Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

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 $\left( V,\,Q \right)$ , 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 Conf $\left( V,\,Q \right)$ , the conformal group of the pair $\left( V,\,Q \right)$ , in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak{g}$ , and furthermore a real form ${{\mathfrak{g}}_{\mathbb{R}}}$ . The connected and simply connected Lie group ${{G}_{\mathbb{R}}}$ with $\text{Lie}\left( {{G}_{\mathbb{R}}} \right)\,=\,{{\mathfrak{g}}_{\mathbb{R}}}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi $ .