Analysis of the Brylinski-Kostant model for spherical minimal representations

Abstract

We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi $ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\goth g$, and furthermore a real form ${\goth g}_{\bboard R}$. The connected and simply connected Lie group $G_{\bboard R}$ with ${\rm Lie}(G_{\bboard R})={\goth g}_{\bboard R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi $