A formal notion of a Typ T of a self-dual linear code over a finite left R-module V is introduced which allows to give explicit generators of a finite complex matrix group, the associated Clifford-Weil group C(T) ≤ GL_|V| (C), such that the complete weight enumerators of self-dual isotropic codes of Type T span the ring of invariants of [Cscr ](T). This generalizes Gleason’s 1970 theorem to a very wide class of rings and also includes multiple weight enumerators (see Section 2.7), as these are the complete weight enumerators cwe_m (C) = cwe(R^m ⊗ C) of R^{m × m} -linear self-dual codes R^m ⊗ C ≤ (V^m)^N of Type T^m with associated Clifford-Weil group [Cscr ]_m(T) = [Cscr ](T^m). The finite Siegel Φ-operator mapping cwe_m(C) to cwe_m−1(C) hence defines a ring epimorphism Φ_m : Inv([Cscr ]_m(T)) ⇒ Inv([Cscr ]_m−1(T)) between invariant rings of complex matrix groups of different degrees. If R = V is a finite field, then the structure of [Cscr ]_m(T) allows to define a commutative algebra of [Cscr ]_m (T) double cosets, called a Hecke algebra in analogy to the one in the theory of lattices and modular forms. This algebra consists of self-adjoint linear operators on Inv([Cscr ]_m (T)) commuting with Φ_m. The Hecke-eigenspaces yield explicit linear relations among the cwe_m of self-dual codes C ≤ V^N.