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Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

Published online by Cambridge University Press:  20 November 2018

Dehbia Achab  and
Jacques Faraut
Dehbia Achab
Affiliation:
Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, case 247, 75252 Paris email: [email protected]@math.jussieu.fr
Jacques Faraut
Affiliation:
Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, case 247, 75252 Paris email: [email protected]@math.jussieu.fr
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Abstract

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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 $.

Keywords

17C3622E4632M1533C80minimal representationKantor–Koecher–Tits constructionJordan algebraBernstein identityMeijer G-function
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 4 , 01 August 2012 , pp. 721 - 754
Copyright
Copyright © Canadian Mathematical Society 2012

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