Hermitian theta series and Maaß spaces under the action of the maximal discrete extension of the Hermitian modular group

Abstract

Let ${\Gamma }_n({\mathcal{O}}_{\mathbb{K}})$ denote the Hermitian modular group of degree $n$ over an imaginary quadratic number field $\mathbb{K}$ and ${\Delta }_{n,\mathbb{K}}^{\ast }$ its maximal discrete extension in the special unitary group $SU(n,n;\mathbb{C})$. In this paper we study the action of ${\Delta }_{n,\mathbb{K}}^{\ast }$ on Hermitian theta series and Maass spaces. For $n=2$ we will find theta lattices such that the corresponding theta series are modular forms with respect to ${\Delta }_{2,\mathbb{K}}^{\ast }$ as well as examples where this is not the case. Our second focus lies on studying two different Maass spaces. We will see that the new found group ${\Delta }_{2,\mathbb{K}}^{\ast }$ consolidates the different definitions of the spaces.