Hecke-type algebras and cohomology of flag varieties
Student No.:50
Time:15:30-16:30,2017-05-31; 13:30-14:30, 2017-06-02
Instructor:Changlong Zhong  [Stata University of New York at Albany]
Place:Conference Room 3, Floor 2, 2017-05-31; Conference Room 1, Floor 1,06-02
Starting Date:2017-5-31
Ending Date:2017-6-2



Duration: 2017-05-31

Time: WED 15:30-16:30

Place:Room 3, Floor 2,Jin Chun Yuan West Building


Duration: 2017-06-02

Time: FRI 13:30-14:30

Place:Room 1, Floor 1,Jin Chun Yuan West Building



The algebraic/combinatorial method in the study of cohomology of flag varieties was started by Demazure and Bernstein-Gelfand-Gelfand in 1970s (for ordinary Chow groups), and were continued by Arabia, Kostant-Kumar, Bressler-Evens in 1980s-1990s (for equivariant singular cohomology, equivariant K-theory and complex cobordism). It was generalized to general oriented cohomology theory by Calmes-Petrov-Zainoulline, and later by myself with Calmes and Zainoulline.


Such method is based on the Bruhat decomposition of flag variety, and the fact that the convolution action of divided difference operators on the fundamental class of identity point generates the whole cohomology ring. The dual of the algebra generated by divided difference operators will be the algebraic model of equivariant oriented cohomology of flag varieties, and many important structures (Bott-Samelson classes, push-pull maps, characteristic map) can be seen from this model.


The tentative plan of the talks is as follows:


Talk 1: I will recall  Kostant-Kumar's construction of affine nil-Hecke algebra. It is some algebra generated by divided-difference operators (also called Demazure operators or BGG operators). I will state the basic properties of this algebra, and the structure theorem. I will then recall the construction of formal affine Demazure algebra and then define its dual.

I will define the push-pull elements and state their properties. The Bott-Samelson classes will be defined, and duality between various classes will be constructed. I will also state their relation with equivariant oriented cohomology of flag varieties.


Talk 2: I will introduce the so-called hyperbolic formal group law and study the relationship between Kazhdan-Lusztig basis of classical Iwahori-Hecke algebra and Schubert classes of flag varieties. If time permits, I will also introduce the formal affine Hecke algebra, which generalizes the Iwahori-Hecke algebra and is closely related with Steinberg variety.