Tohoku Mathematical Journal

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 $\mathcal{AV}(\pi)$. 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.

Article information

Source
Tohoku Math. J. (2), Volume 68, Number 2 (2016), 199-239.

Dates
First available in Project Euclid: 17 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1466172770

Digital Object Identifier
doi:10.2748/tmj/1466172770

Mathematical Reviews number (MathSciNet)
MR3514699

Zentralblatt MATH identifier
1354.22017

Citation

Nishiyama, Kyo; Trapa, Peter; Wachi, Akihito. Codimension one connectedness of the graph of associated varieties. Tohoku Math. J. (2) 68 (2016), no. 2, 199--239. doi:10.2748/tmj/1466172770. https://projecteuclid.org/euclid.tmj/1466172770

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