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



James W. Cogdell (Department of Mathematics, Ohio State University, Columbus, Ohio, U.S.A.)

Nadir Matringe (Laboratoire de Mathématiques et Applications, Université de Poitiers, Futuroscope Chasseneuil, France)


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.

Accepted 27 November 2014

Published 20 May 2015