{
  "id": "2106.01929",
  "version": "v1",
  "published": "2021-06-03T15:29:00.000Z",
  "updated": "2021-06-03T15:29:00.000Z",
  "title": "Orthogonality of invariant vectors",
  "authors": [
    "U. K. Anandavardhanan",
    "Arindam Jana"
  ],
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Let $G$ be a finite group with given subgroups $H$ and $K$. Let $\\pi$ be an irreducible complex representation of $G$ such that its space of $H$-invariant vectors as well as the space of $K$-invariant vectors are both one dimensional. Let $v_H$ (resp. $v_K$) denote an $H$-invariant (resp. $K$-invariant) vector of unit norm in the standard $G$-invariant inner product $\\langle ~,~ \\rangle_\\pi$ on $\\pi$. Our interest is in computing the square of the absolute value of $\\langle v_H,v_K \\rangle_\\pi$. This is the correlation constant $c(\\pi;H,K)$ defined by Gross. In this paper, we give a sufficient condition for $\\langle v_H, v_K \\rangle_\\pi$ to be zero and a sufficient condition for it to be non-zero (i.e., $H$ and $K$ are correlated with respect to $\\pi$), when $G={\\rm GL}_2(\\mathbb F_q)$, where $\\mathbb F_q$ is the finite field of $q=p^f$ elements of odd characteristic $p$, $H$ is its split torus and $K$ is a non-split torus. The key idea in our proof is to analyse the mod $p$ reduction of $\\pi$. We give an explicit formula for $|\\langle v_H,v_K \\rangle_\\pi|^2$ modulo $p$. Finally, we study the behaviour of $\\langle v_H,v_K \\rangle_\\pi$ under the Shintani base change and give a sufficient condition for $\\langle v_H,v_K \\rangle_\\pi$ to vanish for an irreducible representation $\\pi={\\rm BC}(\\tau)$ of ${\\rm PGL}_2(\\mathbb E)$, in terms of the epsilon factor of the base changing representation $\\tau$ of ${\\rm PGL}_2(\\mathbb F)$, where $\\mathbb E/\\mathbb F$ is a finite extension of finite fields. This is reminiscent of the vanishing of $L(1/2, {\\rm BC}(\\tau))$, in the theory of automorphic forms, when the global root number of $\\tau$ is $-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-03T15:29:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C15",
      "20C20",
      "20C33"
    ],
    "keywords": [
      "invariant vectors",
      "sufficient condition",
      "finite field",
      "orthogonality",
      "invariant inner product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}