Graded Multiplicities in the Kostant-Rallis Setting

TL;DR

This paper develops combinatorial rules for branching in classical groups and derives formulas for graded multiplicities of K-types, linking representation theory, crystal combinatorics, and Hodge characters.

Contribution

It extends Littlewood restriction rules to new settings and connects these to graded multiplicities and Hodge characters in a unified framework.

Findings

Provides new combinatorial branching rules for GL to O and Sp groups.

Derives a formula for graded multiplicities of K-types in regular functions.

Links graded multiplicities to Hodge K-characters of spherical principal series.

Abstract

This paper contains two main results. First, we provide combinatorial branching rules for $\text{GL}_n \downarrow \text{O}_n$ and $\text{GL}_{2n} \downarrow \text{Sp}_{2n}$ extending the Littlewood restriction rules. Second, we use these branching rules and the combinatorics of $\text{GL}_n$-crystals to derive a formula for the graded multiplicity of a $K$-type in the regular functions on the $K$-nilpotent cone for $\text{GL}(n, \mathbb{R})$, $\text{GL}(n, \mathbb{C})$ and $\text{GL}(n, \mathbb{H})$. Due to work of Schmid and Vilonen, these graded multiplicities determine the Hodge $K$-character of the spherical principal series with infinitesimal character 0.