Parameters of Hecke algebras for Bernstein components of p-adic groups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866910787222634496 |
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| author | Solleveld, Maarten |
| author_facet | Solleveld, Maarten |
| contents | Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)^s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra H(O,G) whose category of right modules is closely related to Rep(G)^s. In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations.
In this paper we study the q-parameters of the affine Hecke algebras H(O,G). We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups.
Lusztig conjectured that the q-parameters are always integral powers of q_F and that they coincide with the q-parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of simple p-adic groups, and we prove it for most of those. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_13113 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Parameters of Hecke algebras for Bernstein components of p-adic groups Solleveld, Maarten Representation Theory 22E50, 20G25, 20C08 Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)^s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra H(O,G) whose category of right modules is closely related to Rep(G)^s. In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations. In this paper we study the q-parameters of the affine Hecke algebras H(O,G). We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups. Lusztig conjectured that the q-parameters are always integral powers of q_F and that they coincide with the q-parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of simple p-adic groups, and we prove it for most of those. |
| title | Parameters of Hecke algebras for Bernstein components of p-adic groups |
| topic | Representation Theory 22E50, 20G25, 20C08 |
| url | https://arxiv.org/abs/2103.13113 |