it is well-known from weil that the isomorphism classes of rank n vector bundles on an algebraic curve can be written as the set of certain double cosets of gl(n,a), where a is the adeles of the curve. i will introduce such presentation in the realm of algebraic geometry and discuss two of its applications: the first is the tamagawa number formula (proved by gaitsgory-lurie), and the second is the verlinde formula in positive characteristic