Double cosets $NgN$ of normalizers of maximal tori of simple algebraic groups and orbits of partial actions of Cremona subgroups

Abstract

Let G be a simple algebraic group over an algebraically closed field K and let N = N G (T ) be the normalizer of a fixed maximal torus T ≤ G. Further, let U be the unipotent radical of a fixed Borel subgroup B that contains T and let U − be the unipotent radical of the opposite Borel subgroup B − . The Bruhat decomposition implies the decomposition G = N U − U N . The Zariski closed subset U − U ⊂ G is isomorphic to the affine space A m K where m = dim G − dim T is the number of roots in the corresponding root system. Here we construct a subgroup N ≤ Cr m (K) that "acts partially" on A m K ≈ U and we show that there is one-to-one correspondence between the orbits of such a partial action and the set of double cosets {N gN }. Here we also calculate the setThe pairs of maximal tori of simple algebraic groups. In this paper we consider the case when G is a simple algebraic group over an algebraically closed field K. The decomposition of a group G into the union of double cosets G = ∪ g i H 2 g i H 1 is a very important construction in the theory of algebraic groups, especially in the case when H 1 , H 2 are parabolic subgroups. For these cases the decomposition is finite. Here we consider the case when H 1 = H 2 = N = N G (T ) is the normalizer of a fixed maximal torus T . Now let X be the set of all maximal tori of G. The group G acts on X by conjugation. Then X is just one G-orbit of T ∈ X and N := N G (T ) = St T . Thus, we have one-to-one correspondence between the set of G-orbits of the set X × X and the set of double cosets {Ng α N} α∈A . Further, we have the decomposition