Affine Beilinson-Bernstein localization at the critical level for $GL_2$

TL;DR

This paper proves a key case of a conjecture linking critical level Kac-Moody representations with geometric localization on the affine Grassmannian for $GL_2$, using methods inspired by local geometric Langlands theory.

Contribution

It establishes the rank 1 case of the Frenkel-Gaitsgory conjecture, connecting representation theory and geometric localization at the critical level for $GL_2$.

Findings

Localization of critical level representations onto the affine Grassmannian.

Use of local geometric Langlands analogue of Whittaker models.

Proof of the conjecture for the rank 1 case.

Abstract

We prove the rank 1 case of a conjecture of Frenkel-Gaitsgory: critical level Kac-Moody representations with regular central characters localize onto the affine Grassmannian. The method uses an analogue in local geometric Langlands of the existence of Whittaker models for most representations of $GL_2$ over a non-Archimedean field.