Kalender Department Mathematik

Fri, 12.01.2018, 14:30

Fri, 12.01.2018, 15:30
The Bruhat order on hermitian symmetric varieties and on abelian unipotent radicals

Referent: Jacopo Gandini
Veranstalter: Knop
Raum: SR 04.363

Let G be a semisimple algebraic group over an algebraically closed field of characteristic different from 2, let P be a parabolic subgroup of G with abelian unipotent radical P^u and let L be a Levi factor of P. If B is a Borel subgroup of G contained in P, then B acts with finitely many orbits both in P^u and in G/L, which is called a symmetric variety of Hermitian type. In the talk, I will describe the Bruhat order of these B-orbits (that is, the partial order defined by the inclusions of orbit closures) in terms of suitable Weyl group elements, proving two related conjectures of Panyushev (concerning the B-orbits in P^u) and of Richardson-Ryan (concerning the B-orbits in G/L). Then I will discuss a generalization of Panyushev conjecture to the case of any abelian ideal A of the Lie algebra of B: in this case as well, B acts with finitely many orbits on A, and I will explain how the Bruhat order of the B-orbits on A should be controlled by suitable involutions of the affine Weyl group. The talk is based on a joint work with Andrea Maffei, and on a joint project with Pierluigi Moseneder-Frajria and Paolo Papi.