Abstract
We generalize results of P. Schneider and U. Stuhler for G L 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 an arbitrary ring 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.
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