Some reductive anisotropic groups that admit no non-trivial split spherical BN-pairs

TL;DR

This paper proves that for infinite fields, certain reductive groups have no non-trivial split spherical BN-pairs, especially when the group is anisotropic, supporting a conjecture related to group structure.

Contribution

It establishes that virtually trivial split spherical BN-pairs are trivial over infinite fields and shows anisotropic groups often lack non-trivial split spherical BN-pairs, advancing the conjecture of Caprace and Marquis.

Findings

Virtually trivial split spherical BN-pairs are trivial over infinite fields.

Anisotropic groups often admit no non-trivial split spherical BN-pairs.

Supports a conjecture relating group anisotropy to BN-pair triviality.

Abstract

We prove, for any infinite field k, that any virtually trivial split spherical BN-pair in the group G(k) of k-rational points of a reductive k-group G is already trivial. We then inspect the case when G is k-anisotropic and show that in many situations G(k) admits no non-trivial split spherical BN-pairs. This improves results and contributes to a conjecture of Caprace and Marquis, which can be viewed as a converse to a well-known result of Borel and Tits.