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Espaces préhomogènes de type parabolique. (French) Zbl 0561.17005
Harmonic analysis on Lie groups and related topics, Lect. Math., Dep. Math. Kyoto Univ. 14, 189-221 (1982).
[For the entire collection see Zbl 0533.00008.]
This work studies certain prehomogeneous vector spaces that arise in a natural way in the structure of complex semi-simple Lie algebras. The associated zeta functions are important objects in this study. Let $${\mathfrak g}$$ be a semi-simple Lie algebra and let $${\mathfrak g}_ i$$ (i in $${\mathbb{Z}})$$ be a $${\mathbb{Z}}$$-gradation of $${\mathfrak g}$$, i.e., a sequence of vector subspaces such that [$${\mathfrak g}_ i,{\mathfrak g}_ j]\subset {\mathfrak g}_{i+j}$$. Let $$G_ 0$$ be the subgroup (assumed to be parabolic) of the adjoint group G of $${\mathfrak g}$$ corresponding to $${\mathfrak g}_ 0$$. The natural action of $$G_ 0$$ on $${\mathfrak g}_ 1$$ is prehomogeneous, i.e., there exists a $$G_ 0$$-orbit in $${\mathfrak g}_ 1$$ which is open in the Zariski sense.
The author’s first result (obtained also by Vinberg as follows) is: $${\mathfrak g}_ 1$$ decomposes into a finite number of $$G_ 0$$-orbits. The author goes on to demonstrate the equivalence (in the irreducible case) of the notion of regularity and the existence of certain $$s\ell_ 2$$- triplets. Thus he finds the classification of regular, prehomogeneous spaces of this type. Also it is shown that when [$${\mathfrak g}_ 1,{\mathfrak g}_ 1]=0$$ the local zeta function associated with the prehomogeneous space is interpreted as an intertwining integral of a degenerate principal series of representations of G. Certain orbits are also shown to be symmetric spaces.
Reviewer: M.Walter

##### MSC:
 17B20 Simple, semisimple, reductive (super)algebras 14M17 Homogeneous spaces and generalizations