Centralizer of the elementary subgroup of an isotropic reductive group

Abstract

Let R be a commutative ring with 1, and let G be an isotropic reductive algebraic group over R. In [5] Victor Petrov and the second author gave a definition of an elementary subgroup E(R) of the group of points G(R). This definition of E(R) generalized the well known definition of the elementary subgroup of a Chevalley group, as well as several other definitions of the elementary subgroup of isotropic classical groups and simple algebraic groups over fields, see the references in [5]. The definition is as follows. Assume that G is isotropic in the following strong sense: it possesses a par abolic subgroup that intersects properly any semisimple normal subgroup of G. Such a parabolic subgroup P is called strictly proper. Denote by EP(R) the subgroup of G(R) generated by the R points of the unipotent radicals of P and of an opposite parabolic subgroup P–. The main theorem of [5] states that EP(R) does not depend on the choice of P, as soon as for any maximal ideal M of R all irreducible components of the rel ative root system of (see [2, Exp. XXVI, §7] for the definition) are of rank ≥2. Under this assumption,