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

Abstract

Let F F be a non-Archimedean local field and n 1 n_{1} , n 2 n_{2} positive integers. For i = 1 , 2 i=1,2 , let G i = G L n i ( F ) G_{i}=\mathrm {GL}_{n_{i}}(F) and let π i \pi _{i} be an irreducible supercuspidal representation of G i G_{i} . Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant ε ( π 1 × π 2 , s , ψ ) \varepsilon (\pi _{1}\times \pi _{2},s,\psi ) to the π i \pi _{i} and an additive character ψ \psi of F F . This object is of central importance in the study of the local Langlands conjecture. It takes the form ε ( π 1 × π 2 , s , ψ ) = q − f s ε ( π 1 × π 2 , 0 , ψ ) , \begin{equation*}\varepsilon (\pi _{1}\times \pi _{2},s,\psi ) = q^{-fs}\varepsilon (\pi _{1} \times \pi _{2},0,\psi ), \end{equation*} where f = f ( π 1 × π 2 , ψ ) f=f(\pi _{1}\times \pi _{2},\psi ) is an integer. The irreducible supercuspidal representations of G = G L n ( F ) G=\mathrm {GL}_{n}(F) have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of G G . This paper gives an explicit formula for f ( π 1 × π 2 , ψ ) f(\pi _{1} \times \pi _{2},\psi ) in terms of the inducing data for the π i \pi _{i} . It uses, on the one hand, the alternative approach to the local constant due to Shahidi, and, on the other, the general theory of types along with powerful existence theorems for types in G L ( n ) \mathrm {GL}(n) , developed by Bushnell and Kutzko.

On the degree of certain localL-functions

“…The constant c(β) is defined such that C(β) = q c(β) . Now we state the conductor formula of [Bushnell et al 1998].…”

“…In the second case of the above, we have made use of the fact that E/F is tamely ramified, which is true since p is odd by our assumption. Since the formula of Bushnell, Henniart, and Kutzko [1998] computes the left hand side, in order to derive a formula for the Asai lift, it suffices to compute f (r (ρ π )) − f (r (ρ π ) ⊗ ω E/F ).…”

Sur le comportement, par torsion, des facteurs epsilon de paires

“…où ψ 0 est un caractère de F de niveau 0. Les calculs de [10] donnent la valeur de a(π × π ), ce qui détermine la valuation de c K (π,π, ψ). On a a(π × π ) = a(π ×π) si χ est non ramifié, a(π × π ) = a(π ×π) + 1 si χ est ramifié.…”

Local Langlands correspondence and ramification for Carayol representations

“…The function is piecewise linear, strictly increasing and convex. It is given by an explicit formula [BH17, (4.4.1)] derived from the conductor formula of [BHK98, 6.5 Theorem].…”

“…One of these is invariably the first. If is of Carayol type, the other is the last: this follows from an application of the conductor formula of [BHK98, 6.5 Theorem], which also gives a relation between the first and last jumps. One may then assume that has the symmetry property and proceed by induction on dimension.…”