Duke Mathematical Journal
- Duke Math. J.
- Volume 163, Number 12 (2014), 2179-2216.
On special representations of -adic reductive groups
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.
Duke Math. J., Volume 163, Number 12 (2014), 2179-2216.
First available in Project Euclid: 15 September 2014
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Grosse-Klönne, Elmar. On special representations of $p$ -adic reductive groups. Duke Math. J. 163 (2014), no. 12, 2179--2216. doi:10.1215/00127094-2785697. https://projecteuclid.org/euclid.dmj/1410789513