Conductors and local newforms for the metaplectic group of rank 1

TL;DR

This paper defines and computes conductors and local newforms for irreducible genuine representations of the rank 1 metaplectic group over non-archimedean local fields, extending previous work and establishing explicit formulas and compatibility results.

Contribution

It introduces explicit formulas for conductors and dimensions of local newforms for the rank 1 metaplectic group, and demonstrates their compatibility with the local theta correspondence.

Findings

Explicit formulas for conductors of representations.

Dimension calculations for spaces of local newforms.

Compatibility with local theta correspondence.

Abstract

In an earlier paper of W. Casselman, the theory of local newforms and conductors was initiated. Later, Roberts and Schmidt studied local newforms for the metaplectic group of rank 1. In this paper we define and calculate conductors of irreducible genuine representations of the metaplectic group of rank 1 over non-archimedean local field of characteristic zero and of odd residual characteristic. Moreover, we shall give an explicit formulae for dimensions of spaces of local newforms, and show a compatibility with the local theta correspondence.