Nilpotent Invariants for Generic Discrete Series of Real Groups

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.