For a Hermitian vector bundle $$E$$ on $$X$$, the $$\xi$$-operator is defined as   ξ=ξE=∗¯E∘∂¯:Γ(X,Ea,b⊗E) ⟶ Γ(X,EN−a,N−b−1⊗E∗). (1.16) For a finite-dimensional representation $$(\sigma,\mathcal V_\sigma)$$ of $$\mathrm{GL}_n({\mathbb C})$$ with an admissible Hermitian norm, there is an associated homogeneous Hermitian vector bundle $$\mathcal L_\sigma$$ on $$X$$. For example, for an integer $$r$$, $$\mathcal L_r:= \mathcal L_{(\det)^{-r}}$$ is the line bundle whose sections correspond to functions on $$\mathfrak H_n$$ that transform like Siegel modular forms of weight $$r$$ with respect to $$\Gamma_0$$. In particular, $$\mathcal E^{N,0} \simeq \mathcal L_{n+1}$$. If $$\mathcal F_\nu$$ is a flat Hermitian bundle associated with a unitary representation $$(\nu, \mathcal F_{\nu})$$ of $$\Gamma_0$$, and $$\kappa$$ is an integer, then sections of $$\mathcal F_\nu\otimes \mathcal E^{0,N-1}\otimes\mathcal L_{n+1-\kappa}$$ can be viewed as $$\mathcal F_\nu$$-valued $$(0,N-1)$$-forms of weight $$n+1-\kappa$$, and $$\xi$$ carries such sections to sections of   Fν∨⊗EN,0⊗Lκ−n−1≃Fν∨⊗Lκ. For $$n=1$$ and $$\Gamma_0$$ a subgroup of finite index in $$\mathrm{SL}_2({\mathbb Z})$$, this reduces to the $$\xi$$-operator defined in [6] where $$(\nu, \mathcal F_\nu)$$ is a finite Weil representation. For simplicity, we now omit the bundle $$\mathcal F_\nu$$. Motivated by the construction of [5], we consider the $$\xi$$-operator applied to a space of Whittaker forms   ξ: W−T(E0,N−1⊗Ln+1−κ)⟶WT(Lκ). Here we take $$T\in \mathrm{Sym}_n({\mathbb Z})_{>0}^\vee$$ and $$\Gamma_0 = \Gamma^u_\infty=\mathrm {Sp}_n({\mathbb Z})\cap N$$. Lifted to $$G$$, this amounts to   ξ: [W−T(G)⊗∧N−1(p−∗)⊗C(κ−n−1) ]K⟶[WT(G)⊗C(−κ) ]K. A family of Whittaker forms in the space on the left here can be constructed by means of our Whittaker functional. Note that   σ∨=∧N−1(p−∗)⊗C(κ−n−1) is an irreducible representation of $$K$$. Since the $$K$$-types of $$I(s)$$ occur with multiplicity 1, we see that the space   [ I(s)⊗∧N−1(p−∗)⊗C(κ−n−1) ]K has dimension $$1$$. Let $$\phi_{s,\sigma}$$ be a basis vector. We then obtain a diagram   $$\$$ (1.17) and define the Whittaker form   fs,σ −T:=ω1−T(ϕs,σ). We finally determine the behavior of this family of Whittaker forms under the $$\xi$$-operator. Theorem C. The Whittaker form $$\boldsymbol{f}_{s,\sigma}^{\ -T}$$ has the following properties. (i) The form $$\boldsymbol{f}_{s,\sigma}^{\ -T}$$ is an eigenfunction of the center of the universal enveloping algebra of $$\mathfrak g$$ with eigencharacter the infinitesimal character of $$I(s)$$. In particular, for the Casimir operator $$C$$,   C⋅fs,σ −T=18(s+ρ)(s−ρ)fs,σ −T. (ii) For the $$\xi$$-operator,   ξ(fs,σ −T)=n(s¯−ρ+κ)fs,−κ−T¯, where   fs,−κ−T=ω1−T(ϕs,−κ). (iii) At $$s=s_0=\kappa-\rho$$,   ξ(fs0,σ −T)(g)=c(n,s0)WκT(g), where   WκT(n(b)m(a)k)=det(k)κdet(a)κe(tr(Tτ))=j(g,i)−κqT, (1.18) and $$c(n,s_0) = 2^{n(\kappa-2\rho+1)}$$. Here $$\tau = b+ \mathrm{i} a{}^ta$$ and $$q^{\mathrm{T}} = e(\mathrm{tr}(T\tau))$$.□ Thus, the $$\xi$$-operator carries $$\boldsymbol{f}^{\ -T}_{s_0,\sigma}$$ to the standard holomorphic Whittaker function of weight $$\kappa$$. Here note that, for $$\Gamma = \mathrm {Sp}_n({\mathbb Z})$$ and for $$\kappa>2n$$, the Poincaré series defined by   PΓ(WκT)(g)=∑γ∈Γ∞u∖ΓWκT(γg) is termwise absolutely convergent and defines a cusp form of weight $$\kappa$$. We define the global $$\xi$$-operator   ξΓ=PΓ∘ξ :Hn+1−κ(G)⟶Sκ(Γ), where $$\mathbb H_{n+1-\kappa}$$ is the subspace of   [ C∞(Γ∞u∖G)⊗∧N−1(p−∗) ]K spanned by the $$\boldsymbol{f}_{s_0,\sigma}^{\ -T}$$ for $$T\in \mathrm{Sym}_n({\mathbb Z})^\vee_{>0}$$. Since the Poincaré series spans $$S_\kappa(\Gamma)$$, the conjugate linear map $$\xi_\Gamma$$ is surjective and we obtain a kind of “resolution”   ker⁡(ξΓ)⟶Hn+1−κ⟶Sκ(Γ) (1.19) of the space of cusp forms which, for $$n>1$$, might be viewed as a kind of replacement for harmonic weak Maass forms in the higher genus case. It is our hope that the “resolution” of the space of cusp forms resulting from (1.19) will have interesting arithmetic applications. In particular, we will consider the Borcherds lift/regularized theta lift of such forms in a sequel to this paper. We remark that there are two points where our results could be extended. First, we have only determined the Whittaker function $$\omega_1^{\mathrm{T}}(\phi_{s,\ell})$$ for definite $$T$$. As mentioned above and explained in Section 4, our calculation depends on this assumption in an essential way, although it may be that some variant could be used for $$T$$ of arbitrary signature. This case distinction also occurs in [23] where the case of arbitrary signature requires a more elaborate argument. Second, we have not treated the Whittaker functions $$\boldsymbol{\omega}_{\underline{w}}^{\mathrm{T}}(f_{s,\ell})$$ arising from the other conical vector $$c_{\underline{w}}$$. There are two reasons for this. On the one hand, we do not need them for the applications we have in mind, and, on the other hand, already in the case $$n=1$$, some additional complications arise which we did not see how to handle for general $$n$$. We now briefly describe the contents of the various sections. In Section 2, we review background material about the degenerate principal series representation $$I(s)$$. In Section 3, we begin with a sketch of the theory of Goodman–Wallach operators relevant to our situation, intended to summarize some of the basic ideas from [12, 20] for nonspecialists (like the authors). We then state and prove our first main result, Theorem 3.1 (Theorem A). Its proof depends on some basic facts about matrix argument Bessel functions and Bessel operators from [22]. Everything up to this point could just as well have been formulated in terms of analysis on symmetric cones associated with Euclidean Jordan algebras, as in [11, 22, 23], and it should be possible to prove the analogue of Theorem A in this generality. We plan to do this in a sequel. In Section 4, we compute the “bad” Whittaker function with scalar $$K$$-type explicitly via an elaborate exercise with special functions of matrix argument. As the final answer is quite simple, we wonder if there is not a more direct derivation of it but did not succeed in finding one. In Section 5, we begin by defining the $$\xi$$-operator in some generality. We then show that its action on Whittaker forms can be determined from that of the corresponding operator on a complex associated with the degenerate principal series, cf. (5.16). We then construct certain Whittaker forms whose images under the $$\xi$$-operator interpolate, in the variable $$s$$, the standard Whittaker function $$W^{\mathrm{T}}_\kappa$$ occurring in the Fourier expansion of holomorphic Siegel cusp forms of weight $$\kappa$$, as explained in Theorem C. In Section 6, we briefly discuss the global $$\xi$$-operator and in the Appendix, we review some notation from [11], the inversion formula used in the proof of Theorem A, and an estimate for the growth of $${}_1F_1$$. 2 Background 2.1 Notation Let $$W$$, $$\langle \,{},{}\,\rangle$$ be a symplectic vector space of dimension $$2n$$ over $${\mathbb Q}$$ with standard basis $$e_1, \ldots, e_n, f_1,\ldots, f_n$$ with $$\langle \,{e_i},{f_j}\,\rangle = \delta_{ij}$$ and $$\langle \,{e_i},{e_j}\,\rangle= \langle \,{f_i},{f_j}\,\rangle=0$$. Let $$G= \mathrm {Sp}(W) \simeq \mathrm {Sp}_n/{\mathbb Q}$$. Following the tradition of [25, 26], we view $$W$$ as a space of row vectors with $$G$$ acting on the right. The Siegel parabolic $$P$$ is the stabilizer of the subspace spanned by the $$f_j$$’s, and we write $$P= M N$$ with Levi subgroup   M={m(a)=(ata−1)∣a∈GLn}, (2.1) and unipotent radical   N={n(b)=(1b1)∣b∈Symn} (2.2) and its opposite   N¯={n−(b)=(1b1)∣b∈Symn}. (2.3) For $$X\in \mathrm{Sym}_n$$, we write   n(X)=(X), and n−(X)=(X), for elements of $$\mathrm{Lie}(N)$$ and $$\mathrm{Lie}(\bar N)$$, respectively. We frequently write $$S = \mathrm{Sym}_n(\mathbb R)$$ and $$S_{\mathbb C}= \mathrm{Sym}_n({\mathbb C})$$. The stabilizer in $$G(\mathbb R)= \mathrm {Sp}_n(\mathbb R)$$ of the point $$i\cdot 1_n\in \mathfrak H_n$$, the Siegel space of genus $$n$$, is the maximal compact subgroup   K={k=(AB−BA)∣k=A+iB∈U(n)}. (2.4) If $$g = n m(a) k$$, then $$g = n m(a\mathbf k_0) k_0^{-1}k$$, where $$\mathbf k_0\in O(n)$$. In particular, in such a decomposition, we can always assume that $$\det(a)>0$$. Let   w_=(1n−1n), (2.5) so that $$\underline{w}$$ corresponds to $$i\,1_n\in U(n)$$ and lies in the center of $$K$$. For $$\tau\in \mathfrak H_n$$ and $$g\in G$$, $g = \begin{pmatrix} a&b\\c&d\end{pmatrix}$, we let $$j(g,\tau) = \det(c\tau+d)$$ be the standard scalar automorphy factor. Note that $$j(gk,i) = j(g,i)\,\det(\mathbf k)^{-1}$$. 2.2 Weil representations Let $$V$$, $$(\ ,\ )$$ be a non-degenerate inner product space over $${\mathbb Q}$$ of signature $$(p,q)$$. If $$\dim V = m = p+q$$ is even, $$\mathrm {Sp}_n(\mathbb R)\times\mathrm{ O}(V(\mathbb R))$$ acts on the space of Schwartz functions $$\mathcal S(V(\mathbb R)^n)$$ via the Weil representation:   ω(m(a))φ(x) =χV(det(a))|det(a)|m2φ(xa)ω(n(b))φ(x) =e(tr(Q(x)b))φ(x)ω(w_)φ(x) =γ(V)∫V(R)ne(tr(x,y))φ(y)dy, and $$\omega(h)\varphi(x) = \varphi(h^{-1}x)$$ for $$h\in O(V)(\mathbb R)$$. Here $$\chi_V(t) = (\mathrm{sgn}(t))^{\frac12(p-q)}$$ and $$\gamma(V) = e(\frac18(p-q))$$. Let $$D(V)$$ be the space of oriented negative $$q$$-planes in $$V(\mathbb R)$$. For $$z\in D$$, let $$(\,,\,)_z$$ be the majorant of $$(\,,\,)$$ defined by   (x,x)z=(x,x)−2(prz(x),prz(x)), and let $$\varphi_0(\cdot,z)\in \mathcal S(V(\mathbb R)^n)$$, given by   φ0(x,z)=exp⁡(−πtr((x,x)z)), be the associated Gaussian. It is an eigenfunction for $$K$$ with   ω(k)φ0(⋅,z)=det(k)p−q2φ0(⋅,z). 2.3 The degenerate principal series For $$G=\mathrm {Sp}_n(\mathbb R)$$ and the Siegel parabolic $$P=NM$$, with notation as in Section 2.1, let $$I^{\mathrm{ sm}}(s,\chi)$$ be the degenerate principal series representation given by right multiplication on the space of smooth functions $$\phi$$ on $$G$$ with   ϕ(n(b)m(a)g)=χ(det(a))|det(a)|s+ρϕ(g), (2.6) where $$\rho = \rho_n = \frac12(n+1)$$. In the case of interest to us, $$\chi(t) = \mathrm{sgn}(t)^{\nu}$$ for $$\nu=0$$, $$1$$. We let $$I(s)=I(s,\chi)$$ be the space of $$K$$-finite functions; it is the $$(\mathfrak g,K)$$-module associated with $$I^{\mathrm{ sm}}(s)$$. The structure of $$I(s)$$ is known [16–18]. We review the facts that we need and refer the reader to these papers for more information. 2.4 The infinitesimal character Let $$\mathfrak h\subset \mathfrak g= \mathrm{Lie}(G)_{\mathbb C}$$ be a Cartan subalgebra of $$\mathfrak k = \mathrm{Lie}(K)_{\mathbb C}$$ and hence also of $$\mathfrak g$$. Let $$\mathfrak z(\mathfrak g)$$ be the center of the universal enveloping algebra $$U(\mathfrak g)$$ and let   γ:z(g) ⟶∼ S(h)W be the Harish–Chandra isomorphism [9]. For $$\lambda\in \mathfrak h^*$$, let $$\chi_\lambda$$ be the character of $$\mathfrak z(\mathfrak g)$$ given by $$\chi_\lambda(Z) = \gamma(Z)(\lambda)$$. Following the notation of [16], for $$x=(x_1,\ldots, x_n)\in {\mathbb C}^n$$, we write   d(x)=diag(x1,…,xn),h(x)=(−id(x)id(x)), and take $$\mathfrak h = \{ h(x)\mid x\in {\mathbb C}^n\}$$. Then $$H_j = h(e_j)$$ is a basis for $$\mathfrak h$$ with dual basis $$\epsilon_j\in \mathfrak h^*$$, $$1\le j\le n$$. Then the infinitesimal character of $$I(s)$$ is $$\chi_{\lambda(s)+\rho_G}$$, where   λ(s)=(s−ρ)∑jϵj (2.7) and $$\rho_G= \sum_j (n-j+1)\epsilon_j$$, cf. [9, Theorem 4, p. 76], for example. Let $$C$$ be the Casimir operator of $$\mathfrak g$$. Then $$C$$ acts in $$I(s)$$ by the scalar   χλ(s)+ρG(C)=⟨λ(s)+ρG,λ(s)+ρG⟩−⟨ρG,ρG⟩=18(s+ρ)(s−ρ). (2.8) This is consistent with the fact that the trivial representation of $$G$$ is a constituent of $$I(s)$$ at the points $$s=\pm\rho$$. The Killing form on $$\mathfrak g\subset M_{2n}({\mathbb C})$$ is given by   ⟨X,Y⟩g=4ntr(XY), so that, since $$\mathrm{tr}(p_+(x)p_-(y)) = \mathrm{tr}(xy)$$,   C=Ck+14n∑αp+(eα)p−(eα∨)+p−(eα∨)p+(eα), (2.9) where $$C_{\mathfrak k}$$ is the $$\mathfrak k$$ component of $$C$$. 2.5 $$K$$-types For further details, cf. [16]. As a representation of $$K$$, we have   I(s)≃IndM∩KK(χ)≃IndO(n)U(n)sgn(det)ν. Thus the $$K$$-types of $$I(s)$$ have multiplicity 1 and an irreducible representation $$(\sigma, \mathcal V_\sigma)$$ of $$K$$ occurs precisely when its highest weight has the form   (ℓ1,…,ℓn),ℓ1≥…≥ℓn,ℓj∈ν+2Z, or, equivalently, precisely when its restriction to $$O(n) \simeq M\cap K$$ contains the representation $$(\det)^\nu$$. For such $$\sigma$$,   dim⁡HomK(σ,I(s))=dim⁡[I(s)⊗σ∨]K=1. (2.10) Suppose that $$v_0\in \sigma^\vee$$ is an eigenvector for $$O(n)$$, so that $$\sigma^\vee(k) v_0 = \det(k)^\nu v_0$$ for all $$k\in O(n)$$. The vector $$v_0$$ is unique up to a non-zero scalar. A standard basis element for $$[I(s)\otimes \sigma^\vee]^K$$ is then given by   ϕs,σ(nm(a)k)=χ(det(a))|det(a)|s+ρσ∨(k−1)v0. For example, for an integer $$\ell$$, with $$\ell\equiv \nu\mod 2$$, the unique function $$\phi_{s,\ell}\in I(s)$$ with scalar $$K$$-type $$\det(k)^\ell$$ is given by   ϕs,ℓ(n(b)m(a)k)=χ(det(a))|det(a)|s+ρdet(k)ℓ. (2.11) 2.6 Submodules For $$s\notin \nu+2{\mathbb Z}$$, the $$(\mathfrak g,K)$$-module $$I(s)$$ is irreducible. At points $$s\in \nu+2{\mathbb Z}$$, nontrivial submodules arise via the coinvariants for the Weil representation. For a quadratic space $$V$$ over $$\mathbb R$$ of signature $$(p,q)$$, $$p+q$$ even, with associated Weil representation $$(\omega, S(V^n))$$, for the additive character $$x \mapsto e(x)$$, there is an equivariant map   λV:S(Vn)⟶I(s0),φ↦(ω(g)φ)(0), where $$s_0 = \frac12(p+q) - \rho$$ and $$\nu \equiv \frac12(p-q) \mod 2$$. The image, $$R(p,q)$$, is the $$(\mathfrak g,K)$$-submodule of $$I(s_0)$$ generated by the scalar $$K$$-type $$(\det)^\ell$$ with $$\ell = \frac12(p-q)$$. Moreover, the vector $$\phi_{s_0,\frac12(p-q)}$$ is the image of the Gaussian $$\varphi^0_V \in S(V^n)$$, where   ωV(k)φV0=det(k)12(p−q)φV0. For example, for signature $$(m+2,0)$$ with $$m+2>2n+2$$, $$R(m+2,0)\subset I(s_0)$$ is a holomorphic discrete series representation with scalar $$K$$-type $$(\det)^{\kappa}$$, $$\kappa = \frac{m}2+1$$. In particular, the vector $$\phi_{s_0, \kappa}$$ for $$s_0= \kappa -\rho$$ is killed by $$\mathfrak p_-$$. On the other hand, for signature $$(m,2)$$, the vector   φKM∈[S(Vn)⊗∧(n,n)(pH∗)]KH, satisfies   ω(k)φKM=det(k)κφKM. The image of this vector under the map $$\lambda_{m,2}$$ is again $$\phi_{s_0,\kappa}$$, so that we have submodules   R(m+2,0)⊂R(m,2)⊂I(s0),s0=κ−ρ. Note that, by [16], $$R(p,q)$$ is the largest quotient of $$S(V^n)$$ on which the orthogonal group $$O(V)= O(p,q)$$ acts trivially, that is, the space of $$O(V)$$-coinvariants. 3 Goodman–Wallach Operators In our discussion of both conical and Whittaker vectors, we will only consider the degenerate principal series representation $$I(s)$$ and the Siegel parabolic $$P=NM$$. In this case, the simple example for $$\mathrm{SL}_2(\mathbb R)$$ worked out in the introduction of Goodman–Wallach [12] provides an adequate template. An essential feature is that the various classical special functions occurring there are replaced by their matrix argument generalizations. These results seem to be new; at least we could not find such explicit formulas in the literature. Since we work with intertwining operators which express our functions via integral representations, we derive, as a consequence, the behavior of our functions under the differential operators coming from the center of the enveloping algebra $$\mathfrak z(\mathfrak g)$$, whereas, in the case of $$\mathrm{SL}_2(\mathbb R)$$ one can work with classical solutions of the second-order ode satisfied by the radial part. For discussion of the more general theory including general results about the existence of Goodman–Wallach operators, cf. [20]. 3.1 Conical and Whittaker vectors Suppose that $$(\pi,\mathcal V)$$ is a continuous representation of $$G$$ on a Banach space $$\mathcal V$$ and that   Fcon:V⟶Ian(s) is a $$G$$-equivariant linear map. Here $$I^{\mathrm{ an}}(s)$$ is the space of real analytic functions on $$G$$ satisfying (2.6). The linear functional $$\mu_{\mathrm{ con}}\in \mathcal V^*$$ defined by $$\mu_{\mathrm{ con}}(v) = F_{\mathrm{ con}}(v)(e)$$ satisfies   μcon(π(nm(a))v)=χ(det(a))|det(a)|s+ρμcon(v). We refer to such a vector as a conical vector in $$\mathcal V^*$$ of type $$(P,s+\rho)$$. Conical vectors in the dual $$I^{\text{an}}(s)^*$$ are given by   c1(ϕ)=ϕ(e), (3.1) and   cw_(ϕ)=(A(s,w_)ϕ)(e)=∫Nϕ(w_n(b))db, (3.2) where, for $$\mathrm{Re}(s)>\rho$$, the intertwining operator $$A(s,\underline{w}):I(s) \longrightarrow I(-s)$$ is defined by the integral   (A(s,w_)ϕ)(g)=∫Nϕ(w_ng)dn, (3.3) for $$\underline{w}$$ given by (2.5). It has a meromorphic analytic continuation. According to our terminology, these are of type $$(P,s+\rho)$$ and $$(P,-s+\rho)$$, respectively. ({Here, we are introducing a compressed version of the standard terminology [12, 20] convenient for our special case.}) Similarly, for $$T\in \mathrm{Sym}_n({\mathbb C})$$, suppose that   Fwh:V⟶WT(G)an is a $$G$$-equivariant linear map, where $$\mathcal W^{\mathrm{T}}(G)^{\mathrm{ an}}$$ is the space of real analytic functions on $$G$$ satisfying (1.4). The linear functional $$\mu_{\mathrm{ wh}} \in \mathcal V^*$$ defined by $$\mu_{\mathrm{ wh}}(v) = F_{\mathrm{ wh}}(v)(e)$$ satisfies   μwh(π(n(b))v)=e(tr(Tb))μwh(v). We refer to such a vector in $$\mathcal V^*$$ as a Whittaker vector of type $$(N,T)$$. We can make analogous definitions for $$\mathcal V$$ an irreducible $$U(\mathfrak g)$$-module, and now explain an essential idea of [12] relating conical and Whittaker vectors in $$\mathcal V^*$$. Our goal is to motivate the explicit construction given below; for a more careful treatment cf. [12, 20]. Let $$\mathfrak n=\mathrm{Lie}(N)_{\mathbb C}$$, $$\mathfrak m = \mathrm{Lie}(M)_{\mathbb C}$$, and let $$\bar{\mathfrak n} = \mathrm{Lie}(\bar N)_{\mathbb C}$$ where $$\bar N$$ is the unipotent radical of the opposite maximal parabolic $$\bar P = M \bar N$$. Let $${\mathbb C}(s+\rho)$$ be the one-dimensional representation of $$\mathfrak m$$ determined by the representation $$|\det \,|^{s+\rho}$$ of $$M$$ and extend it to a representation of $$\mathfrak m+\mathfrak n$$, trivial on $$\mathfrak n$$. Define the generalized Verma modules   V(P,s+ρ)=U(g)⊗U(m+n)C(s+ρ) and   V¯(P¯,−s−ρ)=C(−s−ρ)⊗U(m+n¯)U(g). By the Poincaré–Birkoff–Witt theorem,   V(P,s+ρ)=U(n¯)⊗CC(s+ρ),andV¯(P¯,−s−ρ)=C(−s−ρ)⊗CU(n). There is a natural pairing   ⟨⟨ , ⟩⟩:V¯(P¯,−s−ρ)⊗CV(P,s+ρ)⟶V¯(P¯,−s−ρ)⊗U(g)V(P,s+ρ) ⟶∼ C, and, for $$Z\in U(\mathfrak g)$$, $$\bar u\in \bar V(\bar P,-s-\rho)$$, and $$u\in V(P,s+\rho)$$,   ⟨⟨u∗Z,u⟩⟩=⟨⟨u∗,Zu⟩⟩. This pairing is non-degenerate precisely when $$V(P,s+\rho)$$ is irreducible, [20, Section 3.1]. For the rest of our discussion, we suppose that this is the case. Since $$\mathfrak n$$ and $$\bar{\mathfrak n}$$ are abelian   U(n¯)=S(n¯)=⨁∂≥0S(n¯)d (3.4) is the symmetric algebra on $$\bar{\mathfrak n}$$, graded by degree, and the completion   S(n¯)[n¯]=U(n¯)[n¯]=lim⟵U(n¯)/n¯kU(n¯)≃∏∂≥0S(n¯)d, (3.5) is the ring of formal power series in such elements. Let   V^(P,s+ρ)=V(P,s+ρ)[n¯]=U(n¯)[n]¯⊗CC(s+ρ) be the $$\bar{\mathfrak n}$$-completion of $$V(P,s+\rho)$$. The pairing $$\langle\langle{\,},{}\rangle\rangle$$ extends to this space and induces an isomorphism   V^(P,s+ρ) ⟶∼ V¯(P¯,−s−ρ)∗. (3.6) Then, following the notation of [20], for an admissible character $$\psi$$ of $$\mathfrak n$$, the spaces   Whn,ψ∗(V¯(P¯,−s−ρ))={w∈V¯(P¯,−s−ρ)∗∣X⋅w=ψ(X)w, ∀X∈n}, (3.7) and   Whn,ψ(V^(P,s+ρ))={w∈V^(P,s+ρ)∣X⋅w=ψ(X)w, ∀X∈n} (3.8) are isomorphic via (3.6). ({In our situation, for $$X\in \mathfrak n\simeq \mathrm{Sym}_n({\mathbb C})$$, and $$\psi(X) = \mathrm{tr}(TX)$$, admissibility simply means that $$\det(T)\ne0$$.}) If we extend $$\psi$$ to an algebra homomorphism $$\psi: U(\mathfrak n)\longrightarrow {\mathbb C}$$, then there is an obvious basis for the space (3.7) given by the functional $$1\otimes\psi$$ on $$\bar V(\bar P,-s-\rho)\simeq {\mathbb C}(-s-\rho)\otimes_{\mathbb C} U(\mathfrak n)$$. We write $$\mathrm{gw}_s^\psi$$ for the corresponding element of $$\hat V(P,s+\rho)$$, viewed as a formal power series in the elements of $$\bar{\mathfrak n}$$, and we refer to it as the Goodman–Wallach element. It is characterized by   ⟨⟨A,gwsψ⟩⟩=ψ(A),for all A∈U(n). Returning to the irreducible $$U(\mathfrak g)$$-module $$(\pi, \mathcal V)$$, we let $$U(\mathfrak g)$$ act on $$\mathcal V^*$$ on the left by $$Z\cdot \mu = \mu\cdot {}^tZ$$, where $${}^t:U(\mathfrak g) \longrightarrow U(\mathfrak g)$$ is the involution restricting to $$X\mapsto -X$$ on $$\mathfrak g$$. We assume that $$\mathcal V$$ is finitely generated as a $$U(\mathfrak n)$$-module. For a conical vector $$\mu\in \mathcal V^*$$ of type $$(P,s+\rho)$$, we have a homomorphism   V^(P,−s−ρ)⟶V∗,Z↦Z⋅μ. Then, for the Goodman–Wallach element $$\mathrm{gw}_s^\psi\in \hat V(P, -s-\rho)$$, and for $$X\in \mathfrak n$$, we have   (gwsψ⋅μ)⋅X=(μ⋅tgwsψ)⋅X=μ⋅t(−Xgwsψ)=−ψ(X)gwsψ⋅μ. Thus, $$\mathrm{gw}_s^{\psi}\cdot \mu$$ is a Whittaker vector of type $$(N,-\psi)$$. Of course, we have only given a rough sketch of the idea here, including the %unrealistic restriction to the case where $$V(P,s+\rho)$$ is irreducible. After removing this restriction, a main point of the Goodman–Wallach theory is to give estimates on the growth of the components of the power series $$\mathrm{gw}_s^\psi$$ so that it can be used to define $$G$$-intertwining operators—the Goodman–Wallach operators—from principal series representations to spaces of Whittaker functions preserving Gevrey classes, that is, certain function spaces between real analytic and $$C^\infty$$, cf. the remark on p. 228 of [20] for a more precise statement. 3.2 An explicit formula for the Goodman–Wallach operator for $$\mathrm {Sp}_n(\mathbb R)$$ We now describe the matrix argument Bessel function giving the Goodman–Wallach operator. For $$\phi\in I(s)$$, the function $$\Psi(\phi)$$ defined by (1.12) and its Fourier transform (1.13) satisfy   Ψ(x;π(m(a))ϕ)=det(a)s+ρΨ(taxa;ϕ), (3.9) and   Ψ^(y;π(m(a))ϕ)=det(a)s−ρΨ^(a−1yta−1;ϕ). (3.10) The conical vectors $$c_1$$ and $$c_{\underline{w}}$$ defined by (3.1) and (3.2) can be written as   ⟨c1,ϕ⟩=∫SΨ^(y;ϕ)dy=Ψ(0;ϕ), (3.11) and   ⟨cw_,ϕ⟩=∫SΨ(y;π(w_)ϕ)dy=Ψ^(0;π(w_)ϕ). (3.12) Here note that   ∫Sϕ(w_n(b))db=∫Sϕ(n−(−b)w_)db=∫Sϕ(n−(b)w_)db. We next define the relevant Bessel type function on $$S$$, specializing some of the notation of [11, 22] to the present case. This notation is summarized in the Appendix, which the reader should consult for things not explained here. For $$z$$ and $$w\in S_{\mathbb C}$$, the function $$\mathrm{GW}_s(z,w)$$ defined by (1.15) coincides with the $$J$$-Bessel function   GWs(z,w)=Js+ρ(z,w), (3.13) in the notation of [22, p. 818 and p. 823]. Note that $$\Phi_{\mathbf m}(z,w)$$ is the function on $$S_{\mathbb C}\times S_{\mathbb C}$$ described in [22, Lemma 1.11]. In particular, this function is holomorphic in $$z$$, antiholomorphic in $$w$$, and satisfies $$\Phi_{\mathbf m}(z,1_n) = \Phi_{\mathbf m}(z)$$, where $$\Phi_{\mathbf m}(z)$$ is the spherical polynomial in [11, Chapter XI, Section 3]. Also recall that $$\Phi_{\mathbf m}(z)$$ is homogeneous of degree $$|\mathbf m| = \sum_i m_i$$. Moreover, for $$a\in \mathrm{GL}_n({\mathbb C})$$,   Φm(a⋅z,w)=Φm(z,ta¯⋅w), (3.14) and $$\overline{\Phi_{\mathbf m}(z,w)} = \Phi_{\mathbf m}(w,z)$$. The invariance (3.14) is inherited by $$\mathrm{GW}_s(z,w)$$. Theorem 3.1. For $$\phi\in I(s)$$, let   ω1(ϕ)=∫SGWs(2πy,2πw)Ψ^(y;ϕ)dy. (3.15) Recall that for $$X\in \mathrm{Sym}_n(\mathbb R)$$,   n(X)=(0X0). Then,   ω1(π(n(X))ϕ)=2πitr(Xw¯)ω1(ϕ). (3.16)□ Corollary 3.2. For $$Y\in \mathrm{Sym}_n({\mathbb C}) \simeq \bar{\mathfrak n}$$, view $$\mathrm{GW}_s(2\pi Y,2\pi w)$$ as a power series in $$S(\bar{\mathfrak n})_{[\bar{\mathfrak n}]}$$. Then this power series defines the Goodman–Wallach–Matumoto operator in $$\hat V(P,-s-\rho)$$ for the Siegel parabolic $$P$$ of $$G=\mathrm {Sp}_n(\mathbb R)$$ and the character $${\frak n}(X)\mapsto 2\pi \mathrm{i}\,\mathrm{tr}(X\bar w)$$ of $$\mathfrak n$$. (Concretely, write $$Y = \sum_{\alpha} Y_\alpha\, e_\alpha^\vee$$ and view $$\mathrm{GW}_s(2\pi Y,w)$$ as a power series in the $$Y_\alpha$$’s.) □ Definition 3.3. For $$T\in \mathrm{Sym}_n(\mathbb R)$$ with $$\det(T)\ne0$$, let $$\omega^{\mathrm{T}}_1$$ be the Whittaker functional constructed from the conical vector $$c_1$$, so that   ω1T(ϕ)=∫SGWs(2πy,2πT)Ψ^(y;ϕ)dy. (3.17) and   ω1T(ϕ)(n(b)g)=e(tr(Tb))ω1T(ϕ)(g). □ 3.3 Proof of the Goodman-Wallach identity We want to prove (3.16) and so we consider   ω1(n(X)ϕ)=∫SGWs(2πy,2πw)Ψ^(y;n(X)ϕ)dy. (3.18) We adopt some of the notation and setup from [11, 22]. Recall that $$S$$ is a simple Euclidean Jordan algebra with product $$x\cdot y = \frac12(xy+yx)$$. Endomorphisms $$P(x)$$ and $$P(x,y)$$ of $$S$$ are defined by, [22, p.794],   P(x)z=xzx,P(x,y)z=xzy+yzx. Let $$e_\alpha$$ be a basis for $$S$$ and let $$e_\alpha^\vee$$ be the dual basis with respect to the trace form, cf. Appendix. In particular, we write $$x = \sum_\alpha x_\alpha e_\alpha$$. Define vector-valued differential operators as follows. The gradient operator   ∂∂x=∑α∂∂xαeα∨ is characterized by   ∂a=tr(a∂∂x), where, for $$a\in S$$, $$\partial_a$$ is the directional derivative associated with the constant vector field $$a$$. For a complex scalar $$\lambda$$, the Bessel operator $$\mathcal B_\lambda$$ is defined by   Bλ=P(∂∂x)x+λ∂∂x=∂∂xx∂∂x+λ∂∂x. Thus, if $$f$$ is a $$C^2$$ function on $$S$$, then $$\mathcal B_\lambda f$$ is an $$S$$-valued function on $$S$$. The key fact is the following, see Proposition 3.3 in [14] and also [15],: Proposition 3.4. For $$X$$ and $$Y\in \mathrm{Sym}_n(\mathbb R)$$, (i)   Ψ(x;π(n(X))ϕ)=−tr(X(sx+x∂∂xx)Ψ(x;ϕ)), (ii)   Ψ(x;π(n−(Y))ϕ)=tr(Y∂∂x)Ψ(x;ϕ), (iii)   Ψ^(y;π(n(X))ϕ)=12πitr(XB−sΨ^(y;ϕ)), (iv)   Ψ^(y;π(n−(Y))ϕ)=−2πitr(Yy)Ψ^(y;ϕ). □ Proof First we note that $$\exp({\frak n}(X)) = n(X)$$, and that   (1x1)(1X1)=(1Xx1+xX)=(1∗1)(tA−1A)(1A−1x1), where $$A = 1+xX$$ and $$* = X(1+xX)^{-1}$$. Here, we are going to take $$tX$$ in place of $$X$$ so that $$1+xX$$ will be invertible for $$t$$ sufficiently small. Thus, in this range,   ϕ(n−(x)n(X))=det(1+xX)−s−ρϕ(n−((1+xX)−1x)). We now replace $$X$$ by $$tX$$ and take $$\mathrm{d}/\mathrm{d}t\vert_{t=0}$$. First we have   ddtdet(1+txX)−s−ρ|t=0=−(s+ρ)tr(xX), where we note that   det(1+txX)=1+tr(txX)+O(t2). Next, we let $$z = (1+xtX)^{-1}x$$ and compute   0=ddt((1+xtX)z)|t=0=xXx+dzdt|t=0, so that, writing $$z = \sum_\alpha z_\alpha e_\alpha$$, we have   dzdt|t=0=∑αdzαdteα=−∑α(xXx)αeα. Therefore   ddtϕ(z)|t=0=∑α∂ϕ∂zαdzαdt|t=0=−tr(xXx∂∂x)ϕ. Thus we have proved that   Ψ(x;π(n(X))ϕ)=−((s+ρ)tr(Xx)+tr(xXx∂∂x))Ψ(x;ϕ). But we have   tr(xXx∂∂x)=tr(Xx∂∂xx)−ρtr(xX). Indeed, writing $$\bullet$$ for the “evaluation” product, we have   (∑α∂∂xαeα∨)∙(∑αxαeα)=∑αeα∨eα=12(n+1)∑ieii. This gives (i). Now we consider the Fourier transform   −∫Se(tr(xy))(sx+x∂∂xx)Ψ(x;ϕ)dx. If we write $$y = \sum_\alpha y_\alpha e_\alpha^\vee$$, then $$(\partial/\partial y) = \sum_\alpha (\partial/\partial y_\alpha)\,e_\alpha$$, and we have   ∂∂ye(tr(xy))=(∑α∂∂yαeα)(e(∑αxαyα))=2πi∑αxαeα=2πix. Then the factor $$s\, x$$ (resp. $$x^2$$) can be obtained by applying   s2πi∂∂y,(resp. (2πi)−2(∂∂y)2) outside the integral. Also   ∫Se(tr(xy))∂∂xΨ(x;ϕ)dx=−2πiy∫Se(tr(xy))Ψ(x;ϕ)dx. We obtain   − ∫Se(tr(xy))(sx+x∂∂xx)Ψ(x;ϕ)dx =−12πi(s∂∂y−∂∂yy∂∂y)∫Se(tr(xy))Ψ(x;ϕ)dx =12πiB−sΨ^(y;ϕ). This proves (iii) of Proposition 3.4. The proofs of (ii) and (iv) are easy and omitted. ■ Now we return to (3.18) and, using (ii) of the previous proposition, obtain   ω1(π(n(X))ϕ) =∫SGWs(2πy,2πw)Ψ^(y;π(n(X))ϕ)dy =12πitr(X∫SGWs(2πy,2πw)B−sΨ^(y;ϕ)dy) =12πitr(X∫SB−s∗GWs(2πy,2πw)Ψ^(y;ϕ)dy ) =12πitr(X∫SBsGWs(2πy,2πw)Ψ^(y;ϕ)dy). Here we use the fact that the adjoint of $$\mathcal B_{-s}$$ is $$\mathcal B_{s}$$. But now by (3.13) and Proposition 3.6 of [22], we have   BsGWs(2πy,2πw)=−(2π)2w¯⋅GWs(2πy,2πw) so that we obtain   ω1(n(X)ϕ)=2πitr(Xw¯)ω1(ϕ), as required. This proves Theorem 3.1. 3.4 Proof of Corollary 3.2 To show that $$\mathrm{GW}_s(2\pi Y,2\pi w)$$ is indeed the Goodman–Wallach element, as claimed, we proceed as follows. The pairing   ⟨⟨,⟩⟩:V¯(P¯,s+ρ)⊗CV^(P,−s−ρ)⟶C is characterized by   ⟨⟨A⋅Z,B⟩⟩=⟨⟨A,Z⋅B⟩⟩, for $$A\in \bar V(\bar P, s+\rho)$$, $$B\in \hat V(P,-s-\rho)$$, and $$Z\in U(\mathfrak g)$$, and   ⟨⟨1⊗n(X),n−(Y)⊗1⟩⟩=(s+ρ)tr(XY). For any function $$\phi\in I(s)$$, and for $$A\in S(\mathfrak n)$$ and $$B\in U(\mathfrak g)$$, we define   ⟨⟨A,B⟩⟩ϕ=−∫SΨ^(y;t(A⋅B)⋅ϕ)dy=−Ψ(0;t(A⋅B)⋅ϕ). As a function of $$B$$ this map factors through $$V(P,-s-\rho)$$ and $$\langle\langle{A},{B}\rangle\rangle_\phi = \langle\langle{1},{A\cdot B}\rangle\rangle_\phi$$. Recall here that, as in 3.1, $$A\mapsto {}^tA$$ is the involution of $$U(\mathfrak g)$$ which is $$-1$$ on $$\mathfrak g$$. Moreover, we have   ⟨⟨n(X),n−(Y)⟩⟩ϕ =−Ψ(0;n−(−Y)n(−X)⋅ϕ) =−Ψ(0;([n−(−Y),n(−X)]+n(−X)n−(−Y))⋅ϕ) =(s+ρ)tr(XY)Ψ(0;ϕ). Taking $$\phi$$ with $$\Psi(0;\phi)=\phi(e)=1$$, we have $$\langle\langle{\,},{}\rangle\rangle_\phi = \langle\langle{\,},{}\rangle\rangle$$ on $$U(\mathfrak n)\times V(P,-s-\rho)$$ . Now by (iv) of Proposition 3.4, we take a power series $$\mathrm{ gw}_s \in S(\bar{\mathfrak n})_{[\bar{\mathfrak n}]}$$ so that, for all $$\phi$$,   Ψ^(y;tgws⋅ϕ)=GWs(2πy,2πw)Ψ^(y;ϕ). Then   ⟨⟨A,gws⟩⟩ =−∫SΨ^(y;t(A⋅gws)⋅ϕ)dy =−∫SGWs(2πy,2πw)Ψ^(y;tAϕ)dy =−ω1(tAϕ) =−ψ−2πiw¯(A)ω1(ϕ) =ψ−2πiw¯(A)⟨⟨1,gws⟩⟩, where $$\psi_{-2\pi \mathrm{i}\bar w}: S(\mathfrak n)\longrightarrow {\mathbb C}$$ is the character determined by $${\frak n}(X)\mapsto -2\pi \mathrm{i}\,\mathrm{tr}(X\bar w)$$. This in fact shows that the Goodman–Wallach element is actually given by   gws♮=⟨⟨1,gws⟩⟩−1gws. 4 Calculation of the “bad” Whittaker Function: the Scalar Case In this section, we determine the Whittaker function $$\omega_1^{ T}(\phi_{s,\ell})$$ for $$\phi_{s,\ell}\in I(s)$$ with a scalar $$K$$-type and for $$\epsilon\,T\in \mathrm{Sym}_n(\mathbb R)_{>0}$$ and $$\epsilon=\pm1$$. Whittaker functions for other $$K$$-types can then be obtained by applying differential operators. In the case of $$\mathrm{SL}_2(\mathbb R)$$, that is, for $$n=1$$, the radial part of the Whittaker function we obtain is essentially the classical confluent hypergeometric function $$M(a,b;z)$$ in the notation of [1], for example. On the other hand, again for $$n=1$$, the Whittaker functional obtained by applying the Goodman–Wallach operator to the conical vector $$c_w$$ yields a Whittaker function whose radial part is the classical function $$U(a,b;z)$$. Thus, this traditional basis for the solution space to the Whittaker ode arises in a natural way from the pair of conical vectors $$c_1$$ and $$c_w$$. The calculation of this section constructs the analogue of the $$M$$-Whittaker function for $$\mathrm {Sp}_n(\mathbb R)$$. As noted earlier, we do not have a corresponding evaluation for $$T$$ of arbitrary signature, and we will indicate in the course of the calculation where the assumption that $$T$$ is definite is used. For the standard vector $$\phi_{s,\ell}$$ with scalar $$K$$-type defined in (2.11), the Whittaker function   fs,ℓT(g):=ω1T(π(g)ϕs,ℓ)=∫SGWs(2πy,2πT)Ψ^(y;π(g)ϕs,ℓ)dy (4.1) satisfies   fs,ℓT(n(b)m(a)k)=e(tr(Tb))fs,ℓT(m(a))det(k)ℓ, (4.2) and hence is determined by its restriction to $$\mathrm{GL}_n(\mathbb R)$$. We will frequently omitted the $$T$$ as a superscript to lighten the notation. By analogy with the one variable case, for $$z\in S_{\mathbb C}$$ and the $$a$$ and $$b\in {\mathbb C}$$ with $$\mathrm{Re}(a)>\rho-1$$, $$\mathrm{Re}(b)>\rho-1$$, we let   Mn(a,b;z)=Γn(b)Γn(a)Γn(b−a)∫t>01−t>0etr(zt)det(t)a−ρdet(1−t)b−a−ρdt. (4.3) This is the standard matrix argument hypergeometric function   Mn(a,b;z)=1F1(a;b;z) (4.4) as in [11, 13], etc. Our first main result is the following. Theorem 4.1. Let $$v=a{}^ta$$, $$\alpha= \frac12(s+\rho- \epsilon \ell)$$ and $$\beta = \frac12(s+\rho+\epsilon \ell)$$. Then, writing $$\epsilon\,T= {}^tc c$$,   fs,ℓ(m(a))=2n(s−ρ+1)det(v)12(s+ρ)exp⁡(−2πtr(ϵTv))Mn(α,α+β;4πcvtc).□ For future reference, we give the full formula   fs,ℓ(n(b)m(a)k)=c(n,s)det(v)12(s+ρ)e(tr(Tb))exp⁡(−2πtr(ϵTv))Mn(α,α+β;4πcvtc)det(k)ℓ, (4.5) where   c(n,s)=2n(s−ρ+1). (4.6) Remark 4.2. The hypergeometric function $$M_n(\alpha,\alpha+\beta; z)$$ is given by a power series which is everywhere convergent in $$z$$ provided the values of $$s$$ for which some factor $$(\alpha+\beta)_\mathbf{m} = (s+\rho)_{\mathbf{m}}$$ in the denominator vanishes are excluded, [11]. The excluded values are thus $${\mathbb Z}_{<0}$$ for $$n=1$$ and   Z<0∪(−12+Z<0) for $$n\ge 2$$. □ By construction, the Whittaker function $$f_{s,\ell}$$ is an eigenfunction for the center $$\mathfrak z(\mathfrak g)$$ of the universal enveloping algebra $$U(\mathfrak g)$$ with the same eigencharacter as $$I(s)$$. Corollary 4.3. For all $$Z\in \mathfrak z(\mathfrak g)$$,   Z⋅fs,ℓ=χλ(s)+ρG(Z)fs,ℓ, where $$\chi_{\lambda(s)+\rho_G}$$ is the character of $$\mathfrak z(\mathfrak g)$$ given by (2.7). In particular, for the Casimir operator $$C$$, by (2.8),   C⋅fs,ℓ=18(s+ρ)(s−ρ)fs,ℓ.□ Proof of Theorem 4.1. This amounts to a long calculation. We first simplify by eliminating $$T$$. Writing $$2\pi\,\epsilon\, T = c^2$$ for $$c={}^tc >0$$, we have, by (3.14),   GWs(2πy,2πT)=GWs(2πϵtcyc,1n). Then, using (3.10), and setting $$\mathrm{GW}_s(z) = \mathrm{GW}_s(z,1_n)$$, we have   ∫SGWs(2πy,2πT)Ψ^(y;π(m(a))ϕ)dy=|det(2πT)|−12(s+ρ)∫SGWs(2πϵy)Ψ^(y;π(m(ca))ϕ)dy. Remark 4.4. It is at this point that we use the fact that $$T$$ is definite in an essential way. □ Therefore, it suffices to compute   ω1ϵ(π(m(a))ϕ):=∫SGWs(2πϵy)Ψ^(y;π(m(a))ϕ)dy. (4.7) We write this as   ω1ϵ(π(m(a))ϕ)=∑p+q=nω1ϵ(π(m(a))ϕ)p,q, where   ω1ϵ(π(m(a))ϕ)p,q=∫Sp,qGWs(2πϵy)Ψ^(y;π(m(a))ϕ)dy (4.8) for $$S_{p,q}$$ the subset of invertible matrices in $$S$$ of signature $$(p,q)$$. Specializing to the case $$\phi= \phi_{s,\ell}$$, the first step is the following. Proposition 4.5. For $$y\in S_{p,q}$$,   Ψ^(y;π(m(a))ϕℓ,s)=(2π)nρ2−n(ρ−1)Γn(α)Γn(β)det(v)−12(s+ρ)|det(y)|s(2π)ns2ns ×∫x+ϵp>0x+ϵq′>0e−2πtr(tcv−1c(1+2x))det(x+ϵp)α−ρdet(x+ϵq′)β−ρdx, (4.9) where $$y = c \epsilon_{p,q} {}^tc$$ and $$v= a{}^ta$$. Here, as in [23], $$\alpha = \frac12(s+\rho+\ell)$$, $$\beta= \frac12(s+\rho-\ell)$$, $$\epsilon_{p} = \mathrm{diag}(1_p,0)$$, and $$\epsilon'_q = \mathrm{diag}(0,1_q)$$. □ Proof If we write $$n_-(x) = n m k$$ for $$k=k(n_-(x))\in K$$, then   Ψ(x;ϕ)=det(1+x2)−12(s+ρ)ϕ(k(n−(x))) and   Ψ(x;ϕℓ,s)=det(1+ix)−αdet(1−ix)−β, where we are now using Shimura’s convention where $$\alpha=\frac12(s+\rho+\ell)$$ and $$\beta=\frac12(s+\rho-\ell)$$, so that, for example, $$\alpha+\beta = s+\rho$$. Then   Ψ^(y;ϕℓ,s)=∫Se(tr(xy))det(1+ix)−αdet(1−ix)−βdx, and, following the standard manipulations on pp. 274–5 of [23], we have   Ψ^(y;π(m(a))ϕℓ,s) =det(a)s−ρ∫Se(tr(xa−1yta−1))det(1+ix)−αdet(1−ix)−βdx =det(a)s−ρ∫Se(−tr(xa−1yta−1))det(1−ix)−αdet(1+ix)−βdx =in(α−β)det(a)s−ρξ(1,h;α,β) =det(a)s−ρ(2π)nρ2−n(ρ−1)Γn(α)−1Γn(β)−1 ×∫u>0,u>2πhe2tr(πh−u)det(u)α−ρdet(u−2πh)β−ρdu =det(a)s−ρ(2π)nρ2−n(ρ−1)Γn(α)−1Γn(β)−1η(2,πh;α,β), where $$h= a^{-1}y{}^ta^{-1}$$, cf. the top of p. 275 of [23]. Here $$\eta$$ is the function defined in (1.26) of loc.cit. Recalling (3.1) of loc. cit., for any $$a'\in \mathrm{GL}_n(\mathbb R)^+$$,   η(g,a′hta′;α,β)=det(a′)2sη(ta′ga′,h;α,β), and writing $$\pi y = c \epsilon_{p,q}{}^tc$$ so that $$\pi h = a^{-1} c \epsilon_{p,q}{}^tc{}^ta^{-1}$$, we have   η(2,πa−1yta−1;α,β)=det(a)−2s|det(πy)|sη(2tcta−1a−1c,ϵp,q;α,β). Next recall that Shimura writes, p. 288,   η(g,ϵp,q;α,β)=2n(α+β−ρ)ζp,q(2g;α,β), where   ζp,q(g;α,β)=e−12tr(g)∫x+ϵp>0x+ϵq′>0e−tr(gx)det(x+ϵp)α−ρdet(x+ϵq′)β−ρdx. Altogether this gives the claimed expression. ■ Now we return to the integral (4.8), using the expression just given for $$\hat\Psi(y;\pi(m(a))\phi_{\ell,s})$$. If we substitute the series expansion for $$\mathrm{GW}_s(2\pi\epsilon y)$$ and switch the order of integration, we obtain the expression   ω1ϵ(π(m(a))ϕℓ,s)p,q =(2π)nρ2−n(ρ−1)Γn(α)Γn(β)det(v)−12(s+ρ)(2π)ns2ns ×∫x+ϵp>0x+ϵq′>0det(x+ϵp)α−ρdet(x+ϵq′)β−ρ ×(∑m≥0dm(−2πϵ)|m|(ρ)m(s+ρ)m ×∫Sp,qe−2πtr(tcv−1c(1+2x))|det(y)|sΦm(y)dy)dx. (4.10) Before proceeding, we observe that, in the expression $$y = c \epsilon_{p,q} {}^tc$$, there is an ambiguity in the choice of $$c$$, that is, only the coset $$c\, O(p,q)$$ is well defined. More precisely, we have the following basic structural observations where, in particular, (a) implies that the ambiguity in the choice of $$c$$ has no effect on the double integral. The value of the inner integral does depend on the choice of $$c$$, however! Lemma 4.6. (a) There is a bijection   Xp,q={x∈S∣x+ϵp>0, x+ϵq′>0}↓Zp,q={z∈S∣z+ϵp,q>0, z−ϵp,q>0} given by $$x\mapsto 2x+1=z$$. The action of $$G= \mathrm{GL}_n(\mathbb R)$$ on $$S$$ induces an action of the group $$O(p,q)$$ on $$Z_{p,q}$$. Since $$z+ \epsilon_{p,q} = 2(x+\epsilon_p)$$ and $$z-\epsilon_{p,q}= 2(x+\epsilon'_q)$$, the quantities $$\det(x+\epsilon_p)$$ and $$\det(x+e'_q)$$ are constant on the $$O(p,q)$$-orbits in $$Z_{p,q}$$. (b) Let   Wp,q={w∈Sp,q∣1−w>0,w+1>0}=Sp,q∩S(−1,1), where   S(−1,1)={x∈S∣1−x>0and1+x>0}. The action of $$G= \mathrm{GL}_n(\mathbb R)$$ on $$S$$ induces an action of the group $$O(n)$$ on $$W_{p,q}$$. Moreover, there is a bijection on orbits   O(p,q)∖Zp,q⟷O(n)∖Wp,q defined as follows. For $$z \in Z_{p,q}$$, write $$z = \zeta{}^t\zeta$$ and let $$w=\zeta^{-1}\epsilon_{p,q}{}^t\zeta^{-1}$$. Then $$w\in W_{p,q}$$, $$w$$ depends only on the $$O(p,q)$$-orbit of $$z$$, and the $$O(n)$$-orbit of $$w$$ is independent of the choice of $$\zeta$$. Conversely, for $$w\in W_{p,q}$$, write $$w = \eta \epsilon_{p,q}{}^t\eta$$ and let $$z = \eta^{-1}{}^t\eta^{-1}$$. Then $$z\in Z_{p,q}$$, $$z$$ depends only on the $$O(n)$$-orbit of $$w$$, and the $$O(p,q)$$-orbit of $$z$$ is independent of the choice of $$\eta$$. □ Proof First note that   x+ϵp+x+ϵq′=2x+1, so that $$2x+1$$ is automatically positive definite. This also follows from the given conditions on $$z$$, viz.   z+ϵp,q+z−ϵp,q=2z>0. Now, given $$x\in X_{p,q}$$, we have   2x+1+ϵp,q=2(x+ϵp)>0,2x+1−ϵp,q=2(x+ϵq′)>0, so that $$2x+1$$ lies in $$Z_{p,q}$$. Conversely, if $$z \in Z_{p,q}$$, then   12(z−1)+ϵp=12(z+ϵp,q)>0,12(z−1)+ϵq′=12(z−ϵp,q)>0, so that $$\frac12(z-1)$$ lies in $$X_{p,q}$$. This proves (a). To prove (b), we first note that $$w \in S_{p,q}$$, by construction. We have   1±w=1±ζ−1ϵp,qζ−1=ζ−1(ζtζ±ϵp,q)tζ−1=ζ−1(z±ϵp,q)tζ−1>0, so that $$w\in S_{p,q}\cap S_{(-1,1)}$$, as claimed. The other direction is analogous.■ Remark 4.7. It will be useful to note that, under the bijection of part (b),   2ndet(x+ϵp) =det(z+ϵp,q),2ndet(x+ϵq′) =det(z−ϵp,q), and   det(z±ϵp,q) =|det(w)|−1det(1±w). □ We can make one simplification in the inner integral in (4.10) as follows. Writing $$v = a{}^ta=a k {}^tk{}^ta$$ with $$k\in O(n)$$ and setting $$z=1+2x$$, as in (a) of the previous lemma, we have    ∫Sp,qe−2πtr(tcta−1a−1cz)|det(y)|sΦm(y)dy =det(a)2(s+ρ)∫Sp,qe−2πtr(tccz)|det(y)|sΦm(ak⋅y)dy. (4.11) Since the whole expression is independent of $$k$$, integrating over $$O(n)$$ has no effect, but bringing the $$O(n)$$ integration inside the $$S_{p,q}$$-integral, we have   ∫O(n)Φm(ak⋅y)dk=Φm(a⋅1n)Φm(y)=Φm(v)Φm(y), by Corollary XI.3.2 in [11]. Thus (4.11) is equal to   det(v)s+ρΦm(v)∫Sp,qe−2πtr(tccz)|det(y)|sΦm(y)dy. Noting that $$\mathrm{d}z = 2^{n\rho}\,\mathrm{d}x$$, for (4.10) we have   ω1ϵ(π(m(a))ϕℓ,s)p,q =(2π)nρ2−n(ρ−1)Γn(α)Γn(β)det(v)12(s+ρ)(2π)ns ×∫Zp.qdet(z+ϵp,q)α−ρdet(z−ϵp,q)β−ρ (×∑m≥0dm(−2πϵ)|m|(ρ)m(s+ρ)mΦm(v) ×∫Sp,qe−2πtr(tccz)|det(y)|sΦm(y)dy)dz. (4.12) We need some additional structural information. Let   Rp,q+={δ=diag(δ1,…,δn)∣δ1>⋯>δp>0,δn>⋯>δp+1>0}. There is then a map   O(n)×Rp,q+⟶Sp,q,(u,δ)↦u⋅δϵp,q=uδϵp,qtu=y (4.13) with open dense image, and, by TheoremVI.2.3 of [11],   dy=Ξp,q(δ)dδdu, where   Ξp,q(δ)=c00∏1≤i<j≤porp<i<j≤n|δi−δj|∏i≤p<j(δi+δj), and   dδ=dδ1…dδn. Here $$c_{00}$$ is a certain positive constant depending only on $$n$$. The map (4.13) is $$2^n$$ to $$1$$, due to the fact that the stabilizer in $$O(n)$$ of an element $$\delta\epsilon_{p,q}$$ is the diagonal subgroup, isomorphic to $$(\mu_2)^n$$. Let   Ap,q+={δ12∣δ∈Rp,q+}. Then we have a map   O(n)×Ap,q+×O(p,q)⟶G,(u,a,h)↦uah=g with open dense image, and a left invariant measure $$\mathrm{d}g$$ on $$G$$ has pullback   dg=det(g)−2ρΞp,q(δ)dδdudh, (4.14) where $$a^2=\delta$$ and $$\mathrm{d}h$$ is a Haar measure on $$H=O(p,q)$$. Let   Rp,q+0={δ∈Rp,q+∣δj∈(0,1) for all j }. Then (4.13) restricts to a map   O(n)×Rp,q+0⟶Wp,q with open dense image and we have an injection $$R^{+0}_{p,q} \hookrightarrow O(n)\backslash W_{p,q}$$. Similarly, we have a map   O(p,q)×Rp,q+0⟶Zp,q,(h,δ)↦h⋅δ−1=hδ−1th, with open dense image and an injection $$R^{+0}_{p,q}\hookrightarrow O(p,q)\backslash Z_{p,q}$$. We now return to (4.12) and write $$z = h\cdot \delta_w^{-1}$$ with $$h\in O(p,q)$$ so that   dz=Ξp,q(δw−1)d(δw−1)dh=det(δw)−2ρΞp,q(δw)dδwdh. The double integral becomes    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)−2ρΞp,q(δw) ×∫O(p,q)∫Sp,qe−2πtr(ch⋅δw−1)|det(y)|sΦm(c⋅ϵp,q,1n)dydδwdh. Writing $$g = ch$$, we have $$\mathrm{d}y\,\mathrm{d}h = (\det g)^{2\rho}\,\mathrm{d}g$$ and this becomes    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)−2ρΞp,q(δw) ×∫Ge−2πtr(g⋅δw−1)|det(g)|2sΦm(g⋅ϵp,q,1n)(detg)2ρdgdδw. Now we put $$g\delta_w^{\frac12}$$ for $$g$$ and have    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)s−ρΞp,q(δw) ×∫Ge−2πtr(gtg)|det(g)|2sΦm(g⋅δwϵp,q,1n)(detg)2ρdgdδw. This is    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)s−ρΞp,q(δw) ×∫Ge−2πtr(tgg)|det(g)|2sΦm(δwϵp,q,tgg)(detg)2ρdgdδw, and so, setting $$y^\vee = {}^tgg$$, we arrive at the expression    ∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρdet(δw)s−ρΞp,q(δw) ×∫Sn,0e−2πtr(y∨)det(y∨)sΦm(δwϵp,q,y∨)dy∨dδw. By the inversion formula, cf. Lemma XI.2.3 of [11], we have    ∫Sn,0e−2πtr(y∨)det(y∨)sΦm(δwϵp,q,y∨)dy∨ =Γm(s+ρ)(2π)−n(s+ρ)(2π)−|m|Φm(δwϵp,q,1n). Returning to the inner sum in (4.12) and canceling the gamma factor, we have   Γn(s+ρ)(2π)−n(s+ρ)∑m≥0(−ϵ)|m|dm(ρ)mΦm(v)Φm(δwϵp,q). (4.15) Now, for $$v=a{}^ta$$, we write   Φm(v)Φm(δwϵp,q)=∫O(n)Φm(ak⋅δwϵp,q)dk, so that (4.15) becomes the integral over $$O(n)$$ of   Γn(s+ρ)(2π)−n(s+ρ)∑m≥0(−ϵ)|m|dm(ρ)mΦm(ak⋅δwϵp,q) =Γn(s+ρ)(2π)−n(s+ρ)exp⁡(−ϵtr(ak⋅δwϵp,q)), via the standard expansion of $$\exp(\mathrm{tr}(z))$$, Proposition XII.1.3 (i) of [11]. But now the last three lines of (4.12) amount to   Γn(s+ρ)(2π)−n(s+ρ)∫O(n)∫Rp,q+0det(δw−1+ϵp,q)α−ρdet(δw−1−ϵp,q)β−ρexp⁡(−ϵtr(ak⋅δwϵp,q)) ×det(δw)s−ρΞp,q(δw)dkdδw=Γn(s+ρ)(2π)−n(s+ρ)∫Wp,q|det(w)|−(α+β−2ρ)det(1+w)α−ρdet(1−w)β−ρexp⁡(−ϵtr(a⋅w)) ×|det(w)|s−ρdw. Here the exponent of $$|\det(w)|$$ is   2ρ−α−β+s−ρ=0.(!!!) Taking $$a$$ with $$a={}^ta$$, we get simply   Γn(s+ρ)(2π)−n(s+ρ)∫Wp,qdet(1+w)α−ρdet(1−w)β−ρexp⁡(−ϵtr(vw))dw, and, altogether:   ω1ϵ(π(m(a))ϕℓ,s)p,q =Bn(α,β)−12−n(ρ−1)det(v)12(s+ρ) ×∫Wp,qdet(1+w)α−ρdet(1−w)β−ρexp⁡(−ϵtr(vw))dw. (4.16) Summing over the signatures, we have   ω1ϵ(π(m(a))ϕs,ℓ) =Bn(α,β)−12−n(ρ−1)det(v)12(s+ρ) ×∫1±w>0det(1+w)α−ρdet(1−w)β−ρexp⁡(−ϵtr(vw))dw. In the integral here we put $$2r = 1+w$$ and obtain   2nsexp⁡(ϵtr(v))∫r>01−r>0exp⁡(−2ϵtr(vr))det(r)α−ρdet(1−r)β−ρdr. For $$\epsilon=-1$$, this gives   ω1ϵ(π(m(a))ϕs,ℓ)=2n(s−ρ+1)det(v)12(s+ρ)exp⁡(−tr(v))Mn(α,α+β;2v) and for $$\epsilon=+1$$, this is   2nsexp⁡(−tr(v))∫r>01−r>0exp⁡(tr(2v(1−r))det(r)α−ρdet(1−r)β−ρdr, and hence   ω1ϵ(π(m(a))ϕs,ℓ)=2n(s−ρ+1)det(v)12(s+ρ)exp⁡(−tr(v))Mn(β,α+β;2v). Finally, taking into account the scaling transformation used to eliminate $$T$$, we obtain the claimed expression. This completes the proof of Theorem 4.1. ■ 5 The $$\xi$$-Operator In this section, we construct the $$\xi$$-operator, analogous to that defined in [5, 6], in our present situation. This operator is a slight modification of the $$\bar\partial$$-operator and is best expressed in terms of differential forms and the Hodge $$*$$-operator for homogeneous vector bundles on the Siegel space $$\mathfrak H_n$$. Here, as before, we write $$\tau = u+iv$$, $$v>0$$, for an element of $$\mathfrak H_n$$. 5.1 Homogeneous bundles and differential forms For a representation $$(\mu,\mathcal V_\mu)$$ of $$K$$, let $$\mathcal L_\mu$$ be the homogeneous vector bundle   Lμ=(G×Vμ)/K⟶G/K=Hn. Here $$K$$ acts by $$(g,v)\cdot k = (gk, \mu(k)^{-1}v)$$, and the $$C^\infty$$-sections are given by   Γ(Hn,Lμ)≃[C∞(G)⊗Vμ]K,μ(k)ϕ(gk)=ϕ(g). (5.1) If $$\Gamma$$ is a discrete subgroup $$Sp_n(R)$$—the main cases of interest will be $$\Gamma\subset \mathrm {Sp}_n({\mathbb Z})$$ an arithmetic subgroup, the intersection of such a subgroup with $$N$$, or the trivial subgroup — we use the same notation for the quotient bundle on $$X=\Gamma\backslash \mathfrak H_n$$. ({Here we should use orbifolds/stacks.}) We write $$\mathcal L_r = \mathcal L_{\det^{-r}}$$ with $$\mathcal V_{\det^{-r}}={\mathbb C}(-r)$$, so that sections of $$\mathcal L_r$$ satisfy $$\phi(gk) = \det(\mathbf k)^r\,\phi(g)$$. The function $$j(g,i)^{-r}$$ defines a smooth section of $$\mathcal L_r$$ on $$\mathfrak H_n$$. If $$\phi$$ is any section, we can write   ϕ(g)=j(g,i)−rf(τ), where $$f(g(i)) = j(g,i)^{r}\,\phi(g)$$, and invariance of $$\phi$$ under left multiplication by an element $$\gamma\in \Gamma$$ is equivalent to the invariance of $$f$$ under the corresponding weight $$r$$ slash operator for $$\gamma$$. The Petersson metric on $$\mathcal L_r$$ is given by $$|\phi(g)|^2 = |f(\tau)|^2\,\det(v)^r$$. More generally, suppose that the representation $$\mu$$ of $$K$$ on $$\mathcal V_\mu$$ extends to a representation of $$K_{\mathbb C} \simeq \mathrm{GL}_n({\mathbb C})$$. Let $$J(g,\tau) = c\tau+d$$ be the canonical automorphy factor $$J: G\times \mathfrak H_n \longrightarrow K_{\mathbb C}$$. Note that   J(k,i)=A−iB=k¯=tk−1. A general smooth section of $$\mathcal L_\mu$$ on $$\mathfrak H_n$$ can be written as   ϕ(g)=μ(tJ(g,i))f(τ), where $$f$$ is a smooth $$\mathcal V_\mu$$-valued function of $$\mathfrak H_n$$. The left invariance of the section $$\phi$$ under $$\gamma\in G$$ is equivalent to the invariance   f(γτ)=μ(tJ(γ,τ)−1)f(τ). (5.2) Now suppose, moreover, that $$\langle \,{},{}\,\rangle_\mu$$ is a Hermitian inner product on $$\mathcal V_\mu$$ such that $$\mu(a)^* = \mu({}^t\bar a)$$ for all $$a\in \mathrm{GL}_n({\mathbb C})$$. Such an inner product is “admissible” in the terminology of [3, p. 47]. Then we can define the Petersson metric on $$\mathcal L_\mu$$ by   ||ϕ(g)||μ2=⟨ϕ(g),ϕ(g)⟩μ=⟨f(τ),μ(v−1)f(τ)⟩μ,v=ℑ(τ)=ℑ(g(i)). (5.3) Since   v(γ(τ))=t(cτ+d)−1v(cτ¯+d)−1, the right side of (5.3) is $$\gamma$$-invariant if $$f$$ satisfies (5.2). If $$(\lambda, F_\lambda)$$ is a finite-dimensional unitary representation of $$\Gamma$$, there is an associated flat bundle $$\mathcal F_\lambda$$ on $$X$$ defined by   Fλ=Γ∖(Hn×Fλ), with Hermitian metric given by the norm on $$F_\lambda$$. The bundle $$\Omega^{N}$$ of top-degree holomorphic differential forms on $$\mathfrak H_n$$ is $$\mathcal L_{n+1}$$. Here $$N=\frac12n(n+1)$$. Writing $$\tau = \sum_\alpha \tau_\alpha \, e_\alpha$$, we let   dμ(τ)=∧αdτα and note that   dμ(g(τ))=j(g,τ)−2ρdμ(τ),ρ=12(n+1). If $$\phi$$ is a section of $$\mathcal L_{n+1}$$, the corresponding section of $$\Omega^N$$ is   ϕ(g)j(g,i)2ρdμ(τ)=f(τ)dμ(τ). Let $$\mathcal E^{a,b}$$ be the bundle of differential forms of type $$(a,b)$$ on $$\mathfrak H_n$$. We use the same notation for the corresponding bundle on $$X$$. Let   g=k+p++p− be the Harish-Chandra decomposition of $${\mathfrak g} = \mathrm{Lie}(G)\otimes_\mathbb R{\mathbb C}$$. Then we have an isomorphism   Γ(Hn,Ea,b)=A(a,b)(Hn) ⟶∼ [C∞(G)⊗∧a(p+∗)⊗∧b(p−∗) ]K. (5.4) More explicit coordinates can be given as follows. Let $$S= \mathrm{Sym}_n(\mathbb R)$$ with basis $$e_\alpha$$ and dual basis $$e_\alpha^\vee$$ with respect to the trace form. ({Recall that we take $$e_{jj}$$, $$1\le j \le n$$ and $$e_{ij}+e_{ji}$$, $$1\le i<j\le n$$, as basis for $$S$$ with dual basis is $$e_{jj}$$ and $$\frac12(e_{ij}+e_{ji})$$.}) There are isomorphisms   p±:SC ⟶∼ p±,p±(X)=12(X±iX±iX−X). (5.5) Then we have a basis $$L_\alpha = p_-(e_\alpha^\vee)$$ for $${\frak p}_-$$, and we write $$\eta'_\alpha\in {\frak p}_-^*$$ for the dual basis. The operator on the right side of (5.4) corresponding to $$\bar\partial$$ is then   ∂¯=∑αp−(eα∨)⊗ηα′, (5.6) where $$\eta'_\alpha$$ acts on $$\wedge^\bullet({\frak p}^*)$$ by exterior multiplication. Suppose that $$E$$ is any Hermitian vector bundle on $$X$$, and let $$\nu: E \ {\overset{\sim}{\longrightarrow}}\ E^*$$ be the conjugate linear isomorphism determined by the Hermitian inner product. Recall that the Hodge $$*$$-operator gives a conjugate linear operator [27, Chapter V, Section 2],   ∗¯E:Ea,b⊗E⟶EN−a,N−b⊗E∗,α⊗h↦(∗α¯)⊗ν(h). Definition. For a Hermitian vector bundle $$E$$ on $$X$$, the $$\xi$$-operator is defined as   ξ=ξE=∗¯E∂¯:Γ(X,Ea,b⊗E) ⟶ Γ(X,EN−a,N−b−1⊗E∗). (5.7) If $$E=\mathcal F_\lambda\otimes \mathcal L_\mu$$ for a unitary flat bundle $$\mathcal F_\lambda$$ and for $$\mathcal L_\mu$$ with the Petersson metric defined by (5.3), then $$E^* \simeq \mathcal F_{\lambda^\vee} \otimes \mathcal L_{\mu^\vee}$$ where $$\lambda^\vee$$ and $$\mu^\vee$$ are the contragradients of $$\lambda$$ and $$\mu$$. For example, for an integer $$\kappa$$, a $$C^\infty$$-section $$f$$ of the bundle   Fλ⊗E0,N−1⊗Ln+1−κ (5.8) can be viewed as an $$\mathcal F_\lambda$$-valued $$(0,N-1)$$-form on $$\mathfrak H_n$$ of weight $$n+1-\kappa$$. Then $$\xi(f)$$ is a section of   Fλ∨⊗EN,0⊗Lκ−n−1 ≃ Fλ∨⊗Lκ. For $$n=1$$, this coincides with the $$\xi$$-operator defined in [6]. From now on, to simplify things slightly, we will omit the flat bundle $$\mathcal F_\lambda$$. It is useful to note that we have the diagram   $$\$$ and that the two maps on the right are given explicitly by (5.6) and   ∗¯:ϕ⊗x⊗ω⟼ϕ¯⊗ν(x)⊗∗ω¯, (5.9) where $$\nu:\mathcal V_\mu \ {\overset{\sim}{\longrightarrow}}\ \mathcal V_{\mu^\vee}$$ is the conjugate linear isomorphism determined by $$\langle \,{},{}\,\rangle_\mu$$. 5.2 Whittaker forms As explained earlier, we consider a version of these operators involving Whittaker forms. For $$T\in\mathrm{Sym}_n(\mathbb R)$$, recall that $$\mathcal W^{\mathrm{T}}(G)$$ is the space of smooth functions $$\phi$$ on $$G$$ such that $$\phi(n(b)g) = e(\mathrm{tr}(Tb))\,\phi(g)$$. Define the space of Whittaker forms valued in $$\mathcal L_\mu$$ as   W−T(Ea,b⊗Lμ) ⟶∼ [W−T(G)⊗Vμ⊗∧a(p+∗)⊗∧b(p−∗) ]K. (5.10) There is a corresponding $$\xi$$-operator described by the diagram   $$\$$ where the maps in the right column are given by (5.6) and (5.9). Our Whittaker functionals (3.17) provide a supply of elements in these spaces via the diagram   $$\$$ Here the maps in the left column are again given by (5.6) and (5.9). We can utilize the fact that the $$K$$-spectrum of $$I(s)$$ is multiplicity free to produce various examples. 5.3 Some particular vectors In the construction of Whittaker forms, we will be interested in the following functions in $$I(s)$$. The isomorphism (5.5) satisfies   Ad(k)p+(X)=p+(k⋅X),k⋅X=kXtk. (5.11) Similarly, $$p_-(X) = \overline{p_+(X)}$$ and   Ad(k)p−(X)=p−(k¯⋅X)=p−(tk−1⋅X). The trace pairing   ⟨p+(X),p−(Y)⟩=tr(XY) is then invariant under the adjoint action of $$K$$, so that $$\mathfrak p_{\pm}^* \simeq \mathfrak p_{\mp}$$ as $$K$$-modules. Note that   ∧N(p+) ⟶∼ C(2ρ) as $$K$$-modules, where $$N= n\rho = \dim \mathfrak p_{\pm}$$. For the fixed integer $$\kappa$$, let $$r= \kappa - n-1$$, and consider the space   [ I(s)⊗∧N−1(p−∗)⊗C(r) ]K. (5.12) Fixing a basis vector $$\bar\omega$$ for $$\wedge^N(\mathfrak p_-^*)$$, we have a pairing   ∧N−1(p−∗)⊗p−∗⟶∧N(p−∗) ⟶∼ C(2ρ), and hence an isomorphism   ∧N−1(p−∗) ⟶∼ p−⊗C(2ρ). (5.13) Thus   σ∨:=∧N−1(p−∗)⊗C(r)≃Symn(C)⊗C(κ). The vector $$v_0= 1_n\in \mathrm{Sym}_n({\mathbb C})$$ is $$O(n)$$-invariant, and so, via (5.13), we have the standard function   ϕs,σ(nm(a)k)=χ(det(a))|det(a)|s+ρdet(k)−κtkk (5.14) in (5.12). By (2.10), it is characterized by the invariance property   π(k)ϕs,σ=det(k)−κtk⋅ϕs,σ, together with the normalization $$\phi_{s,\sigma}(e) = 1_n$$. For generic $$s$$, the function $$\phi=\phi_{s,\sigma}$$ can also be obtained by applying a certain differential operator to $$\phi_{s,-\kappa}$$. Here we use the conventions described in more detail in Section 3.3. Let $$e_\alpha$$ be a basis for $$S= \mathrm{Sym}_n(\mathbb R)$$ and let $$e_\alpha^\vee$$ be the dual basis with respect to the trace form. Let   D=∑αp+(eα)⊗eα∨ ∈ p+⊗S⊂ U(g)⊗S. (5.15) This operator has the following invariance property. Lemma 5.1. For $$k\in K$$,   Ad(k)D=tk⋅D=tkDk.□ Proof We compute using (5.11)   Ad(k)D =∑αp+(k⋅eα)⊗eα∨=∑α∑βtr((k⋅eα)eβ∨)p+(eβ)⊗eα∨ =∑βp+(eβ)⊗∑αtr(eα(tk⋅eβ∨))eα∨=∑βp+(eβ)⊗tk⋅eβ∨=tk⋅D.■ Now consider the function $$\pi(D)\phi_{s,-\kappa} \in I(s)\otimes \mathrm{Sym}_n({\mathbb C})$$. For $$k\in K$$, we have   π(k)π(D)ϕs,−κ =π(Ad(k)D)π(k)ϕs,−κ =det(k)−κtk⋅π(D)ϕs,−κ. Thus $$\pi(D)\phi_{s,-\kappa}$$ is a multiple of $$\phi_{s,\sigma}$$, and it remains to calculate the constant of proportionality. Lemma 5.2.   π(D)ϕs,−κ=12(s+ρ−κ)ϕs,σ=α(s)ϕs,σ.□ Proof An easy calculation shows that   p+(X)ϕs,ℓ(e)=12(s+ρ+ℓ)tr(X). Therefore   Dϕs,−κ=12(s+ρ−κ)1n.■ In particular, for $$s_0 = \kappa-\rho$$, we have $$p_+(X)\,\phi_{s_0,-\kappa} =0$$, so that $$\phi_{s_0,-\kappa}$$ is a highest weight vector. But this was already clear since this vector is the generator of $$R(0,m+2)$$, that is, the image in $$I(s_0)$$ of the Gaussian for the negative definite space of dimension $$m+2$$, cf. Section 2.6. Thus, we have basis vectors $$\phi_{s,-\kappa}$$ and $$\phi_{s,\sigma}$$ in the one-dimensional spaces on the upper left side of the diagram   $$\$$ (5.16) 5.4 Calculation of $$\bar\partial$$ and $$\xi$$ We now compute the image of $$\phi_{s,\sigma}$$ under the operator $$\xi$$ on the left side of (5.16). Proposition 5.3.   ∂¯ϕs,σ=n(s−ρ+κ)ϕs,−κ⋅dμ(τ¯), and   ξ(ϕs,σ)=n(s¯−ρ+κ)ϕs¯,κ.□ Here, we are slightly abusing notation and writing $$\mathrm{d}\mu(\bar\tau)$$ for the basis element of $$\wedge^{N}(\mathfrak p_-^*)$$ arising as the restriction of this global form to the tangent space at $$i$$. Proof First we apply $$\bar\partial$$:   [ I(s)⊗∧N−1(p−∗)⊗C(n+1−κ) ]K⟶∂¯[ I(s)⊗∧N(p−∗)⊗C(n+1−κ) ]K, noting that both spaces are one dimensional. Using (5.6), and (5.15),   ∂¯ϕs,σ=α(s)−1∂¯⋅Dϕs,−κ=α(s)−1∑αp−(eα∨)p+(eα)ϕs,−κ⋅dμ(τ¯). The second-order operator occurring here has the following expression in terms of the Casimir operator (2.9). We have   ∑αp+(eα)p−(eα∨)+p−(eα∨)p+(eα) =2∑αp−(eα∨)p+(eα)+[p+(eα),p−(eα∨)] =ρH+2∑αp−(eα∨)p+(eα), where $$H = \sum_j H_j$$, for $$H_j$$ as in Section 2.4. On the other hand, a short calculation shows that $$C_{\mathfrak k}$$ acts by $$\frac18\,\kappa^2$$, whereas $$H$$ acts by $$-n\kappa$$. Thus we have   2∑αp−(eα∨)p+(eα)ϕs,−κ =(4n(C−Ck)−ρH)ϕs,−κ =12n(s2−ρ2+2ρκ−κ2)ϕs,−κ =12n(s−ρ+κ)(s+ρ−κ)ϕs,−κ. This gives the first identity. Then   ∗¯∂¯ϕs,σ=n(s¯−ρ+κ)ϕs¯,κ⋅dμ(τ), so the second identity is immediate. ■ 5.5 Vector-valued Whittaker functions Now we can apply the Whittaker functionals to obtain Whittaker forms on the right side of (5.16). Recall that $$f_{s,-\kappa}(g) = \omega_1^{-T}(\pi(m(g))\phi_{s,-\kappa})$$ and let   fs,σ(g)=ω1−T(π(m(g))ϕs,σ). (5.17) Then by Lemma 5.2,   α(s)fs,σ(g)=Dfs,−κ(g)=∑αp+(eα)fs,−κ(g)eα∨, (5.18) where $$\alpha(s) = \frac12(s-s_0) = \frac12(s+\rho-\kappa)$$. Thus $$\boldsymbol{f}_{s,\sigma}$$ is $$S$$-valued and, by Lemma 5.1, satisfies   fs,σ(gk)=det(k)−κtkfs,σ(g)k. (5.19) The scaling relation   fs,σ −T(m(c)g)=det(c)s+ρfs,σ −cTtc(g),c∈GLn(R)+, (5.20) holds for $$\boldsymbol{f}_{s,\sigma}=\boldsymbol{f}_{s,\sigma}^{\ -T}$$, where we include the normally omitted superscript. As a consequence of our constructions, we have obtained the following. Theorem 5.4. The Whittaker forms $$\boldsymbol{f}_{s,\sigma}$$ defined by (5.18) lie in the space   [W−T(G)⊗∧N−1(p−∗)⊗C(r) ]K. (i) The infinitesimal character of $$\boldsymbol{f}_{s,\sigma}$$ is $$\chi_{\lambda(s)+\rho_G}$$. In particular, for the Casimir operator $$C$$,   C⋅fs,σ=18(s+ρ)(s−ρ)fs,σ. (ii) For the $$\xi$$-operator,   ξ(fs,σ)=n(s¯−ρ+κ)fs,−κ¯. (iii) At $$s=s_0=\kappa-\rho$$,   ξ(fs0,σ)(g)=c(n,s0)WκT(g), where   WκT(n(b)m(a)k)=det(k)κdet(a)κe(tr(Tτ))=j(g,i)−κqT, (5.21)$$v= a{}^ta$$, $$\tau = b+{\rm i}v$$, and   c(n,s0)=2n(κ−2ρ+1). (Here note that, although $$\alpha=\alpha(s_0)=0$$, the calculations of Section 5.6 show that the right side of (5.18) is divisible by $$\alpha$$ and that $$\boldsymbol{f}_{s_0,\sigma}$$ is given by a nice convergent power series in $$v$$.) □ Proof The first two statements follow from the commutativity of (5.16) and the corresponding results for the vectors on the left side. In particular,   ξ(fs,σ)=n(s¯−ρ+κ)fs,−κ¯. By (4.5), we have   fs,−κ¯(n(b)m(a)k)=c(n,s¯)qTdet(v)12(s¯+ρ)det(k)κMn(α,α+β;4πcvtc)¯. But, at $$s=s_0=\kappa-\rho$$, $$\alpha=0$$ and $$M_n(0,s_0;4\pi cv{}^tc)=1$$, so we get the claimed expression. ■ 5.6 Some explicit formulas In this section, we calculate the function $$\boldsymbol{f}_{s,\sigma}$$ more explicitly. In view of the scaling relation (5.20), it suffices to consider the case $$T=1_n$$. Recall that   Mn(α,α+β;z)=∑m≥0(α)m(s+ρ)mdm(ρ)mΦm(z), where $$\mathbf{m} = (m_1,\ldots,m_n)$$ with $$m_1\ge m_2\ge \cdots\ge m_m\ge0$$. Since   (α)m=(α)m1(α−12)m2…(α−12(n−1))mn, we have $$(\alpha)_0 =1$$ and $$(0)_{\mathbf{m}}=0$$ for $$\mathbf{m} \ne0$$. In particular, $$M_n(0,\beta;z) = 1$$. For $$\mathbf{m}\ne 0$$, let   [0]m=(α−1(α)m))|α=0. (5.22) Then   [0]m={(1)m1−1(−12)m2(−12)r(r+1)!,if m=(m1,m2,1,…,1,0,…,0),0,otherwise.  Here, in the first case, $$\mathbf{m}>0$$ with $$m_3\le 1$$ and with a string of $$r$$$$1$$’s following $$m_2$$. With this notation, we can state our result. Proposition 5.5. (i) For $$a\in \mathrm{GL}_n(\mathbb R)^+$$,   fs,σ(m(a)) =c(n,s)e−2πtr(v)det(v)12(s+ρ)Bn(α,β)−1 ×∫t>01−t>0(1n+α(s)−14πtata)e4πtr(tv)det(t)α−ρdet(1−t)β−ρdt. (5.23) Recall that $$c(n,s) = 2^{n(s-\rho+1)}$$. (ii) For $$n=1$$,   fs0,σ(m(a))=2κ−14πe2πvv12κ+1∫01e−4πtvtκ−1dt. (iii) For $$n\ge 2$$,   fs0,σ(m(a))=2n(κ−n)e−2πtr(v)det(v)12κ( 1n+∑m>0[0]m(κ)mdm(ρ)mta∂∂v{ Φm(4πv) }a). Here $$v = a{}^ta$$, as usual. □ Remark 5.6. In the case $$n=1$$, we recover the basic formula from [5]. Notice that, in the case $$n\ge 2$$, we do not yet have a complete evaluation of $$\boldsymbol{f}_{s_0,\sigma}(m(a))$$; it would be interesting to have a nicer closed formula. □ Proof Recalling (5.18), we begin by computing some derivatives. Lemma 5.7. For $$X\in S$$,   p+(X)fs,ℓ(m(a))=12(DX+ℓtr(X)+4πtr(aXta))⋅fs,ℓ(m(a)). Here, for a function $$h$$ on $$\mathrm{GL}_n(\mathbb R)$$,   DXh(a)=ddt(h(aexp⁡(tX)) )|t=0. In particular, for a function of $$v = a{}^ta$$,   DX=2tr(aXta∂∂v).□ This follows from a simple direct calculation. Then, recalling the expression of the function $$f_{s,\ell}$$ given in Theorem 4.1, we have the nice expression for our Whittaker form   α(s)fs,σ(m(a)) =∑αp+(eα)fs,−κ(m(a))eα∨ =c(n,s)e−2πtr(v)det(v)12(s+ρ)Bn(α,β)−1 ×∫t>01−t>0(α(s)1n+4πtata)e4πtr(tv)det(t)α−ρdet(1−t)β−ρdt. (5.24) This gives statement (i). When $$n=1$$, (5.24) amounts to   fs,σ(m(a)) =c(1,s)e−2πvv12(s+1)α(s)−1B(α,β)−1 ×∫01(α(s)+4πtv)e4πtvtα−1(1−t)β−1dt. (5.25) Lemma 5.8.   α−1B(α,β)−1∫01(α(s)+4πtv)e4πtvtα−1(1−t)β−1dt=M(α+1,s+1;4πv).□ Proof Initially, we have   α−1B(α,β)−1∫01(α(s)+4πtv)e4πtvtα−1(1−t)β−1dt =M(α,s+1;4πv)+α−1v∂∂v{M(α,s+1;4πv)}. But by 13.4.10 of [1],   4πvM′(α,s+1;4πv)=−αM(α,s+1;4πv)+αM(α+1,s+1;4πv).■ Thus   fs,σ(m(a))=2se−2πvv12(s+1)M(α+1,s+1;4πv). This agrees with the Whittaker form for $$m=1$$ in [5], up to a simple factor. To evaluate at $$s=s_0$$ we return to the original expression (5.25). Since   α−1B(α,β)−1=Γ(s+1)Γ(α+1)Γ(β), we can plug in $$s_0= \kappa-1$$ so that $$\alpha_0=0$$ and $$\beta_0= \kappa$$, and obtain the following expression   fs0,σ(g) =c0e−2πvv12κ∫01(4πtv)e4πtvt−1(1−t)κ−1dt =c04πe−2πvv12κ+1∫01e4πtv(1−t)κ−1dt =c04πe2πvv12κ+1∫01e−4πtvtκ−1dt =c04πe2πvv12κ+1[ (4πv)−κΓ(κ)−∫1∞e−4πtvtκ−1dt ], where $$c_0 = c(1,s_0) = 2^{\kappa-1}$$. The last expression here is in some ways more enlightening, although we do not record it in the statement (ii) of the proposition. Finally, for general $$n\ge 2$$, we want to evaluate (5.23) at $$s_0= \kappa-\rho$$. The term associated with $$1_n$$ is given by the scalar matrix   c(n,s)e−2πtr(v)det(v)12(s+ρ)Bn(α,β)−1 ×∫t>01−t>0e4πtr(tv)det(t)α−ρdet(1−t)β−ρdt⋅1n =c(n,s)e−2πtr(v)det(v)12(s+ρ)Mn(α,α+β;4πv)⋅1n. Since $$M_n(0,\kappa;z)=1$$, we obtain a contribution   2n(κ−n)e−2πtr(v)det(v)12κ⋅1n. The other contribution is the value of   c(n,s)e−2πtr(v)det(v)12(s+ρ) ×ta( α−1Bn(α,β)−1∫t>01−t>04πte4πtr(tv)det(t)α−ρdet(1−t)β−ρdt )a at $$s_0$$. The inner integral here is just   α−1∂∂v{ Mn(α,α+β;4πv) }=∑m>0α−1(α)m(s+ρ)mdm(ρ)m∂∂v{ Φm(4πv) }, where we can omit the term $$\mathbf{m}=0$$ since $$\Phi_{0}(z)=1$$ is killed by $$\partial/\partial v$$. For $$\mathbf{m}\ne 0$$, $$\alpha^{-1}(\alpha)_{\mathbf{m}}$$ is finite at $$\alpha=0$$ and vanishes if $$m_3>1$$. With the notation explained in (5.22), we arrive at the expression given in (iii). ■ 6 A Global Construction In this section, we define the space of Whittaker forms and discuss the “global” $$\xi$$-operator. For simplicity, we restrict to the case of $$\Gamma= \mathrm {Sp}_n({\mathbb Z})$$ and a positive even integer $$\kappa$$. For $$T\in \mathrm{Sym}_n({\mathbb Z})^\vee_{>0}$$, we consider the basic Whittaker form   fs,σ −T∈[W−T(G)⊗∧N−1(p−∗)⊗C(κ−n−1) ]K. (6.1) These forms are the analogues of those considered in Section 4 of [5] in the case of $$\mathrm{SL}_2(\mathbb R)$$, that is, $$n=1$$. By construction, they are invariant under the translation subgroup $$\Gamma_\infty^u$$ of $$\mathrm {Sp}_n({\mathbb Z})$$. Setting $$s=s_0$$, we have Whittaker forms $$\boldsymbol{f}_{s_0,\sigma}^{\ -T}$$ satisfying   C⋅fs0,σ −T=18κ(κ−n−1)fs0,σ −T, where $$C$$ is the Casimir operator for $$G$$. Let   Hn+1−κ(G) ⊂ [C∞(G)⊗∧N−1(p−∗)⊗C(κ−n−1) ]K be the subspace spanned by these forms as $$T$$ varies over $$\mathrm{Sym}_n({\mathbb Z})^\vee_{>0}$$. Clearly, the basic forms $$\boldsymbol{f}^{\ -T}_{s_0,\sigma}$$ are linearly independent. Let $$\mathbb M_\kappa(G)$$ be the subspace of $$C^\infty(G)$$ spanned by the functions $$W_T^{\kappa}$$ given by (5.21) as $$T$$ varies over $$\mathrm{Sym}_n({\mathbb Z})^\vee_{>0}$$. By (iii) of Theorem 5.4, the $$\xi$$-operator induces an isomorphism   ξ:Hn+1−κ(G) ⟶∼ Mκ(G). On the other hand, if $$\kappa > 2n$$, the classical theory of Poincaré series [19] implies that the series   PκT(g)=∑Γ∞u∖ΓWκT(γg) converges absolutely and uniformly on compact subsets of $$G$$. The resulting functions are “holomorphic” cusp forms and span the space $$S_\kappa(\Gamma)$$ of such forms. Thus we have constructed a diagram   $$\$$ (6.2) For example, assume that $$S_\kappa(\Gamma)$$ is one-dimensional and let $$\chi$$ be a generator (this is, e.g., the case for $$n=2$$ and $$\kappa=10$$ or $$12$$, where $$S_\kappa(\Gamma)$$ is spanned by the Igusa cusp form $$\chi_{10}$$ or $$\chi_{12}$$). Then, using the fundamental formula (7) of [19], it is easily seen that   ξΓ(fs0,σ −T)=A⋅(detT)n+12−κε(T)aT(χ)¯⋅χ‖χ‖2, where $$A$$ is a non-zero constant independent of $$T$$, and $$\varepsilon(T)$$ is the order of the stabilizer of $$T$$ in $$\mathrm{GL}_n({\mathbb Z})$$. Moreover, $$a_T(\chi)$$ denotes the $$T$$th Fourier coefficient of $$\chi$$, and $$\|\chi\|$$ its Petersson norm. Funding This work was supported by an Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant and Deutsche Forschungsgemeinshaft (DFG) grant BR-2163/4-1 within the research unit Symmetry, Geometry, and Arithmetic. Acknowledgements This study is part of an ongoing joint project begun during a visit by the third author to Darmstadt in December of 2011. The third author would like to acknowledge the support of an Oberwolfach Simons Visiting Professorship for a two-week visit to Darmstadt in connection with the Oberwolfach meeting on Modular Forms in April of 2014 as well as a Simons Visiting Professorship at MSRI, August–December 2014 for the Program on New Geometric Methods in Number Theory and Automorphic Forms. He also benefited from several additional productive visits to Darmstadt in 2012–2013. All authors benefited from stays at the ESI in Vienna during the program on Arithmetic Geometry and Automorphic Representations in April–May 2015, and much appreciate the excellent and congenial working conditions at all of these institutions. Finally, the authors would like to thank the referee for useful comments which clarified the exposition. Appendix A.1 Notation We summarize the slight variation of the notation from [11, 22] used in this paper. In particular, we specialize to the case of the formally real Jordan algebra $$S = \mathrm{Sym}_n(\mathbb R)$$. Here is a list of notation:   G =Spn(R),S =Symn(R). For $$g\in \mathrm{GL}_n({\mathbb C})$$ and $$x\in S_{\mathbb C}= \mathrm{Sym}_n({\mathbb C})$$, $$g\cdot x = g x \,{}^tg$$.   ρ =12(n+1)P(SC) =polynomial functions,ℓ(g)f(z)=f(g−1⋅z),Δj(z) =principal j×j-minor of z∈SC, Δn(z)=det(z) ,m =(m1≥m2≥⋯≥mn),mj∈Z,  Δm(z) =Δ1(z)m1−m2…Δj(z)mn−1−mnΔn(z)mn,Pm(SC) =subspace generated by GLn(C)-translates of Δm, m≥0,dm =dimC⁡Pm(SC),Φm(z) =∫O(n)Δm(k⋅z)dk,Γn(s) =(2π)14n(n−1)Γ(s)Γ(s−12)…Γ(s−12(n−1)),Bn(α,β) =Γn(α)Γn(β)Γn(α+β),Γm(λ) =Γn(λ+m)=(2π)14n(n−1)∏i=1nΓ(λ+mi−12(i−1)),Γm(λ) =(λ)mΓn(λ),(λ)m =∏i=1n(λ−12(i−1))mi,(s)m =s(s+1)…(s+m−1)=Γ(s+m)Γ(s). The notation $$\Gamma_{\mathbf m}(\lambda)$$ does not seem to be standard, but it is frequently convenient. The function $$\Phi_{\mathbf m}(z,w)$$ on $$S_{\mathbb C}\times S_{\mathbb C}$$ defined in [22, Lemma 1.11] is characterized by the following two properties: (i) $$\Phi_{\mathbf m}(z,1_n) = \Phi_{\mathbf m}(z).$$ (ii) For $$a\in \mathrm{GL}_n({\mathbb C})$$,   Φm(a⋅z,w)=Φm(z,ta¯⋅w). For the trace form $$\langle \,{x},{y}\,\rangle = \mathrm{tr}(xy)$$ on $$S$$ and the standard basis   {eα}={eii,eij+eji∣1≤i≤n,i<j}, the dual basis is   {eα∨}={eii,12(eij+eji)∣1≤i≤n,i<j}. At several points in the calculations, we need the following inversion formula [11, Chapter XI, Lemma XI.2.3]. For $$p\in \mathcal P_{\mathbf m}(S_{\mathbb C})$$, and $$\mathrm{Re}(\lambda)>\rho-1$$,   ∫x>0e−tr(xy)p(x)det(x)λ−ρdx=Γn(λ+m)det(y)−λp(y−1). We also recall that   etr(zw¯)=∑m≥0dm(ρ)mΦm(z,w), and that, for $$\lambda\in{\mathbb C}$$ with $$(\lambda)_{\mathbf m}\ne 0$$ for all $$\mathbf m\ge 0$$, $$z$$, $$w\in S_{\mathbb C}$$ the J-Bessel function is defined by [22, p.818],   Jλ(z,w)=∑m≥0(−1)|m|dm(λ)m(ρ)mΦm(z,w). A.2 An estimate The following classical estimate will be useful. Lemma A.2. (i) For $$\mathrm{Re}(\alpha)>\rho$$ and $$\mathrm{Re}(\beta-\alpha)>\rho$$,   |1F1(α,β;v)|≤etr(v)det(v)Re(α−β)Γn(Re(β−α))|Bn(α,β−α)|−1. (ii) Suppose that $$\alpha$$ and $$\beta$$ are real with $$\alpha>\rho$$ and $$\beta-\alpha>\rho$$. Then for any $$\epsilon$$ with $$0<\epsilon<1$$,   |1F1(α,β;v)|≥Cϵe(1−ϵ)tr(v), where $$C_\epsilon>0$$ depends on $$\epsilon$$, $$\alpha$$, and $$\beta$$ and can be taken uniformly for $$\alpha$$ and $$\beta$$ in a compact set.□ Proof Suppose that $$v = a^2$$ where $$a={}^ta$$, and, using (4.3), consider   1F1(α,β;v) =Bn(α,β−α)−1∫t>01−t>0etr(vt)det(t)α−ρdet(1−t)β−α−ρdt =Bn(α,β−α)−1etr(v)∫t>01−t>0e−tr(v(1−t))det(t)α−ρdet(1−t)β−α−ρdt =Bn(α,β−α)−1etr(v)∫r>01−r>0e−tr(vr)det(1−r)α−ρdet(r)β−α−ρdr =Bn(α,β−α)−1etr(v)det(v)α−β∫r>0v−r>0e−tr(r)det(1−a−1ra−1)α−ρdet(r)β−α−ρdr. Now $$1\ge a^{-1}ra^{-1} >0$$ so that, for $$\mathrm{Re}(\alpha)>\rho$$, the factor $$\det(1-a^{-1}ra^{-1})^{\alpha-\rho}$$ lies in $$(0,1)$$ and we have   |1F1(α,β;v)|≤|Bn(α,β−α)|−1etr(v)det(v)Re(α−β)∫r>0v−r>0e−tr(r)det(r)Re(β−α)−ρdr. The integral here is bounded by   ∫r>0e−tr(r)det(r)Re(β−α)−ρdr=Γn(Re(β−α)). 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International Mathematics Research NoticesOxford University Press

Published: Jan 1, 2018

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