A birational invariant for algebraic group actions

We construct a birational invariant for certain algebraic group actions. We use this invariant to classify linear repre-sentations of finite abelian groups up to birational equivalence, thus answering, in a special case, a question of E.B. Vinberg and giving a family of counterexamples to a related conjec-ture of P.I. Katsylo. We also give a new proof of a theorem of M. Lorenz on birational equivalence of quantum tori (in a slightly expanded form) by applying our invariant in the setting of PGLn-varieties. 1. Introduction. Let G be an algebraic group and let X be a smooth projective G-variety (i.e., an algebraic variety with a G-action) defined over an algebraically closed base field of characteristic zero. It is shown in [RY1] that for each finite abelian subgroup H, the presence of an H-fixed points is a birationa