Eigenvarieties for classical groups and complex conjugations in Galois representations

TL;DR

This paper removes the irreducibility assumption in a theorem about the image of complex conjugations in Galois representations linked to automorphic forms, using eigenvarieties and recent automorphic spectrum results.

Contribution

It extends Taylor's theorem to reducible cases and to certain representations with odd characters, employing p-adic deformation on eigenvarieties for classical groups.

Findings

Eigenvarieties contain many points with (quasi-)irreducible Galois representations

Extension of Taylor's theorem to reducible and odd-character cases

Application of Arthur's automorphic spectrum results

Abstract

The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual, cuspidal automorphic representations of $\GL_{2n+1}$ over a totally real number field $F$. We also extend it to the case of representations of $\GL_{2n}/F$ whose multiplicative character is "odd". We use a $p$-adic deformation argument, more precisely we prove that on the eigenvarieties for symplectic and even orthogonal groups, there are "many" points corresponding to (quasi-)irreducible Galois representations. The recent work of James Arthur describing the automorphic spectrum for these groups is used to define these Galois representations, and also to transfer self-dual automorphic representations of the general linear group to these classical groups.