The ($Γ$-asymptotic) wavefront sets: $GL_n$
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arXiv
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| Format: | Preprint |
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2024
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| author | Ciubotaru, Dan Kim, Ju-Lee |
| author_facet | Ciubotaru, Dan Kim, Ju-Lee |
| contents | Let $G$ be a connected reductive $p$-adic group. As verified for unipotent representations, it is expected that there is a close relation between the (Harish-Chandra-Howe) wavefronts sets of irreducible smooth representations and their Langlands parameters in the local Langlands correspondence via the Lusztig-Spaltenstein duality and the Aubert-Zelevinsky duality. In this paper, we define the $Γ$-asymptotic wavefront sets generalizing the notion of wavefront sets via the $Γ$-asymptotic expansions (in the sense of Kim-Murnaghan), and then study the their relation with the Langlands parameters. When $G=GL_n$, it turns out that this reduces to the corresponding relation of unipotent representations of the appropriate twisted Levi subgroups via Hecke algebra isomorphisms. For unipotent representations of $GL_n$, we also describe the Harish-Chandra-Howe (HCH) local character expansions of irreducible smooth representations using Kazhdan-Lusztig theory, and give another computation of the coefficients in the HCH expansion and the wavefront sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07581 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The ($Γ$-asymptotic) wavefront sets: $GL_n$ Ciubotaru, Dan Kim, Ju-Lee Representation Theory 22E50 Let $G$ be a connected reductive $p$-adic group. As verified for unipotent representations, it is expected that there is a close relation between the (Harish-Chandra-Howe) wavefronts sets of irreducible smooth representations and their Langlands parameters in the local Langlands correspondence via the Lusztig-Spaltenstein duality and the Aubert-Zelevinsky duality. In this paper, we define the $Γ$-asymptotic wavefront sets generalizing the notion of wavefront sets via the $Γ$-asymptotic expansions (in the sense of Kim-Murnaghan), and then study the their relation with the Langlands parameters. When $G=GL_n$, it turns out that this reduces to the corresponding relation of unipotent representations of the appropriate twisted Levi subgroups via Hecke algebra isomorphisms. For unipotent representations of $GL_n$, we also describe the Harish-Chandra-Howe (HCH) local character expansions of irreducible smooth representations using Kazhdan-Lusztig theory, and give another computation of the coefficients in the HCH expansion and the wavefront sets. |
| title | The ($Γ$-asymptotic) wavefront sets: $GL_n$ |
| topic | Representation Theory 22E50 |
| url | https://arxiv.org/abs/2408.07581 |