Lifting quasianalytic mappings over invariants (Q2882890)

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 \(V/\!/G\).NEWLINENEWLINEThe 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 [\textit{E. Bierstone} and \textit{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 \((x_1,\dots,x_q)\rightarrow (\pm\, x_1^{n_1},\dots,\pm\, x_q^{n_q})\)).NEWLINENEWLINEThen 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).NEWLINENEWLINEThe 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 \(E\).