Number Theory Seminar: Avner Segal

In the theory of automorphic representations the study of L-functions plays a key role. A common method to study the analytic behavior of such functions (and, in fact, proving that they are meromorphic functions) is the Rankin-Selberg method. In this method an integral representation, with good analytic properties, is attached to the L-function. Many examples of Rankin-Selberg integrals were studied along the years. However most examples rely on the uniqueness of certain models of the representation (most popular in use is the Whittaker model but many other, such as the Peterson bilinear form and Bessel model, are used). In a pioneering paper ("A new-way to get Euler products", Krelle, 1988) I. Piatetski-Shapiro and S. Rallis suggested a remarkable mechanism that makes it possible to use integrals containing a "non-unique model" by a slight strengthening of the unramified computations. In the first part of my talk we will have a crash-course on cuspidal automorphic representations and the new-way mechanism via the classical example of Hecke's integral representation for L-functions of cuspidal representations of GL_2. In the second part of my talk I will present a joint work with N. Gurevich in which we proved that a family of Rankin-Selberg integrals representing the standard twisted L-function of a cuspidal representation of the exceptional group of type G_2. In its unfolded form (a term which will be explained in the talk), the integrals contain a non-unique model and we apply the new-way mechanism. The unramified computation gives rise to two interesting objects: the generating function of the L-function and its approximations. If time permits, I will discuss the possible poles of this L-function and some applications to the theory of cuspidal representations of G_2.