On vector valued automorphic forms for the Weil representation

Abstract

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.