{
  "id": "1410.3274",
  "version": "v1",
  "published": "2014-10-13T12:12:52.000Z",
  "updated": "2014-10-13T12:12:52.000Z",
  "title": "Crossed $S$-matrices and Character Sheaves on Unipotent Groups",
  "authors": [
    "Tanmay Deshpande"
  ],
  "comment": "29 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathtt{k}$ be an algebraic closure of a finite field $\\mathbb{F}_{q}$ of characteristic $p$. Let $G$ be an algebraic group over $\\mathtt{k}$ equipped with an $\\mathbb{F}_q$-structure given by a Frobenius map $F:G\\to G$. We will denote the corresponding algebraic group defined over $\\mathbb{F}_q$ by $G_0$. Character sheaves on $G$ are supposed to be certain objects in the triangulated braided monoidal category $\\mathscr{D}_G(G)$ of bounded conjugation equivariant $\\bar{\\mathbb{Q}}_l$-complexes (where $l\\neq p$ is a prime number) on $G$. If $C\\in \\mathscr{D}_G(G)$ is any object equipped with an isomorphism $\\psi:F^*(C){\\cong} C$ and $g\\in G$ then using Grothendieck's sheaf-function correspondence we can define the \"trace of Frobenius\" class function $t^g_{C,\\psi}:G^g_0(\\mathbb{F}_q)\\to \\bar{\\mathbb{Q}}_l$ on each pure inner form $G^g_0$ of $G_0$ corresponding to the modified Frobenius, $ad(g)\\circ F:G\\to G$. Boyarchenko has proved that if the neutral connected component $G^\\circ$ is unipotent, then the functions associated with $F$-stable character sheaves on $G$ form an orthonormal basis of the space of class functions on all pure inner forms $G^g_0(\\mathbb{F}_q)$ and that the matrix relating this basis to the basis formed by the irreducible characters of the pure inner forms $G^g_0(\\mathbb{F}_q)$ is block diagonal with \"small\" blocks. In this paper we describe these block matrices and interpret them as certain \"crossed $S$-matrices\".",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-13T12:12:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "character sheaves",
      "pure inner form",
      "unipotent groups",
      "class function",
      "algebraic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.3274D"
    }
  }
}