Distinguished representations for $\rm{SL}(n,F)$

TL;DR

The paper develops a multiplicity formula for $H_{\flat}$-distinguished representations of $G_{\flat}=\mathrm{SL}_{2n}(F)$ over a finite field, in terms of data lifted from representations of $G=\mathrm{GL}_{2n}(F)$. It introduces the intermediate group $H^{+}=G_{\flat}H$ and leverages Prasad’s cuspidal multiplicity-one results, along with restriction theory for finite groups, to connect $\dim\mathrm{Hom}_{H_{\flat}}(\pi_{\flat},1)$ to $X_{\pi}$, $Z_{\pi}$ and associated multiplicities $m_{\alpha}$. The main result expresses the dimension as $\frac{\sum_{\alpha\in X_{\pi}} m_{\alpha}}{|Z_{\pi}|}$, with a simpler $|X_{\pi}|/|Z_{\pi}|$ formula in the cuspidal case, providing a practical tool for determining distinguished representations in this finite-field setting. The paper also analyzes explicit GL$_4(F)$ principal-series cases to illustrate multiplicity-one behavior and to identify instances where multiplicity-one may fail outside cuspidal scenarios.

Abstract

Let $F$ be a finite field, and let $\mathbb{E}$ be either a quadratic field extension $E/F$ or the split algebra $F \oplus F$. We study distinguished representations of $\rm{SL}_{2n}(F)$ by the subgroup $H_{\flat} := \rm{SL}_{2n}(F) \cap \rm{GL}_{n}(\mathbb{E})$, which is a variation of the work of Anandavardhanan and Prasad on distinguished representations of $\rm{SL}_{n}(\mathbb{E})$ by the subgroup $\rm{SL}_n(F)$. This is in a similar framework of our earlier work of a $p$-adic non-split variation of Anandavardhanan-Prasad over finite fields. We give a formula for the dimension of the complex vector space $\rm{Hom}_{H_{\flat}}(π_{\flat}, 1)$ in terms of certain characters of $F^{\times}$, where $π_{\flat}$ is an irreducible representation which is also distinguished by $H_{\flat}$.

Theorems & Definitions (21)

• Theorem 1.1
• Conjecture 1.2
• Proposition 2.1: Lehrer
• Theorem 2.2: Henderson
• Theorem 2.3: Prasad
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof