Unipotent representations and the dual pair correspondence

TL;DR

This paper constructs unipotent representations for classical complex groups using the Theta correspondence, linking them to orbit philosophy and providing parametrizations for Spin groups.

Contribution

It introduces a new construction of unipotent representations via the Theta correspondence and details their K-type structure and parametrization for Spin groups.

Findings

Unipotent representations are constructed through the Theta correspondence.

K-type structures are derived from character formulas.

Parametrization for Spin groups is established.

Abstract

This paper provides a construction of the unipotent representations for classical complex groups in terms of the Theta correspondence as introduced and studied by R. Howe. The K-type structure of unipotent representations is obtained as a consequence of the character formulas for unipotent representations of D. Vogan and the author. This provides a tight link between unipotent representations and the orbit philosophy. A parametrization of unipotent representations for the Spin groups is obtained.