Tame local Betti geometric Langlands

TL;DR

This paper establishes a monoidal equivalence confirming the tamely ramified local Betti geometric Langlands correspondence, connecting spectral and automorphic perspectives, and offers an alternative proof of a key theorem in the case of unipotent monodromy.

Contribution

It proves the conjectured equivalence in the tamely ramified local Betti geometric Langlands program, advancing understanding of the correspondence.

Findings

Proves monoidal equivalence between spectral and automorphic categories.

Confirms the tamely ramified local Betti geometric Langlands conjecture.

Provides an alternative proof of Bezrukavnikov's fundamental theorem for unipotent monodromy.

Abstract

We prove a monoidal equivalence between spectral and automorphic realizations of the universal affine Hecke category, thereby proving the tamely ramified local Betti geometric Langlands correspondence, as conjectured by Ben-Zvi--Nadler [BZN07, BZN18]. Specializing to the case of unipotent monodromy, this provides another argument for a fundamental theorem of Bezrukavnikov [B16].