Holomorphic quantization on the torus and finite quantum mechanics

We explicitly construct the quantization of classical linear maps of SL(2, R) on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that finite quantum mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of SL(2, Z) to the subgroup SL(2, Z)/Gamma(l), Gamma(l) being the principal congruent subgroup modl, on a finite dimensional Hilbert space. The generators of the 'rotation group' modl, O-l(2) subset of SL(2, l), for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.