The Pseudo-Character of the Weil Representation and its Relation with the Conley-Zehnder Index

TL;DR

This paper computes the character of the Weil representation by linking the Weyl symbol of metaplectic operators with the symplectic Cayley transform and the Conley--Zehnder index, revealing deep geometric connections.

Contribution

It introduces a novel method to calculate the Weil representation's character using symplectic geometry tools and the Conley--Zehnder index, enhancing understanding of metaplectic operators.

Findings

Explicit formula for the Weil representation's character.

Connection established between Weyl symbols and symplectic invariants.

Enhanced understanding of the geometric structure of metaplectic operators.

Abstract

We calculate the character of the Weil representation using previous results which express the Weyl symbol of metaplectic operators in terms of the symplectic Cayley transform and the Conley--Zehnder index.