College of Liberal Arts & Sciences

# Representation Theory Seminar

**Abstract: **

In the pioneering work of Jacquet, Piatetski-Shapiro and Shalika, they defined and described the local Rankin-Selberg *L*-functions for *GL*(n) x *GL*(m) in terms of *L*-functions for inducing datum at finite places. Later Jacquet and Shalika established integral representations for exterior square *L*-functions. Cogdell and Piatetski-Shapiro revisited Rankin-Selberg *L*-functions to develop the method of exceptional poles attached to derivatives due to Bernstein and Zelevinsky, independent of multiplicativity of gamma factors.

In the first talk, we review Langlands classification and the local Rankin-Selberg and exterior square factors at ramified places using integral representations. As an application of their results, we closely follow the approach of Rudnick and Sarnak to derive the estimate for Laglands parameter appearing in a local constituent of an irreducible cuspidal automorphic representation of *GL*(n).