Shalika newforms for GL$(n)$

Takeo Okazaki

Vol. 340 (2026), No. 1, 139–177

DOI: 10.2140/pjm.2026.340.139

Abstract

Let $(\pi ,V)$ be a generic irreducible representation of a general linear group over a $p$-adic field. Jacquet, Piatetski-Shapiro, and Shalika gave an open compact subgroup $K$, so that the subspace $V^K$ consisting of $v\in V$ fixed by $K$ is one-dimensional. If $\pi$ has a Shalika model $\Lambda$, then we call vectors in $\Lambda (V)$ the Shalika forms of $\pi$, and those in $\Lambda (V^K)$ the Shalika newforms. In this article, in the case where $\pi$ is supercuspidal, we show the nonvanishing of Shalika newforms at a minimal point in a sense. This point is not the identity, and the Shalika newform vanishes at the identity if the character defining the Shalika model is ramified. In view of this result, in this case, we give another Shalika form with nice properties.

Keywords

newform, Shalika period

Mathematical Subject Classification

Primary: 11F55

Secondary: 11F70

Milestones

Received: 18 March 2024

Revised: 15 September 2025

Accepted: 16 September 2025

Published: 31 October 2025