Duke Mathematical Journal

On special representations of p-adic reductive groups

Elmar Grosse-Klönne

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Abstract

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.

Article information

Source
Duke Math. J., Volume 163, Number 12 (2014), 2179-2216.

Dates
First available in Project Euclid: 15 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1410789513

Digital Object Identifier
doi:10.1215/00127094-2785697

Mathematical Reviews number (MathSciNet)
MR3263032

Zentralblatt MATH identifier
1298.22018

Subjects
Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Secondary: 11S99: None of the above, but in this section

Citation

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


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