Codimension one connectedness of the graph of associated varieties

Abstract

Let $\pi$ be an irreducible Harish-Chandra $(\mathfrak{g},K)$-module, and denote its associated variety by $AV(\pi )$. If $AV(\pi )$ 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 $\pi$, 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 ${\Gamma }_K(O_{\lambda })$ thus defined, there is an irreducible Harish-Chandra module $\pi$ whose associated graph is exactly equal to the connceted component.

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

References in corpus (1)

• Asymptotic cone of semisimple orbits for symmetric pairs