Finite Weil representations and associated Hecke algebras

An algebra H(Gm) of double cosets is constructed for every finite Weil representation Gm. For the Clifford-Weil groups Gm=Cm(ρ) associated to some classical Type ρ of selfdual codes over a finite field, this algebra is shown to be commutative. Then the eigenspace decomposition of H(Cm(ρ)) acting on the space of degree N invariants of Cm(ρ) may be obtained from the kernels of powers of the coding theory analogue of the Siegel Φ-operator.