The restricted Siegel disc as coadjoint orbit

TL;DR

This paper demonstrates that the restricted Siegel disc can be understood as a coadjoint orbit of a universal central extension of the restricted symplectic group, linking complex analysis, geometry, and group theory.

Contribution

It establishes the restricted Siegel disc as a coadjoint orbit of the universal central extension of the restricted symplectic group, revealing new geometric structure.

Findings

Identifies the restricted Siegel disc as a coadjoint orbit.

Connects the Siegel disc to the universal central extension.

Uses the Schwinger term as the cocycle in the orbit description.

Abstract

The restricted Siegel disc is a homogeneous space related to the connected component $T_0(1)$ of the Universal Teichm\"uller space via the period mapping. In this paper we show that it is a coadjoint orbit of the universal central extension of the restricted symplectic group or, equivalently, an affine coadjoint orbit of the restricted symplectic group with cocycle given by the Schwinger term.