Upper Triangularity for Unipotent Representations

TL;DR

This paper investigates the structure of unipotent representations of real reductive groups, showing that a smaller subset of these representations can generate all unipotent representations through induction, advancing understanding of their classification.

Contribution

It proves that unipotent representations attached to non-induced nilpotent orbits can generate all unipotent representations via induction, revealing a new structural insight.

Findings

Smaller set of unipotent representations generates all unipotent representations.

Provides evidence for a generative process via induction.

Advances classification of unipotent representations.

Abstract

Suppose $G$ is a real reductive group. The determination of the irreducible unitary representations of $G$ is one of the major unsolved problem in representation theory. There is evidence to suggest that every irreducible unitary representation of $G$ can be constructed through a sequence of well-understood operations from a finite set of building blocks, called the unipotent representations. These representations are `attached' (in a certain mysterious sense) to the nilpotent orbits of $G$ on the dual space of its Lie algebra. Inside this finite set is a still smaller set, consisting of the unipotent representations attached to non-induced nilpotent orbits. In this paper, we prove that in many cases this smaller set generates (through a suitable kind of induction) all unipotent representations.