{
  "id": "2203.14376",
  "version": "v1",
  "published": "2022-03-27T19:24:09.000Z",
  "updated": "2022-03-27T19:24:09.000Z",
  "title": "A Whittaker category for the Symplectic Lie algebra",
  "authors": [
    "Yang Li",
    "Jun Zhao",
    "Yuanyuan Zhang",
    "Genqiang Liu"
  ],
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "For any $n\\in \\mathbb{Z}_{\\geq 2}$, let $\\mathfrak{m}_n$ be the subalgebra of $\\mathfrak{sp}_{2n}$ spanned by all long negative root vectors $X_{-2\\epsilon_i}$, $i=1,\\dots,n$. An $\\mathfrak{sp}_{2n}$-module $M$ is called a Whittaker module with respect to the Whittaker pair $(\\mathfrak{sp}_{2n},\\mathfrak{m}_n)$ if the action of $\\mathfrak{m}_n$ on $M$ is locally finite, according to a definition of Batra and Mazorchuk. This kind of modules are more general than the classical Whittaker modules defined by Kostant. In this paper, we show that each non-singular block $\\mathcal{WH}_{\\mathbf{a}}^{\\mu}$ with finite dimensional Whittaker vector subspaces is equivalent to a module category $\\mathcal{W}^{\\mathbf{a}}$ of the even Weyl algebra $\\mathcal{D}_n^{ev}$ which is semi-simple. As a corollary, any simple module in the block $\\mathcal{WH}_{\\mathbf{i}}^{-\\frac{1}{2}\\omega_n}$ for the fundamental weight $\\omega_n$ is equivalent to the Nilsson's module $N_{\\mathbf{i}}$ up to an automorphism of $\\mathfrak{sp}_{2n}$. We also characterize all possible algebra homomorphisms from $U(\\mathfrak{sp}_{2n})$ to the Weyl algebra $\\mathcal{D}_n$ under a natural condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-27T19:24:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symplectic lie algebra",
      "whittaker category",
      "finite dimensional whittaker vector subspaces",
      "weyl algebra",
      "whittaker module"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}