Shalika periods and parabolic induction for GL(n) over a non archimedean local field

Abstract

Let $F$ be a non archimedean local field, and $n_1$ and $n_2$ two positive even integers. We prove that if ${\pi }_1$ and ${\pi }_2$ are two smooth representations of $GL(n_1,F)$ and $GL(n_2,F)$ respectively, both admitting a Shalika period, then the normalised parabolically induced representation ${\pi }_1\times {\pi }_2$ also admits a Shalika period. Combining this with the results of \cite{M-localBF}, we obtain as a corollary the classification of generic representations of $GL(n,F)$ admitting a Shalika period when $F$ has characteristic zero. This result is relevant to the study of the Jacquet-Shalika exterior square $L$ factor.

This is the version to appear in Bull. Lond. Math. Soc

References in corpus (2)

• The functional equation of the Jacquet-Shalika integral representation of the local exterior-square $L$-function
• Local newforms and formal exterior square L-functions

Cited by in corpus (2)

• Anticyclotomic Euler systems for unitary groups
• The local exterior square and Asai $L$-functions for $GL(n)$ in odd characteristic