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.
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Michael Bate, David I. Stewart, Irreducible modules for pseudo-reductive groups. J. Eur. Math. Soc. 24 (2022), no. 7, pp. 2533–2553DOI 10.4171/JEMS/1153