Graded Multiplicities in the Kostant-Rallis Setting
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909609323659264 |
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| author | Frohmader, Andrew |
| author_facet | Frohmader, Andrew |
| contents | This paper contains two main results. First, we provide combinatorial branching rules for $\text{GL}_n \downarrow \text{O}_n$ and $\text{GL}_{2n} \downarrow \text{Sp}_{2n}$ extending the Littlewood restriction rules. Second, we use these branching rules and the combinatorics of $\text{GL}_n$-crystals to derive a formula for the graded multiplicity of a $K$-type in the regular functions on the $K$-nilpotent cone for $\text{GL}(n, \mathbb{R})$, $\text{GL}(n, \mathbb{C})$ and $\text{GL}(n, \mathbb{H})$. Due to work of Schmid and Vilonen, these graded multiplicities determine the Hodge $K$-character of the spherical principal series with infinitesimal character 0. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11295 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Graded Multiplicities in the Kostant-Rallis Setting Frohmader, Andrew Representation Theory Combinatorics This paper contains two main results. First, we provide combinatorial branching rules for $\text{GL}_n \downarrow \text{O}_n$ and $\text{GL}_{2n} \downarrow \text{Sp}_{2n}$ extending the Littlewood restriction rules. Second, we use these branching rules and the combinatorics of $\text{GL}_n$-crystals to derive a formula for the graded multiplicity of a $K$-type in the regular functions on the $K$-nilpotent cone for $\text{GL}(n, \mathbb{R})$, $\text{GL}(n, \mathbb{C})$ and $\text{GL}(n, \mathbb{H})$. Due to work of Schmid and Vilonen, these graded multiplicities determine the Hodge $K$-character of the spherical principal series with infinitesimal character 0. |
| title | Graded Multiplicities in the Kostant-Rallis Setting |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2312.11295 |