Gamma factors of intertwining periods and distinction for inner forms of $\mathrm{GL(n)}$

TL;DR

This paper studies the gamma factors of intertwining periods for inner forms of GL(n) over p-adic fields, providing conditions for their singularities, computing proportionality constants, and classifying distinguished representations.

Contribution

It introduces a new framework for analyzing intertwining periods and their gamma factors for inner forms of GL(n), extending previous results to more general division algebra cases.

Findings

Identified conditions under which intertwining periods have singularities.

Computed proportionality constants using Asai gamma factors.

Classified distinguished unitary and ladder representations of G.

Abstract

Let $F$ be a $p$-adic field, $E$ be a quadratic extension of $F$, $D$ be an $F$-central division algebra of odd index and let $\theta$ be the Galois involution attached to $E/F$. Set $H=GL(m,D)$, $G=GL(m,D\otimes_F E)$, and let $P=MU$ be a standard parabolic subgroup of $G$. Let $w$ be a Weyl involution stabilizing $M$ and $M^{\theta_w}$ be the subgroup of $M$ fixed by the involution $\theta_w:m\mapsto \theta(wmw)$. We denote by $X(M)^{w,-}$ the complex torus of $w$-anti-invariant unramified characters of $M$. Following the global methods of Jacquet, Lapid and Rogawski, we associate to a finite length representation $\sigma$ of $M$ and to a linear form $L\in Hom_{M^{\theta_w}}(\sigma,\mathbb{C})$ a family of $H$-invariant linear forms called intertwining periods on $Ind_P^G(\chi \sigma)$ for $\chi \in X(M)^{w,-}$, which is meromorphic in the variable $\chi$. Then we give sufficient conditions for some of these intertwining periods, namely the open intertwining periods studied by Blanc and Delorme, to have singularities. By a local/global method, we also compute in terms of Asai gamma factors the proportionality constants involved in their functional equations with respect to certain intertwining operators. As a consequence, we classify distinguished unitary and ladder representations of $G$, extending respectively results the author and Gurevich for $D=F$, which both relied at some crucial step on the theory of Bernstein-Zelevinsky derivatives. We make use of one of a recent result of Beuzart-Plessis which in the case of the group $G$ asserts that the Jacquet-Langlands correspondence preserves distinction. Such a result is for essentially square-integrable representations, but our method in fact allows us to use it only for cuspidal representations of $G$.