Codimension one connectedness of the graph of associated varieties (Q312728)

Take a connected reductive Lie group \(G\), consider a non-trivial involution \(\theta\) and let \(K\) be a fixed point subgroup. The Lie algebra \(g\) is decomposed into a direct sum of \(1\) and \(-1\) eigenspaces which are denoted as \(l\) and \(s\) respectively.NEWLINENEWLINEDenote as \(N(s)\) the set of nilpotent elements in \(s\). Consider \(K\)-orbits in \(N(s)\). Take a graph \(\Gamma_K(O^G)\), whose vertices are \(K\)-orbits and the edges connect two vertices if and only if the corresponding \(K\)-orbits are contained in one \(G\)-orbit \(O^G\) and the intersection of their closures contains a \(K\)-orbit of codimension \(1\).NEWLINENEWLINENilpotent \(K\)-orbits occur as irreducible components of the associated varieties of Harish-Chandra modules. For an irreducible Harish-Chandra module \(X\) let \(AV(X)^{\Gamma}\) be a full subgraph of \(O^G\) with vertices occurring as irreducible components of the associated variety of \(X\). Vogan conjectured that for an irreducible \(X\) the graph \(AV(X)^{\Gamma}\) is connected.NEWLINENEWLINEIn the paper the converse statement is proved. For \(G=GL_{p+q}(\mathbb{C})\), \(K=GL_p(\mathbb{C})\times GL_q(\mathbb{C})\) the authors prove that for any connected component of \(\Gamma_K(O^G)\) there exists a Harish-Chandra module \(X\), such that \(AV(X)^{\Gamma}\) is the chosen connected component.