# Manchester Geometry Seminar 2015/2016

Extra session: Monday 29 February 2016. The Frank Adams Room (Room 1.212), the Alan Turing Building. 5pm

## From Motives of Twisted Flag Varieties to Modular Representations of Hecke-type Algebras Kirill Zaynullin (University of Ottawa)

kirill@uottawa.ca

Let $G$ be a split semisimple linear algebraic group over a field $k$, let $E$ be a $G$-torsor over $k$. Let $h$ be an algebraic oriented cohomology theory in the sense of Levine-Morel (e.g. Chow ring or an algebraic cobordism). Consider a twisted form $E/B$ of the variety of Borel subgroups $G/B$. Following the motivic Galois group approach and the Kostant-Kumar results on equivariant cohomology of flag varieties we establish an equivalence between the $h$-motivic subcategory generated by $E/B$ and the category of ojective modules of certain Hecke-type algebra $H$ which depends on the root system of $G$, its isogeny class, on $E$, and on the formal group law of the theory $h$.

In particular, taking $h$ to be the Chow groups with finite coefficients $F_p$ and $E$ to be a generic torsor we obtain that all irreducible modules of the affine nil-Hecke algebra $H$ of $G$ with coefficients in $F_p$ are isomorphic and correspond to the generalized Rost-Voevodsky motive for $(G,p)$.