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.

To the top of this page

Back to the Contents