Metaplectic and spin representations: a parallel treatment

TL;DR

This paper explores the parallel structures of metaplectic and spin representations, focusing on their group extensions and actions on Fock spaces, highlighting similarities in their cocycles.

Contribution

It provides a unified treatment of Mp$^{c}$ and Spin$^{c}$ groups, emphasizing their analogous cocycles and actions on Gaussian vectors in Fock spaces.

Findings

Identifies similarities between cocycles of Mp$^{c}$ and Spin$^{c}$ extensions.

Develops a parallel framework for analyzing symplectic and orthogonal group representations.

Highlights the role of Gaussian vectors in the representation theory of these groups.

Abstract

The analogies between symplectic and orthogonal groups, regarded as symmetries of real bilinear forms, are manifest in their (metaplectic and spin) projective representations. In finite dimensions, those are true representations of doubly covering groups; but one may also use group extensions by a circle. Here we lay out a parallel treatment of of the Mp$^\mathrm{c}$ and Spin$^\mathrm{c}$ covering groups, acting on the respective Fock spaces by permuting certain Gaussian vectors. The cocycles of these extensions exhibit interesting similarities.