Decomposition of reductive regular prehomogeneous vector spaces

Hubert Rubenthaler (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let (G,V) be a regular prehomogeneous vector space (abbreviated to PV), where G is a connected reductive algebraic group over C. If $V= \oplus_{i=0}^{n}V_{i}$ 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-isotopic 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 quasi-irreducible PV's. Finally we classify the quasi-irreducible PV's of parabolic type.

Domaines: Théorie des représentations [math.RT]; Anneaux et algèbres [math.RA]

Hubert Rubenthaler. Decomposition of reductive regular prehomogeneous vector spaces. Annales de l'Institut Fourier, 2011, 61 (5), pp.2183-2218. ⟨hal-00464602v2⟩