Representations of SL2(F)

Let p be a prime number, F a nonarchimedean local field with residue field k F 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 of GL 2 (F) restricted to SL 2 (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 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 Vignras (2024), which studied the case of inner forms of GL n (F), we show that near identity a nontrivial irreducible smooth R-representation of SL 2 (F) is, up to a finitedimensional 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 in an L-packet of size r and a sufficiently large integer j , the dimension of the invariants of by the j -th congruence subgroup of an Iwahori or a pro-p Iwahori subgroup of SL 2 (F) is equal to a + 2r -1 |k F | j , with a = -1 2 if p is odd and r = 4, otherwise a 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 .