Opérateurs différentiels invariants et problème de Noether

Abstract

Let G be a group and ρ: G → GL(V) a representation of G in a vector space V of dimension n over a commutative field k of characteristic zero. The group ρ(G) acts by automorphisms on the algebra of regular functions k[V], and this action can be canonically extended to the Weyl algebra An(k) of differential operators over k[V] and then to the skewfield of fractions Dn(k) of An(k). The problem studied in this paper is to determine sufficient conditions for the subfield of invariants of Dn(k) under this action to be isomorphic to a Weyl skewfield Dm(K) for some integer 0 ≤ m ≤ n and some purely transcendental extension K of k. We obtain such an isomorphism in two cases: (1) when ρ splits into a sum of representations of dimension one, (2) when ρ is of dimension two. We give some applications of these general results to the actions of tori on Weyl algebras and to differential operators over Kleinian surfaces.