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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

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