Representations of SL₂ (F)

Let p be a prime number, F a non-archimedean local field with residue characteristic p, and R an algebraically closed field of characteristic different from p. We thoroughly investigate the irreducible smooth R-representations of SL₂ (F). The components of an irreducible smooth R-representation of GL₂ (F) restricted to SL₂ (F) form an L-packet L (). We use the classification of such to determine the cardinality of L (), which is 1, 2 or 4. When p=2 we have to use the Langlands correspondence for GL₂ (F). When is a prime number distinct from p and R= Q_^ac, we establish the behaviour of an integral L-packet under reduction modulo. We prove a Langlands correspondence for SL₂ (F), and even an enhanced one when the characteristic of R is not 2. Finally, pursuing a theme of HV23, which studied the case of inner forms of GLₙ (F), we show that near identity an irreducible smooth R-representation of SL₂ (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).