Let G be a reductive algebraic group and let B be an affine variety with an algebraic action of G. Everything is defined over the field C of complex numbers. Consider the trivial G-vector bundle B x S = S over B where S is a G-module. From the endomorphism ring R of the G-vector bundle S a construction of G-vector bundles over B is given. The bundles constructed this way have the property that when added to S they are isomorphic to F + S for a fixed G-module F. For such a bundle E an invariant rho(E) is defined that lies in a quotient of R. This invariant allows us to distinguish nonisomorphic G-vector bundles. This is applied to the case where B is a G-module and, in that case, an invariant of the underlying equivariant variety is given too. These constructions and invariants are used to produce families of inequivalent G-vector bundles over G-modules and families of inequivalent G actions on affine spaces for some finite and some connected semisimple groups.
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机译：令G为一个还原性代数群，令B为一个具有G代数作用的仿射变种。一切都定义在复数场C上。考虑B上的平凡G矢量包B x S = S，其中S是G模块。从G向量束S的内同态环R出发，给出了B之上的G向量束的构造。以这种方式构造的束具有以下性质：当将它们添加到S时，对于固定G模块F，它们与F + S同构。对于这样的束E，定义了不变的rho（E），它等于R的商。不变性使我们能够区分非同构G向量束。这适用于B是G模的情况，在这种情况下，也给出了基本等变品种的不变量。这些构造和不变量用于在某些有限的和某些相连的半简单群的仿射空间上，在G-模块上产生不等价的G-向量束族，以及不等式的G作用族。
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