Shintani relation for base change: unitary and elliptic representations

TL;DR

This paper extends the Shintani relation, originally known for tempered representations, to unitary and elliptic representations in the context of base change for $GL_n$ over cyclic $p$-adic and number field extensions, ensuring character identities hold.

Contribution

It proves that the Shintani relation applies to unitary and elliptic representations, broadening the scope of base change character identities beyond tempered cases.

Findings

Shintani relation holds for unitary representations

Shintani relation holds for elliptic representations

Base change respects Shintani relation at all places for number fields

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 $GL_n(F)$ a smooth irreducible representation of $GL_n(E)$, 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 ad\`ele group $GL_n(A_F)$ respects the Shintani relation at each place of $F$.