NILPOTENT INVARIANTS FOR GENERIC DISCRETE SERIES OF REAL GROUPS

Abstract Let upper G left parenthesis double struck upper R right parenthesis $G(\mathbb {R})$ G ( R ) be a real reductive group. Suppose pi $\pi $ π is an irreducible representation of upper G left parenthesis double struck upper R right parenthesis $G(\mathbb {R})$ G ( R ) having a Whittaker model, and consider three invariants of pi $\pi $ π related to nilpotent 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 $\pi $ π has a Whittaker model. If pi $\pi $ π is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related matters. 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.