## Tohoku Mathematical Journal 2016 June SECOND SERIES VOL. 68, NO. 2

 Tohoku Math. J. 68 (2016), 199-239

Title CODIMENSION ONE CONNECTEDNESS OF THE GRAPH OF ASSOCIATED VARIETIES

Author Kyo Nishiyama, Peter Trapa and Akihito Wachi

(Received March 18, 2014, revised October 7, 2014)
Abstract. Let $\pi$ be an irreducible Harish-Chandra $(\mathfrak{g}, K)$-module, and denote its associated variety by $\mathcal{AV}(\pi)$. If $\mathcal{AV}(\pi)$ 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 connected for even nilpotent orbits.
Finally, for indefinite unitary group $U(p, q)$, we prove that for each connected component of the orbit graph $\Gamma_K(\mathcal{O}^G_\lambda)$ thus defined, there is an irreducible Harish-Chandra module $\pi$ whose associated graph is exactly equal to the connected component.

Mathematics Subject Classification. Primary 22E45; Secondary 22E46, 05E10, 05C50.

Key words and phrases. Nilpotent orbit, orbit graph, signed Young diagram, associated variety, unitary representations, degenerate principal series, derived functor module.