The asymptotic Hecke algebra and rigidity
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arXiv
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| Format: | Preprint |
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2023
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| author | Dawydiak, Stefan |
| author_facet | Dawydiak, Stefan |
| contents | We reprove the surjectivity statement of Braverman-Kazhdan's spectral description of Lusztig's asymptotic Hecke algebra $J$ in the context of $p$-adic groups. The proof is based on Bezrukavnikov-Ostrik's description of $J$ in terms of equivariant $K$-theory. As a porism, we prove that the action of $J$ extends from the non-strictly positive unramified characters to the complement of a finite union of divisors, and that the trace pairing between the Ciubotaru-He rigid cocentre of an affine Hecke algebra with equal parameters and the rigid quotient of its Grothendieck group is perfect whenever the parameter $q$ is not a root of the Poincaré polynomial of the finite Weyl group.
Without recourse to $K$-theory, we prove a weak version of Xi's description of $J$ in type $A$.
As an application of relationship between $J$ and the rigid cocentre, we prove that the formal degree of a unipotent discrete series representation of a connected reductive $p$-adic group $G$ with a split inner form has denominator dividing the Poincaré polynomial of the Weyl group of $G$. Additionally, we give formulas for $t_w$ in terms of inverse and spherical Kazhdan-Lusztig polynomials for $w$ in the lowest cell. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11092 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The asymptotic Hecke algebra and rigidity Dawydiak, Stefan Representation Theory 20C08 (primary), 22E50 (secondary), 19C08 (secondary) We reprove the surjectivity statement of Braverman-Kazhdan's spectral description of Lusztig's asymptotic Hecke algebra $J$ in the context of $p$-adic groups. The proof is based on Bezrukavnikov-Ostrik's description of $J$ in terms of equivariant $K$-theory. As a porism, we prove that the action of $J$ extends from the non-strictly positive unramified characters to the complement of a finite union of divisors, and that the trace pairing between the Ciubotaru-He rigid cocentre of an affine Hecke algebra with equal parameters and the rigid quotient of its Grothendieck group is perfect whenever the parameter $q$ is not a root of the Poincaré polynomial of the finite Weyl group. Without recourse to $K$-theory, we prove a weak version of Xi's description of $J$ in type $A$. As an application of relationship between $J$ and the rigid cocentre, we prove that the formal degree of a unipotent discrete series representation of a connected reductive $p$-adic group $G$ with a split inner form has denominator dividing the Poincaré polynomial of the Weyl group of $G$. Additionally, we give formulas for $t_w$ in terms of inverse and spherical Kazhdan-Lusztig polynomials for $w$ in the lowest cell. |
| title | The asymptotic Hecke algebra and rigidity |
| topic | Representation Theory 20C08 (primary), 22E50 (secondary), 19C08 (secondary) |
| url | https://arxiv.org/abs/2312.11092 |