On vector valued automorphic forms for the Weil representation

TL;DR

This paper develops a theory of vector valued automorphic forms linked to the Weil representation, analyzing their structure and the action of associated Hecke algebras for each prime.

Contribution

It introduces a new framework connecting vector valued automorphic forms with the Weil representation and characterizes the local Hecke algebra structures.

Findings

Determined the structure of vector valued spherical Hecke algebras for each prime.

Established a correspondence between automorphic forms and the Weil representation.

Provided tools for analyzing automorphic forms with vector valued modular transformations.

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.