Shintani relation for base change: unitary and elliptic representations

Abstract

Let $E/F$ be a cyclic extension of $p$-adic fields and $n$ a positive integer. Arthur and Clozel constructed a base change process $\pi \mapsto {\pi }_E$ which associates to a smooth irreducible representation of $G{L}_n(F)$ a smooth irreducible representation of $G{L}_n(E)$, invariant under $Gal(E/F)$. When $\pi$ is tempered, ${\pi }_E$ is tempered and is characterized by an identity (the Shintani character relation) relating the character of $\pi$ to the character of ${\pi }_E$ twisted by the action of $Gal(E/F)$. In this paper we show that the Shintani relation also holds when $\pi$ is unitary or elliptic. We prove similar results for the extension $C/R$. As a corollary we show that for a cyclic extension $E/F$ of number fields the base change for automorphic residual representations of the adèle group $G{L}_n(A_F)$ respects the Shintani relation at each place of $F$.

Advances in the Theory of Automorphic Forms and Their L-functions, in honor of James Cogdell's 60th birthday, Contemporary Mathematics 664, 2016, Editors D. Jiang, F. Shahidi and D. Soudry, pages 23-67