An Upper Bound on Conductors for Pairs Original Research Article

LetFbe 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:cless-than-or-equals, slantna+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).