The local character expansion as branching rules: nilpotent cones and the case of $\mathrm{SL}(2)$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929414547177472 |
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| author | Nevins, Monica |
| author_facet | Nevins, Monica |
| contents | We show there exist representations of each maximal compact subgroup $K$ of the $p$-adic group $G=\mathrm{SL}(2,F)$, attached to each nilpotent coadjoint orbit, such that every irreducible representation of $G$, upon restriction to a suitable subgroup of $K$, is a sum of these five representations in the Grothendieck group. This is a representation-theoretic analogue of the analytic local character expansion due to Harish-Chandra and Howe. Moreover, we show for general connected reductive groups that the wave front set of many irreducible positive-depth representations of $G$ are completely determined by the nilpotent support of their unrefined minimal $K$-types. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_17213 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The local character expansion as branching rules: nilpotent cones and the case of $\mathrm{SL}(2)$ Nevins, Monica Representation Theory 22E50 We show there exist representations of each maximal compact subgroup $K$ of the $p$-adic group $G=\mathrm{SL}(2,F)$, attached to each nilpotent coadjoint orbit, such that every irreducible representation of $G$, upon restriction to a suitable subgroup of $K$, is a sum of these five representations in the Grothendieck group. This is a representation-theoretic analogue of the analytic local character expansion due to Harish-Chandra and Howe. Moreover, we show for general connected reductive groups that the wave front set of many irreducible positive-depth representations of $G$ are completely determined by the nilpotent support of their unrefined minimal $K$-types. |
| title | The local character expansion as branching rules: nilpotent cones and the case of $\mathrm{SL}(2)$ |
| topic | Representation Theory 22E50 |
| url | https://arxiv.org/abs/2309.17213 |