# Quantization of Fields and Automorphic Representations

## TL;DR

This paper explores the use of geometric quantization and the Geometric Langlands Correspondence to realize automorphic representations across various classes of groups, including affine, solvable, nilpotent, and reductive Lie groups.

## Contribution

It introduces a unified framework leveraging geometric quantization and Langlands duality to construct automorphic representations for diverse group types.

## Key findings

- Constructs automorphic representations for affine Heisenberg groups using Fock spaces.
- Shows that representations for solvable and nilpotent groups arise from geometric quantization methods.
- Demonstrates that all automorphic representations of reductive groups can be obtained through iterative geometric quantization and Langlands duality.

## Abstract

In this paper we use the quantization of fields based on Geometric Langlands Correspondence \cite{diep1} to realize the automorphic representations of some concrete series of groups: for the affine Heisenberg (loop) groups it is reduced to the construction of the affine Kac-Moody representation by the Weyl relations in Fock spaces. For the solvable and nilpotent groups following the construction we show that it is the result of applying the constructions of irreducible unitary representation via the geometric quantization and the construction of positive energy representations and finally, for the semi-simple or reductive Lie groups, using the Geometric Langlands Correspondence, we show that a repeated application of the construction give all the automorphic representations of reductive Lie groups: first we show that every representation of the fundamental group of Riemann surface into the dual Langlands groups ${}^LG$ of $G$ corresponds to a representation of the fundamental group of the surface into the reductive group $G$, what is corresponding to a quantum inducing bundle of the geometric quantization of finite dimensional reductive Lie groups and then apply the construction of positive energy representation of loop groups.