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On reducibility of induced representations of odd unitary groups: the depth zero case

Subha Sandeep Repaka

Abstract

We study a problem concerning parabolic induction in certain $p$-adic unitary groups. More precisely, for $E/F$ a quadratic extension of $p$-adic fields the associated unitary group $G=\mathrm{U}(n,n+1)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $\mathrm{GL}_n(E) \times \mathrm{U}_1(E)$. Let $π$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $ι_P^G π$ is reducible.

On reducibility of induced representations of odd unitary groups: the depth zero case

Abstract

We study a problem concerning parabolic induction in certain -adic unitary groups. More precisely, for a quadratic extension of -adic fields the associated unitary group contains a parabolic subgroup with Levi component isomorphic to . Let be an irreducible supercuspidal representation of of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation is reducible.

Paper Structure

This paper contains 28 sections, 22 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Question
  3. Preliminaries
  4. Bernstein Decomposition
  5. Types
  6. Hecke algebras
  7. Covers
  8. Depth zero supercuspidal representations
  9. Representation $\rho$ of $\mathfrak{P}$
  10. Calculation of $N_G(\mathfrak{P}_0)$
  11. Calculation of $N_G(\rho_0)$
  12. Unramified case
  13. Ramified case:
  14. Structure of $\mathcal{H}(G,\rho)$
  15. Unramified case:
  16. ...and 13 more sections

Key Result

Theorem 1

Let $G= \mathrm{U}(n,n+1)$. Let $P$ be the Siegel parabolic subgroup of $G$ and $L$ be the Siegel Levi component of $P$. Let $\pi$= $c$-$Ind_{Z(L)\mathfrak{P}_0}^L \widetilde{\rho_0}$ be a smooth irreducible supercuspidal depth zero representation of $L \cong \mathrm{GL}_n(E) \times \mathrm{U}_1(E)$

Theorems & Definitions (42)

  • Theorem 1
  • Definition 1
  • Proposition 1: Bushnell and Kutzko, Proposition 6.3 MR1643417
  • Definition 2
  • Definition 3
  • Proposition 2: Bushnell and Kutzko, Lemma 6.14 MR1643417, Proposition 7.1, MR1643417
  • Proposition 3: Bushnell and Kutzko, Theorem 7.2.i MR1643417
  • Definition 4
  • Proposition 4: Bushnell and Kutzko, Theorem 8.3 MR1643417
  • Proposition 5
  • ...and 32 more