ANALYSE OF THE BRYLINSKI-KOSTANT MODEL FOR SPHERICAL MINIMAL REPRESENTATIONS

AbstractWe 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 Ξ 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 Ξ