On the degree of certain local L-functions

Abstract

Let π be an irreducible supercuspidal representation of GL_n(F),where F is a p-adic field. By a result of Bushnell and Kutzko, the group of unramified self-twists of π has cardinality n/e where e is the oF -period of the principal oF -order in M_n(F) attached to π. This is the degree of the local Rankin-Selberg L-function L(s, π × π^∨). In this paper, we compute the degree of the Asai, symmetric square and exterior square L-functions associated to π. As an application, assuming p is odd, we compute the conductor of the Asai lift of a supercuspidal representation, where we also make use of the conductor formula for pairs of supercuspidal representations due to Bushnell, Henniart, and Kutzko [BHK98].