Analysis of minimal representations of SL(n,R)

TL;DR

This paper explores two different Hilbert space models for minimal representations of SL(n,R), describing their construction, interrelation, and an analogue of the Bargmann transform connecting them.

Contribution

It introduces and compares two models of minimal representations of SL(n,R), providing a transformation analogous to the Bargmann transform between them.

Findings

Two models of minimal representations are described.

A transformation connecting the models is constructed.

The models are shown to be analogous to classical representations.

Abstract

Some minimal representations of SL(n,R) can be realized on a Hilbert space of holomorphic functions. This is the analogue of the Brylinski-Kostant model. They can also be realized on a Hilbert space of homogeneous functions on ${\bboard R}^n$. This is the analogue of the Kobayashi-Orsted model. We will describe the two realizations and a transformation which maps one model to the other. It can be seen as an analogue of the classical Bargmann transform.