Abstract:

The talk is based on a joint work with Nikita Geldhauser.

Chow rings with finite coefficients for split reductive groups we computed by Victor Kac in 1985,
based on the previous computation of singular cohomology of compact Lie groups. It follows from
the Kunneth theorem that these rings are finite dimensional Hopf algebras, hence have a presentation
as quotients of polynomial algebras modulo some powers of the variables, and these powers were explicitly
calculated by Kac. Later, building on this computation, Geldhauser, Petrov and Zainoulline introduced
J-invariant for algebraic groups of inner type, which is an important motivic invariant used to study
such algebraic groups and their homogeneous varieties. In order to extend J-invariant to arbitrary
linear algebraic groups one has to understand the Chow rings of non-split quasi-split groups, which we
do in a joint work with Nikita Geldhauser. It turns out that in general a Chow ring of a quasi-split
algebraic group fails to be a Hopf algebra (Kunneth theorem does not hold) and that the usual technical
tool, the characteristic sequence of Grothendieck, fails to be exact. As a way to circumvent these
issues we introduce and study (equivariant) conormed Chow groups, and analyze the pushforward morphism
for Ch(G_K) -> Ch(G), where K is the splitting field of the quasi-split group G. Combining this with
previous computations by Geldhauser and Zhykhovich we compute Chow rings of quasi-split geometrically
almost simple algebraic groups.