Invariant Differential Operators in Symmetrical Spaces. II.Generalized Harish-Chandra Homomorphism

We construct in this article an injective homomorphism, γ, from the algebra of invariant differential operators on a general symmetric space in the field of invariant rational functions on the cotangent space at the origin. We conjecture that this homomorphism is an isomorphism into the invariant polynomial functions. This fact would prove a conjecture of Duflo. For describing γ, we introduce, in the first part, a class of symmetric spaces for which the symmetrisation is an isomorphism.