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

TL;DR

This paper proves that the property of admitting a Shalika period is preserved under parabolic induction for certain representations of GL(n) over non-archimedean local fields, aiding classification and L-factor studies.

Contribution

It establishes that representations with Shalika periods maintain this property after parabolic induction, and classifies generic representations with Shalika periods in characteristic zero fields.

Findings

Shalika periods are preserved under parabolic induction for GL(n) representations.

Classification of generic representations with Shalika periods in characteristic zero.

Implications for the study of the Jacquet-Shalika L-factor.

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.