Local Rankin-Selberg convolutions for $\mathrm{GL}_{n}$: explicit conductor formula

Let $F$ be a non-Archimedean local field and $n_{1}$, $n_{2}$ positive integers. For $i=1,2$, let $G_{i}=\mathrm{GL}_{n_{i}}(F)$ and let $\pi _{i}$ be an irreducible supercuspidal representation of $G_{i}$. Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant $\varepsilon (\pi _{1}\times \pi _{2},s,\psi )$ to the $\pi _{i}$ and an additive character $\psi$ of $F$. This object is of central importance in the study of the local Langlands conjecture. It takes the form

where $f=f(\pi _{1}\times \pi _{2},\psi )$ is an integer. The irreducible supercuspidal representations of $G=\mathrm{GL}_{n}(F)$ have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of $G$. This paper gives an explicit formula for $f(\pi _{1} \times \pi _{2},\psi )$ in terms of the inducing data for the $\pi _{i}$. 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 $\mathrm{GL}(n)$, developed by Bushnell and Kutzko.