On Bessel distributions for quasi-split groups

We show that the Bessel distribution attached by Gelfand and Kazhdan and by Shalika to a generic representation of a quasi-split reductive group over a local field is given by a function when it is restricted to the open Bruhat cell. As in the case of the character distribution, this function is real analytic for archimedean fields and locally constant for non-archimidean fields.