Opérateurs différentiels invariants et problème de Noether

Abstract

Let G be a group and ≤ : 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 ≤(G) acts by automorphisms on the algebra of regular functions k[V], and this action can be canonically extended to theWeyl algebra A n (k) of differential operators over k[V] and then to the skewfield of fractions D n(k) of A n(k). The problem studied in this paper is to determine sufficient conditions for the subfield of invariants of D n(k) under this action to be isomorphic to a Weyl skewfield D m(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 ≤ splits into a sum of representations of dimension one, (2) when ≤ 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.