Distinguished representations and poles of twisted tensor $L$-functions

Abstract:

Let $E/F$ be a quadratic extension of $p$-adic fields. If $\pi$ is an admissible representation of $GL_n(E)$ that is parabolically induced from discrete series representations, then we prove that the space of $P_n(F)$-invariant linear functionals on $\pi$ has dimension one, where $P_n(F)$ is the mirabolic subgroup. As a corollary, it is deduced that if $\pi$ is distinguished by $GL_n(F)$, then the twisted tensor $L$-function associated to $\pi$ has a pole at $s=0$. It follows that if $\pi$ is a discrete series representation, then at most one of the representations $\pi$ and $\pi \otimes \chi$ is distinguished, where $\chi$ is an extension of the local class field theory character associated to $E/F$. This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.