Upper Triangularity for 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 wellunderstood 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.