Isotropy subgroups of transformation groups (Q2473320)

The paper studies in great detail three examples of transitive actions of Lie groups on analytical manifolds. Then the manifolds can be identified with homogeneous spaces. The detailed study of the actions and their isotropy groups is motivated by the forthcoming study of invariant differential operators on the manifold under consideration. First, the authors consider the analytical manifold \(M\) defined by \(M = \{ x \in {\mathbb C}^n \times k \colon xx^T =0\), \(\text{rank\,} x = n\}\) for \(k> 2n\) and two actions of \(G=SO(k, {\mathbb C})\) and \(G' = GL(n, {\mathbb C})\). The second manifold under consideration is \(M = \{ x \in {\mathbb C}^{n \times 2k} \colon xs_kx^T = 0, \; \text{rank\,}x =n \}\) where \(s_k =\left(\begin{smallmatrix} 0 &-I_k\\ I_k &0\end{smallmatrix}\right)\). In this case the groups are \(G = Sp(k, {\mathbb C})\) and \(G' = GL(n, {\mathbb C})\). Finally, the authors investigate \(M = \{ (\xi , x ) \in {\mathbb C}^{n \times k} \times {\mathbb C}^{k \times n} \colon \xi x =0,\;\text{rank\,}\xi = \text{rank\,}x = n\}\) and the groups \(G = GL(k, {\mathbb C })\) acting diagonally on \(M\) and \(G' = GL(k, {\mathbb C }) \times GL(k, {\mathbb C })\). The paper contains very detailed and explicit calculations which will be useful in any future applications.