Gelfand–Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $\boldsymbol{{SU}(p,q)}$

Abstract

Let (G, K) be an irreducible Hermitian symmetric pair of noncompact type with G = SU (p, q), and let λ be an integral weight such that the simple highest weight module L(λ) is a Harish-Chandra (g, K)-module. We give a combinatorial algorithm for the Gelfand-Kirillov dimension of L(λ). This enables us to prove that the Gelfand-Kirillov dimension of L(λ) decreases as the integer λ + ρ, β ∨ increases, where ρ is the half sum of positive roots and β is the maximal noncompact root. Finally by the combinatorial algorithm, we obtain a description of the associated variety of L(λ).

A classification of Kazhdan-Lusztig right cells containing smooth elements

“…The Young tableaux with only two columns always appear in representation theory of Lie algebras and Lie groups, see for example [2]. Now we pay attention on KL right cells corresponding to these special tableaux.…”

Gelfand-Kirillov dimensions and associated varieties of highest weight modules

“…In [17, §1], Lusztig gives a formula connecting the Gelfand-Kirillov dimension and the Lusztig's a-function. The following proposition is a generalized form of Lusztig's formula, see [2,Prop.3.8].…”

“…By Schensted insertion, the Young tableau Y a (x) associated to x has at most two columns. By Proposition 5.3 and Example 5.4 of [2], there is an algorithm of determining entries in the second column of Y a (x). Let us recall it.…”

“…This motivates us to investigate GK dimensions and associated varieties of arbitrary highest weight HC modules. This aim was achieved in [2] for the Lie group SU (p, q) by applying a formula by Lusztig [17] connecting GK dimensions and a-functions and an algorithm for the a-function of type A, see also [11]. This paper will give an answer for other Hermitian symmetric pairs of classical types.…”

A characterization of some highest weight modules for type A with irreducible associated varieties

“…Sometime we will write it as O m (p, q) to emphasize that it comes from SU (p, q). From Bai-Xie [2] or NOTYK [15], we know dim O j = j(n − j).…”