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.

arXiv commentRevised version incorporating changes made in response to the referees' comments, together with further corrections and clarifications. In particular, the covering group (\widetilde{\mathcal G}(\mathbb A)) has been incorporated to make the definition of automorphic forms compatible with the square-root factor in the Weil representation. The main results remain unchanged