Codimension one connectedness of the graph of associated varieties

Abstract

Let $ π$ be an irreducible Harish-Chandra $ (\mathfrak{g}, K) $-module, and denote its associated variety by $ AV(π) $. If $ AV(π) $ is reducible, then each irreducible component must contain codimension one boundary component. Thus we are interested in the codimension one adjacency of nilpotent orbits for a symmetric pair $ (G, K) $. We define the notion of orbit graph and associated graph for $ π$, and study its structure for classical symmetric pairs; number of vertices, edges, connected components, etc. As a result, we prove that the orbit graph is connecetd for even nilpotent orbits. Finally, for indefinite unitary group $ U(p, q) $, we prove that for each connected component of the orbit graph $ Γ_K(O_λ) $ thus defined, there is an irreducible Harish-Chandra module $ π$ whose associated graph is exactly equal to the connceted component.

47 pages. Appendix is added. to appear in Tohoku Math. J