Amenability and representation theory of pro-Lie groups

TL;DR

This paper introduces a semigroup-based method for studying the representation theory of pro-Lie groups under amenability conditions, establishing a correspondence between irreducible representations and coadjoint orbits for certain classes of groups.

Contribution

It develops a novel semigroup approach to representation theory for pro-Lie groups and links irreducible representations with coadjoint orbits for groups including nilpotent and infinite products.

Findings

Established a one-to-one correspondence between irreducible representations and coadjoint orbits.

Extended the orbit method to classes of pro-Lie groups including infinite products.

Provided a framework where traditional $C^*$-algebraic methods fail.

Abstract

We develop a semigroup approach to representation theory for pro-Lie groups satisfying suitable amenability conditions. As an application of our approach, we establish a one-to-one correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits for a class of pro-Lie groups including all connected locally compact nilpotent groups and arbitrary infinite direct products of nilpotent Lie groups. The usual $C^*$-algebraic approach to group representation theory positivey breaks down for infinite direct products of non-compact locally compact groups, hence the description of their unitary duals in terms of coadjoint orbits is particularly important whenever it is available, being the only description known so far.