Lifting smooth curves over invariants

Armin Rainer
(University of Vienna)

Abstract: Consider a smooth curve of polynomials with all roots real. Then it is known that the roots can be parameterized twice differentiably, but not better. This problem can be viewed as lifting problem of smooth curves (of polynomials) in the orbit space of the standard representation of the symmetric group which acts on $\mathbb{R}^n$ by permuting the coordinates (the roots) to the representation space. Now I am interested in the following generalization: Given a smooth curve in the orbit space of a compact Lie group representation, is it possible to find twice differentiable lifts to the representation space? In fact it is possible for all orthogonal representations of finite groups.