Codimension one connectedness of the graph of associated varieties

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

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