A combinatorial interpretation of the Bernstein degree of unitary highest weight modules
William Q. Erickson, Markus Hunziker
TL;DR
The paper provides a uniform combinatorial interpretation of the Bernstein degree for unitary highest weight modules in the Howe duality setting, expressing $\operatorname{Deg} L_{\lambda(\sigma)}$ as $|\mathcal{Q}_k(\sigma)\times\mathcal{P}_k|$ and decomposing multiplicities via $m_k(\sigma)=|\mathcal{Q}_k(\sigma)|$, with $|\mathcal{P}_k|$ encoding the degree of the associated orbit closure. It introduces semistandard tableaux constraints $\mathcal{Q}_k(\sigma)$ and plane partitions within diagrams $D_k$ to model multiplicities and Hilbert-series data, and establishes a pivotal bijection $\mathrm{id}\times\Theta$ between $\mathcal{Q}_k(\sigma)\times\mathcal{P}_k$ and maximal jellyfish indexing the Stanley decomposition of $L_{\lambda(\sigma)}$. The work connects the Bernstein degree to counts of maximal nonintersecting lattice-path families and plane partitions, unifying the interpretation across $G_R\in\{U(p,q),\mathrm{Mp}(2n),O^*(2n)\}$ and generalizing to all Hermitian types via generalized diagrams $D_k$, with extensions to the exceptional groups as future goals. It also clarifies the role of the metaplectic case, where certain $k$-ranges remain conjectural, and lays groundwork for a uniform combinatorial framework beyond dual-pair settings. Overall, the paper advances a concrete, diagrammatic bridge between representation-theoretic invariants and enumerative combinatorics in Hermitian symmetric contexts.
Abstract
Consider the $(\mathfrak{g}, K)$-modules $L_λ$ for unitary highest weight representations of the real reductive group $G_{\mathbb{R}} = \operatorname{U}(p,q)$, $\operatorname{Mp}(2n, \mathbb{R})$, or $\operatorname{O}^*(2n)$, where $\operatorname{Mp}(2n,\mathbb{R})$ denotes the metaplectic double cover of $\operatorname{Sp}(2n,\mathbb{R})$. Let $k$ be a positive integer. Corresponding to $G_{\mathbb{R}}$ via Howe duality is the compact group $\operatorname{U}(k)$, $\operatorname{O}(k)$, or $\operatorname{Sp}(k)$, respectively, for which every irreducible representation $σ$ corresponds to a unique $L_λ = L_{λ(σ)}$. Nishiyama-Ochiai-Taniguchi (2001) expressed the Bernstein degree $\operatorname{Deg} L_{λ(σ)}$ as the product of $\dim σ$ and the degree of the associated variety of $L_{λ(σ)}$; this result is valid when $k \leq r :=$ the real rank of $G_{\mathbb{R}}$. In this paper, for arbitrary $k$, we give a new 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, whose cardinality (for $k \geq r$) interpolates between $\dim σ$ and the dimension of the simple $K$-module with highest weight $λ(σ)$. The set $\mathcal{P}_k$ consists of certain plane partitions that encode the Hilbert series of the associated variety. We exhibit analogous sets $\mathcal{P}_k$ of plane partitions for all real reductive groups of Hermitian type, including the exceptional groups.