### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 041, 31 pages      arXiv:1807.07703      https://doi.org/10.3842/SIGMA.2019.041
Contribution to the Special Issue on Moonshine and String Theory

### Hecke Operators on Vector-Valued Modular Forms

Vincent Bouchard a, Thomas Creutzig ab and Aniket Joshi a
a) Department of Mathematical & Statistical Sciences, University of Alberta, 632 Central Academic Building, Edmonton T6G 2G1, Canada
b) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Received September 26, 2018, in final form May 13, 2019; Published online May 25, 2019

Abstract
We study Hecke operators on vector-valued modular forms for the Weil representation $\rho_L$ of a lattice $L$. We first construct Hecke operators $\mathcal{T}_r$ that map vector-valued modular forms of type $\rho_L$ into vector-valued modular forms of type $\rho_{L(r)}$, where $L(r)$ is the lattice $L$ with rescaled bilinear form $(\cdot, \cdot)_r = r (\cdot, \cdot)$, by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vector-valued Hecke operators $\mathcal{T}_r$ 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 $\mathcal{T}_r$. In the particular case when $r=n^2$ for some positive integer $n$, we compose $\mathcal{T}_{n^2}$ with a projection operator to construct new Hecke operators $\mathcal{H}_{n^2}$ that map vector-valued modular forms of type $\rho_L$ into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators $\mathcal{H}_{n^2}$, 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].

Key words: Hecke operators; vector-valued modular forms; Weil representation.

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