# Manchester Geometry Seminar 2015/2016

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