Scott Larson: Small Resolutions of Closures of K-Orbits in Flag Varieties I

14th September, 2020

The geometry of closures of K-orbits in the flag variety governs key properties in representation theory of real reductive groups. For example, Kazhdan-Lusztig-Vogan polynomials and characteristic cycles of Harish-Chandra modules are of current interest but difficult to compute. Barbasch-Evens constructed resolutions for K-orbits on grassmannian flag varieties and found some small resolutions. We do the same thing for isotropic flag varieties of the symplectic group, where K=GLn. This leads us to describe natural resolutions for K-orbits, generalizing many constructions found in the literature.

Andrei Rapinchu: On the notion of genus for division algebras and algebraic groups

2nd November, 2017

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) where D' is a central division K-algebra of degree n having the same isomorphism classes of maximal subfields as D. I will review the results on gen(D) obtained in the last several years, in particular the finiteness theorem for gen(D) when K is finitely generated of characteristic not dividing n. I will then discuss how the notion of genus can be extended to arbitrary absolutely almost simple algebraic K-groups using maximal K-tori in place of maximal subfields, and report on some recent progress in this direction. (Joint work with V. Chernousov and I. Rapinchuk)