Digital Object Identiﬁer (DOI) 10.1007/s00220-017-2967-x
Commun. Math. Phys. 355, 1209–1241 (2017)
New Singular Gelfand–Tsetlin gl
-Modules of Index 2
, Dimitar Grantcharov
, Luis Enrique Ramirez
Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, Brazil.
University of Texas at Arlington, Arlington, TX 76019, USA. E-mail: firstname.lastname@example.org
Universidade Federal do ABC, Santo André, SP, Brazil. E-mail: email@example.com
Received: 9 June 2016 / Accepted: 30 May 2017
Published online: 2 August 2017 – © Springer-Verlag GmbH Germany 2017
Abstract: Singular Gelfand–Tsetlin modules of index 2 are modules whose tableaux
bases may have singular pairs but no singular triples of entries on each row. In this paper
we construct singular Gelfand–Tsetlin modules for arbitrary singular character of index
2. Explicit bases of derivative tableaux and the action of the generators of gl(n) are given
for these modules. Our construction leads to new families of irreducible Gelfand–Tsetlin
modules and also provides tableaux bases for some simple Verma modules.
Gelfand–Tsetlin bases are among the most remarkable discoveries of the representation
theory of classical Lie algebras. Originally introduced in , these bases provide a
convenient tableaux realization of every simple ﬁnite-dimensional representation of the
Lie algebra gl(n), as well as explicit formulas for the action of the generators of gl(n).
The explicit nature of the Gelfand–Tsetlin formulas inevitably raises the question of what
inﬁnite-dimensional modules admit tableaux bases. This question naturally initiated the
theory of Gelfand–Tsetlin modules, a theory that has attracted considerable attention in
the last 30 years and has been studied in [1,2,20–22,26], among others. Gelfand–Tsetlin
bases and modules are also related to Gelfand–Tsetlin integrable systems that were ﬁrst
introduced for the unitary Lie algebra u(n) by Guillemin and Sternberg in , and later
for the general linear Lie algebra gl(n) by Kostant and Wallach in [15,16].
We now deﬁne the main object of study in this paper. Consider a chain of embeddings
gl(1) ⊂ gl(2) ⊂ ···⊂gl(n).
The choice of embeddings is not essential but for simplicity we chose embeddings of
principal submatrices. Let U = U (gl(n)) be the universal enveloping algebra of gl(n),
and let be the Gelfand–Tsetlin subalgebra of U , i.e. the subalgebra generated by the
centers of universal enveloping algebras of all gl(i). Then is a maximal commutative