Analysis of the Brylinski-Kostant model for spherical minimal representations

2011/01/23 by Achab, Dehbia, Faraut, Jacques
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1101.4402

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 Ξ 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 Ξ

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