Irreducible tori in semi-simple groups

Abstract

Let G be an absolutely simple simply connected algebraic group over a global field k. In this note, we analyze arithmetic properties of the maximal k-tori of G .We establish density (in the variety of maximal tori) of the set of maximal k-tori which do not contain proper k-subtori (we call such tori k-irreducible). Using the fact that the k-irreducible tori we construct have the weak approximation property, we extend some of our previous results (contained in Publ. Math. Inst. Haute Etudes Sci. 84 (1996), 91–187, § 9) to global function fields. This allows one to establish the congruence subgroup property for the groups of points of G over semi-local subrings of k. Finally, we examine the strong approximation property in the maximal k-tori of G with respect to generalized arithmetic progressions.