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Lifting quasianalytic mappings over invariants. (English) Zbl 1243.14039

Let \(V\) be a finite dimensional representation of a complex reductive linear algebraic group \(G\). The author is interested in lifting mappings \(f:U\subset \mathbb{R}^q\rightarrow V/\!/G\) to maps \(\bar{f}:U\rightarrow V\), where \(U\subset \mathbb{R}^q\) is an open subset and \(\pi:V\rightarrow V/\!/G=\mathrm{Spec}(C[V]^G)\) is the categorical quotient. One can think of \(V/\!/G\) as a closed subset of a complex space by choosing a system of homogeneous generators of \(C[V]^G\). The author requires that \(\pi\circ \bar{f}=f\), that the image of \(\bar{f}\) is contained in the union of the closed \(G\)-orbits plus some conditions on the regularity of \(\bar{f}\) depending on the regularity of \(f\). Note that there is a one-to-one correspondence given by \(\pi\) between the set of closed \(G\)-orbits in \(V\) and the points of \(V/\!/G\).
The author considers a class \(C\) of \(C^\infty\) functions which contains the real analytic class \(C^\omega\) and is stable under composition, derivation, division by coordinates and taking the inverse (for example \(C^\omega\)). Then, by [E. Bierstone and P. D. Milman, Invent. Math. 128, No. 2, 207–302 (1997; Zbl 0896.14006); Sel. Math., New Ser. 10, No. 1, 1–28 (2004; Zbl 1078.14087)], the category of \(C\)-manifolds and \(C\)-maps admits resolution of singularities. Given a \(C\) mapping \(f:M\rightarrow V/\!/G\) from a \(C\)-manifold and a compact \(K\subset M\) there is a open covering \(K\subset\bigcup U_k\) of \(K\) such that \(f|U_k\) can be \(C\)-lifted to \(V\) after composing with finitely many blow-up and power substitutions (i.e. mapping given in local coordinates by \((x_1,\dots,x_q)\rightarrow (\pm\, x_1^{n_1},\dots,\pm\, x_q^{n_q})\)).
Then he proves that a \(C\) map \(f: U\subset \mathbb{R}^q\rightarrow V/\!/G\) admits a lift \(\bar{f}\in W^C_{\mathrm{loc}}\), i.e. \(\bar{f}\) is of class \(C\) outside a nullset \(E\) of finite \((q-1)\) measure such that its classical derivative is locally integrable. Moreover \(f\in SBV_{\mathrm{loc}}\) (\(SBV\) stands for special function of bounded variation).
The author shows that the regularity of \(\bar{f}\) is best possible, but is not known if the assumption on the regularity of \(f\) are optimales. Finally he prove for real polar representation of compact lie group that \(\bar{f}\) is “piecewise locally Lipschitz”, i.e. its classical derivative is locally bounded outside the exceptional set \(E\).

MSC:

14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
13A50 Actions of groups on commutative rings; invariant theory
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