Graded Multiplicities in the Kostant-Rallis Setting

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.