Title: Gamma factors of intertwining periods and distinction for inner forms of $\GL(n)$,
Abstract: 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 $G$.