ИСТИНА

For a connected reductive algebraic group G defined over the field of real numbers R, the group of real points G(R) is a real Lie group, not necessarily connected; look at GL_n(R) or SO_{p,q}(R) for example. A natural problem is to determine the component group of G(R). It turns out that this problem is related to another important problem in the theory of algebraic groups: to compute the Galois cohomology H^1(R,G). We give a uniform solution to both problems in terms of combinatorial data which determine the reductive group G over R, such as the affine Dynkin diagram with a nonnegative integer labeling of its vertices and the cocharacter lattice of a maximal torus equipped with an involution. Though the answer is purely algebraic and combinatorial, the proofs involve transcendental Lie-theoretic and differential-geometric methods such as the exponential mapping on algebraic tori and an action of the affine Weyl group on their Lie algebras. This is a joint work with Mikhail Borovoi.