I am interested in harmonic Maass form. I would like to know the historical background of hyperbolic Laplacian $$ \Delta_{k}=-y^2(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})+iky(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}). $$ I know that $\Delta_{0}$ is the same as the Laplace-Beltrami operator of the hyperbolic plane $\{x+iy \mid x,y \in \mathbb{R},\ y>0\}$ with metric $ds^2 = \frac{dx^2+dy^2}{y^2}.$ Then, what is the second term $iky(\frac{\partial}{\partial x}+\frac{\partial}{\partial y})$? In what kind of study was $\Delta_{k}$ discoverd?
I know that Maass form is named after Hans Maass. But I do not know much about his work because I cannot read German well. I read "On two geometric theta lifts" (Bruinier and Funke), but I could not understand why $\Delta_{k}$ was introdued in that paper.
Please tell me the background of $\Delta_{k}$ or some good reference about it.