Andrzej Weber, University of Warsaw, Poland – Geometric Methods in Representation Theory Seminar

Title: Characteristic classes of Schubert varieties and Hecke-type algebras

Abstract: We study various types of cohomological invariants of Schubert varieties in the generalized flag variety G/B. According to Bernstein-Gelfand-Gelfand the fundamental classes in cohomology can be computed via the action of the nil-Hecke algebra. It was shown by Aluffi and Mihalcea that the Chern-Schwartz-MacPherson classes are obtained by the action of the group ring of the Weyl group. Similarly, the motivic Chern classes in K-theory are related to the classical Hecke algebra, as announced by Aluffi-Mihalcea-Schürmann-Su. We will concentrate on the equivariant elliptic classes in the sense of Borisov-Libgober, which depend on an additional parameter – an auxiliary line bundle. We show that these classes are related to a Hecke-type elliptic algebra. The proof uses basic properties of the canonical divisor of the Schubert varieties and its Bott-Samelson resolutions. As a corollary we show that for G=GL_n the Borisov-Libgober elliptic classes are represented by Rimanyi-Tarasov-Varchenko elliptic weight function.