Vol. 276, No. 1, 2015

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On the degree of certain local $L$-functions

U. K. Anandavardhanan and Amiya Kumar Mondal

Vol. 276 (2015), No. 1, 1–17
Abstract

Let $\pi$ be an irreducible supercuspidal representation of ${GL}_{n}\left(F\right)$, where $F$ is a $p$-adic field. By a result of Bushnell and Kutzko, the group of unramified self-twists of $\pi$ has cardinality $n∕e$, where $e$ is the ${\mathfrak{o}}_{F}$-period of the principal ${\mathfrak{o}}_{F}$-order in ${M}_{n}\left(F\right)$ attached to $\pi$. This is the degree of the local Rankin–Selberg $L$-function $L\left(s,\pi ×{\pi }^{\vee }\right)$. In this paper, we compute the degree of the Asai, symmetric square, and exterior square $L$-functions associated to $\pi$. As an application, assuming $p$ is odd, we compute the conductor of the Asai lift of a supercuspidal representation, where we also make use of the conductor formula for pairs of supercuspidal representations due to Bushnell, Henniart, and Kutzko (1998).

Keywords
Asai $L$-function, symmetric square $L$-function, exterior square $L$-function, degree of a local $L$-function
Mathematical Subject Classification 2010
Primary: 22E50
Secondary: 11F33, 11F70, 11F85