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We develop a theory of vector valued automorphic forms associated to the Weilrepresentation $\omega_f$ and corresponding to vector valued modular formstransforming with the "finite" Weil representation $\rho_L$. For each prime $p$we determine the structure of a vector valued spherical Hecke algebra dependingon $\omega_f$, which acts on the space of automorphic forms. We finally definea standard $L$-function of such an automorphic form an prove that it can becontinued meromorphically to the whole complex $s$-plane.

On vector valued automorphic forms and the standard $L$-function attached to them