A note on bessel functions for supercuspidal representations of GL(2) over a p-adic field

We compute the Bessel functions which give the action of the Weyl group element in the Kirillov model of supercuspidal representations of GL ₂(K), where Kis a p-adic field. Together with the known action of the Borel subgroup, this gives the full action of GL₂(K) for such representations. We consider supercuspidal representations constructed by Jacquet and Langlands using the Weil representation. When the residual characteristic is odd, these are all the supercuspidal representations, hence we get a full description of all supercuspidal representations in this case.