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# Mathematical Research Letters

## Volume 22 (2015)

### Number 3

### The functional equation of the Jacquet-Shalika integral representation of the local exterior-square $L$-function

Pages: 697 – 717

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n3.a4

#### Authors

#### Abstract

An integral representation for the exterior square $L$-function for $GL_n$ was given by Jacquet and Shalika in 1990. Recently there has been renewed interest in both the local and global theory of the exterior square $L$-function via this integral representation. In an earlier work, the second author used his results on the connection between linear periods and Shalika periods to analyze the local exterior square $L$-functions via Bernstein-Zelevinsky derivatives and prove the local functional equation in the case of $GL_{2m}(F)$, for $F$ a nonarchimedean local field. In this paper we complete this work and derive the local functional equation for the exterior square $L$-function for $GL_{2m+1}(F)$ by similar methods, and extending the functional equation in both cases to non-generic representations. With these results, we have the local functional equation of the exterior square $L$-function for irreducible admissible representations of $GL_n(F)$, for any $n$, for use in future applications.

Published 20 May 2015