Local Rankin-Selberg convolutions for _{}: Explicit conductor formula

In this paper, F denotes a non-Archimedean local field with finite residue field of q elements. The discrete valuation ring of F is denoted o, and ψ is a non-trivial, continuous character of the additive group of F . We write p for the maximal ideal of o and c(ψ) for the largest integer c such that p−c ⊂ Kerψ. For i = 1, 2, let ni be a positive integer, write Gi = GLni(F ), and let πi be an irreducible smooth representation of Gi. To the data π1, π2 and ψ, Jacquet, Piatetskii-Shapiro and Shalika [14] attach an L-function L(π1 × π2, s) and a local constant ε(π1 × π2, s, ψ), where s denotes a complex variable. The alternative approach of Shahidi [19] places these objects in a more general context; either way, they are absolutely central to the study of the local Langlands Conjecture [12]. The L-function has the form L(π1 × π2, s) = P (q−s)−1, where P (X) ∈ C[X ] satisfies P (0) = 1. On the other hand, ε(π1 × π2, s, ψ) = ε(π1 × π2, 0, ψ) q−f(π1×π2,ψ)s, for some integer f(π1 × π2, ψ). In fact, f(π1 × π2, ψ) = n1n2c(ψ) + f(π1 × π2), where f(π1 × π2) is independent of ψ. There is a full description of the function L(π1 × π2, s) in [14] which, at the same time, reduces the study of the local constant to the case where both πi are supercuspidal. The aim of this paper is to give an explicit formula for f(π1 × π2) when the πi are supercuspidal. This formula (Theorem 6.5 below) contains substantial arithmetic information about the representations πi and their relationship to each other, in terms of their description as induced representations [5]. As one of the consequences of this formula, we obtain sharp upper and lower bounds for f(π1 × π2), as we shall now explain. For this purpose, we need to recall the local constant ε(π, s, ψ) of a supercuspidal representation π of GLn(F ), in the sense of Godement and Jacquet [10]. This takes the form ε(π, s, ψ) = ε(π, 0, ψ) q−f(π,ψ)s,