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Representations of $\mathrm{SL}_2(F)$

Guy Henniart and Marie-France Vignéras

Vol. 335 (2025), No. 2, 229–286
Abstract

Let p be a prime number, F a nonarchimedean local field with residue field kF of characteristic p, and R an algebraically closed field of characteristic different from p. We investigate the irreducible smooth R-representations of SL 2(F). The components of an irreducible smooth R-representation Π of GL 2(F) restricted to SL 2(F) form an L-packet L(Π). We use the classification of such Π to determine the cardinality of L(Π), which is 1,2 or 4. When p = 2 we have to use the Langlands correspondence for GL 2(F). When is a prime number distinct from p and R = ac, we determine the behaviour of an integral L-packet under reduction modulo . We prove a Langlands correspondence for SL 2(F), and an enhanced one when the characteristic of R is not 2. Finally, pursuing a theme of Henniart and Vignéras (2024), which studied the case of inner forms of GL n(F), we show that near identity a nontrivial irreducible smooth R-representation π of SL 2(F) is, up to a finite-dimensional representation, isomorphic to a sum of 1,2 or 4 representations in an L-packet of size 4 (when p is odd there is only one such L-packet). We show that for π in an L-packet of size rπ and a sufficiently large integer j, the dimension of the invariants of π by the j-th congruence subgroup of an Iwahori or a pro-p Iwahori subgroup of SL 2(F) is equal to aπ + 2rπ1|kF|j, with aπ = 1 2 if p is odd and rπ = 4, otherwise aπ is an integer. We also study the fixed points by the j-th congruence subgroups of the maximal compact subgroups of SL 2(F) where the answer depends on the parity of j.

Keywords
modular irreducible representations, $L$-packets, Whittaker spaces, local Langlands correspondence
Mathematical Subject Classification
Primary: 22E50
Secondary: 11F70
Milestones
Received: 9 April 2024
Revised: 3 January 2025
Accepted: 6 February 2025
Published: 18 April 2025
Authors
Guy Henniart
Laboratoire de Mathématiques d’Orsay
Université Paris-Saclay
Orsay
France
Marie-France Vignéras
Institut de Mathématiques de Jussieu
Université Paris Cité
Paris
France

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