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15 September 2014 On special representations of p-adic reductive groups
Elmar Grosse-Klönne
Duke Math. J. 163(12): 2179-2216 (15 September 2014). DOI: 10.1215/00127094-2785697


Let F be a non-Archimedean locally compact field, and let G be a split connected reductive group over F. For a parabolic subgroup QG and a ring L, we consider the G-representation on the L-module C(G/Q,L)/Q'QC(G/Q',L).() Let IG denote an Iwahori subgroup. We define a certain free finite rank-L module M (depending on Q; if Q is a Borel subgroup, then (∗) is the Steinberg representation and M is of rank 1) and construct an I-equivariant embedding of (∗) into C(I,M). This allows the computation of the I-invariants in (∗). We then prove that if L is a field with characteristic equal to the residue characteristic of F and if G is a classical group, then the G-representation (∗) is irreducible. This is the analogue of a theorem of Casselman (which says the same for L=C); it had been conjectured by Vignéras.

Herzig (for G=GLn(F)) and Abe (for general G) have given classification theorems for irreducible admissible modulo p representations of G in terms of supersingular representations. Some of their arguments rely on the present work.


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Elmar Grosse-Klönne. "On special representations of p-adic reductive groups." Duke Math. J. 163 (12) 2179 - 2216, 15 September 2014.


Published: 15 September 2014
First available in Project Euclid: 15 September 2014

zbMATH: 1298.22018
MathSciNet: MR3263032
Digital Object Identifier: 10.1215/00127094-2785697

Primary: 22E50
Secondary: 11S99

Rights: Copyright © 2014 Duke University Press


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Vol.163 • No. 12 • 15 September 2014
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