The value of the Kac polynomial at one

The value of the Kac polynomial at one Abstract We establish a formula for the value of the Kac polynomial at one in terms of Kac polynomials, evaluated at one, of the universal (abelian) covering quiver by applying torus localization methods to quiver varieties introduced by Hausel–Letellier–Rodriguez-Villegas. 1. Introduction Given a quiver Γ and a dimension vector α, Kac defines in [12] the function aΓ,α(q) that counts the number of isomorphism classes of absolutely indecomposable representations of Γ of dimension vector α over the finite field with q elements. He shows that, regarded as a function in q, this defines a polynomial with integer coefficients—it is called the Kac polynomial. In fact the coefficients of the Kac polynomial are non-negative as shown by Hausel–Letellier–Rodriguez-Villegas in [8]—confirming a conjecture of Kac. The main objective of this paper is to study the value of the Kac polynomial at one which has many geometric interpretations. A rather bold conjecture of Kac [12, Conjecture 9] says that the set of isomorphism classes of indecomposable representations of a fixed dimension admits a cell decomposition into affine spaces. The number of cells of a fixed dimension would then be given by the corresponding coefficient of the Kac polynomial. This would imply that the Kac polynomial at one is just the number of cells of this cell decomposition. Another interpretation is the one as the dimension of the middle cohomology of character varieties corresponding to star-shaped quivers which is explained in [9]. We give a description of the number aΓ,α(1) in terms of the values aΓ^,β(1) where Γ^ is the universal abelian covering quiver of Γ. More precisely, the main result of the paper states: Theorem 1.1 The value of the Kac polynomial at one of Γ attached to a dimension vector α is the sum  aΓ,α(1)=∑βaΓ^,β(1),which ranges over a complete system of representatives of equivalence classes of dimension vectors β of Γ^that are compatible with α. Theorem 1.1 can be proved for an indivisible dimension vector α by applying torus localization to the moduli space Mλ(Γ,α) of representations of the deformed preprojective algebra Πλ(Γ) (see [24, Section 3.2.2]). When λ is generic, the Poincaré polynomial of Mλ(Γ,α) is shown to be equal to aΓ,α(q) by Crawley-Boevey–Van den Bergh [5], whence its Euler characteristic equals aΓ,α(1). The fixed points under a suitable torus action can be identified with moduli Mλ(Γ^,β). The localization principle then proves the theorem in the indivisible case. The proof of the general case uses the varieties Mσ(Γ,α) introduced in the proof of the Kac conjecture in [8]. Their cohomology carries a natural action of the Weyl group W of a maximal torus of GLα. The generating series of the anti-invariant part of the cohomology equals the Kac polynomial. We find a similar torus action on Mσ(Γ,α) which commutes with the W-action on cohomology. Again we describe the components of the fixed point locus (see Theorem 5.4) in terms of Mσ(Γ^,β) and prove that the cohomology of the fixed point locus identifies—as a W-representation—with a sum of induced representations from the Weyl groups of the coverings (see Theorem 5.7). Arguing that the localization isomorphism is compatible with the Weyl group action in our setup (this is a bit of an awkward business as W does not act on the quiver variety, but only on its cohomology; see Proposition 6.1), we conclude that Theorem 1.1 holds for arbitrary dimension vectors. Applying Theorem 1.1 iteratively we obtain the following corollary: Corollary 1.2 The value of the Kac polynomial at one of Γ attached to a dimension vector α is the sum  aΓ,α(1)=∑βaΓ˜,β(1),which ranges over a complete system of representatives of equivalence classes of dimension vectors β of the universal covering quiver Γ˜of Γ that are compatible with α. The value aΓ,α(1) is closely related to the number of indecomposable tree modules. A consequence of Theorem 1.1 is Corollary 8.2 stating that the number of indecomposable tree modules of dimension α equals aΓ,α(1), provided that all compatible roots β of the universal covering quiver are exceptional. As every finite connected subquiver of the universal covering naturally defines an indecomposable tree module which is exceptional as a representation of Γ˜, the number of such subquivers of a fixed dimension type gives a lower bound for the Kac polynomial at one. This is a consequence of the fact that the Kac polynomial is simply 1 for exceptional roots. Indecomposable tree modules of this kind are also called cover-thin. We apply these considerations to the generalized Kronecker quiver K(m). Statements about the growth behavior of the number of cover-thin tree modules of K(m) then yield that aK(m),n(d,e)(1) grows at least exponentially in n if (d,e) is a root. Kinser and Derksen sketch a proof of the above theorem in an unpublished note using entirely different methods. The coprime case was also treated in the habilitation thesis [24] of the second author. 2. Terminology and preliminary results Let Γ be a quiver. Let Γ0 be its set of vertices and Γ1 be the set of arrows; both assumed to be finite. A representation M of Γ over the field k consists of a tuple of finite-dimensional k-vector spaces (Mi)i∈Γ0 and k-linear maps Ma:Mi→Mj for every arrow a:i→j. A morphism f:M→N of representations of Γ is a collection fi:Mi→Ni of k-linear maps such that fjMa=Nafi for every a:i→j. We thus obtain an abelian category. A representation M over k is called indecomposable if it cannot be decomposed as the direct sum of two proper subrepresentations. We call M absolutely indecomposable if M⊗kK is an indecomposable representation for every finite extension K∣k. If k=Fq then the number aΓ,α(q) of absolutely indecomposable representations of Γ over Fq of dimension vector α—the dimension vector of a representation M is the tuple (dimkMi)i∈Γ0—is a finite number. Kac shows in [12, Section 1.15] that aΓ,α(q) is a polynomial in q with integer coefficients. It is called the Kac polynomial of (Γ,α). Given α∈Z≥0Γ0, we define the vector space   R(Γ,α)=⊕a:i→jHomk(kαi,kαj).On R(Γ,α), the group GLα=∏iGLαi acts by change of basis. The set of GLα-orbits on R(Γ,α) is in natural bijection with the set of isomorphism classes of representations of Γ of dimension α over k. The double quiver Γ¯ of Γ is obtained as follows: the set Γ¯0 is just Γ0, but Γ¯1 is obtained from Γ1 by adding a new arrow a*:j→i for every arrow a:i→j in Γ. Then   R(Γ¯,α)=⊕(a:i→j)∈Γ1(Hom(kαi,kαj)⊕Hom(kαj,kαi)).An element φ∈R(Γ¯,α) consists of linear maps φa:kαi→kαj and φa*:kαj→kαi. Let glα be the Lie algebra of GLα and let glα0 be the Lie subalgebra consisting of elements X whose total trace tr(X)=∑itr(Xi) is zero. The moment map μ:R(Γ¯,α)→glα0 is defined by   μ(φ)=∑a∈Γ1[φa,φa*].Let λ∈ZΓ0 with λ·α=∑iλiαi=0 which we regard as a central element of glα0. Elements of the fiber μ−1(λ) are representations of Πλ(Γ)=kΓ¯/(∑a∈Γ1[a,a*]−∑iλiei), the so-called deformed preprojective algebra of Γ. As μ is GLα-equivariant and λ is central the fiber μ−1(λ) carries an action of GLα. The GLα-orbits on μ−1(λ) are in bijection with isomorphism classes of representations of Πλ(Γ) of dimension vector α. We refer to [4] for more details. Given a quiver Γ we define the (infinite) quiver Γ^ by   Γ^0=Γ0×ZΓ1,Γ^1=Γ1×ZΓ1,where for an arrow a:i→j in Γ and χ∈ZΓ1, the arrow (a,χ)∈Γ1 has source (i,χ) and target (j,χ+ea) (the element ea is the respective unit vector in ZΓ1), i.e. pictorially   (a,χ):(i,χ)→(j,χ+ea).The quiver Γ^ is called the universal abelian covering quiver of Γ (see [23, Section 3.1]). We also recall the notion of the universal covering quiver. We denote by WΓ the free group generated by Γ1. The universal covering quiver Γ˜ of Γ is given by the vertex set   Γ˜0=Γ0×WΓ,Γ˜1=Γ1×WΓ,where (a,w):(i,w)→(j,wa) for a:i→j. Finally, we consider iterated covering quivers and define the kth universal abelian covering quiver Γ^k recursively by Γ^0=Γ and   Γ^0k=Γ^0k−1×ZΓ^1k−1,Γ^1k=Γ^1k−1×ZΓ^1k−1.As explained in [23, Section 3.4], there exist natural surjective morphisms ck:Γ˜→Γ^k which become injective on finite subquivers of Γ˜ for k≫0, see [23, Proposition 3.13]. There is a natural morphism of quivers c:Γ^→Γ which projects along ZΓ1. Let Λ=ZΓ0 and let Λ^ be the sublattice of ZΓ^0 of those vectors β=(βi,χ) with finite support. We extend the map c linearly to a map c:Λ^→Λ, concretely   c(β)i=∑χβi,χ.A dimension vector of Γ^ is defined to be an element of Λ^ whose entries are non-negative. We say that a dimension vector β of Γ^ is compatible with a dimension vector α of Γ if c(β)=α. We define an action of the group ZΓ1 on Λ^ by letting ξ∈ZΓ1 act on β∈Λ^ by (ξ.β)i,χ=βi,χ+ξ. Dimension vectors which lie in the same ZΓ1-orbit are called equivalent. The map c is ZΓ1-invariant, and it is clear that up to equivalence only finitely many dimension vectors of Γ^ with connected support are compatible with a given dimension vector α of Γ. There is also a natural morphism c:Γ˜→Γ with the same properties. Finally, the morphisms c and ck extend to natural functors ck:Rep(Γ^k)→Rep(Γ) and c:Rep(Γ˜)→Rep(Γ) between the representation categories preserving indecomposability. We refer to [7] for more details on covering theory. The Kac polynomial at one aΓ,α(1) is closely related to the number of indecomposable tree modules of dimension α. We investigate this connection in Section 8 in greater detail. Thus, let us recall the notion of coefficient quivers and tree modules. For a fixed representation M, we choose a basis B of ⊕i∈Γ0Mi such that Bi≔B∩Mi is a basis of Mi for every vertex i∈Γ0. For every arrow a:i→j, we write Ma as an (αj×αi)-matrix Ma,B with coefficients in k such that the rows and columns are indexed by Bj and Bi, respectively. The coefficient quiver Γ(M,B) of a representation M with a fixed basis B has vertex set B and arrows between vertices are defined by the following condition: if (Ma,B)b′,b≠0, there exists an arrow (a,b,b′):b→b′, where b∈Bi, b′∈Bj and a:i→j. A representation M is called a tree module if there exists a basis B for M such that the corresponding coefficient quiver is a tree. Remark 2.1 The existence of indecomposable tree modules is known for Schur roots [19, 22], but in general it is an open question. Heuristically, [12, Conjecture 9], which we already mentioned in the introduction, and also the results of this paper suggest that every tree module spans an affine space of indecomposables. This is also because every tree module can be lifted to a representation of the universal (abelian) covering quiver. Thus, every indecomposable tree module is a torus fixed point under the torus action which we introduce in these notes. But in general not every torus fixed point is a tree module, and note also that there are cases where there exist more than aΓ,α(1) indecomposable tree modules. 3. The coprime case If α is coprime, then the proof of Theorem 1.1 is easier. It suffices to consider moduli spaces of complex representations of the deformed preprojetive algebra. Let k=C from now on. Consider the moment map μ:R(Γ¯,α)→glα0. As α is coprime, we find λ∈ZΓ0 with λ·α=0 such that λ·α′≠0 for every 0≤α′≤α, unless α′ equals 0 or α. Such a λ is called generic for α. Let Mλ(Γ,α)=μ−1(λ)//GLα. By using [5], Formula (2.7), Corollary 2.3.2, we have   aΓ,α(q)=∑i=0ddimHc2d+2i(Mλ(Γ,α);C)qi,where we consider singular cohomology with compact supports and where d is half of the complex dimension of Mλ(Γ,α). Since Mλ(Γ,α) is cohomologically pure, the existence of a polynomial with integer coefficients which counts the rational points yields that the odd cohomology vanishes, see [5, Appendix A]. In particular, we obtain aΓ,α(1)=χc(Mλ(Γ,α)). By a well-known result, we have χc(X)=χc(XT) for any complex variety with a torus action, see for instance [3, Section 2.5] or [6, Appendix B]. Here XT denotes the fixed point set. It is straightforward to transfer the results of [23] to the case of the moduli spaces Mλ(Γ,α). Note that representations of Πλ(Γ) are simple if λ is generic. This enables us to understand the corresponding fixed point components as moduli spaces attached to the universal abelian covering of Γ. More precisely, let T≔(C×)Γ1 act on R(Γ¯,α) by   (ta)a∗(φa,φa*)a∈Γ1=(taφa,ta−1φa*).This descends to an action on μ−1(λ) which commutes with the usual base change action of GLα. Now the same proofs as those of [23, Sections 3.1, 3.2] apply to show the following: Theorem 3.1 The set of torus fixed points Mλ(Γ,α)Tis isomorphic to the disjoint union of moduli spaces  ⨆βMλ(Γ^,β),where β ranges over all equivalence classes of dimension vectors of Γ^compatible with α. Note that λ, which we regard as an element of ZΓ^0 by setting λi,χ=λi, is generic for every β that is compatible with α. This shows that Theorem 1.1 holds for α coprime. Note further that every β for which Mλ(Γ^,β) is non-empty must have connected support by genericity of λ. Finally, Corollary 1.2 follows by Remark 5.8. 4. Construction of the moduli space in the general case We recall the construction of Hausel–Letellier–Rodriguez-Villegas from [8]. Consider again μ:R(Γ¯,α)→glα0 over the complex numbers. Let Tα be the maximal torus of GLα of tuples of invertible diagonal matrices. Let tα be the Lie algebra of Tα. A semi-simple element of glα is called regular if its centralizer is a maximal torus. Therefore the centralizer of a regular element of tα is Tα. An element σ∈tα is called generic if tr(σ)=0 and if tr(σ∣V)≠0 for all non-trivial Γ0-graded subspaces V⊆Cα which are stable under σ. Let tαgen be the (non-empty) open subset of regular generic elements of tα0. The variety   M=M(Γ,α)={(φ,hTα,σ)∈R(Γ¯,α)×GLα/Tα×tαgen∣μ(φ)=hσh−1}//GLαis the quotient by the GLα-action defined by g(φ,hTα,σ)=(g·φ,ghTα,σ). Note that the diagonally embedded C× acts trivially and the induced action of GLα/C× is free. The map π:M→tαgen arising by projecting onto the third factor is surjective. Theorem 4.1 ([8, Theorem 2.1]) The fibers Mσare smooth and their cohomology vanishes in odd degrees. The Weyl group W=Wα=NGLα(Tα)/Tα≅∏iSαi acts on M via   w.(φ,hTα,σ)=(φ,hw˙−1Tα,w˙σw˙−1),where w˙ is the permutation matrix defined by w (or any other representative of w in NGLα(Tα)). We will drop the dot in the notation for convenience. This gives isomorphisms   w:Mσ→Mwσw−1. Theorem 4.2 ([8, Theorem 2.3]) For any σ∈tαgen, the cohomology group Hci(Mσ;C)becomes in a natural way a representation of W which is up to isomorphism independent of σ∈tαgen. In [8], this result is stated for the cases that the ground field has large positive characteristic or is the complex numbers and with coefficients in Q¯ℓ. For our purposes, it will be sufficient to consider the complex case and C-coefficients. The above theorem follows from a result of Maffei [15, Lemma 48] which shows that Riπ!Z, and hence also Riπ!C, is constant. As it is useful for our purposes we will explain how the W-representation arises. Remark 4.3 Let f:Y→X be a continuous map of locally compact topological spaces. Let W be a finite group which acts on both X and Y such that f is W-equivariant. Let F be a sheaf of complex vector spaces on Y with a W-linearization φ:act*F→≅pr2*F (where act:W×Y→Y is the action map and pr2:W×Y→Y is the projection); see [17, Section 1.3] for the definition of a linearization. We obtain isomorphisms φw:w*F→F with φw1w2=φw2◦w2*φw1. Consider the higher direct images Rif!F with compact supports. The definitions and results on cohomology of sheaves can, for instance, be found in [13, Chapters II and III]. Suppose that Rif!F is constant, say with fiber F. For every x∈X, we are thus given an isomorphism F→(Rif!F)x=Hci(Yx;F). This yields an isomorphism ix,x′:Hci(Yx;F)→Hci(Yx′;F) for every two points x,x′∈X. The induced W-linearization on Rif!F yields isomorphisms   φw,x:Hci(Ywx;F)=Hci(Yx;w*F)→Hci(Yx;F).It is easy to verify from the cocyle conditions and the compatibility of the isomorphisms ix,x′ that ρi:W→GL(Hci(Yx;F)) defined by   ρi(w):Hci(Yx;F)⟶φw−1,wxHci(Ywx;F)⟶iwx,xHci(Yx;F)is a representation of W. If F is constructible, the assumption Rif*F be constant induces a representation W→GL(Hi(Yx;F)) in the same way. The central result of [8] is the description of the Kac polynomial as the generating series of the alternating part of the graded W-representation Hc*(Mσ;C). More precisely: Theorem 4.4 [8, Theorem 1.4] The Kac polynomial aΓ,α(q)coincides with  ∑i=0ddim(Hc2i+2d(Mσ;C)sign)qi,where d is half of the complex dimension of Mσand the subscript ‘ sign’ denotes the alternating component of the cohomology regarded as a W-representation. 5. Torus action Let T=(C×)Γ1 act on R(Γ¯,α) via   (t·φ)a=taφa(t·φ)a*=ta−1φa*for any t=(ta)a∈Γ1∈T and φ∈R(Γ¯,α). This T-action commutes with the action of GLα which implies that we get an action of T on M by   t.(φ,hTα,σ)=(t·φ,hTα,σ).The T-action on M commutes with the W-action, whence W acts on the fixed point locus MT. We analyze the fixed point locus and the W-action on it. Let (φ,hTα,σ)∈MT. This means for all t∈T, there exists g∈GLα with   (t·φ,hTα,σ)=(g·φ,ghTα,σ),or, in other words, taφa=gjφagi−1 for all a:i→j and hi−1gihi∈Tαi for all i∈Γ0. As GLα/C× acts freely on the total space of the quotient M, the element g is uniquely determined by t up to a scalar. Using the arguments from [23], we deduce that there exists a homomorphism ψ:T→GLα, unique up to the diagonally embedded C×, with t.(φ,hTα)=ψ(t)·(φ,hTα). The ith component ψi:T→GLαi of ψ induces a T-action on Vi=Cαi and, therefore, a weight space decomposition   Vi=⊕χ∈X(T)Vi,χ.The character group X(T) of T is precisely ZΓ1. For a weight vector vχ∈Vi,χ and an arrow a:i→j, we get   taφa(vχ)=(t·φ)a(vχ)=ψj(t)φaψi(t)−1(vχ)=χ(t)−1ψj(t)φa(vχ),or, in other words, φa(vχ)∈Vj,χ+ea. It is shown analogously that φa*(Vj,χ)⊆Vi,χ−ea. These considerations show that φ can be regarded as a representation of the double of the covering quiver Γ^ of dimension vector β with βi,χ=dimVi,χ. Let ei,r be the rth unit vector in Vi=Cαi (we keep the index i in the notation for bookkeeping reasons). Let χ1,…,χN be those characters for which there exists an i such that the weight space Vi,χk is non-zero. Embed Cβi,χk as the subspace of Cαi spanned by the unit vectors   ei,(βi,χ1+⋯+βi,χk−1+1),…,ei,(βi,χ1+⋯+βi,χk), (5.1)and consider GLβ=∏i,χGLβi,χ as a subgroup of GLα via this direct sum decomposition. As there exists g∈GLα with g(Vi,χk)=Cβi,χk which is unique up to a (unique) element of GLβ, we may, by passing from (φ,hTα,σ) to (g·φ,ghTα,σ), assume without loss of generality that Vi,χk=Cβi,χk. For a number r∈{1,…,αi}, let k be the unique index with ei,r∈Vi,χk. Write hi(ei,r) as ∑χwχ with wχ∈Vi,χ. As hi−1ψi(t)hi lies in Tαi, the vector   (hi−1ψi(t)hi)(ei,r)=hi−1(∑χχ(t)wχ)lies in the span of ei,r. Precisely one summand wχ is non-zero. To see this, assume otherwise. Take a character ξ for which wξ≠0 and observe that hi(ei,r) would by assumption not be a multiple of wξ. Choose a one-parameter subgroup λ of T which vanishes on ξ and is positive on all other χ which occur in the summation ∑χwχ. Then   (hi−1ψi(λ(z))hi)(ei,r)=hi−1(∑χz⟨λ,χ⟩wχ)⟶z→0hi−1(wξ),which does not lie in the span of ei,r. A contradiction. We deduce that for every r as above there exists an l such that hi(ei,r)∈Vi,χl. This implies the existence of a w∈W such that hw˙−1∈GLβ (and this Weyl group element w is unique up to an element of Wβ=NGLβ(Tα)/Tα). These considerations show that the fixed point locus MT is contained in the disjoint union   ⨆β⨆w¯∈W/WβMβ,w¯over a full system of representatives of dimension vectors β which are compatible with α, where   Mβ,w¯={(φ,hTα,σ)∈R(Γ^¯,β)×GLα/Tα×tαgen∣hw˙−1∈GLβ,μ(φ)=hσh−1}//GLβ,and where w˙∈NGLα(Tα) is a representative of w¯∈W/Wβ≅NGLα(Tα)/NGLβ(Tα). To show conversely that every element of Mβ,w¯ is a T-fixed point, we take a triple (φ,hTα,σ) as above and show that to every t∈T there exists g∈GLα such that (t·φ,hTα,σ)=(g·φ,ghTα,σ).To interpret the representation φ of the double of Γ^ as a representation of the double of Γ, let again χ1,…,χN be the characters for which there exists an i with βi,χk≠0 and identify Cβi,χk with the span Vi,χk of the unit vectors as in (5.1). Then φa(Vi,χ)⊆Vj,χ+ea for every arrow a:i→j of Γ and φa*(Vi,χ)⊆Vj,χ−ea. Let ψ:T→GLα be the homomorphism that corresponds to the decompositions Vi=⊕χVi,χ. A computation as above shows that ψj(t)φaψi(t)−1=taφa.What remains to show is that hTα=ψ(t)hTα. Let ei,r∈Vi,χk. Then ei,w(r) lies in, say, Vi,χl, whence   hi(ei,r)=hw˙−1(ei,w(r))∈Vi,χl,as hw˙−1∈GLβ. Therefore, ψi(t)hi(ei,r)=χl(t)hi(ei,r), which implies ψi(t)hiTαi=hiTαi. We have shown Proposition 5.1 As a closed subset of M(Γ,α), the fixed point locus M(Γ,α)Tagrees with the disjoint union  ⨆β⨆w¯∈W/WβMβ,w¯over a full system of representatives of dimension vectors β which are compatible with α. The fixed point locus MT is smooth as M is. In particular, MT is reduced. As Mβ,id¯ is isomorphic to M(Γ^,β), it is smooth as well. The component Mβ,w¯ is isomorphic to Mβ,id¯, so the disjoint union ⨆β⨆w¯∈W/WβMβ,w¯ is smooth. Hence, MT and ⨆β⨆w¯∈W/WβMβ,w¯ do not only agree as closed subsets, but in fact as closed subschemes. An element w′∈W applied to (φ,hTα,σ)∈Mβ,w¯ yields (φ,hw′−1Tα,w′σw′−1) which lies in Mβ,ww′−1¯. Lemma 5.2 shows that the disjoint union ⨆w¯∈W/WβMβ,w¯ agrees—as a W-scheme—with the associated fiber bundle W×WβM(Γ^,β). Lemma 5.2 Let W be a finite group, let U⊆Wbe a subgroup of W. Suppose that we are given schemes Xa¯for every a¯∈W/Uand isomorphisms  φw,a¯:Xa¯→≅Xaw−1¯satisfying φe,a¯=idand φw,aw′−1¯◦φw′,a¯=φww′,a¯. Then the disjoint union  ⨆a¯∈W/UXa¯carries an action of W via w·x=φw,a¯(x) (where w∈Wand x∈Xa¯) for which Xe¯is a U-invariant subset. Equipped with these actions, ⨆a¯∈W/UXa¯is isomorphic to the associated fiber bundle W×UXe¯as a W-scheme. We show this using the following result which seems to be well known, but for which we could not find a reference; for the convenience of the reader, we give a proof here. Lemma 5.3 Let G be an algebraic group, let X be a scheme equipped with a G-action, let H be a closed subgroup of G and let f:X→G/Hbe a G-equivariant morphism. Denote Z=f−1(eH). Then Z is H-invariant and X is isomorphic to G×HZas a G-scheme. Proof The associated fiber bundle G×HZ is defined as follows: equip G×Z with an action of G×H via (g,h)·(a,z)=(gah−1,h·z). The actions of the two factors commute, whence the geometric quotient G×HZ≔(G×Z)/H (which exists) carries a G-action. For a more detailed introduction to associated fiber bundles, see [20, Sections 3.3 and 3.5] or [21, Section 2.1]. In our concrete situation, we consider the map φ:G×Z→X defined by φ(a,z)=a·z. As φ fulfills   φ((g,h)·(a,z))=φ(gah−1,h·z)=ga·z=g·φ(a,z),it gives rise to a morphism φ¯:G×HZ→X which is apparently G-equivariant. Conversely, we define the map ψ:X→G×HX as the composition of X→G/H×X,x↦(f(x),x) with the G-equivariant isomorphism G/H×X→≅G×HX (with G acting on G/H×X simultaneously on both factors). This isomorphism comes from the isomorphism G×X→≅G×X,(a,x)↦(a,a−1·x). The map ψ factors through the closed subscheme G×HZ as for x∈X with f(x)=aH, we obtain   ψ(x)=[a,a−1·x],and a−1·x∈Z, as f is G-equivariant. So ψ yields a G-equivariant morphism ψ¯:X→G×HZ. It is easy to show that φ¯ and ψ¯ are mutually inverse.□ Proof of Lemma 5.2 The claim follows from the previous lemma by the observation that the map f:⨆a¯∈W/UXa¯→W/U which assigns f(x)=a−1¯ to every x∈Xa¯ is W-equivariant.□ We have proved. Theorem 5.4 The fixed point locus M(Γ,α)Tis, as a W-scheme, isomorphic to the disjoint union  ⨆βW×WβM(Γ^,β)over all dimension vectors β up to equivalence which are compatible with α. This isomorphism commutes with the natural maps to tαgen. As a next step, we would like to relate the W-module structure on the stalk of the sheaf Ri(π∣MT)!C (which will turn out to be constant) with the Wβ-module-structure on the stalk of Ri(πβ)!C, where πβ:M(Γ^,β)→tαgen. We need two general lemmas for this. Lemma 5.5 Let X be a scheme with an action of a finite group W. Let U be a subgroup of W and consider X as a U-scheme with the restricted action. Let Fbe the constant sheaf with stalk F on W×UXand let φ:act*F→pr2*Fbe a W-linearization. Let i:X→W×UXbe the closed embedding given by i(x)=[e,x]; it is U-equivariant. The sheaf i*Fis also constant with stalk F and the induced U-linearization i*φgives F the structure of a U-representation FU. The action map gives rise to a W-equivariant map act¯:W×UX→X. The push-forward act¯!Fis constant and the W-module structure on its stalk is IndUWFU. Proof To prove the lemma, we identify W×UX with W/U×X. Under this identification, act¯ corresponds to the projection W/U×X→X to the second factor and the map i tallies with the embedding x↦(e¯,x). It is hence clear that act¯!F is constant with stalk   (act¯!F)x=⊕a¯∈W/UF(a¯,x)≅FW/U.The action of w∈W on the stalk is defined by   ⊕a¯∈W/UF(a¯,x)⟶⨁φw,(a¯,x)−1⊕a¯∈W/UF(wa¯,wx)⟶⨁i(wa¯,wx),(wa¯,x)⊕a¯∈W/UF(wa¯,x)so under the identification with FW/U the action of w on a tuple (va¯)a¯ is given by (w·vwa¯)a¯. Here, the expression w·vwa¯ stands for the W-action on F(wa¯,x)=F which is induced by φ. Noting that FU is precisely ResUWFW, we have shown that FU is a U-invariant subspace of the W-representation FW/U defined above and this W-representation decomposes, as a vector space, into the w-translates of the subspace FU. This shows that FW/U=IndUWFU.□ Lemma 5.6 Let W be a finite group which acts on a scheme X and let Y be a scheme equipped with the action of a subgroup U of W. Let f:Y→Xbe a U-equivariant morphism of schemes. Let f˜:W×UY→W×UXbe the induced morphism and let f¯:W×UY→Xbe the composition act¯◦f˜. Let Fbe a sheaf on W×UYwith a W-linearization such that Rif˜!Fis constant. Consider the closed embedding iY:Y→W×UYand the sheaf iY*Fwith the natural U-linearization. Then, also Rif!iY*Fand Rif¯!Fare constant sheaves and we get  Hci((W×UY)x;F)≅IndUWHci(Yx;iY*F)for the associated W-representation structures on the cohomology groups of the stalks of x. Proof Base change yields an isomorphism Rif!iY*F≅iX*Rif˜!F, where iX is the closed embedding X→W×UX. On the other hand, Rif¯!F=act¯!Rif˜!F because the functor act¯! (which agrees with act¯*) is exact, as act¯ is (quasi-)finite. The sheaf Rif˜!F is assumed to be constant. We are hence in the situation of Lemma 5.5. It tells us that Rif¯!F is constant (and Rif!iY*F is, too) and that the W-module structures on their stalks—at, say, the point x∈X—are related by   (Rif¯!F)x≅IndUW(Rif!iY*F)x.This concludes the proof of the lemma.□ We apply Lemma 5.6 for every β to Y=M(Γ^,β), U=Wβ, X=tαgen, f=πβ, and F the constant sheaf C on W×WβM(Γ^,β) with the trivial W-linearization. This shows Theorem 5.7 The cohomology group Hci(Mσ(Γ,α)T;C)has a structure of a W-representation which is independent of σ and which is isomorphic to the direct sum  ⊕βIndWβWHci(Mσ(Γ^,β);C)over all dimension vectors β up to equivalence which are compatible with α. Remark 5.8 When applying iterated localization, in a first step, we can replace Γ^ by Γ^k in Theorem 5.7. In a second step, we can replace Γ^k by Γ˜. Indeed, every root β of Γ^k which is compatible with α defines a finite connected subquiver of Γ^k. As recalled in Section 2, every finite subquiver of the universal covering quiver embeds into Γ^k for k≫0. As the maps ck defined there are also surjective, this means that we can choose k in such a way that the supports of all compatible roots β are finite subquivers of Γ˜. We refer also to [23, Section 3.4] for more details. 6. Localization isomorphisms Consider a smooth morphism f:Y→X of smooth complex varieties which is assumed to be equivariant with respect to a finite group W that acts on both X and Y. Now suppose that a complex torus T=(C×)n acts on Y in a way that f is T-invariant and such that the T- and the W-action on Y commute. We assume further that f arises by base change from a (topologically) locally trivial fibration Y′→X′ whose basis X′ is contractible (this holds in our concrete situation, see the proof of [8, Theorem 2.3] or [15, Lemma 48]). In particular f is also a locally trivial fibration. Consider the constant sheaf C on Y. Then the higher direct images Rif!C and Rif*C are constant by contractibility of X′. Remark 4.3 ensures that Hci(Yx;C) and Hi(Yx;C) have a natural structure of a W-representation. Moreover, every connected component of the fixed point set YT is smooth. Suppose further that the induced map fT:YT→X satisfies the same properties as f (this is true in our setup, see Theorem 5.4). We show: Proposition 6.1 The classes [Hc*(Yx;C)]and [Hc*(YxT;C)]agree in the Grothendieck group K0(CW)of finitely generated (complex) W-representations. In the above context, the class [V*] in K0(CW) of a finite-dimensional graded W-representation V*=⊕i∈ZVi is defined as [V*]=∑i(−1)i[Vi]. We have to show that the localization isomorphism in [3, Section 2.5] is compatible with the W-action. But as the W-action on cohomology of the fiber does not arise from a W-action on the fiber but, from monodromy, we have to adjust the proof given in [3] relative to the basis X. Proof Let ET be a contractible space on which T acts freely and let BT=ET/T. Consider the cartesian diagrams   By base change we see that π*RifT,*C≅p*Rif*C and, as π is faithfully flat, we obtain RifT,*C≅pr*Rif*C where pr:X×BT→X is the projection. The composition   Y×TET→fTX×BT→prXyields a spectral sequence E2p,q=(Rppr*)(RqfT,*)C⇒Rp+q(pr◦fT)*C. By the above considerations, we conclude that   E2p,q≅(Rppr*)pr*(Rqf*)C≅Hp(BT;C)⊗Rqf*C.This is a spectral sequence in the category of W-equivariant sheaves on X. Note that as fT is a locally trivial bundle, the composition pr◦fT is, too, and as BT is contractible all Ri(pr◦fT)*C are constant. Then the stalk Hi(Yx×TBT;C)=HTi(Yx;C) inherits the structure of a W-representation (cf. Remark 4.3). Applying the stalk functor to the spectral sequence E from above yields the spectral sequence   HTp(pt)⊗Hq(Yx)⇒HTp+q(Yx)(all cohomology groups are taken with complex coefficients). This is the spectral sequence associated with the fibration ET×TYx→BT, but by this detour, we have shown that the differentials of this spectral sequence are W-linear. This shows that [HT*⊗H*(Yx)]=[HT*(Yx)] in the Grothendieck group K0((HT*)W) of finitely generated (HT*)W-modules. On the other hand, the inclusion YT→Y of the fixed point locus induces a natural morphism   Ri(fT◦pr)*C→Ri((fT×idBT)◦pr)*C,which, after taking fibers, is the pull-back HTi(Yx)→HTi(YxT) in equivariant cohomology. This shows that this map is also W-equivariant. The HT*-linear map HT*(Yx)→HT*(YxT) becomes an isomorphism after localizing finitely many non-trivial characters. This isomorphism doesn’t preserve the grading, but only the parity. Let S⊆HT* be the multiplicative subset arising from the aforementioned characters. Then [S−1HT*⊗HT*HT*(Yx)]=[S−1HT*⊗H*(YxT)] in the Grothendieck group K0((S−1HT*)W), and thus   [S−1HT*⊗H*(Yx)]=[S−1HT*⊗H*(YxT)]in the same group. But this implies that [H*(Yx)]=[H*(YxT)] already in K0(CW). Finally, we apply Poincaré duality. As Yx and all connected components of YxT are smooth varieties, the classes [H*(Yx)] and [H*(YxT)] agree with [Hc*(Yx)] and [Hc*(YxT)], respectively, in the Grothendieck group K0(C). But Poincaré duality in this case comes from Verdier duality RHom̲(Rf!C,ωX)≅Rf*RHom̲(C,ωY), which is W-equivariant (by interpreting it as an identity in the bounded derived category of W-equivariant sheaves on X, which is the same as DWb(X) the W-equivariant bounded derived category in the sense of Bernstein–Lunts [2] as W is finite). Therefore, the identity [Hc*(Yx)]=[H*(Yx)] holds also in the Grothendieck group K0(CW). The same argument applies for the equality [Hc*(YxT)]=[H*(YxT)].□ 7. Finishing the proof of the main result Applying Proposition 6.1 and the fact that Mσ has no odd cohomology, we observe that Hc*(MσT;C)≅Hc*(Mσ;C) as ungraded W-representations (we define H*(X;C) as the direct sum ⊕iHi(X;C)). The W-representation Hc*(MσT;C) decomposes by Theorem 5.7 as   ⊕βIndWβWHc*(Mσ(Γ^,β);C).The sign-isotypical component of the induced W-representation IndWβWHci(Mσ(Γ^,β);C) is the sign-isotypical component of Hci(Mσ(Γ^,β);C) (with respect to the Wβ-action). We know by [8, Theorem 1.4] that   aΓ,α(1)=dimHc*(Mσ(Γ,α);C)signaΓ^,β(1)=dimHc*(Mσ(Γ^,β);C)sign,the sum of the dimensions of all sign-isotypical components of the cohomology groups (note that the odd-degree cohomology groups of Mσ(Γ,α) and Mσ(Γ^,β) vanish). Using that taking isotypical components and taking classes in the Grothendieck group commute, we have proved aΓ,α(1)=∑βaΓ^,β(1), as asserted in Theorem 1.1. Now Corollary 1.2 is an immediate consequence of Theorem 1.1 when taking Remark 5.8 into account. 8. First consequences of the main theorem The main result of this paper has plenty of interesting consequences which were also mentioned in [24] for coprime dimension vectors. For instance, the number of indecomposable tree modules, as defined in [19], can be related to the Kac polynomial at one. Recall that a tree module of Γ is already a representation of the universal covering quiver Γ˜ of Γ. We call an indecomposable tree module of Γ cover-thin if its dimension vector α is of type one as a representation of Γ˜, i.e. if αi∈{0,1} for all i∈Γ˜0. We denote the number of indecomposable tree modules of dimension α by tα and the number of cover-thin tree modules by ctα. Let σi be the BGP-reflection at a source or sink i∈Γ0 introduced in [1]. Lemma 8.1 Every indecomposable tree module M which is cover-thin is exceptional as a representation of the universal covering quiver. In particular, we have that σiMis also an indecomposable tree module for any sink (resp. source) i∈Γ0. Proof The first part follows because every exceptional representation M is Schurian and, moreover, because ⟨dim̲M,dim̲M⟩=dimkHom(M,M)−dimkExt(M,M)=1. In particular, σiM is also exceptional as a representation of the universal covering quiver, and thus a tree module by the main result of [19].□ Corollary 8.2 Let α be a dimension vector such that all equivalence classes of compatible dimension vectors with connected support consist of exceptional roots. Then the number of indecomposable tree modules of dimension vector α is equal to the Kac polynomial at one. Proof For the Kac polynomial of a real root α˜, we have aΓ˜,α˜(q)=aΓ˜,α˜(1)=1. Thus, it suffices to show that the number of indecomposable tree modules is equal to the number of compatible roots. By the main result of [19], every exceptional representation is an indecomposable tree module, and thus a representation of the universal covering quiver, say of dimension α˜. Thus, every compatible dimension vector gives rise to an indecomposable tree module. Conversely, every indecomposable tree module T yields a root α˜ of Γ˜ which is compatible with α. Since α˜ is exceptional by assumption, T is up to isomorphism the only representation of dimension α˜.□ It was pointed out to us by Jie Xiao that Corollary 1.2 implies at once: Corollary 8.3 Every real root representation is already a representation of the universal covering quiver. Proof As aΓ,α(1)=1=∑βaΓ˜,β(1) for a real root α, the unique indecomposable representation of dimension α already lives on the universal covering Γ˜. Alternatively, note that the corresponding root of the universal covering can be obtained via reflections of the Weyl group of Γ˜; they correspond precisely to those reflections of the Weyl group of Γ which are used to obtain α from a simple root.□Using iterated localization we also re-obtain the following result: Corollary 8.4 ([16, Corollary 4.4]) If α is a dimension vector of Γ such that αi=1for all i∈Γ0, the Kac polynomial at one is equal to the number of spanning trees of Γ. Let K(m) be the generalized Kronecker quiver with two vertices i,j and m arrows al:i→j. In this case, the main result, together with the following proposition, enables us to investigate the asymptotic behavior of the Kac polynomial at one. Proposition 8.5 ([24, Proposition 3.2.6]) Let n≔m−1. The number of indecomposable cover-thin tree modules ct(d,e)of K(m)which are of dimension (d,e)is  1d∑i=1m(mi)(ned−1)(n(d−1)e−i)ie. Example 8.6 If m=3 and (d,e)=(d,d+1), it is straightforward to check that we have   ct(d,d+1)=3(d+2)(d+3)(2dd)(2(d+1)d+1).The respective sequence of natural numbers appears as sequence A186266 in [18]. It seems that there was no combinatorial interpretation of this sequence before. Applying Lemma 8.1, we thus obtain: Corollary 8.7 The number of indecomposable tree modules tn(d,e)of K(m)grows (at least) exponentially with the dimension vector, i.e. for every imaginary Schur root (d,e)of K(m)there exists a real number K(d,e)>1such that tn(d,e)>K(d,e)n. As a consequence, we obtain the following: Corollary 8.8 Let (d,e)be a root of the generalized Kronecker quiver K(m). Then the Kac polynomial at one grows (at least) exponentially with the dimension vector, i.e. there exists a real number K(d,e)>1such that  aK(m),n(d,e)(1)>K(d,e)n. Actually, Corollary 8.7 can also be used to show that the number of indecomposable tree modules which have an imaginary Schur root as the dimension vector grows exponentially with the dimension vector, see [24, Theorem 3.2.8]. We conclude with the following natural question, which was for instance asked in [14, Question 7], but also posed to the second author by Crawley-Boevey and Hubery: Question 8.9 Do we always have tα≥aΓ,α(1)? The main result of the paper implies that this needs to be checked only for quivers which are trees. But actually, similar to the question of the existence of tree modules, it seems that this does not make things much easier. Many examples which can be found in the literature suggest that this is true. In the case of extended Dynkin quivers of type D˜n, this can be checked by hand. We also conjecture that equality holds if and only if the assumptions of Corollary 8.2 hold. 9. Concrete examples Consider the generalized Kronecker quiver K(3) and the dimension vector (2,3). By use of Hua’s formula [11], we obtain   aK(3),(2,3)=q6+q5+3q4+4q3+5q2+3q+2,and thus aK(3),(2,3)(1)=19, see [11, Section 5]. There are 18 cover-thin tree modules of K(3) of dimension (2,3) which are given by   Here the arrows mi∈{a1,a2,a3} satisfy the conditions m1≠m2≠m3≠m4 in the first case and the conditions m1≠m2 and m2,m3,m4 pairwise distinct in the second case. Finally, there is one tree module which is not cover-thin, and whose coefficient quiver is the one on the left-hand side in the case when m2=m3 and m1,m2,m4 are pairwise distinct. More precisely, its dimension vector is given by   i.e. the real root of D4 of dimension (2,1,1,1). Thus, the number of indecomposable tree modules is 19. We consider the generalized Kronecker quiver K(4) and the root (2,4). In this case, the number of indecomposable tree modules, which is 126 (120 cover-thin tree modules and six others), is greater than a(2,4)(1). Using Hua’s formula [11] we obtain   aK(4),(2,4)(q)=q13+q12+3q11+4q10+8q9+9q8+15q7+16q6+20q5+17q4+15q3+9q2+5q+2,and thus a(2,4)(1)=125. Up to coloring the arrows, we obtain the following subquivers and dimension vectors (where the dots indicate one-dimensional vector spaces) which give a contribution to the Kac polynomial at one:   It is straightforward to check that there exist (up to translation) 108 possibilities to embed the first quiver into K(4)˜ (resp. to color the arrows with four different colors m1,m2,m3,m4 such that m1,m2,m3 are pairwise distinct and m3≠m4≠m5). For the second one, we have 12 possibilities and, for the last one, there exists only one possible embedding. Since the first two dimension vectors are real roots, the Kac polynomials are 1. The quiver and the dimension vector considered in the last case is a quiver of type D˜4 together with the unique imaginary Schur root δ. Thus, the Kac polynomial is q+4. This can be checked by classifying the absolutely indecomposable representations, which coincide with the indecomposables in this case, up to isomorphism. In summary, we obtain   aK(4),(2,4)(1)=108·1+12·1+1·5=125. Finally, we consider the quiver Lg with only one vertex i and g loops. If αi≤5, all non-empty moduli spaces appearing are points and the Kac polynomials of the compatible dimension vectors are one. In particular, the number of indecomposable tree modules coincides with aLg,αi(1). The first non-trivial moduli space appears for αi=6. Also in this case, we need to consider the Kac polynomial of the imaginary Schur root δ=(2,1,1,1,1) of D˜4 with Kac polynomial aL˜g,δ(q)=q+4. Taking into account the different possibilities of coloring the arrows of D˜4 with the colors {1,…,g} and orienting the arrows, this gives the contribution   5·(24(g4)+3·22(g3)+(g2))to the Kac polynomial at one (we can choose four, three or two different arrows out of the g arrows of the original quiver). Note that, in the universal covering, the following orientations do not appear   Since there are six indecomposable tree modules of dimension δ of D˜4, one checks that this indeed fills the gap between aLg,6(1), see [10, Section 1], and the number of indecomposable tree modules of dimension αi=6, see [14, Section 4.1]. 10. Conjecture on the asymptotic behavior of the Kac polynomial at one The following is based on a conjecture of M. Douglas concerning moduli spaces of stable representations of generalized Kronecker quivers, see [23, Section 6.1]. It generalizes [24, Conjecture 4.1.2] to arbitrary dimension vectors: Conjecture 10.1 There exists a continuous function f:RΓ0→R such that   f(α)=limn→∞ln(aΓ,nα(1))nfor all dimension vectors α∈NΓ0. If a function as predicted in Conjecture 10.1 exists, Proposition 8.5 immediately yields a lower bound in the case of the Kronecker quiver and for coprime dimension vectors (d, e) such that d≤e≤(m−1)d+1. Lemma 10.2 Let (d,e)be a root of the Kronecker quiver such that d≤e≤(m−1)d+1and define k≔e/dand n≔m−1. Then we have  limd→∞aK(m),(d,kd)(1)d≥limd→∞ct(d,kd)d=n(k+1)lnn+k(n−1)lnk−(nk−1)ln(nk−1)−(n−k)ln(n−k). Proof Obviously, we only have to consider one of the m summands of the formula obtained in Proposition 8.5. Then the claim follows straightforwardly when applying the Stirling formula.□ Note that the numbers ct(d,e) are not invariant under the reflection functor. Nevertheless, the reflection functor can clearly be used to obtain a lower bound for aK(m),(d,e)(1) for every dimension vector (d,e). Furthermore, it would be interesting to know if there are tuples (d,e) for which equality holds or to know more about the contribution of cover-thin tree modules to the Kac polynomial at one. Acknowledgements The authors would like to thank R. Kinser, S. Mozgovoy, M. Reineke and J. Xiao for valuable discussions and remarks. The referee’s comments have greatly helped to improve the exposition and were much appreciated. While doing this research, H.F. was supported by the DFG SFB/Transregio 45 ‘Perioden, Modulräume und Arithmetik algebraischer Varietäten’. References 1 I. N. Bernstein, I. M. Gel’fand and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk  28 ( 1973), 19– 33. 2 J. Bernstein and V. Lunts, Equivariant Sheaves and Functors, Vol. 1578, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994. 3 N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry , Birkhäuser Boston Inc., Boston, MA, 1997. 4 W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compos. Math.  126 ( 2001), 257– 293. Google Scholar CrossRef Search ADS   5 W. Crawley-Boevey and M. Van den Bergh, Absolutely indecomposable representations and Kac–Moody Lie algebras, Invent. Math.  155 ( 2004), 537– 559. With an appendix by Hiraku Nakajima. Google Scholar CrossRef Search ADS   6 S. Evens and I. Mirković, Fourier transform and the Iwahori–Matsumoto involution, Duke Math. J.  86 ( 1997), 435– 464. Google Scholar CrossRef Search ADS   7 P. Gabriel, The Universal Cover of a Representation-finite Algebra. In Representations of Algebras (Puebla, 1980), Vol. 903, Lecture Notes in Mathematics, Springer, Berlin, New York, 1981, 68–105. 8 T. Hausel, E. Letellier and F. Rodriguez-Villegas, Positivity for Kac polynomials and DT-invariants of quivers, Ann. Math.  177 ( 2013), 1147– 1168. 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The value of the Kac polynomial at one

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Abstract

Abstract We establish a formula for the value of the Kac polynomial at one in terms of Kac polynomials, evaluated at one, of the universal (abelian) covering quiver by applying torus localization methods to quiver varieties introduced by Hausel–Letellier–Rodriguez-Villegas. 1. Introduction Given a quiver Γ and a dimension vector α, Kac defines in [12] the function aΓ,α(q) that counts the number of isomorphism classes of absolutely indecomposable representations of Γ of dimension vector α over the finite field with q elements. He shows that, regarded as a function in q, this defines a polynomial with integer coefficients—it is called the Kac polynomial. In fact the coefficients of the Kac polynomial are non-negative as shown by Hausel–Letellier–Rodriguez-Villegas in [8]—confirming a conjecture of Kac. The main objective of this paper is to study the value of the Kac polynomial at one which has many geometric interpretations. A rather bold conjecture of Kac [12, Conjecture 9] says that the set of isomorphism classes of indecomposable representations of a fixed dimension admits a cell decomposition into affine spaces. The number of cells of a fixed dimension would then be given by the corresponding coefficient of the Kac polynomial. This would imply that the Kac polynomial at one is just the number of cells of this cell decomposition. Another interpretation is the one as the dimension of the middle cohomology of character varieties corresponding to star-shaped quivers which is explained in [9]. We give a description of the number aΓ,α(1) in terms of the values aΓ^,β(1) where Γ^ is the universal abelian covering quiver of Γ. More precisely, the main result of the paper states: Theorem 1.1 The value of the Kac polynomial at one of Γ attached to a dimension vector α is the sum  aΓ,α(1)=∑βaΓ^,β(1),which ranges over a complete system of representatives of equivalence classes of dimension vectors β of Γ^that are compatible with α. Theorem 1.1 can be proved for an indivisible dimension vector α by applying torus localization to the moduli space Mλ(Γ,α) of representations of the deformed preprojective algebra Πλ(Γ) (see [24, Section 3.2.2]). When λ is generic, the Poincaré polynomial of Mλ(Γ,α) is shown to be equal to aΓ,α(q) by Crawley-Boevey–Van den Bergh [5], whence its Euler characteristic equals aΓ,α(1). The fixed points under a suitable torus action can be identified with moduli Mλ(Γ^,β). The localization principle then proves the theorem in the indivisible case. The proof of the general case uses the varieties Mσ(Γ,α) introduced in the proof of the Kac conjecture in [8]. Their cohomology carries a natural action of the Weyl group W of a maximal torus of GLα. The generating series of the anti-invariant part of the cohomology equals the Kac polynomial. We find a similar torus action on Mσ(Γ,α) which commutes with the W-action on cohomology. Again we describe the components of the fixed point locus (see Theorem 5.4) in terms of Mσ(Γ^,β) and prove that the cohomology of the fixed point locus identifies—as a W-representation—with a sum of induced representations from the Weyl groups of the coverings (see Theorem 5.7). Arguing that the localization isomorphism is compatible with the Weyl group action in our setup (this is a bit of an awkward business as W does not act on the quiver variety, but only on its cohomology; see Proposition 6.1), we conclude that Theorem 1.1 holds for arbitrary dimension vectors. Applying Theorem 1.1 iteratively we obtain the following corollary: Corollary 1.2 The value of the Kac polynomial at one of Γ attached to a dimension vector α is the sum  aΓ,α(1)=∑βaΓ˜,β(1),which ranges over a complete system of representatives of equivalence classes of dimension vectors β of the universal covering quiver Γ˜of Γ that are compatible with α. The value aΓ,α(1) is closely related to the number of indecomposable tree modules. A consequence of Theorem 1.1 is Corollary 8.2 stating that the number of indecomposable tree modules of dimension α equals aΓ,α(1), provided that all compatible roots β of the universal covering quiver are exceptional. As every finite connected subquiver of the universal covering naturally defines an indecomposable tree module which is exceptional as a representation of Γ˜, the number of such subquivers of a fixed dimension type gives a lower bound for the Kac polynomial at one. This is a consequence of the fact that the Kac polynomial is simply 1 for exceptional roots. Indecomposable tree modules of this kind are also called cover-thin. We apply these considerations to the generalized Kronecker quiver K(m). Statements about the growth behavior of the number of cover-thin tree modules of K(m) then yield that aK(m),n(d,e)(1) grows at least exponentially in n if (d,e) is a root. Kinser and Derksen sketch a proof of the above theorem in an unpublished note using entirely different methods. The coprime case was also treated in the habilitation thesis [24] of the second author. 2. Terminology and preliminary results Let Γ be a quiver. Let Γ0 be its set of vertices and Γ1 be the set of arrows; both assumed to be finite. A representation M of Γ over the field k consists of a tuple of finite-dimensional k-vector spaces (Mi)i∈Γ0 and k-linear maps Ma:Mi→Mj for every arrow a:i→j. A morphism f:M→N of representations of Γ is a collection fi:Mi→Ni of k-linear maps such that fjMa=Nafi for every a:i→j. We thus obtain an abelian category. A representation M over k is called indecomposable if it cannot be decomposed as the direct sum of two proper subrepresentations. We call M absolutely indecomposable if M⊗kK is an indecomposable representation for every finite extension K∣k. If k=Fq then the number aΓ,α(q) of absolutely indecomposable representations of Γ over Fq of dimension vector α—the dimension vector of a representation M is the tuple (dimkMi)i∈Γ0—is a finite number. Kac shows in [12, Section 1.15] that aΓ,α(q) is a polynomial in q with integer coefficients. It is called the Kac polynomial of (Γ,α). Given α∈Z≥0Γ0, we define the vector space   R(Γ,α)=⊕a:i→jHomk(kαi,kαj).On R(Γ,α), the group GLα=∏iGLαi acts by change of basis. The set of GLα-orbits on R(Γ,α) is in natural bijection with the set of isomorphism classes of representations of Γ of dimension α over k. The double quiver Γ¯ of Γ is obtained as follows: the set Γ¯0 is just Γ0, but Γ¯1 is obtained from Γ1 by adding a new arrow a*:j→i for every arrow a:i→j in Γ. Then   R(Γ¯,α)=⊕(a:i→j)∈Γ1(Hom(kαi,kαj)⊕Hom(kαj,kαi)).An element φ∈R(Γ¯,α) consists of linear maps φa:kαi→kαj and φa*:kαj→kαi. Let glα be the Lie algebra of GLα and let glα0 be the Lie subalgebra consisting of elements X whose total trace tr(X)=∑itr(Xi) is zero. The moment map μ:R(Γ¯,α)→glα0 is defined by   μ(φ)=∑a∈Γ1[φa,φa*].Let λ∈ZΓ0 with λ·α=∑iλiαi=0 which we regard as a central element of glα0. Elements of the fiber μ−1(λ) are representations of Πλ(Γ)=kΓ¯/(∑a∈Γ1[a,a*]−∑iλiei), the so-called deformed preprojective algebra of Γ. As μ is GLα-equivariant and λ is central the fiber μ−1(λ) carries an action of GLα. The GLα-orbits on μ−1(λ) are in bijection with isomorphism classes of representations of Πλ(Γ) of dimension vector α. We refer to [4] for more details. Given a quiver Γ we define the (infinite) quiver Γ^ by   Γ^0=Γ0×ZΓ1,Γ^1=Γ1×ZΓ1,where for an arrow a:i→j in Γ and χ∈ZΓ1, the arrow (a,χ)∈Γ1 has source (i,χ) and target (j,χ+ea) (the element ea is the respective unit vector in ZΓ1), i.e. pictorially   (a,χ):(i,χ)→(j,χ+ea).The quiver Γ^ is called the universal abelian covering quiver of Γ (see [23, Section 3.1]). We also recall the notion of the universal covering quiver. We denote by WΓ the free group generated by Γ1. The universal covering quiver Γ˜ of Γ is given by the vertex set   Γ˜0=Γ0×WΓ,Γ˜1=Γ1×WΓ,where (a,w):(i,w)→(j,wa) for a:i→j. Finally, we consider iterated covering quivers and define the kth universal abelian covering quiver Γ^k recursively by Γ^0=Γ and   Γ^0k=Γ^0k−1×ZΓ^1k−1,Γ^1k=Γ^1k−1×ZΓ^1k−1.As explained in [23, Section 3.4], there exist natural surjective morphisms ck:Γ˜→Γ^k which become injective on finite subquivers of Γ˜ for k≫0, see [23, Proposition 3.13]. There is a natural morphism of quivers c:Γ^→Γ which projects along ZΓ1. Let Λ=ZΓ0 and let Λ^ be the sublattice of ZΓ^0 of those vectors β=(βi,χ) with finite support. We extend the map c linearly to a map c:Λ^→Λ, concretely   c(β)i=∑χβi,χ.A dimension vector of Γ^ is defined to be an element of Λ^ whose entries are non-negative. We say that a dimension vector β of Γ^ is compatible with a dimension vector α of Γ if c(β)=α. We define an action of the group ZΓ1 on Λ^ by letting ξ∈ZΓ1 act on β∈Λ^ by (ξ.β)i,χ=βi,χ+ξ. Dimension vectors which lie in the same ZΓ1-orbit are called equivalent. The map c is ZΓ1-invariant, and it is clear that up to equivalence only finitely many dimension vectors of Γ^ with connected support are compatible with a given dimension vector α of Γ. There is also a natural morphism c:Γ˜→Γ with the same properties. Finally, the morphisms c and ck extend to natural functors ck:Rep(Γ^k)→Rep(Γ) and c:Rep(Γ˜)→Rep(Γ) between the representation categories preserving indecomposability. We refer to [7] for more details on covering theory. The Kac polynomial at one aΓ,α(1) is closely related to the number of indecomposable tree modules of dimension α. We investigate this connection in Section 8 in greater detail. Thus, let us recall the notion of coefficient quivers and tree modules. For a fixed representation M, we choose a basis B of ⊕i∈Γ0Mi such that Bi≔B∩Mi is a basis of Mi for every vertex i∈Γ0. For every arrow a:i→j, we write Ma as an (αj×αi)-matrix Ma,B with coefficients in k such that the rows and columns are indexed by Bj and Bi, respectively. The coefficient quiver Γ(M,B) of a representation M with a fixed basis B has vertex set B and arrows between vertices are defined by the following condition: if (Ma,B)b′,b≠0, there exists an arrow (a,b,b′):b→b′, where b∈Bi, b′∈Bj and a:i→j. A representation M is called a tree module if there exists a basis B for M such that the corresponding coefficient quiver is a tree. Remark 2.1 The existence of indecomposable tree modules is known for Schur roots [19, 22], but in general it is an open question. Heuristically, [12, Conjecture 9], which we already mentioned in the introduction, and also the results of this paper suggest that every tree module spans an affine space of indecomposables. This is also because every tree module can be lifted to a representation of the universal (abelian) covering quiver. Thus, every indecomposable tree module is a torus fixed point under the torus action which we introduce in these notes. But in general not every torus fixed point is a tree module, and note also that there are cases where there exist more than aΓ,α(1) indecomposable tree modules. 3. The coprime case If α is coprime, then the proof of Theorem 1.1 is easier. It suffices to consider moduli spaces of complex representations of the deformed preprojetive algebra. Let k=C from now on. Consider the moment map μ:R(Γ¯,α)→glα0. As α is coprime, we find λ∈ZΓ0 with λ·α=0 such that λ·α′≠0 for every 0≤α′≤α, unless α′ equals 0 or α. Such a λ is called generic for α. Let Mλ(Γ,α)=μ−1(λ)//GLα. By using [5], Formula (2.7), Corollary 2.3.2, we have   aΓ,α(q)=∑i=0ddimHc2d+2i(Mλ(Γ,α);C)qi,where we consider singular cohomology with compact supports and where d is half of the complex dimension of Mλ(Γ,α). Since Mλ(Γ,α) is cohomologically pure, the existence of a polynomial with integer coefficients which counts the rational points yields that the odd cohomology vanishes, see [5, Appendix A]. In particular, we obtain aΓ,α(1)=χc(Mλ(Γ,α)). By a well-known result, we have χc(X)=χc(XT) for any complex variety with a torus action, see for instance [3, Section 2.5] or [6, Appendix B]. Here XT denotes the fixed point set. It is straightforward to transfer the results of [23] to the case of the moduli spaces Mλ(Γ,α). Note that representations of Πλ(Γ) are simple if λ is generic. This enables us to understand the corresponding fixed point components as moduli spaces attached to the universal abelian covering of Γ. More precisely, let T≔(C×)Γ1 act on R(Γ¯,α) by   (ta)a∗(φa,φa*)a∈Γ1=(taφa,ta−1φa*).This descends to an action on μ−1(λ) which commutes with the usual base change action of GLα. Now the same proofs as those of [23, Sections 3.1, 3.2] apply to show the following: Theorem 3.1 The set of torus fixed points Mλ(Γ,α)Tis isomorphic to the disjoint union of moduli spaces  ⨆βMλ(Γ^,β),where β ranges over all equivalence classes of dimension vectors of Γ^compatible with α. Note that λ, which we regard as an element of ZΓ^0 by setting λi,χ=λi, is generic for every β that is compatible with α. This shows that Theorem 1.1 holds for α coprime. Note further that every β for which Mλ(Γ^,β) is non-empty must have connected support by genericity of λ. Finally, Corollary 1.2 follows by Remark 5.8. 4. Construction of the moduli space in the general case We recall the construction of Hausel–Letellier–Rodriguez-Villegas from [8]. Consider again μ:R(Γ¯,α)→glα0 over the complex numbers. Let Tα be the maximal torus of GLα of tuples of invertible diagonal matrices. Let tα be the Lie algebra of Tα. A semi-simple element of glα is called regular if its centralizer is a maximal torus. Therefore the centralizer of a regular element of tα is Tα. An element σ∈tα is called generic if tr(σ)=0 and if tr(σ∣V)≠0 for all non-trivial Γ0-graded subspaces V⊆Cα which are stable under σ. Let tαgen be the (non-empty) open subset of regular generic elements of tα0. The variety   M=M(Γ,α)={(φ,hTα,σ)∈R(Γ¯,α)×GLα/Tα×tαgen∣μ(φ)=hσh−1}//GLαis the quotient by the GLα-action defined by g(φ,hTα,σ)=(g·φ,ghTα,σ). Note that the diagonally embedded C× acts trivially and the induced action of GLα/C× is free. The map π:M→tαgen arising by projecting onto the third factor is surjective. Theorem 4.1 ([8, Theorem 2.1]) The fibers Mσare smooth and their cohomology vanishes in odd degrees. The Weyl group W=Wα=NGLα(Tα)/Tα≅∏iSαi acts on M via   w.(φ,hTα,σ)=(φ,hw˙−1Tα,w˙σw˙−1),where w˙ is the permutation matrix defined by w (or any other representative of w in NGLα(Tα)). We will drop the dot in the notation for convenience. This gives isomorphisms   w:Mσ→Mwσw−1. Theorem 4.2 ([8, Theorem 2.3]) For any σ∈tαgen, the cohomology group Hci(Mσ;C)becomes in a natural way a representation of W which is up to isomorphism independent of σ∈tαgen. In [8], this result is stated for the cases that the ground field has large positive characteristic or is the complex numbers and with coefficients in Q¯ℓ. For our purposes, it will be sufficient to consider the complex case and C-coefficients. The above theorem follows from a result of Maffei [15, Lemma 48] which shows that Riπ!Z, and hence also Riπ!C, is constant. As it is useful for our purposes we will explain how the W-representation arises. Remark 4.3 Let f:Y→X be a continuous map of locally compact topological spaces. Let W be a finite group which acts on both X and Y such that f is W-equivariant. Let F be a sheaf of complex vector spaces on Y with a W-linearization φ:act*F→≅pr2*F (where act:W×Y→Y is the action map and pr2:W×Y→Y is the projection); see [17, Section 1.3] for the definition of a linearization. We obtain isomorphisms φw:w*F→F with φw1w2=φw2◦w2*φw1. Consider the higher direct images Rif!F with compact supports. The definitions and results on cohomology of sheaves can, for instance, be found in [13, Chapters II and III]. Suppose that Rif!F is constant, say with fiber F. For every x∈X, we are thus given an isomorphism F→(Rif!F)x=Hci(Yx;F). This yields an isomorphism ix,x′:Hci(Yx;F)→Hci(Yx′;F) for every two points x,x′∈X. The induced W-linearization on Rif!F yields isomorphisms   φw,x:Hci(Ywx;F)=Hci(Yx;w*F)→Hci(Yx;F).It is easy to verify from the cocyle conditions and the compatibility of the isomorphisms ix,x′ that ρi:W→GL(Hci(Yx;F)) defined by   ρi(w):Hci(Yx;F)⟶φw−1,wxHci(Ywx;F)⟶iwx,xHci(Yx;F)is a representation of W. If F is constructible, the assumption Rif*F be constant induces a representation W→GL(Hi(Yx;F)) in the same way. The central result of [8] is the description of the Kac polynomial as the generating series of the alternating part of the graded W-representation Hc*(Mσ;C). More precisely: Theorem 4.4 [8, Theorem 1.4] The Kac polynomial aΓ,α(q)coincides with  ∑i=0ddim(Hc2i+2d(Mσ;C)sign)qi,where d is half of the complex dimension of Mσand the subscript ‘ sign’ denotes the alternating component of the cohomology regarded as a W-representation. 5. Torus action Let T=(C×)Γ1 act on R(Γ¯,α) via   (t·φ)a=taφa(t·φ)a*=ta−1φa*for any t=(ta)a∈Γ1∈T and φ∈R(Γ¯,α). This T-action commutes with the action of GLα which implies that we get an action of T on M by   t.(φ,hTα,σ)=(t·φ,hTα,σ).The T-action on M commutes with the W-action, whence W acts on the fixed point locus MT. We analyze the fixed point locus and the W-action on it. Let (φ,hTα,σ)∈MT. This means for all t∈T, there exists g∈GLα with   (t·φ,hTα,σ)=(g·φ,ghTα,σ),or, in other words, taφa=gjφagi−1 for all a:i→j and hi−1gihi∈Tαi for all i∈Γ0. As GLα/C× acts freely on the total space of the quotient M, the element g is uniquely determined by t up to a scalar. Using the arguments from [23], we deduce that there exists a homomorphism ψ:T→GLα, unique up to the diagonally embedded C×, with t.(φ,hTα)=ψ(t)·(φ,hTα). The ith component ψi:T→GLαi of ψ induces a T-action on Vi=Cαi and, therefore, a weight space decomposition   Vi=⊕χ∈X(T)Vi,χ.The character group X(T) of T is precisely ZΓ1. For a weight vector vχ∈Vi,χ and an arrow a:i→j, we get   taφa(vχ)=(t·φ)a(vχ)=ψj(t)φaψi(t)−1(vχ)=χ(t)−1ψj(t)φa(vχ),or, in other words, φa(vχ)∈Vj,χ+ea. It is shown analogously that φa*(Vj,χ)⊆Vi,χ−ea. These considerations show that φ can be regarded as a representation of the double of the covering quiver Γ^ of dimension vector β with βi,χ=dimVi,χ. Let ei,r be the rth unit vector in Vi=Cαi (we keep the index i in the notation for bookkeeping reasons). Let χ1,…,χN be those characters for which there exists an i such that the weight space Vi,χk is non-zero. Embed Cβi,χk as the subspace of Cαi spanned by the unit vectors   ei,(βi,χ1+⋯+βi,χk−1+1),…,ei,(βi,χ1+⋯+βi,χk), (5.1)and consider GLβ=∏i,χGLβi,χ as a subgroup of GLα via this direct sum decomposition. As there exists g∈GLα with g(Vi,χk)=Cβi,χk which is unique up to a (unique) element of GLβ, we may, by passing from (φ,hTα,σ) to (g·φ,ghTα,σ), assume without loss of generality that Vi,χk=Cβi,χk. For a number r∈{1,…,αi}, let k be the unique index with ei,r∈Vi,χk. Write hi(ei,r) as ∑χwχ with wχ∈Vi,χ. As hi−1ψi(t)hi lies in Tαi, the vector   (hi−1ψi(t)hi)(ei,r)=hi−1(∑χχ(t)wχ)lies in the span of ei,r. Precisely one summand wχ is non-zero. To see this, assume otherwise. Take a character ξ for which wξ≠0 and observe that hi(ei,r) would by assumption not be a multiple of wξ. Choose a one-parameter subgroup λ of T which vanishes on ξ and is positive on all other χ which occur in the summation ∑χwχ. Then   (hi−1ψi(λ(z))hi)(ei,r)=hi−1(∑χz⟨λ,χ⟩wχ)⟶z→0hi−1(wξ),which does not lie in the span of ei,r. A contradiction. We deduce that for every r as above there exists an l such that hi(ei,r)∈Vi,χl. This implies the existence of a w∈W such that hw˙−1∈GLβ (and this Weyl group element w is unique up to an element of Wβ=NGLβ(Tα)/Tα). These considerations show that the fixed point locus MT is contained in the disjoint union   ⨆β⨆w¯∈W/WβMβ,w¯over a full system of representatives of dimension vectors β which are compatible with α, where   Mβ,w¯={(φ,hTα,σ)∈R(Γ^¯,β)×GLα/Tα×tαgen∣hw˙−1∈GLβ,μ(φ)=hσh−1}//GLβ,and where w˙∈NGLα(Tα) is a representative of w¯∈W/Wβ≅NGLα(Tα)/NGLβ(Tα). To show conversely that every element of Mβ,w¯ is a T-fixed point, we take a triple (φ,hTα,σ) as above and show that to every t∈T there exists g∈GLα such that (t·φ,hTα,σ)=(g·φ,ghTα,σ).To interpret the representation φ of the double of Γ^ as a representation of the double of Γ, let again χ1,…,χN be the characters for which there exists an i with βi,χk≠0 and identify Cβi,χk with the span Vi,χk of the unit vectors as in (5.1). Then φa(Vi,χ)⊆Vj,χ+ea for every arrow a:i→j of Γ and φa*(Vi,χ)⊆Vj,χ−ea. Let ψ:T→GLα be the homomorphism that corresponds to the decompositions Vi=⊕χVi,χ. A computation as above shows that ψj(t)φaψi(t)−1=taφa.What remains to show is that hTα=ψ(t)hTα. Let ei,r∈Vi,χk. Then ei,w(r) lies in, say, Vi,χl, whence   hi(ei,r)=hw˙−1(ei,w(r))∈Vi,χl,as hw˙−1∈GLβ. Therefore, ψi(t)hi(ei,r)=χl(t)hi(ei,r), which implies ψi(t)hiTαi=hiTαi. We have shown Proposition 5.1 As a closed subset of M(Γ,α), the fixed point locus M(Γ,α)Tagrees with the disjoint union  ⨆β⨆w¯∈W/WβMβ,w¯over a full system of representatives of dimension vectors β which are compatible with α. The fixed point locus MT is smooth as M is. In particular, MT is reduced. As Mβ,id¯ is isomorphic to M(Γ^,β), it is smooth as well. The component Mβ,w¯ is isomorphic to Mβ,id¯, so the disjoint union ⨆β⨆w¯∈W/WβMβ,w¯ is smooth. Hence, MT and ⨆β⨆w¯∈W/WβMβ,w¯ do not only agree as closed subsets, but in fact as closed subschemes. An element w′∈W applied to (φ,hTα,σ)∈Mβ,w¯ yields (φ,hw′−1Tα,w′σw′−1) which lies in Mβ,ww′−1¯. Lemma 5.2 shows that the disjoint union ⨆w¯∈W/WβMβ,w¯ agrees—as a W-scheme—with the associated fiber bundle W×WβM(Γ^,β). Lemma 5.2 Let W be a finite group, let U⊆Wbe a subgroup of W. Suppose that we are given schemes Xa¯for every a¯∈W/Uand isomorphisms  φw,a¯:Xa¯→≅Xaw−1¯satisfying φe,a¯=idand φw,aw′−1¯◦φw′,a¯=φww′,a¯. Then the disjoint union  ⨆a¯∈W/UXa¯carries an action of W via w·x=φw,a¯(x) (where w∈Wand x∈Xa¯) for which Xe¯is a U-invariant subset. Equipped with these actions, ⨆a¯∈W/UXa¯is isomorphic to the associated fiber bundle W×UXe¯as a W-scheme. We show this using the following result which seems to be well known, but for which we could not find a reference; for the convenience of the reader, we give a proof here. Lemma 5.3 Let G be an algebraic group, let X be a scheme equipped with a G-action, let H be a closed subgroup of G and let f:X→G/Hbe a G-equivariant morphism. Denote Z=f−1(eH). Then Z is H-invariant and X is isomorphic to G×HZas a G-scheme. Proof The associated fiber bundle G×HZ is defined as follows: equip G×Z with an action of G×H via (g,h)·(a,z)=(gah−1,h·z). The actions of the two factors commute, whence the geometric quotient G×HZ≔(G×Z)/H (which exists) carries a G-action. For a more detailed introduction to associated fiber bundles, see [20, Sections 3.3 and 3.5] or [21, Section 2.1]. In our concrete situation, we consider the map φ:G×Z→X defined by φ(a,z)=a·z. As φ fulfills   φ((g,h)·(a,z))=φ(gah−1,h·z)=ga·z=g·φ(a,z),it gives rise to a morphism φ¯:G×HZ→X which is apparently G-equivariant. Conversely, we define the map ψ:X→G×HX as the composition of X→G/H×X,x↦(f(x),x) with the G-equivariant isomorphism G/H×X→≅G×HX (with G acting on G/H×X simultaneously on both factors). This isomorphism comes from the isomorphism G×X→≅G×X,(a,x)↦(a,a−1·x). The map ψ factors through the closed subscheme G×HZ as for x∈X with f(x)=aH, we obtain   ψ(x)=[a,a−1·x],and a−1·x∈Z, as f is G-equivariant. So ψ yields a G-equivariant morphism ψ¯:X→G×HZ. It is easy to show that φ¯ and ψ¯ are mutually inverse.□ Proof of Lemma 5.2 The claim follows from the previous lemma by the observation that the map f:⨆a¯∈W/UXa¯→W/U which assigns f(x)=a−1¯ to every x∈Xa¯ is W-equivariant.□ We have proved. Theorem 5.4 The fixed point locus M(Γ,α)Tis, as a W-scheme, isomorphic to the disjoint union  ⨆βW×WβM(Γ^,β)over all dimension vectors β up to equivalence which are compatible with α. This isomorphism commutes with the natural maps to tαgen. As a next step, we would like to relate the W-module structure on the stalk of the sheaf Ri(π∣MT)!C (which will turn out to be constant) with the Wβ-module-structure on the stalk of Ri(πβ)!C, where πβ:M(Γ^,β)→tαgen. We need two general lemmas for this. Lemma 5.5 Let X be a scheme with an action of a finite group W. Let U be a subgroup of W and consider X as a U-scheme with the restricted action. Let Fbe the constant sheaf with stalk F on W×UXand let φ:act*F→pr2*Fbe a W-linearization. Let i:X→W×UXbe the closed embedding given by i(x)=[e,x]; it is U-equivariant. The sheaf i*Fis also constant with stalk F and the induced U-linearization i*φgives F the structure of a U-representation FU. The action map gives rise to a W-equivariant map act¯:W×UX→X. The push-forward act¯!Fis constant and the W-module structure on its stalk is IndUWFU. Proof To prove the lemma, we identify W×UX with W/U×X. Under this identification, act¯ corresponds to the projection W/U×X→X to the second factor and the map i tallies with the embedding x↦(e¯,x). It is hence clear that act¯!F is constant with stalk   (act¯!F)x=⊕a¯∈W/UF(a¯,x)≅FW/U.The action of w∈W on the stalk is defined by   ⊕a¯∈W/UF(a¯,x)⟶⨁φw,(a¯,x)−1⊕a¯∈W/UF(wa¯,wx)⟶⨁i(wa¯,wx),(wa¯,x)⊕a¯∈W/UF(wa¯,x)so under the identification with FW/U the action of w on a tuple (va¯)a¯ is given by (w·vwa¯)a¯. Here, the expression w·vwa¯ stands for the W-action on F(wa¯,x)=F which is induced by φ. Noting that FU is precisely ResUWFW, we have shown that FU is a U-invariant subspace of the W-representation FW/U defined above and this W-representation decomposes, as a vector space, into the w-translates of the subspace FU. This shows that FW/U=IndUWFU.□ Lemma 5.6 Let W be a finite group which acts on a scheme X and let Y be a scheme equipped with the action of a subgroup U of W. Let f:Y→Xbe a U-equivariant morphism of schemes. Let f˜:W×UY→W×UXbe the induced morphism and let f¯:W×UY→Xbe the composition act¯◦f˜. Let Fbe a sheaf on W×UYwith a W-linearization such that Rif˜!Fis constant. Consider the closed embedding iY:Y→W×UYand the sheaf iY*Fwith the natural U-linearization. Then, also Rif!iY*Fand Rif¯!Fare constant sheaves and we get  Hci((W×UY)x;F)≅IndUWHci(Yx;iY*F)for the associated W-representation structures on the cohomology groups of the stalks of x. Proof Base change yields an isomorphism Rif!iY*F≅iX*Rif˜!F, where iX is the closed embedding X→W×UX. On the other hand, Rif¯!F=act¯!Rif˜!F because the functor act¯! (which agrees with act¯*) is exact, as act¯ is (quasi-)finite. The sheaf Rif˜!F is assumed to be constant. We are hence in the situation of Lemma 5.5. It tells us that Rif¯!F is constant (and Rif!iY*F is, too) and that the W-module structures on their stalks—at, say, the point x∈X—are related by   (Rif¯!F)x≅IndUW(Rif!iY*F)x.This concludes the proof of the lemma.□ We apply Lemma 5.6 for every β to Y=M(Γ^,β), U=Wβ, X=tαgen, f=πβ, and F the constant sheaf C on W×WβM(Γ^,β) with the trivial W-linearization. This shows Theorem 5.7 The cohomology group Hci(Mσ(Γ,α)T;C)has a structure of a W-representation which is independent of σ and which is isomorphic to the direct sum  ⊕βIndWβWHci(Mσ(Γ^,β);C)over all dimension vectors β up to equivalence which are compatible with α. Remark 5.8 When applying iterated localization, in a first step, we can replace Γ^ by Γ^k in Theorem 5.7. In a second step, we can replace Γ^k by Γ˜. Indeed, every root β of Γ^k which is compatible with α defines a finite connected subquiver of Γ^k. As recalled in Section 2, every finite subquiver of the universal covering quiver embeds into Γ^k for k≫0. As the maps ck defined there are also surjective, this means that we can choose k in such a way that the supports of all compatible roots β are finite subquivers of Γ˜. We refer also to [23, Section 3.4] for more details. 6. Localization isomorphisms Consider a smooth morphism f:Y→X of smooth complex varieties which is assumed to be equivariant with respect to a finite group W that acts on both X and Y. Now suppose that a complex torus T=(C×)n acts on Y in a way that f is T-invariant and such that the T- and the W-action on Y commute. We assume further that f arises by base change from a (topologically) locally trivial fibration Y′→X′ whose basis X′ is contractible (this holds in our concrete situation, see the proof of [8, Theorem 2.3] or [15, Lemma 48]). In particular f is also a locally trivial fibration. Consider the constant sheaf C on Y. Then the higher direct images Rif!C and Rif*C are constant by contractibility of X′. Remark 4.3 ensures that Hci(Yx;C) and Hi(Yx;C) have a natural structure of a W-representation. Moreover, every connected component of the fixed point set YT is smooth. Suppose further that the induced map fT:YT→X satisfies the same properties as f (this is true in our setup, see Theorem 5.4). We show: Proposition 6.1 The classes [Hc*(Yx;C)]and [Hc*(YxT;C)]agree in the Grothendieck group K0(CW)of finitely generated (complex) W-representations. In the above context, the class [V*] in K0(CW) of a finite-dimensional graded W-representation V*=⊕i∈ZVi is defined as [V*]=∑i(−1)i[Vi]. We have to show that the localization isomorphism in [3, Section 2.5] is compatible with the W-action. But as the W-action on cohomology of the fiber does not arise from a W-action on the fiber but, from monodromy, we have to adjust the proof given in [3] relative to the basis X. Proof Let ET be a contractible space on which T acts freely and let BT=ET/T. Consider the cartesian diagrams   By base change we see that π*RifT,*C≅p*Rif*C and, as π is faithfully flat, we obtain RifT,*C≅pr*Rif*C where pr:X×BT→X is the projection. The composition   Y×TET→fTX×BT→prXyields a spectral sequence E2p,q=(Rppr*)(RqfT,*)C⇒Rp+q(pr◦fT)*C. By the above considerations, we conclude that   E2p,q≅(Rppr*)pr*(Rqf*)C≅Hp(BT;C)⊗Rqf*C.This is a spectral sequence in the category of W-equivariant sheaves on X. Note that as fT is a locally trivial bundle, the composition pr◦fT is, too, and as BT is contractible all Ri(pr◦fT)*C are constant. Then the stalk Hi(Yx×TBT;C)=HTi(Yx;C) inherits the structure of a W-representation (cf. Remark 4.3). Applying the stalk functor to the spectral sequence E from above yields the spectral sequence   HTp(pt)⊗Hq(Yx)⇒HTp+q(Yx)(all cohomology groups are taken with complex coefficients). This is the spectral sequence associated with the fibration ET×TYx→BT, but by this detour, we have shown that the differentials of this spectral sequence are W-linear. This shows that [HT*⊗H*(Yx)]=[HT*(Yx)] in the Grothendieck group K0((HT*)W) of finitely generated (HT*)W-modules. On the other hand, the inclusion YT→Y of the fixed point locus induces a natural morphism   Ri(fT◦pr)*C→Ri((fT×idBT)◦pr)*C,which, after taking fibers, is the pull-back HTi(Yx)→HTi(YxT) in equivariant cohomology. This shows that this map is also W-equivariant. The HT*-linear map HT*(Yx)→HT*(YxT) becomes an isomorphism after localizing finitely many non-trivial characters. This isomorphism doesn’t preserve the grading, but only the parity. Let S⊆HT* be the multiplicative subset arising from the aforementioned characters. Then [S−1HT*⊗HT*HT*(Yx)]=[S−1HT*⊗H*(YxT)] in the Grothendieck group K0((S−1HT*)W), and thus   [S−1HT*⊗H*(Yx)]=[S−1HT*⊗H*(YxT)]in the same group. But this implies that [H*(Yx)]=[H*(YxT)] already in K0(CW). Finally, we apply Poincaré duality. As Yx and all connected components of YxT are smooth varieties, the classes [H*(Yx)] and [H*(YxT)] agree with [Hc*(Yx)] and [Hc*(YxT)], respectively, in the Grothendieck group K0(C). But Poincaré duality in this case comes from Verdier duality RHom̲(Rf!C,ωX)≅Rf*RHom̲(C,ωY), which is W-equivariant (by interpreting it as an identity in the bounded derived category of W-equivariant sheaves on X, which is the same as DWb(X) the W-equivariant bounded derived category in the sense of Bernstein–Lunts [2] as W is finite). Therefore, the identity [Hc*(Yx)]=[H*(Yx)] holds also in the Grothendieck group K0(CW). The same argument applies for the equality [Hc*(YxT)]=[H*(YxT)].□ 7. Finishing the proof of the main result Applying Proposition 6.1 and the fact that Mσ has no odd cohomology, we observe that Hc*(MσT;C)≅Hc*(Mσ;C) as ungraded W-representations (we define H*(X;C) as the direct sum ⊕iHi(X;C)). The W-representation Hc*(MσT;C) decomposes by Theorem 5.7 as   ⊕βIndWβWHc*(Mσ(Γ^,β);C).The sign-isotypical component of the induced W-representation IndWβWHci(Mσ(Γ^,β);C) is the sign-isotypical component of Hci(Mσ(Γ^,β);C) (with respect to the Wβ-action). We know by [8, Theorem 1.4] that   aΓ,α(1)=dimHc*(Mσ(Γ,α);C)signaΓ^,β(1)=dimHc*(Mσ(Γ^,β);C)sign,the sum of the dimensions of all sign-isotypical components of the cohomology groups (note that the odd-degree cohomology groups of Mσ(Γ,α) and Mσ(Γ^,β) vanish). Using that taking isotypical components and taking classes in the Grothendieck group commute, we have proved aΓ,α(1)=∑βaΓ^,β(1), as asserted in Theorem 1.1. Now Corollary 1.2 is an immediate consequence of Theorem 1.1 when taking Remark 5.8 into account. 8. First consequences of the main theorem The main result of this paper has plenty of interesting consequences which were also mentioned in [24] for coprime dimension vectors. For instance, the number of indecomposable tree modules, as defined in [19], can be related to the Kac polynomial at one. Recall that a tree module of Γ is already a representation of the universal covering quiver Γ˜ of Γ. We call an indecomposable tree module of Γ cover-thin if its dimension vector α is of type one as a representation of Γ˜, i.e. if αi∈{0,1} for all i∈Γ˜0. We denote the number of indecomposable tree modules of dimension α by tα and the number of cover-thin tree modules by ctα. Let σi be the BGP-reflection at a source or sink i∈Γ0 introduced in [1]. Lemma 8.1 Every indecomposable tree module M which is cover-thin is exceptional as a representation of the universal covering quiver. In particular, we have that σiMis also an indecomposable tree module for any sink (resp. source) i∈Γ0. Proof The first part follows because every exceptional representation M is Schurian and, moreover, because ⟨dim̲M,dim̲M⟩=dimkHom(M,M)−dimkExt(M,M)=1. In particular, σiM is also exceptional as a representation of the universal covering quiver, and thus a tree module by the main result of [19].□ Corollary 8.2 Let α be a dimension vector such that all equivalence classes of compatible dimension vectors with connected support consist of exceptional roots. Then the number of indecomposable tree modules of dimension vector α is equal to the Kac polynomial at one. Proof For the Kac polynomial of a real root α˜, we have aΓ˜,α˜(q)=aΓ˜,α˜(1)=1. Thus, it suffices to show that the number of indecomposable tree modules is equal to the number of compatible roots. By the main result of [19], every exceptional representation is an indecomposable tree module, and thus a representation of the universal covering quiver, say of dimension α˜. Thus, every compatible dimension vector gives rise to an indecomposable tree module. Conversely, every indecomposable tree module T yields a root α˜ of Γ˜ which is compatible with α. Since α˜ is exceptional by assumption, T is up to isomorphism the only representation of dimension α˜.□ It was pointed out to us by Jie Xiao that Corollary 1.2 implies at once: Corollary 8.3 Every real root representation is already a representation of the universal covering quiver. Proof As aΓ,α(1)=1=∑βaΓ˜,β(1) for a real root α, the unique indecomposable representation of dimension α already lives on the universal covering Γ˜. Alternatively, note that the corresponding root of the universal covering can be obtained via reflections of the Weyl group of Γ˜; they correspond precisely to those reflections of the Weyl group of Γ which are used to obtain α from a simple root.□Using iterated localization we also re-obtain the following result: Corollary 8.4 ([16, Corollary 4.4]) If α is a dimension vector of Γ such that αi=1for all i∈Γ0, the Kac polynomial at one is equal to the number of spanning trees of Γ. Let K(m) be the generalized Kronecker quiver with two vertices i,j and m arrows al:i→j. In this case, the main result, together with the following proposition, enables us to investigate the asymptotic behavior of the Kac polynomial at one. Proposition 8.5 ([24, Proposition 3.2.6]) Let n≔m−1. The number of indecomposable cover-thin tree modules ct(d,e)of K(m)which are of dimension (d,e)is  1d∑i=1m(mi)(ned−1)(n(d−1)e−i)ie. Example 8.6 If m=3 and (d,e)=(d,d+1), it is straightforward to check that we have   ct(d,d+1)=3(d+2)(d+3)(2dd)(2(d+1)d+1).The respective sequence of natural numbers appears as sequence A186266 in [18]. It seems that there was no combinatorial interpretation of this sequence before. Applying Lemma 8.1, we thus obtain: Corollary 8.7 The number of indecomposable tree modules tn(d,e)of K(m)grows (at least) exponentially with the dimension vector, i.e. for every imaginary Schur root (d,e)of K(m)there exists a real number K(d,e)>1such that tn(d,e)>K(d,e)n. As a consequence, we obtain the following: Corollary 8.8 Let (d,e)be a root of the generalized Kronecker quiver K(m). Then the Kac polynomial at one grows (at least) exponentially with the dimension vector, i.e. there exists a real number K(d,e)>1such that  aK(m),n(d,e)(1)>K(d,e)n. Actually, Corollary 8.7 can also be used to show that the number of indecomposable tree modules which have an imaginary Schur root as the dimension vector grows exponentially with the dimension vector, see [24, Theorem 3.2.8]. We conclude with the following natural question, which was for instance asked in [14, Question 7], but also posed to the second author by Crawley-Boevey and Hubery: Question 8.9 Do we always have tα≥aΓ,α(1)? The main result of the paper implies that this needs to be checked only for quivers which are trees. But actually, similar to the question of the existence of tree modules, it seems that this does not make things much easier. Many examples which can be found in the literature suggest that this is true. In the case of extended Dynkin quivers of type D˜n, this can be checked by hand. We also conjecture that equality holds if and only if the assumptions of Corollary 8.2 hold. 9. Concrete examples Consider the generalized Kronecker quiver K(3) and the dimension vector (2,3). By use of Hua’s formula [11], we obtain   aK(3),(2,3)=q6+q5+3q4+4q3+5q2+3q+2,and thus aK(3),(2,3)(1)=19, see [11, Section 5]. There are 18 cover-thin tree modules of K(3) of dimension (2,3) which are given by   Here the arrows mi∈{a1,a2,a3} satisfy the conditions m1≠m2≠m3≠m4 in the first case and the conditions m1≠m2 and m2,m3,m4 pairwise distinct in the second case. Finally, there is one tree module which is not cover-thin, and whose coefficient quiver is the one on the left-hand side in the case when m2=m3 and m1,m2,m4 are pairwise distinct. More precisely, its dimension vector is given by   i.e. the real root of D4 of dimension (2,1,1,1). Thus, the number of indecomposable tree modules is 19. We consider the generalized Kronecker quiver K(4) and the root (2,4). In this case, the number of indecomposable tree modules, which is 126 (120 cover-thin tree modules and six others), is greater than a(2,4)(1). Using Hua’s formula [11] we obtain   aK(4),(2,4)(q)=q13+q12+3q11+4q10+8q9+9q8+15q7+16q6+20q5+17q4+15q3+9q2+5q+2,and thus a(2,4)(1)=125. Up to coloring the arrows, we obtain the following subquivers and dimension vectors (where the dots indicate one-dimensional vector spaces) which give a contribution to the Kac polynomial at one:   It is straightforward to check that there exist (up to translation) 108 possibilities to embed the first quiver into K(4)˜ (resp. to color the arrows with four different colors m1,m2,m3,m4 such that m1,m2,m3 are pairwise distinct and m3≠m4≠m5). For the second one, we have 12 possibilities and, for the last one, there exists only one possible embedding. Since the first two dimension vectors are real roots, the Kac polynomials are 1. The quiver and the dimension vector considered in the last case is a quiver of type D˜4 together with the unique imaginary Schur root δ. Thus, the Kac polynomial is q+4. This can be checked by classifying the absolutely indecomposable representations, which coincide with the indecomposables in this case, up to isomorphism. In summary, we obtain   aK(4),(2,4)(1)=108·1+12·1+1·5=125. Finally, we consider the quiver Lg with only one vertex i and g loops. If αi≤5, all non-empty moduli spaces appearing are points and the Kac polynomials of the compatible dimension vectors are one. In particular, the number of indecomposable tree modules coincides with aLg,αi(1). The first non-trivial moduli space appears for αi=6. Also in this case, we need to consider the Kac polynomial of the imaginary Schur root δ=(2,1,1,1,1) of D˜4 with Kac polynomial aL˜g,δ(q)=q+4. Taking into account the different possibilities of coloring the arrows of D˜4 with the colors {1,…,g} and orienting the arrows, this gives the contribution   5·(24(g4)+3·22(g3)+(g2))to the Kac polynomial at one (we can choose four, three or two different arrows out of the g arrows of the original quiver). Note that, in the universal covering, the following orientations do not appear   Since there are six indecomposable tree modules of dimension δ of D˜4, one checks that this indeed fills the gap between aLg,6(1), see [10, Section 1], and the number of indecomposable tree modules of dimension αi=6, see [14, Section 4.1]. 10. Conjecture on the asymptotic behavior of the Kac polynomial at one The following is based on a conjecture of M. Douglas concerning moduli spaces of stable representations of generalized Kronecker quivers, see [23, Section 6.1]. It generalizes [24, Conjecture 4.1.2] to arbitrary dimension vectors: Conjecture 10.1 There exists a continuous function f:RΓ0→R such that   f(α)=limn→∞ln(aΓ,nα(1))nfor all dimension vectors α∈NΓ0. If a function as predicted in Conjecture 10.1 exists, Proposition 8.5 immediately yields a lower bound in the case of the Kronecker quiver and for coprime dimension vectors (d, e) such that d≤e≤(m−1)d+1. Lemma 10.2 Let (d,e)be a root of the Kronecker quiver such that d≤e≤(m−1)d+1and define k≔e/dand n≔m−1. Then we have  limd→∞aK(m),(d,kd)(1)d≥limd→∞ct(d,kd)d=n(k+1)lnn+k(n−1)lnk−(nk−1)ln(nk−1)−(n−k)ln(n−k). Proof Obviously, we only have to consider one of the m summands of the formula obtained in Proposition 8.5. Then the claim follows straightforwardly when applying the Stirling formula.□ Note that the numbers ct(d,e) are not invariant under the reflection functor. Nevertheless, the reflection functor can clearly be used to obtain a lower bound for aK(m),(d,e)(1) for every dimension vector (d,e). Furthermore, it would be interesting to know if there are tuples (d,e) for which equality holds or to know more about the contribution of cover-thin tree modules to the Kac polynomial at one. Acknowledgements The authors would like to thank R. Kinser, S. Mozgovoy, M. Reineke and J. Xiao for valuable discussions and remarks. The referee’s comments have greatly helped to improve the exposition and were much appreciated. While doing this research, H.F. was supported by the DFG SFB/Transregio 45 ‘Perioden, Modulräume und Arithmetik algebraischer Varietäten’. References 1 I. N. Bernstein, I. M. Gel’fand and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk  28 ( 1973), 19– 33. 2 J. Bernstein and V. Lunts, Equivariant Sheaves and Functors, Vol. 1578, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994. 3 N. Chriss and V. 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