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

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 $σ$ of $H_{\mathbb{R}}(k)$ corresponds to a unitary highest weight module $L_{λ(σ)}$ for $G_{\mathbb{R}}$. A landmark result of Nishiyama-Ochiai-Taniguchi (2001) expressed $\operatorname{Deg} L_{λ(σ)}$ as a product of two quantities: the dimension of $σ$ 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 $\operatorname{Deg} L_{λ(σ)} = #( \mathcal{Q}_k(σ) \times \mathcal{P}_k)$, where $\mathcal{Q}_k(σ)$ 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 $\operatorname{E}_7$.

💡 Research Summary

The paper investigates the Bernstein degree Deg, an invariant measuring the asymptotic growth of admissible representations of a real reductive Lie group (G_{\mathbb R}). For unitary highest‑weight modules (L_{\lambda(\sigma)}) arising from classical Howe dual pairs ((U(p,q),U(k))), ((Mp(2n,\mathbb R),O(k))), and ((O^{*}(2n),Sp(k))), a celebrated result of Nishiyama–Ochiai–Taniguchi (2001) expressed (\operatorname{Deg} L_{\lambda(\sigma)}) as (\dim U_{\sigma}\cdot\deg\mathcal O_{k}) but only when the parameter (k) does not exceed the real rank (r) of (G_{\mathbb R}). The present work removes this restriction and provides a uniform combinatorial formula valid for every positive integer (k).

The authors introduce two families of combinatorial objects. For each irreducible representation (\sigma) of the compact group (H(k)) (the second member of the dual pair), they define (\mathcal Q_{k}(\sigma)), a set of semistandard Young tableaux whose initial column entries are constrained by (k). As (k) grows the constraints relax: when (k\le r) the tableaux model a basis of the original (U_{\sigma}); when (k\ge s) (the threshold beyond which (L_{\lambda(\sigma)}) becomes a free module over (\mathbb C