$K$-type multiplicities in degenerate principal series via Howe duality

Abstract

Let $K$ be one of the complex classical groups ${\mathrm{O}}_k$, ${\mathrm{GL}}_k$, or ${\mathrm{Sp}}_{2k}$. Let $M\subseteq K$ be the block diagonal embedding ${\mathrm{O}}_{k_1}\times \cdots \times {\mathrm{O}}_{k_r}$ or ${\mathrm{GL}}_{k_1}\times \cdots \times {\mathrm{GL}}_{k_r}$ or ${\mathrm{Sp}}_{2k_1}\times \cdots \times {\mathrm{Sp}}_{2k_r}$, respectively. By using Howe duality and seesaw reciprocity as a unified conceptual framework, we prove a formula for the branching multiplicities from $K$ to $M$ which is expressed as a sum of generalized Littlewood-Richardson coefficients, valid within a certain stable range. By viewing $K$ as the complexification of the maximal compact subgroup $K_{\mathbb{R}}$ of the real group $G_{\mathbb{R}}=\mathrm{GL}(k,\mathbb{R})$, $\mathrm{GL}(k,\mathbb{C})$, or $\mathrm{GL}(k,\mathbb{H})$, respectively, one can interpret our branching multiplicities as $K_{\mathbb{R}}$-type multiplicities in degenerate principal series representations of $G_{\mathbb{R}}$. Upon specializing to the minimal $M$, where $k_1=\cdots =k_r=1$, we establish a fully general tableau-theoretic interpretation of the branching multiplicities, corresponding to the $K_{\mathbb{R}}$-type multiplicities in the principal series.

28 pages