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Prehomogeneous vector spaces and field extensions. II. (English) Zbl 0889.12004

The purpose of this paper is to give a correspondence between the orbits of a prehomogeneous vector space and the conjugacy classes of homomorphisms from the Galois group of field extensions to the permutation group. In this paper the authors deal with the following three types of prehomogeneous vector spaces \((G,V)\) defined over an infinite field \(k\). (1) \(G=\text{GL}(2)_{k_1} \times \text{GL}(2)_k\) where \(k_1/k\) is a fixed quadratic extension and \(V\) is the space of pairs of binary Hermitian forms. (2) \(G=\text{GL}(1)_k \times \text{GL} (2)_{k_1}\) where \(k_1/k\) is a fixed cubic extension and \(V\) is an eight-dimensional representation of \(G\), which becomes the \(D_4\) cases of the previous paper [D. J. Wright and A. Yukie, Invent. Math. 110, 283–314 (1992; Zbl 0803.12004)] after a suitable field extension of \(k\). (3) \(G=\text{GL}(3)_{k_1} \times \text{GL}(2)_k\) where \(k_1/k\) is a fixed quadratic extension and \(V\) is the space of pairs of ternary Hermitian forms.
Reviewer: M.Muro (Yanagido)

MSC:

12F10 Separable extensions, Galois theory
11S90 Prehomogeneous vector spaces
14M17 Homogeneous spaces and generalizations

Citations:

Zbl 0803.12004
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