Tag - Algebraic groups

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

21st 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. We recall how small resolutions have been used to compute these invariants, describe fibres of the resolutions from last week, and describe more small resolutions for the real reductive groups Sp2n(ℝ), U(p,q), and complex groups. Along the way we consider an application to Schubert varieties.

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)