On traces of operators associated with the actions of compact Lie groups

Let M be a compact smooth closed manifold, and G a discrete group. A G-operator on M is an operator of finite sum defined by D=sumlimits_{gin G} D_gPhi_g:H^s(M)to H^{s-m}(M), where the D_g are (pseudo)differential operators with orders m and gto Phi_g is a representation of the group G by operators which operate on functions on M. par The main aim of this paper is to study a new class of elliptic G-operators, associated with a representation of the group G by quantum canonical transformations Phi_g. par The following theorem is the main theorem of this paper: par Theorem 2. Let the G-operator 1+D:L^2(M)to L^2(M) be elliptic. Then 1+D is Fredholm.