Automorphic Representations of SL(2,ℝ) and Quantization of Fields
1 Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang, Quoc Viet road,
Cau Giay district, 10307 Hanoi, Vietnam
2 Medicine University College, Thai Nguyen University, Thai Nguyen City, Vietnam
Abstract
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 Langlands picture of
automorphic representations is precised by using the Poisson summation formula.
