Irreducible modules for pseudo-reductive groups

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.

To appear in JEMS; significant error noted by referee and corrected from previous version: Lemma 3.1 wrongly claimed Q_G(λ) was M-isotypic; see Rk 7.4(ii) in this version