Irreducible modules for pseudo-reductive groups

TL;DR

This paper classifies irreducible representations of pseudo-reductive algebraic groups over fields, reducing the problem to known cases and providing new results for rank one pseudo-split pseudo-reductive groups.

Contribution

It offers a classification of irreducible modules for pseudo-reductive groups and advances understanding of their structure, especially in the pseudo-split and rank one cases.

Findings

Reduced dimension calculation to split reductive groups

Provided new results for pseudo-split pseudo-reductive commutative groups

Described the rank one case comprehensively

Abstract

We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudo-split pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case.