{
  "id": "2103.10009",
  "version": "v1",
  "published": "2021-03-18T04:03:06.000Z",
  "updated": "2021-03-18T04:03:06.000Z",
  "title": "Categorification of DAHA and Macdonald polynomials",
  "authors": [
    "Syu Kato",
    "Anton Khoroshkin",
    "Ievgen Makedonskyi"
  ],
  "comment": "37 pages, comments are very welcome",
  "categories": [
    "math.RT",
    "math.AG",
    "math.QA"
  ],
  "abstract": "We describe a categorification of the Double Affine Hecke Algebra ${\\mathcal{H}\\kern -.4em\\mathcal{H}}$ associated with an affine Lie algebra $\\widehat{\\mathfrak{g}}$, a categorification of the polynomial representation and a categorification of Macdonald polynomials. All categorification results are given in the derived setting. That is, we consider the derived category associated with graded modules over the Lie superalgera ${\\mathfrak I}[\\xi]$, where ${\\mathfrak I}\\subset\\widehat{\\mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie algebra and $\\xi$ is a formal odd variable. The Euler characteristic of graded characters of a complex of ${\\mathfrak I}[\\xi]$-modules is considered as an element of a polynomial representation. First, we show that the compositions of induction and restriction functors associated with minimal parabolic subalgebras ${\\mathfrak{p}}_{i}$ categorify Demazure operators $T_i+1\\in{\\mathcal{H}\\kern -.4em\\mathcal{H}}$, meaning that all algebraic relations of $T_i$ have categorical meanings. Second, we describe a natural collection of complexes ${\\mathbb{EM}}_{\\lambda}$ of ${\\mathfrak I}[\\xi]$-modules whose Euler characteristic is equal to nonsymmetric Macdonald polynomials $E_\\lambda$ for dominant $\\lambda$ and a natural collection of complexes of $\\mathfrak{g}[z,\\xi]$-modules ${\\mathbb{PM}}_{\\lambda}$ whose Euler characteristic is equal to the symmetric Macdonald polynomial $P_{\\lambda}$. We illustrate our theory with the example $\\mathfrak{g}=\\mathfrak{sl}_2$ where we construct the cyclic representations of Lie superalgebra ${\\mathfrak I}[\\xi]$ such that their supercharacters coincide with renormalizations of nonsymmetric Macdonald polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-18T04:03:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "categorification",
      "euler characteristic",
      "nonsymmetric macdonald polynomials",
      "affine lie algebra",
      "polynomial representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}