On equality of ranks of local components of automorphic representations

TL;DR

This paper proves that all local components of an automorphic representation of an adelic semisimple group share the same rank, extending previous results and providing a uniform proof applicable to all classical and exceptional groups.

Contribution

It establishes a uniform proof that local components of automorphic representations have equal rank across all semisimple groups, including exceptional cases.

Findings

All local components of an automorphic representation have equal rank.

The proof applies uniformly to classical and exceptional groups.

Extends known results about minimal representations to all local components.

Abstract

We prove that the local components of an automorphic representation of an adelic semisimple group have equal rank in the sense defined earlier by the second author. Our theorem is an analogue of the results previously obtained by Howe, Li, Dvorsky--Sahi, and Kobayashi--Savin. Unlike previous works which are based on explicit matrix realizations and existence of parabolic subgroups with abelian unipotent radicals, our proof works uniformly for all of the (classical as well as exceptional) groups under consideration. Our result is an extension of the statement known for several semisimple groups that if at least one local component of an automorphic representation is a minimal representation, then all of its local components are minimal.