Date:	Mon, September 23, 2013
Time:	11:15
Place:	East Hall 8

Abstract: Let G be a complex reductive algebraic group, T a maximal torus in G, B a Borel subgroup of G containing T, and W the Weyl group of G with respect to T. For example, if G = GL_n(ℂ), then one can put T to be the group of diagonal matrices and B to be the group of upper-triangular matrices. In this case, W = S_n, the symmetric group in n letters. Denote by F = G/B the flag variety. Let X_w be the Schubert subvariety of F corresponding to an element w ∈ W.
The Bruhat order on W plays a fundamental role in a multitude contexts. For example, it encodes incidences among Schubert varieties, i.e., X_v ⊆ X_w if and only if v ≤ w. An interesting subposet of the Bruhat order is induced by the involutions, i.e., the elements of order 2 of W. We denote this subposet by I(W). Activity around I(W) was initiated by R. Richardson and T. Springer, who proved that the inverse Bruhat order on I(S_2n+1) encodes the incidences among the closed orbits under the action of the Borel subgroup of the special linear group on the symmetric variety SL_2n+1(ℂ)/SO_2n+1(ℂ).
My goal is to involve coadjoint orbits into the picture. To each involution w ∈ I(W), one can assign the coadjoint B-orbit Ω_w. It turns out that the Bruhat order encodes incidences among the closures of such orbits, i.e., Ω_v ⊆ Ω_w if and only if v ≤ w. I will describe these connections between geometry of coadjoint orbits and combinatorial properties of I(W) in details. I will also formulate some open problems and conjectures.