Invariant tensor fields and orbit varieties for finite algebraic transformation groups
Mark Losik, Peter W. Michor, Vladimir L. Popov
备注：
AmSTeX, revised version with 27 pages. More detailed proofs, small mistakes corrected
发表信息：
In: A Tribute to C.S.Seshadri: Perspectives in Geometry and Representation Theory. Hindustan Book Agency, also Trends Math., Birkhäuser, Basel, 2003, 346--378
Let $X$ be a smooth algebraic variety endowed with an action of a finite group $G$ such that there exists the geometric quotient $π_X:X\to X/G$. We characterize rational tensor fields $τ$ on $X/G$ such that the {\it pull back} of $τ$ is regular on $X$: these are precisely all $τ$ such that $\operatorname{div}_{R_{X/G}}(τ)\ge 0$ where $R_{X/G}$ is the {\it reflection divisor} of $X/G$ and $\operatorname{div}_{R_{X/G}}(τ)$ is the {\it $R_{X/G}$-divisor} of $τ$. We give some applications, in particular to the generalization of Solomon's theorem. In the last section we show that if $V$ is a finite dimensional vector space and $G$ a finite subgroup of $\operatorname{GL}(V)$, then each automorphism $ψ$ of $V/G$ admits a biregular lift $ϕ: V\to V$ provided that $ψ$ maps the regular stratum to itself and $ψ_*(R_{X/G})=R_{X/G}$.
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