Tame local Betti geometric Langlands

TL;DR

The paper proves a Tamely ramified local Betti geometric Langlands equivalence by constructing a monoidal bridge between spectral and automorphic realizations of the universal affine Hecke category, identifying IndCoh_G on the Steinberg-type space with the nilpotent-signal automorphic category. The strategy follows the Kazhdan–Lusztig and Arkhipov–Bezrukavnikov paradigm, realized through a functorial construction that uses the tautological spectral module, Colocalization, and Morita techniques to transfer structures across sides, with Wakimoto sheaves providing key t-structure control. A crucial outcome is that 2-IndCoh_nilp(G/G) is equivalent to Shv_nilp(I\ LG / I)-mod, recovering Bezrukavnikov’s theorem in the unipotent case and supplying a robust framework for tame ramification in the Betti setting. The results open avenues for extensions to de Rham, integral coefficients, and mixed characteristic contexts, and they offer a concrete spectral-automorphic correspondence in families with varying 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].

Theorems & Definitions (24)

• Theorem 1.2.5
• Remark 1.2.6
• Lemma 2.3.7
• Lemma 2.3.9
• proof : Proof of Lemma \ref{['l:verdamp']}
• Theorem 2.5.4
• Theorem 2.5.5
• Proposition 3.1.2
• proof
• Proposition 3.2.2