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

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 $G$.