Generalized Steinberg representations of split reductive linear algebraic groups

TL;DR

This paper extends the understanding of generalized Steinberg representations from GL_{l+1} to broader split reductive algebraic groups over non-archimedean fields, establishing their cyclicity and expressing certain cases via parahoric subgroups.

Contribution

It generalizes previous results to all split reductive groups and provides explicit descriptions for semi-simple adjoint cases, especially of maximal degree.

Findings

Generalized Steinberg representations are cyclic for any abelian coefficient group.

Explicit expressions for maximal degree cases in terms of parahoric subgroups.

Extension of known results from GL_{l+1} to all split reductive groups.

Abstract

We generalize results of P. Schneider and U. Stuhler for GL_l+1 to a reductive algebraic group G defined and split over a non-archimedean local field K. Following their lines, we prove that the generalized Steinberg representations of G with coefficients in any abelian group are cyclic. When G is semi-simple of adjoint type, we give an expression of these representations, whenever it is possible and in particular for those that are of maximal degree, in terms of the parahoric subgroups of G.