Hecke Operators on Vector-Valued Modular Forms

Abstract

We study Hecke operators on vector-valued modular forms for the Weil representation ρL of a lattice L. We first construct Hecke operators Tᵣ that map vector-valued modular forms of type ρL into vector-valued modular forms of type ρL₍ᵣ₎, where L(r) is the lattice L with rescaled bilinear form (⋅,⋅)ᵣ = r(⋅,⋅), by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vector-valued Hecke operators Tᵣ have appeared in [Comm. Math. Phys. 350 (2017), 1069-1121] as generating functions for D4-D2-D0 bound states on K3-fibered Calabi-Yau threefolds. We study algebraic relations satisfied by the Hecke operators Tᵣ. In the particular case when r = n² for some positive integer n, we compose Tn² with a projection operator to construct new Hecke operators Hn² that map vector-valued modular forms of type ρL into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators Hₙ², and compare our operators with the alternative construction of Bruinier-Stein [Math. Z. 264 (2010), 249-270] and Stein [Funct. Approx. Comment. Math. 52 (2015), 229-252].