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

TL;DR

This work addresses the problem of computing $K$-type multiplicities in degenerate principal series by leveraging Howe duality and seesaw reciprocity to translate finite-dimensional restriction problems from $K$ to $M$ into tractable infinite-dimensional branching problems on the dual side. It proves stable formulas expressing the branching multiplicities $b^{\lambda}_{\boldsymbol{\mu}}$ as sums of generalized Littlewood–Richardson coefficients in the orthogonal, general linear, and symplectic families, with precise stable-range conditions tied to the rank parameters. In the minimal-$M$ case, it provides a entirely tableau-based interpretation via $K$-tableaux, yielding a concrete combinatorial model that unifies classical restriction results for ${\rm O}_k\to{\rm O}_{k-1}\times{\rm O}_1$, ${\rm GL}_k\to{\rm GL}_{k-1}\times{\rm GL}_1$, and ${\rm Sp}_{2k}\to{\rm Sp}_{2(k-1)}\times{\rm Sp}_2$. The framework connects to a broad array of tableau theories (orthogonal, rational GL, symplectic) and delivers explicit special-case formulas, thus providing a unified combinatorial approach to branching in degenerate principal series with potential implications for representation theory of real groups and related combinatorics.

Abstract

Let $K$ be one of the complex classical groups ${\rm O}_k$, ${\rm GL}_k$, or ${\rm Sp}_{2k}$. Let $M \subseteq K$ be the block diagonal embedding ${\rm O}_{k_1} \times \cdots \times {\rm O}_{k_r}$ or ${\rm GL}_{k_1} \times \cdots \times {\rm GL}_{k_r}$ or ${\rm Sp}_{2k_1} \times \cdots \times {\rm Sp}_{2k_r}$, 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 $G_{\mathbb{R}} = {\rm GL}(k,\mathbb{R})$, ${\rm GL}(k, \mathbb{C})$, or ${\rm GL}(k,\mathbb{H})$, 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.

Theorems & Definitions (25)

• Theorem 1.1: Stable branching rule from $K$ to $M$
• Theorem 1.2: Branching rule from $K$ to the minimal $M$
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Howe duality Howe89, KV
• Lemma 2.4: $\mathfrak{k}'$-structure in stable range
• proof
• Remark 2.5
• Lemma 2.6
• proof