## Maximal tori in semisimple groups and rational surfaces.(English. Russian original)Zbl 0618.14028

Russ. Math. Surv. 41, No. 4, 177-178 (1986); translation from Usp. Mat. Nauk 41, No. 4(250), 217-218 (1986).
Let G be a finite group and N a G-module of finite type. If $$i=1$$ or 2 then denote by $$\text{Russian{Sh}}^ i_{\omega}(G,N)$$ the kernel of the restriction homomorphism $$H^ i(G,N)\to \prod_{g\in G}H^ i(<g>,N).$$ The authors consider the case when G is a subgroup of the Weyl group $$W(R)$$, where R is the root system $$A_ n$$, $$B_ n$$, $$C_ n$$ or $$D_ n$$, and find the groups $$\text{Russian{Sh}}^ 1_{\omega}(G,P(R))$$ and $$\text{Russian{Sh}}^ 2_{\omega}(G,Q(R))$$, where $$P(R)$$ is the lattice of the weights and $$Q(R)$$ the lattice generated by the roots. As it is explained in this note, these groups provide an arithmetical information about the maximal tori of semisimple adjoint groups and about rational surfaces. No proofs are given.
Reviewer: V.L.Popov

### MSC:

 14M20 Rational and unirational varieties 14L10 Group varieties 20G10 Cohomology theory for linear algebraic groups 20G05 Representation theory for linear algebraic groups 20G15 Linear algebraic groups over arbitrary fields
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