Distinguished representations and poles of twisted tensor 𝐿-functions

Abstract

Let E / F E/F be a quadratic extension of p p -adic fields. If π \pi is an admissible representation of G L n ( E ) GL_n(E) that is parabolically induced from discrete series representations, then we prove that the space of P n ( F ) P_n(F) -invariant linear functionals on π \pi has dimension one, where P n ( F ) P_n(F) is the mirabolic subgroup. As a corollary, it is deduced that if π \pi is distinguished by G L n ( F ) GL_n(F) , then the twisted tensor L L -function associated to π \pi has a pole at s = 0 s=0 . It follows that if π \pi is a discrete series representation, then at most one of the representations π \pi and π ⊗ χ \pi \otimes \chi is distinguished, where χ \chi is an extension of the local class field theory character associated to E / F 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.