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

The Bernstein degree ($\mathrm{Deg}$) 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 $(\mathrm{U}(p,q),\mathrm{U}(k))$, $(\mathrm{Mp}(2n,\mathbb{R}),\mathrm{O}(k))$, and $({\mathrm{O}}^{\ast }(2n),\mathrm{Sp}(k))$, 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 $\mathrm{Deg}{L}_{\lambda (\sigma )}$ 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 $k\le r$, the real rank of $G_{\mathbb{R}}$). The present paper resolves this limitation by introducing, for all $k$, the combinatorial interpretation $\mathrm{Deg}{L}_{\lambda (\sigma )}=\mathrm{\#}({\mathcal{Q}}_k(\sigma )\times {\mathcal{P}}_k)$, 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 $\mathrm{Mp}(2n,\mathbb{R})$ 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 ${\mathrm{E}}_6$ and ${\mathrm{E}}_7$.