A combinatorial interpretation of the Bernstein degree of unitary highest weight modules
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arXiv
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| Natura: | Preprint |
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2024
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| author | Erickson, William Q. Hunziker, Markus |
| author_facet | Erickson, William Q. Hunziker, Markus |
| contents | 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 $\operatorname{E}_7$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_18766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A combinatorial interpretation of the Bernstein degree of unitary highest weight modules Erickson, William Q. Hunziker, Markus Combinatorics Representation Theory 22E46 (Primary) 05E10, 14L30, 32M15 (Secondary) 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 $\operatorname{E}_7$. |
| title | A combinatorial interpretation of the Bernstein degree of unitary highest weight modules |
| topic | Combinatorics Representation Theory 22E46 (Primary) 05E10, 14L30, 32M15 (Secondary) |
| url | https://arxiv.org/abs/2405.18766 |