Theta correspondence and the orbit method

Abstract

The theory of theta correspondence, initiated by R. Howe, provides a powerful method of constructing irreducible admissible representations of classical groups over local fields. For archimedean local fields, a principle of great importance is the orbit method introduced by A. A. Kirillov, and it seeks to describe irreducible unitary representations of a Lie group by its coadjoint orbits. In this article, we examine implications of Howe’s theory for the orbit method and unitary representation theory, with a focus on a recent work of Barbasch, Ma, and the authors on the construction and classification of special unipotent representations of real classical groups (in the sense of Arthur and Barbasch-Vogan).