Decomposition of reductive regular Prehomogeneous Vector Spaces
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 2183-2218.

Let (G,V) be a regular prehomogeneous vector space (abbreviated to PV), where G is a reductive algebraic group over . If V= i=1 n V i is a decomposition of V into irreducible representations, then, in general, the PV’s (G,V i ) are no longer regular. In this paper we introduce the notion of quasi-irreducible PV (abbreviated to Q-irreducible), and show first that for completely Q-reducible PV’s, the Q-isotypic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate sense, any regular PV is a direct sum of Q-irreducible PV’s. Finally we classify the Q-irreducible PV’s of parabolic type.

Soit (G,V) un espace préhomogène (en abrégé PV) régulier, où G est un groupe algébrique réductif, défini sur . Si V= i=1 n V i est une décomposition de V en représentations irréductibles, alors, en général, les espaces préhomogènes (G,V i ) ne sont pas réguliers. Dans cet article nous introduisons la notion de PV quasi-irréductible (en abrégé Q-irréducible), et nous montrons d’abord que pour les PV complètement Q-réductibles, les composantes Q-isotypiques sont définies de manière intrinsèque, comme en théorie ordinaire des représentations. Nous montrons également que, dans un sens approprié, tout PV régulier est une somme directe de PV quasi-irréductibles. Finalement nous classifions les PV de type parabolique qui sont Q-irréductibles.

DOI: 10.5802/aif.2670
Classification: 11S90,  20G05,  17B20
Keywords: reductive groups, prehomogeneous vector spaces, relative invariants, prehomogeneous vector spaces of parabolic type
Rubenthaler, Hubert 1

1 Université de Strasbourg et CNRS Institut de Recherche Mathématique Avancée 7 rue René Descartes 67084 Strasbourg Cedex (France)
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Rubenthaler, Hubert. Decomposition of reductive regular Prehomogeneous Vector Spaces. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 2183-2218. doi : 10.5802/aif.2670. http://archive.numdam.org/articles/10.5802/aif.2670/

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