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# Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

Published:2012-06-09
Printed: Aug 2012
• Dehbia Achab,
Institut de Math\'ematiques de Jussieu, Universit\'e Pierre et Marie Curie, 4 place Jussieu, case 247, 75252 Paris cedex 05
• Jacques Faraut,
Institut de Math\'ematiques de Jussieu, Universit\'e Pierre et Marie Curie, 4 place Jussieu, case 247, 75252 Paris cedex 05
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## 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 $\mathfrak g$, and furthermore a real form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi$.
 Keywords: minimal representation, Kantor-Koecher-Tits construction, Jordan algebra, Bernstein identity, Meijer $G$-function
 MSC Classifications: 17C36 - Associated manifolds 22E46 - Semisimple Lie groups and their representations 32M15 - Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15] 33C80 - Connections with groups and algebras, and related topics