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.