# Twiseted eigenvarities and self-dual representations

## TL;DR

This paper constructs eigenvarieties for reductive groups with automorphisms, showing that self-dual cuspidal Hecke eigensystems for Gln can be deformed within p-adic families containing many classical points.

## Contribution

It introduces a new eigenvariety construction for groups with automorphisms and demonstrates deformation properties of self-dual cuspidal Hecke eigensystems.

## Key findings

- Eigenvarieties parameterize utomorphic representations with automorphisms.
- Self-dual cuspidal Hecke eigensystems can be deformed in p-adic families.
- Dense classical points exist within these p-adic families.

## Abstract

For a reductive group G and a finite order Cartan-type automorphism \iota of G, we construct an eigenvariety parameterizing \iota-invariant cuspidal Hecke eigensystems of G. In particular, for G = Gln, we prove, any self-dual cuspidal Hecke eigensystem can be deformed in a p-adic family of self-dual cuspidal Hecke eigensystems containing a Zariski dense subset of classical points.