# Fundamental local equivalences in quantum geometric Langlands

## TL;DR

This paper proves the conjectural fundamental local equivalence in quantum geometric Langlands, extending the Satake equivalence's role using Soergel module techniques to affine flag varieties.

## Contribution

It establishes the fundamental local equivalence in quantum geometric Langlands and extends it to affine flag varieties, employing novel Soergel module methods.

## Key findings

- Proves the fundamental local equivalence conjecture in quantum geometric Langlands.
- Extends the equivalence to affine flag varieties.
- Uses Soergel module techniques for the proof.

## Abstract

In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory--Lurie proposed a conjectural substitute, later termed the fundamental local equivalence. With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques.