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

TL;DR

This paper explores the connection between automorphic representations of SL(2,R) and quantum field quantization via the Geometric Langlands Correspondence, employing loop group representations and Poisson summation.

Contribution

It establishes a detailed relationship between automorphic representations and quantization, clarifying their geometric and algebraic structures through new realizations.

Findings

Discrete series realized in eigenspaces of Cartan generator

Automorphic representations expressed as induced representations with quantum bundles

Use of loop group representations to realize automorphic representations

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 Lanlands picture of automorphic representations is precised by using the Poisson summation formula.