{
  "id": "1708.01385",
  "version": "v1",
  "published": "2017-08-04T05:53:59.000Z",
  "updated": "2017-08-04T05:53:59.000Z",
  "title": "On dimension growth of modular irreducible representations of semisimple Lie algebras",
  "authors": [
    "Roman Bezrukavnikov",
    "Ivan Losev"
  ],
  "comment": "24 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper we investigate the growth with respect to $p$ of dimensions of irreducible representations of a semisimple Lie algebra $\\mathfrak{g}$ over $\\overline{\\mathbb{F}}_p$. More precisely, it is known that for $p\\gg 0$, the irreducibles with a regular rational central character $\\lambda$ and $p$-character $\\chi$ are indexed by a certain canonical basis in the $K_0$ of the Springer fiber of $\\chi$. This basis is independent of $p$. For a basis element, the dimension of the corresponding module is a polynomial in $p$. We show that the canonical basis is compatible with the two-sided cell filtration for a parabolic subgroup in the affine Weyl group defined by $\\lambda$. We also explain how to read the degree of the dimension polynomial from a filtration component of the basis element. We use these results to establish conjectures of the second author and Ostrik on a classification of the finite dimensional irreducible representations of W-algebras, as well as a strengthening of a result by the first author with Anno and Mirkovic on real variations of stabilities for the derived category of the Springer resolution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-04T05:53:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B20",
      "17B35",
      "17B50"
    ],
    "keywords": [
      "semisimple lie algebra",
      "modular irreducible representations",
      "dimension growth",
      "regular rational central character",
      "basis element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}