Thomas J. Enright

A Classification of Unitary Highest Weight Modules

Let G be a simply connected, connected simple Lie group with center Z. Let K be a closed maximal subgroup of G with K/Z compact and let g be the Lie algebra of G. A unitary representation (π,H) of G such that the underlying (ℊK) — module is an irreducible quotient of a Verma module for ℊℂ is called a unitary highest weight module. Harish-Chandra ([4],[5]) has shown that G admits nontrivial unitary highest weight modules precisely when (G,K) is a Hermitian symmetric pair. In this paper we give a complete classification of the unitary highest weight modules.

Unitary derived functor modules with small spectrum

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Resolutions and Hilbert series of determinantal varieties and unitary highest weight modules

Abstract Let (G,K) be a Hermitian symmetric pair and let g and k denote the corresponding complexified Lie algebras. Let g = k ⊕ p + ⊕ p − be the usual decomposition of g as a k -module. There is a natural correspondence between K C -orbits in p + and a distinguished family of unitarizable highest weight modules for g called the Wallach representations. We denote by Yk the closure of the K C -orbit in p + that is associated to the kth Wallach representation. In this article we give explicit formulas for the numerator polynomials of the Hilbert series of the varieties Yk by using BGG resolutions of unitarizable highest weight modules. A preliminary result gives a new branching formula for a certain two-parameter family of finite dimensional representations of the even orthogonal groups. Our work is an extension of previous work by Enright and Willenbring.

Resolutions and Hilbert series of the unitary highest weight modules of the exceptional groups

We give a sufficient criterion on a highest weight module of a semisimple Lie algebra to admit a resolution in terms of sums of modules induced from a parabolic subalgebra. In particular, we show that all unitary highest weight modules admit such a resolution. As an application of our results we compute (minimal) resolutions and explicit formulas for the Hilbert series of the unitary highest weight modules of the exceptional groups.

Highest weight modules for Hermitian symmetric pairs of exceptional type

We analyze the categories of highest weight modules with a semiregular generalized infinitesimal character for the two exceptional Hermitian symmetric cases. These categories are completely described, and, as a consequence, we see that the combinatorial description of the general (regular integral) categories of highest weight modules previously given in the classical cases holds also in the exceptional cases.

Diagrams of Hermitian type, highest weight modules, and syzygies of determinantal varieties

In this mostly expository paper, a natural generalization of Young diagrams for Hermitian symmetric spaces is used to give a concrete and uniform approach to a wide variety of interconnected topics including posets of noncompact roots, canonical reduced expressions, rational smoothness of Schubert varieties, parabolic Kazhdan–Lusztig polynomials, equivalences of categories of highest weight modules, BGG resolutions of unitary highest weight modules, and finally, syzygies and Hilbert series of determinantal varieties.

Decompositions in Categories of Highest Weight Modules

Let g be a semisimple finite-dimensional subalgebra of a Lie algebra a defined over C. To analyze an a-module X we often restrict the action from a to g and decompose X as a g-module. For example, if X is U(g)-locally finite then X is completely reducible as a g-module. In this article we focus on this decomposition problem not in cases when X is U(g)-locally finite but instead when X admits a nondegenerate g-invariant bilinear form. Irreducible a-modules often admit invariant bilinear or Hermitian forms which by irreducibility are nondegenerate (or zero). This leads us to study a category 9 of g-modules X that are highest weight modules and admit nondegenerate invariant bilinear forms. Our first main result, Theorem 1.8, gives an essentially unique decomposition of X into indecomposable selfdual submodules and classifies these indecomposable self-dual modules. The choice of category studied here is further suggested by the role of the Zuckerman derived functors ri [12]. These functors are fundamental to representation theory and the theory of Harish-Chandra modules. Let A be a real semisimple Lie group with Lie algebra a,, and complex Lie algebra a Let G be a maximal compact subgroup of A with complex Lie algebra g. Let p = m@u be a parabolic subalgebra of g with reductive component m and nilradical u and set s = dim(u). The functors r’ are the right derived functors of the g-finite submodule functor defined on the category of U(m)-locally finite g-modules. Theorem 1.8, recast using derived functors, would assert that any X in 9 has an essentially unique decomposition into special indecomposable submodules A4 which have cohomology PA4 = 0 for all i except possibly i = s.

Hilbert series, Howe duality, and branching rules

Let λ be a partition, with l parts, and let Fλ be the irreducible finite dimensional representation of GL(m) associated to λ when l ≤ m. The Littlewood Restriction Rule describes how Fλ decomposes when restricted to the orthogonal group O(m) or to the symplectic group Sp(m/2) under the condition that l ≤ m/2. In this paper, this result is extended to all partitions λ. Our method combines resolutions of unitary highest weight modules by generalized Verma modules with reciprocity laws from the theory of dual pairs in classical invariant theory. Corollaries include determination of the Gelfand–Kirillov dimension of any unitary highest weight representation occurring in a dual pair setting, and the determination of their Hilbert series (as a graded module for p−). Let L be a unitary highest weight representation of sp(n, R), so*(2n), or u(p, q). When the highest weight of L plus ρ satisfies a partial dominance condition called quasi-dominance, we associate to L a reductive Lie algebra gL and a graded finite dimensional representation BL of gL. The representation BL will have a Hilbert series P(q) that is a polynomial in q with positive integer coefficients. Let δ(L) = δ be the Gelfand–Kirillov dimension of L and set cL equal to the ratio of the dimensions of the zeroth levels in the gradings of L and BL. Then the Hilbert series of L may be expressed in the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\boldsymbol{{\mathit{H}}}_{\boldsymbol{{\mathit{L}}}}(\boldsymbol{{\mathit{q}}})\boldsymbol{{\mathit{=c}}}_{\boldsymbol{{\mathit{L}}}}\boldsymbol{\hspace{.167em}}\frac{\boldsymbol{{\mathit{P}}}(\boldsymbol{{\mathit{q}}})}{(\boldsymbol{{\mathit{1-q}}})^{\boldsymbol{{\mathit{{\delta}}}}}}\boldsymbol{.}\end{equation*}\end{document} In the easiest example of the correspondence L → BL, the two components of the Weil representation of the symplectic group correspond to the two spin representations of an orthogonal group.

Representations of quantum groups defined over commutative rings

We survey some of our old results given in [CE95] and [CE10] and present some new ones in the last three sections.We survey some of our old results given in [CE95] and [CE10] and present some new ones in the last three sections.

Determination of the interwwining operators for holomorphically induced representations ofSU(p,q)

Soient X et Y deux representations holomorphiquement induites pour SU(p,q) avec les poids entiers les plus eleves. On donne des formules pour Hom(X,Y) quand le caractere infinitesimal de X et Y est entier et regulier