An upper bound on conductors for pairs

Let F be a non-Archimedean local field. Let pi be a smooth irreducible complex representation of GL(m)(F), and rho be a smooth irreducible complex representation of GL(n)(F). Denote by a, b, and c the exponents in the conductors of pi, rho, and the pair (pi, rho), respectively. If F has positive characteristic, the following upper bound is a consequence of the Local Langlands correspondence with Galois representations: c less than or equal to na + mb - inf(a, b). We prove this bound directly, regardless of the characteristic of F, using results of Jacquet, Piatetski-Shapiro. and Shalika on the essential (''new'') vector for smooth irreducible generic representations of GL(n)(F). (C) 1997 Academic Press.