ON THE DEGREE OF CERTAIN LOCAL L-FUNCTIONS

Let pi 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 pi has cardinality n/e, where e is the o(F)-period of the principal o(F)-order in M-n(F) attached to pi. This is the degree of the local Rankin-Selberg L-function L(s, pi x pi(boolean OR)). In this paper, we compute the degree of the Asai, symmetric square, and exterior square L-functions associated to pi. 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 (1998).