Graded multiplicities in the Kostant-Rallis setting

Abstract

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