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