Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

Abstract

Abstract We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start froma pair (V,Q), where V is a complex vector space and Q a homogeneous polynomial of degree 4 on V. The manifold is an orbit of a covering of 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 𝔤, and furthermore a real form 𝔤ℝ. The connected and simply connected Lie group Gℝ with Lie(Gℝ) = 𝔤ℝ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold .