{
  "id": "2312.10582",
  "version": "v1",
  "published": "2023-12-17T02:02:31.000Z",
  "updated": "2023-12-17T02:02:31.000Z",
  "title": "A geometric realization of the asymptotic affine Hecke algebra",
  "authors": [
    "Roman Bezrukavnikov",
    "Ivan Karpov",
    "Vasily Krylov"
  ],
  "comment": "32 pages",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "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. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-17T02:02:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "geometric realization",
      "lusztigs asymptotic affine hecke algebra",
      "reprove braverman-kazhdans spectral description",
      "langlands dual group",
      "similar geometric description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}