Waldspurger's Involution and Types

Waldspurger's involution for the genuine irreducible supercuspidal representations of SL 2 ( F ) is parametrized in terms of types in the case F p ‐adic with p odd. In particular, it is shown that the in‐volution is given by conjugating by an element of GL 2 ( F ) and twisting one of the defining parameters of an associated type by a quadratic character, the relevant parameter being a character on the norm one elements of a quadratic extension.