{
  "id": "1511.08157",
  "version": "v1",
  "published": "2015-11-25T19:05:08.000Z",
  "updated": "2015-11-25T19:05:08.000Z",
  "title": "The Lerch zeta function and the Heisenberg group",
  "authors": [
    "Jeffrey C. Lagarias"
  ],
  "comment": "55 pages, preliminary version",
  "categories": [
    "math.NT"
  ],
  "abstract": "This paper gives a representation-theoretic interpretation of the Lerch zeta function and related Lerch $L$-functions twisted by Dirichlet characters. These functions are associated to a four-dimensional solvable real Lie group $H^{J}$, called here the sub-Jacobi group, which is a semi-direct product of $GL(1, {\\mathbb R})$ with the Heisenberg group $H({\\mathbb R})$. The Heisenberg group action on L^2-functions on the Heisenberg nilmanifold $H({\\mathbb Z}) \\backslash H({\\mathbb R})$ decomposes as $\\bigoplus_{N \\in {\\mathbb Z}} H_N$, where each space $H_N~ (N \\neq 0)$ consists of $|N|$ copies of an irreducible representation of $H({\\mathbb R})$ with central character $e^{2 \\pi i Nz}$. The paper shows that show one can further decompose $H_N (N \\ne 0)$ into irreducible $H({\\mathbb R})$-modules $H_{N,d}(\\chi)$ indexed by Dirichlet characters $(\\bmod~ d)$ for $d \\mid N$, each of which carries an irreducible $H^J$-action. On each $H_{N,d}(\\chi)$ there is an action of certain two-variable Hecke operators $\\{T_m: m \\ge 1\\}$; these Hecke operators have a natural global definition on all of $L^2(H({\\mathbb Z})\\backslash H({\\mathbb R}))$, including the space of one-dimensional representations $H_0$, which does not carry an $H^J$-action. For $H_{N,d}(\\chi)$ with $N \\neq 0$ suitable Lerch $L$-functions on the critical line $\\frac{1}{2} + it$ form a complete family of generalized eigenfunctions (purely continuous spectrum) for a certain linear partial differential operator $\\Delta_L$. These Lerch $L$-functions are also simultaneous eigenfunctions for all two-variable Hecke operators $T_m$ and their adjoints $T_m^{\\ast}$, provided $(m, N/d) = 1$. Lerch $L$-functions are characterized by this Hecke eigenfunction property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-25T19:05:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M35",
      "11F55",
      "22E40"
    ],
    "keywords": [
      "lerch zeta function",
      "heisenberg group",
      "two-variable hecke operators",
      "four-dimensional solvable real lie group",
      "linear partial differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}