Structure of prehomogeneous spaces associated with certain graded Lie algebras
Let \(\mathfrak g\) be a complex semisimple Lie algebra. We consider \(\mathbb Z\)-gradations of the following type: \(\mathfrak g={\mathfrak g}_{-1}\oplus {\mathfrak g}_ 0\oplus {\mathfrak g}_ 1\). Let \(G\) be the adjoint group of \(\mathfrak g\), \(G_ 0\) be the connected subgroup corresponding to \({\mathfrak g}_ 0\) and let \(N_ 0\) be a maximal unipotent subgroup of \(G_ 0\). It is known that the natural action of \(G_ 0\) on \({\mathfrak g}_ 1\) is prehomogeneous (i.e. has a Zariski open orbit). We first describe, using the root system of \(\mathfrak g\), the orbital decomposition of the preceding action. If the parabolic subalgebra \({\mathfrak g}_ 0\oplus {\mathfrak g}_ 1\) is auto-opposed we study the \(N_ 0\)-action on \({\mathfrak g}_ 1\) and we prove a generalization of the well known ``Gauss lemma'' concerning reduction of quadratic forms. We also give the structure of the ring of \(N_ 0\)-invariant polynomials. Finally, the preceding results enable us to calculate (via the Sato-Kashiwara-Kimura-Oshima algorithm) the b-functions associated to these prehomogeneous spaces.
scientific article; zbMATH DE number 3906688