Gamma factors of intertwining periods and distinction for inner forms of $\GL(n)$
Let $F$ be a $p$-adic fied, $E$ be a quadratic extension of $F$, and $D$ be an $F$-division algebra of odd index. Set $H=\mathrm{GL}m,D)$ and $G=\mathrm{GL}(m,D\otimes_F E)$, we carry out a fine study of local intertwining open periods attached to $H$-distinguished induced representations of inner forms of $G$. These objects have been studied globally in \cite{JLR} and \cite{LR}, and locally in \cite{BD08}. Here we give sufficient conditions for the local intertwining periods 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 the results of \cite{M14} and \cite{G15} for $D=F$, which both relied at some crucial step on the theory of Bernstein-Zelevinsky derivatives. We make use of one of the main results of \cite{BP17} in our setting, which in the case of the group $G$, asserts that the Jacquet-Langlands correspondence preserves distinction. Such a result is for discrete series representations, but our method in fact allows us to use it only for cuspidal representations of $G$.