Date:	Tue, November 10, 2015
Time:	10:00
Place:	East Hall 8

Abstract: Let \(G\) be a complex reductive algebraic group, \(P\) be a parabolic subgroup of \(G\), \(G_0\) be a real form of \(G\). (For example, one can put \(G = \mathrm{SL}_n(\mathbb{C})\), \(G_0 = \mathrm{SL}_n(\mathbb{R})\).) The group \(G_0\) acts naturally on the flag variety \(\mathcal{Fl} = G/P\). According to J.A. Wolf's results, the number of \(G_0\)-orbits on \(\mathcal{Fl}\) is finite, hence there exists an open orbit. Furthermore, the union of all open orbits is dense in \(\mathcal{Fl}\). On the other hand, there exists a unique closed \(G_0\)-orbit on \(\mathcal{Fl}\), and the real dimension of this orbit is not less than the complex dimension of \(\mathcal{Fl}\).
Let \(G(\infty)\) be a classical ind-group of type \(A(\infty)\), \(B(\infty)\), \(C(\infty)\), or \(D(\infty)\) considered as a subgroup of automorphisms of a countable-dimensional vector space \(V\). Denote by \(E\) a basis of \(V\), by \(\mathcal{F}\) a generalized flag compatible with \(E\), and by \(\mathcal{Fl}(\infty) = \mathcal{Fl}(E, \mathcal{F})\) the ind-variety of generalized flags \(E\)-commensurable with \(\mathcal{F}\). Let \(G_0 (\infty)\) be a real form of \(G(\infty)\) (the real forms of classical ind-groups were classified by A. Baranov). In my talk, I will discuss our joint results with I. Penkov and J.A. Wolf about \(G_0 (\infty)\)-orbits on \(\mathcal{Fl}(\infty)\).