X-Nico

Prehomogeneous vector space

E. B. Vinberg, On the classification of the nilpotent elements of graded Lie algebras, Soviet Math Doklady, volume 16 (1975) 1517-1520.

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This shows in particular (and was noted independently by Vinberg in 1975) that the Levi factor G of P acts prehomogeneously on $V:=\backslash mathfrak\; p^\backslash perp/\backslash mathfrak\; p^\backslash perp,\backslash mathfrak\; p^\backslash perp$.

see also