An Upper Bound on Conductors for Pairs

AbstractLetFbe a non-Archimedean local field. Letπbe a smooth irreducible complex representation ofGLm(F), andρbe a smooth irreducible complex representation ofGLn(F). Denote bya,b, andcthe exponents in the conductors ofπ,ρ, and the pair (π,ρ), respectively. IfFhas positive characteristic, the following upper bound is a consequence of the Local Langlands correspondence with Galois representations:c⩽na+mb−inf(a,b).We prove this bound directly, regardless of the characteristic ofF, using results of Jacquet, Piatetski–Shapiro, and Shalika on the essential (“new”) vector for smooth irreducible generic representations ofGLn(F)