We consider the special magnetic Laplacian given byshow that Δ ], is connected to the sub-Laplacian of a group of Heisenberg type given by C× C realized as a central extension of the real Heisenberg group 2 +1 . We also discuss invariance properties of Δ ], and give some of their explicit spectral properties.

We consider a twisted Laplacian ∆ ν,µ on the n-complex space associated with the sub-Laplacian of the Heisenberg group C× ω 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 ∆ Γ ν,µ when acting on some L 2 space of Γ-automorphic functions of biweight (ν, µ) associated to given cocompat discrete subgroup Γ of the additive group 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.

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We introduce and study a generalization of the classical weighted Bergman and Dirichlet spaces on the unit ball in high dimension, the Bergman-Dirichlet spaces. Their counterparts on the whole n-complex space C n , the Bargmann-Dirichlet spaces, are also introduced and studied. Mainly, we give a complete description of the considered spaces, including orthonormal basis and the explicit formulas for their reproducing kernel functions. Moreover, we investigate their asymptotic behavior when the curvature goes to 0.