The local Rankin-Selberg convolution for GL(n): divisibility of the conductor

Let F be a non-Archimedean local field and n greater than or equal to 2 an integer. Let pi,pi ' be irreducible supercuspidal representations of GL(n) (F) with pi not congruent to pi '. One knows that there exists an irreducible supercuspidal representation rho of GL(m) (F), with m <n, such that the local constants (in the sense of Jacquet, Piatetskii-Shapiro and Shalika) epsilon (pi x rho, s, psi), epsilon(pi ' x rho, s, psi) are distinct. In this paper, we show that. when pi ' is an unramified twist chi pi of pi one may here take in dividing n and less than or equal to n/l for a prime divisor l of n depending on pi and the order of chi: in particular, m less than or equal ton/l(0), where l(0) is the least prime divisor of n. This follows from a result giving control of certain divisibility properties of the conductor of a pair of supercuspidal representations