Automorphic Representations of $\SL(2,\mathbb R)$ and Quantization of Fields

In this paper we make a clear relationship between the automorphic representations and the quantization through the Geometric Langlands Correspondence. We observe that the discrete series representation are realized in the sum of eigenspaces of Cartan generator, and then present the automorphic representations in form of induced representations with inducing quantum bundle over a Riemann surface and then use the loop group representation construction to realize the automorphic representations. The Lanlands picture of automorphic representations is precised by using the Poisson summation formula.

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