A generalization of Puiseux%26apos;s theorem and lifting curves over invariants

作者：Losik Mark; Michor Peter W^*; Rainer Armin

来源：Revista Matematica Complutense, 2012, 25(1): 139-155.

DOI：10.1007/s13163-011-0062-y

• Puiseux%26apos
• s theorem
• Reductive group representations
• Invariants
• Regular lifting
• Ultradifferentiable
• Denjoy-Carleman

Let rho:G -%26gt; GL (V) be a rational representation of a reductive linear algebraic group G defined over a%26quot;, on a finite dimensional complex vector space V. We show that, for any generic smooth (resp. C (M) ) curve c:a%26quot;ea dagger%26apos;V//G in the categorical quotient V//G (viewed as affine variety in some a%26quot;, (n) ) and for any t (0)aa%26quot;e, there exists a positive integer N such that ta dagger broken vertical bar c(t (0)+/-(t-t (0)) (N) ) allows a smooth (resp. C (M) ) lift to the representation space near t (0). (C (M) denotes the Denjoy-Carleman class associated with M=(M (k) ), which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in V//G admits locally absolutely continuous (not better!) lifts. Assume that G is finite. We characterize curves admitting differentiable lifts. We show that any germ of a C (a) curve which represents a lift of a germ of a quasianalytic C (M) curve in V//G is actually C (M) . There are applications to polar representations.

• 出版日期2012-1