Analysis of the minimal representation of Sp(r,R)

TL;DR

This paper explores two different Hilbert space models for the minimal representations of Sp(r,R), describing their relationship via a transformation akin to the Bargmann transform, bridging holomorphic and L^2 function realizations.

Contribution

It introduces a detailed analysis of the two models of minimal representations of Sp(r,R) and constructs a transformation connecting them, analogous to the Bargmann transform.

Findings

Explicit description of the holomorphic model.

Explicit description of the Schrödinger model.

Construction of a transformation linking the two models.

Abstract

The minimal representations of Sp(r,R) can be realized on a Hilbert space of holomorphic functions. This is the analogue of the Brylinski-Kostant model. It can also be realized on a Hilbert space of L^2 functions on R^r. This is the Schr\"oodinger model. We will describe the two realizations and a transformation which maps one model to the other. It involves the classical Bargmann transform and can be seen as its analogue.