On the Centralizer of a Regular, Semi-simple, Stable Conjugacy Class

We describe the isomorphism class of the torus centralizing a regular, semi-simple, stable conjugacy class in a simply-connected, semi-simple group. Let k be a field, and let G be a semi-simple, simply-connected algebraic group, which is quasi-split over k. The theory of semi-simple conjugacy classes in G is well understood, from work of Steinberg [S] and Kottwitz [K]. Any semi-simple conjugacy class s which is defined over k is represented by a semi-simple element γ in G(k). The centralizer Gγ of γ in G is connected and reductive. It is determined by the stable class s up to inner twisting, and one can choose a representative γ so that Gγ is quasi-split over k. In this paper, we will only consider the case when the semi-simple stable class s is regular. Then Gγ is a maximal torus in G, whose k-isomorphism class depends only on the class s. Our aim is to determine the isomorphism class of this torus, which we denote Ts over k, from the data specifying s in the variety of semi-simple stable conjugacy classes. We will first give an abstract description of the character group X(Ts), as an integral representation of the Galois group of k. We will then describe Ts concretely, in some special cases. In particular, for a simple, split group G which is not simplylaced, we use a semi-direct product decomposition of the Weyl group to reduce the problem to a semi-simple, quasi-split subgroup Hs containing Ts and the long root subgroups of G. The concrete description of Ts allows one to compute the terms corresponding to regular classes s in the stable trace formula (cf. [G-P]). For the general semi-simple class, one would like to have a description of the motive M(Gγ) of the centralizer.