Multiple flag ind-varieties with finitely many orbits (Q2674766)

Letting \(\mathbf{G}\) to be one of the ind-groups \(\text{GL}(\infty)\), \(\text{O}(\infty)\), \(\text{Sp}(\infty)\) and letting \(\mathbf{P}_1,\ldots,\mathbf{P}_\ell\) be an arbitrary set of \(\ell\) splitting parabolic subgroups of \(\mathbf{G}\), L. Fresse and I. Penkov determine all such sets with the property that \(\mathbf{G}\) acts with finitely many orbits on the ind-variety \(\mathbf{X}_1\times\cdots\times\mathbf{X}_\ell\) where \(\mathbf{X}_i=\mathbf{G}/\mathbf{P}_i\). An analogous problem has been solved in the case of a finite-dimensional classical linear algebraic group \(G\), i.e., for \(\ell=2\), the condition that \(\mathbf{G}\) acts on \(\mathbf{X}_1\times \mathbf{X}_2\) with finitely many orbits is a restrictive condition on the pair \(\mathbf{P}_1\) and \(\mathbf{P}_2\). Using this description, the authors tackle the most interesting case where \(\ell=3\), and present the answer in the form of a table. For \(\ell\geq 4\) there are infinitely-many \(\mathbf{G}\)-orbits on \(\mathbf{X}_1\times\cdots\times\mathbf{X}_\ell\).