Centralizer of the elementary subgroup of an isotropic reductive group

TL;DR

This paper proves that for certain isotropic reductive groups over a ring, the centralizer of the elementary subgroup aligns with the group's center, extending previous results in algebraic group theory.

Contribution

It generalizes earlier findings by establishing the equality of the centralizer of the elementary subgroup with the group's center under broader conditions.

Findings

Centralizer of E(R) equals the group center for specified groups

Generalizes previous results by Abe and Hurley

Applicable to isotropic reductive groups over rings with rank conditions

Abstract

Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that, for any maximal ideal M of R, the rank of the relative root system of G_{R_M} is greater or equal than 2. We show that under this assumption the centralizer of E(R) in G(R) coincides with the abstract group-theoretic center of G(R) and with Cent(G)(R). This generalizes a result of E. Abe and J. Hurley.