A geometric realization of the asymptotic affine Hecke algebra
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910636009586688 |
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| author | Bezrukavnikov, Roman Karpov, Ivan Krylov, Vasily |
| author_facet | Bezrukavnikov, Roman Karpov, Ivan Krylov, Vasily |
| contents | A key tool for the study of an affine Hecke algebra $\mathcal{H}$ is provided by Springer theory of the Langlands dual group via the realization of $\mathcal{H}$ as equivariant $K$-theory of the Steinberg variety. We prove a similar geometric description for Lusztig's asymptotic affine Hecke algebra $J$ identifying it with the sum of equivariant $K$-groups of the squares of ${\mathbb C}^*$-fixed points in the Springer fibers, as conjectured by Qiu and Xi (the same result was also obtained by Oron Popp using different methods). As an application, we give a new geometric proof of Lusztig's parametrization of irreducible representations of $J$. We also reprove Braverman-Kazhdan's spectral description of $J$. As another application, we prove a description of the cocenters of $\mathcal{H}$ and $J$ conjectured by the first author with Braverman, Kazhdan and Varshavsky. The proof is based on a new algebraic description of $J$, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10582 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A geometric realization of the asymptotic affine Hecke algebra Bezrukavnikov, Roman Karpov, Ivan Krylov, Vasily Representation Theory Algebraic Geometry A key tool for the study of an affine Hecke algebra $\mathcal{H}$ is provided by Springer theory of the Langlands dual group via the realization of $\mathcal{H}$ as equivariant $K$-theory of the Steinberg variety. We prove a similar geometric description for Lusztig's asymptotic affine Hecke algebra $J$ identifying it with the sum of equivariant $K$-groups of the squares of ${\mathbb C}^*$-fixed points in the Springer fibers, as conjectured by Qiu and Xi (the same result was also obtained by Oron Popp using different methods). As an application, we give a new geometric proof of Lusztig's parametrization of irreducible representations of $J$. We also reprove Braverman-Kazhdan's spectral description of $J$. As another application, we prove a description of the cocenters of $\mathcal{H}$ and $J$ conjectured by the first author with Braverman, Kazhdan and Varshavsky. The proof is based on a new algebraic description of $J$, which may be of independent interest. |
| title | A geometric realization of the asymptotic affine Hecke algebra |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2312.10582 |