On the Generalized Steinberg Representation

Steinberg representations for groups of parahoric types: the special case

“…Let T be a maximal torud of G whose parahoric subgroup is contained in K, and let B be a Borel subgroup of G containing T ; the image B of B ∩ K in G is a Borel subgroup of G. We can define the Steinberg representation St B of G the same way as for G; it is irreducible as well; Moreover, it is easy to check that (1 B ) G = 1 B , we have a similar equality for any parabolic subgroup of G containing B, from wich we deduce that (St) G = St B . Similarly, if P is a parahoric subgroup of G, the generalized Steinberg representation St P of G is the reduction to G of a generalized Steinberg representation of G. See for example [9] for a study of such representations.…”