Concrete $L^2$-spectral Analysis of a Bi-weighted $\Gamma$-automorphic Twisted Laplacian

Abstract

We consider a twisted Laplacian $\Delta_{\nu,\mu}$ on the $n$-complex space associated with the sub-Laplacian of the Heisenberg group $\mathbb{C} \times_{\omega} \mathbb{C}^n$ realized as a central extension of the real Heisenberg group $H_{2n+1}$. The main results to which is aimed this paper concern the spectral theory of $\Delta_{\nu,\mu}^{\Gamma}$ when acting on some $L^2$ space of $\Gamma$-automorphic functions of biweight $(\nu,\mu)$ associated to given cocompat discrete subgroup $\Gamma$ of the additive group $\mathbb{C}^n$. We describe its spectrum proving a stability theorem. Using the Selberg's approach, we give the explicit dimension formula for the corresponding $L^2$-eigenspaces. We also construct a concrete basis of such $L^2$-eigenspaces.