Tohoku Mathematical Journal

Codimension one connectedness of the graph of associated varieties

Kyo Nishiyama, Peter Trapa, and Akihito Wachi

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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

Subjects
Primary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}
Secondary: 22E46: Semisimple Lie groups and their representations 05E10: Combinatorial aspects of representation theory [See also 20C30] 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

Keywords
Nilpotent orbit orbit graph signed Young diagram associated variety unitary representations degenerate principal series derived functor module

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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