Twisting of paramodular vectors

TL;DR

This paper introduces a twisting operator for paramodular vectors in representations of GSp(4,F), extending the concept of twisting from GL(2) to a higher rank setting, with explicit level change properties.

Contribution

It defines a new twisting operator for paramodular vectors in GSp(4) representations and establishes its key properties, generalizing the GL(2) case.

Findings

The twisting operator maps paramodular vectors to vectors of increased level.

The operator's properties mirror those of the classical GL(2) twisting operator.

Explicit level change formula for the twisted vectors.

Abstract

Let $F$ be a non-archimedean local field of characteristic zero, let $(\pi,V)$ be an irreducible, admissible representation of $\GSp(4,F)$ with trivial central character, and let $\chi$ be a quadratic character of $F^\times$ with conductor $c(\chi)>1$. We define a twisting operator $T_\chi$ from paramodular vectors for $\pi$ of level $n$ to paramodular vectors for $\chi \otimes \pi$ of level $\max(n+2c(\chi),4c(\chi))$, and prove that this operator has properties analogous to the well-known $\GL(2)$ twisting operator.