2:00 pm, Lecture Hall 4
Introduction to Geometric Representation Theory 2 (Second Lecture)
University of Aarhus, Denmark.
Warm-up: We recall the classical construction of the Hecke algebra given by bi-equivariant functions on GL(n) over a finite field, with the product given by convolution.
We start the geometric story by recalling the presentaiton of the Weyl group by generators and relations, then we introduce the Braid group of the corresponding type. For simplicity, we concentrate on the case of G=SL(n), then the Weyl group is the symmetric group in n letters, the braid group is the classical one, the maximal torus T consists of diagonal matrices with determinant equal to 1, the standard Borel subgroup B consists of upper triangular matrices with determinant 1.
We introduce Bott-Samelson varieties and recall a few basic facts about their geometry.
Then we study the monoidal category of quasicoherent sheaves on B\G/B, with the monoidal structure given by convolution. We show that the structure sheaves of the standard minimal parabolic subgroups P_i in G satisfy the relations of degenerate Hecke type with respect to this monoidal product. Bott-Samelson varieties play the central role in establishing these relations.
Finally given a G-variety X, we construct a categorical action of the degenerate Hecke algebra on the category of quasicoherent sheaves on X/B.