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Distinguished Representations for $\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field

Kwangho Choiy, Shiv Prakash Patel

TL;DR

This work extends the Prasad–Takloo-Bighash framework to non-split inner forms by studying SL_n(D) representations distinguished by SL_n^*(E) with D a quaternion division algebra and E/F quadratic. It establishes a precise multiplicity formula dim Hom_{SL_n^*(E)}(π_flat,1) in terms of Langlands-parameter data and intermediate-restriction data, specifically (|X_π|/|Z_π/Y_π|)·(dim ρ_{π_flat})/C_π^2, where C_π∈{1,2} encodes a restriction multiplicity correction. The paper develops the machinery via GL_n(D)^+-distinguished lifts, L-packets for SL_n(D), and the centralizer representations ρ_{π_flat}, and provides explicit GL_2(D) applications with detailed case-by-case dimensions. It also discusses the split counterpart and highlights how multiplicity-one phenomena arise in the split setting, thereby connecting the non-split and split theories and laying groundwork for further non-quasi-split cases.

Abstract

Let $D$ be a quaternion division algebra over a non-archimedean local field $F$ of characteristic zero. Let $E/F$ be a quadratic extension and $\rm{SL}_{n}^{*}(E) = {\rm{GL}}_{n}(E) \cap \rm{SL}_{n}(D)$. We study distinguished representations of $\rm{SL}_{n}(D)$ by the subgroup $\rm{SL}_{n}^{*}(E)$. Let $π$ be an irreducible admissible representation of $\rm{SL}_{n}(D)$ which is distinguished by $\rm{SL}_{n}^{*}(E)$. We give a multiplicity formula, i.e. a formula for the dimension of the $\mathbb{C}$-vector space ${\rm{Hom}}_{\rm{SL}_{n}^{*}(E)} (π, \mathbbm{1})$, where $\mathbbm{1}$ denotes the trivial representation of $\rm{SL}_{n}^{*}(E)$. This work is a non-split inner form analog of a work by Anandavardhanan-Prasad which gives a multiplicity formula for $\rm{SL}_{n}(F)$-distinguished irreducible admissible representation of $\rm{SL}_{n}(E)$.

Distinguished Representations for $\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field

TL;DR

This work extends the Prasad–Takloo-Bighash framework to non-split inner forms by studying SL_n(D) representations distinguished by SL_n^*(E) with D a quaternion division algebra and E/F quadratic. It establishes a precise multiplicity formula dim Hom_{SL_n^*(E)}(π_flat,1) in terms of Langlands-parameter data and intermediate-restriction data, specifically (|X_π|/|Z_π/Y_π|)·(dim ρ_{π_flat})/C_π^2, where C_π∈{1,2} encodes a restriction multiplicity correction. The paper develops the machinery via GL_n(D)^+-distinguished lifts, L-packets for SL_n(D), and the centralizer representations ρ_{π_flat}, and provides explicit GL_2(D) applications with detailed case-by-case dimensions. It also discusses the split counterpart and highlights how multiplicity-one phenomena arise in the split setting, thereby connecting the non-split and split theories and laying groundwork for further non-quasi-split cases.

Abstract

Let be a quaternion division algebra over a non-archimedean local field of characteristic zero. Let be a quadratic extension and . We study distinguished representations of by the subgroup . Let be an irreducible admissible representation of which is distinguished by . We give a multiplicity formula, i.e. a formula for the dimension of the -vector space , where denotes the trivial representation of . This work is a non-split inner form analog of a work by Anandavardhanan-Prasad which gives a multiplicity formula for -distinguished irreducible admissible representation of .
Paper Structure (15 sections, 12 theorems, 101 equations)

This paper contains 15 sections, 12 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Restriction
  5. General set-up and some known cases
  6. General setting
  7. Some previous works
  8. Our case in question
  9. Main results
  10. An application to ${\mathop{\mathrm{GL}}\limits}_2(D)$
  11. Mackey theory
  12. Case A: $\chi_1\chi_2^{-1} = \omega_{E/F}$
  13. Case B: $(\chi_1\chi_2^{-1})^2=\mathbbm{1}$ but $\chi_1\chi_2^{-1} \notin \{ \omega_{E/F}, \mathbbm{1} \}$
  14. Case C: Either $\chi_{1} \chi_2^{-1} = \mathbbm{1}$ or $(\chi_1 \chi_2^{-1})^2 \neq \mathbbm{1}$
  15. The split counterpart of our main theorem

Key Result

Theorem 1.1

Let $\pi_{\flat}$ be an irreducible admissible representation of $\mathop{\mathrm{SL}}\limits_{n}(D)$ which is $\mathop{\mathrm{SL}}\limits_{n}^{*}(E)$-distinguished. Let $\pi$ be an irreducible admissible representation of ${\mathop{\mathrm{GL}}\limits}_{n}(D)$ which is ${\mathop{\mathrm{GL}}\limit where $X_{\pi}, Y_{\pi}, Z_{\pi}$ are given in Definition def XYZ.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Theorem 5.1
  • Proposition 5.2
  • proof
  • Lemma 5.3
  • proof
  • ...and 15 more