A combinatorial interpretation of the Bernstein degree of unitary highest weight modules

Abstract

The Bernstein degree ($\operatorname{Deg}$) is a fundamental invariant of admissible representations of a real reductive Lie group $G_{\mathbb{R}}$. Our main result concerns the classical dual pairs $(G_{\mathbb{R}}, H_{\mathbb{R}}(k))$, namely $(\operatorname{U}(p,q), \: \operatorname{U}(k))$, $(\operatorname{Mp}(2n, \mathbb{R}), \: \operatorname{O}(k))$, and $(\operatorname{O}^*(2n), \: \operatorname{Sp}(k))$, where $k$ is any positive integer. In this setting, via Howe duality, each irreducible representation $\sigma$ of $H_{\mathbb{R}}(k)$ corresponds to a unitary highest weight module $L_{\lambda(\sigma)}$ for $G_{\mathbb{R}}$. A landmark result of Nishiyama-Ochiai-Taniguchi (2001) expressed $\operatorname{Deg} L_{\lambda(\sigma)}$ as a product of two quantities: the dimension of $\sigma$ and the degree of the associated variety. However, this result was limited to a specific range of the parameter $k$ (namely $k \leq r$, the real rank of $G_{\mathbb{R}}$). The present paper resolves this limitation by introducing, for all $k$, the combinatorial interpretation $\operatorname{Deg} L_{\lambda(\sigma)} = \#( \mathcal{Q}_k(\sigma) \times \mathcal{P}_k)$, where $\mathcal{Q}_k(\sigma)$ is a certain set of semistandard tableaux and $\mathcal{P}_k$ is a set of plane partitions. (The result remains partly conjectural in the $\operatorname{Mp}(2n, \mathbb{R})$ case.) Beyond the dual pair setting, we generalize the set $\mathcal{P}_k$ to all groups $G_{\mathbb{R}}$ of Hermitian type, and we exhibit analogues of the Nishiyama-Ochiai-Taniguchi result for certain families of unitary highest weight modules of $\operatorname{E}_6$ and $\operatorname{E}_7$.