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 Π of GL₂(F) restricted to SL₂(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 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).