Shintani relation for base change: unitary and elliptic representations

• Published in: arXiv.org
• Main Authors: Badulescu, A I, Henniart, G
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.05.2016
• Subjects: Number theory; Representations
• ISSN: 2331-8422

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 adèle group \(GL_n(A_F)\) respects the Shintani relation at each place of \(F\).