Uniqueness of twisted linear periods and twisted Shalika periods

Abstract

Let $\rk$ be a local field of characteristic zero. Let $π$ be an irreducible admissible smooth representation of $\GL_{2n}(\rk)$. We prove that for all but countably many characters $χ$ of $\GL_n(\rk)\times \GL_n(\rk)$, the space of $χ$-equivariant (continuous in the archimedean case) linear functionals on $π$ is at most one dimensional. Using this, we prove the uniqueness of twisted Shalika models.