Nilpotent Invariants for Generic Discrete Series of Real Groups

TL;DR

This paper explores the relationships between nilpotent invariants of discrete series representations of real reductive groups, providing simplified proofs and methods to reconstruct invariants from each other.

Contribution

It offers a self-contained account with elementary proofs of how nilpotent invariants determine each other for generic discrete series, including bijections and reconstruction methods.

Findings

Invariants determine each other for generic discrete series.

Bijections exist between discrete series, Whittaker data, and nilpotent orbits.

Associated variety and wave-front set are related by Kostant-Sekiguchi correspondence.

Abstract

Let $G(\mathbb{R})$ be a real reductive group. Suppose $\pi$ is an irreducible representation of $G(\mathbb{R})$ having a Whittaker model, and consider three invariants of $\pi$ related to nilpotents elements of the Lie algebra of $G$ (or its dual): the associated variety, the wave-front set, and the set of Whittaker data for which $\pi$ has a Whittaker model. If $\pi$ is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related results, including an elementary proof that passage from $\pi$ to the three invariants defines natural bijections between the generic discrete series in an $L$-packet, the possible Whittaker data for $G(\mathbb{R})$, and the appropriate sets of nilpotent orbits. Given one of the three invariants, we also explain how to reconstruct the other two. Many of the results were known: we give simplified proofs for several of them, for instance a simple proof (for generic discrete series) that the associated variety and the wave-front set are related by the Kostant-Sekiguchi correspondence.