On the notion of metaplectic Barbasch-Vogan duality

TL;DR

This paper introduces a duality concept for real metaplectic groups, linking nilpotent orbits, primitive ideals, and Weyl group representations, extending classical theories to the metaplectic setting.

Contribution

It defines a new duality for metaplectic groups, relating nilpotent orbits and primitive ideals, and extends special unipotent representations to this context.

Findings

Defines a duality map on nilpotent orbits for metaplectic groups

Connects the duality with primitive ideals and Weyl group double cells

Extends the notion of special unipotent representations to real metaplectic groups

Abstract

In analogy with the Barbasch-Vogan duality for real reductive linear groups, we introduce a duality notion useful for the representation theory of the real metaplectic groups. This is a map on the set of nilpotent orbits in a complex symplectic Lie algebra, whose range consists of the so-called metaplectic special nilpotent orbits. We relate this duality notion with the theory of primitive ideals and extend the notion of special unipotent representations to the real metaplectic groups. We also interpret the duality map in terms of double cells of Weyl group representations.