Let be a non-Archimedean locally compact field, and let be a split connected reductive group over . For a parabolic subgroup and a ring , we consider the -representation on the -module Let denote an Iwahori subgroup. We define a certain free finite rank- module (depending on ; if is a Borel subgroup, then (∗) is the Steinberg representation and is of rank ) and construct an -equivariant embedding of (∗) into . This allows the computation of the -invariants in (∗). We then prove that if is a field with characteristic equal to the residue characteristic of and if is a classical group, then the -representation (∗) is irreducible. This is the analogue of a theorem of Casselman (which says the same for ); it had been conjectured by Vignéras.
Herzig (for ) and Abe (for general ) have given classification theorems for irreducible admissible modulo representations of in terms of supersingular representations. Some of their arguments rely on the present work.
"On special representations of -adic reductive groups." Duke Math. J. 163 (12) 2179 - 2216, 15 September 2014. https://doi.org/10.1215/00127094-2785697