On abstract homomorphisms of some special unitary groups

TL;DR

This paper investigates the structure of abstract representations of certain special unitary groups over quadratic fields, confirming a conjecture by Borel and Tits under specific conditions.

Contribution

It extends previous methods to analyze abstract representations of quasi-split special unitary groups, providing a standard description aligned with Borel and Tits' conjecture.

Findings

Abstract representations follow a standard form under certain assumptions

Method involves constructing an algebraic ring linked to the representation

Results support the conjecture of Borel and Tits for these groups

Abstract

We analyze the abstract representations of the groups of rational points of even-dimensional quasi-split special unitary groups associated with quadratic field extensions. We show that, under certain assumptions, such representations have a standard description, as predicted by a conjecture of Borel and Tits. Our method extends the approach introduced by the first author to study abstract representations of Chevalley groups and is based on the construction and analysis of a certain algebraic ring associated to a given abstract representation.