Hecke operators on vector-valued modular forms of the Weil representation

Abstract

Vector-valued modular forms of the Weil representation are an indispensable tool in diverse areas of mathematics such as enumerative geometry of Calabi-Yau manifolds and rational conformal field theory. In this thesis, we study Hecke operators on vector-valued modular forms of the Weil representation of a lattice L. We first construct Hecke operators T_r that map vector-valued modular forms of a certain type into vector-valued modular forms of rescaled lattices by lifting standard Hecke operators for scalar-valued modular forms through Siegel theta functions. We also get a set of algebraic relations satisfied by the Hecke operators T_r similar to the scalar-valued case. In the particular case when r is a square number, the Weil representation of the rescaled lattice carries a sub-representation of the Weil representation of the original lattice and we can compose T_r with a projection operator to construct new Hecke operators H_r that map vector-valued modular forms of a certain type into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators H_r , and compare our operators with the Hecke operators of Bruinier and Stein obtained by a different construction.