Jeb F. Willenbring

Stable branching rules for classical symmetric pairs

We approach the problem of obtaining branching rules from the point of view of dual reductive pairs. Specifically, we obtain a stable branching rule for each of 10 classical families of symmetric pairs. In each case, the branching multiplicities are expressed in terms of Littlewood-Richardson coefficients. Some of the formulas are classical and include, for example, Littlewoods restriction rule as a special case.

On Some q-Analogs of a Theorem of Kostant-Rallis

InthefirstpartofthispapergeneralizationsofHesselinks q-analogofKostantsmultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a q-analog of the Kostant-Rallis theorem is given for the real group SL(4, R) (that is SO(4) acting on symmetric 4× 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.

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.

An application of the Littlewood restriction formula to the Kostant-Rallis Theorem

Consider a symmetric pair (G, K) of linear algebraic groups with g ≅ l ○+ p, where l and p are defined as the +1 and -1 eigenspaces of the involution defining K. We view the ring of polynomial functions on p as a representation of K. Moreover, set P(p) = ○+∞ d=0 P d (p), where P d (p) is the space of homogeneous polynomial functions on p of degree d. This decomposition provides a graded K-module structure on P(p). A decomposition of P d (p) is provided for some classical families (G, K) when d is within a certain stable range. The stable range is defined so that the spaces P d (p) are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition of P d (p) is interpreted as a q-analog of the Kostant-Rallis theorem.

A generating function for Blattner’s formula

Let G be a connected, semisimple Lie group with finite center and let K be a maximal compact subgroup. We investigate a method to compute multiplicities of K-types in the discrete series using a rational expression for a generating function obtained from Blattners formula. This expression involves a product with a character of an irreducible finite-dimensional representation of K and is valid for any discrete series system. Other results include a new proof of a symmetry of Blattners formula, and a positivity result for certain low rank examples. We consider in detail the situation for G of type split G 2 . The motivation for this work came from an attempt to understand pictures coming from Blattners formula, some of which we include in the paper.

The Cubic, the Quartic, and the Exceptional Group G2

We study an example first addressed in a 1949 paper of J. A. Todd, in which the author obtains a complete system of generators for the covariants in the polynomial functions on the eight-dimensional space of the double binary form of degree (3,1), under the action of SL2 ×SL2. We reconsider Todd’s result by examining the complexified Cartan complement corresponding to the maximal compact subgroup of simply connected split G 2. A result of this analysis involves a connection with the branching rule from the rank two complex symplectic Lie algebra to a principally embedded \(\mathfrak{s}\mathfrak{l}_{2}\)-subalgebra. Special cases of this branching rule are related to covariants for cubic and quartic binary forms.

Invariant Differential Operators and FCR Factors of Enveloping Algebras

If 𝔤 is a semisimple Lie algebra, we describe the prime factors of 𝒰 (𝔤) that have enough finite dimensional modules. The proof depends on some combinatorial facts about the Weyl group which may be of independent interest. We also determine, which finite dimensional 𝒰 (𝔤)-modules are modules over a given prime factor. As an application we study finite dimensional modules over some rings of invariant differential operators arising from Howe duality.