Residual automorphic forms and spherical unitary representations of exceptional groups

TL;DR

This paper extends the verification of Arthur's conjecture on the unitarity of spherical automorphic representations to all remaining exceptional groups, confirming a deep connection between Langlands parameters and unitarizability.

Contribution

It proves Arthur's conjecture for the spherical constituents of unramified principal series of all remaining exceptional groups E_6, E_7, E_8, and F_4.

Findings

Confirmed unitarity of spherical representations for E_6, E_7, E_8, and F_4

Extended previous results from classical and G_2 groups to all exceptional groups

Validated the link between Langlands parameters and unitarizability in these cases

Abstract

Arthur has conjectured that the unitarity of a number of representations can be shown by finding appropriate automorphic realizations. This has been verified for classical groups by Moeglin and for the exceptional Chevalley group G_2 by Kim. In this paper we extend their results on spherical representations to the remaining exceptional groups E_6, E_7, E_8, and F_4. In particular we prove Arthur's conjecture that the spherical constituent of an unramified principal series of a Chevalley group over any local field of characteristic zero is unitarizable if its Langlands parameter coincides with half the marking of a coadjoint nilpotent orbit of the Langlands dual Lie algebra.