Affine Hecke categories in equal and mixed characteristic

TL;DR

The paper addresses the problem of relating affine Hecke categories in mixed and equal characteristic for quasi-split groups over p-adic fields. It develops a framework based on Bruhat-Tits group schemes, parahoric and Iwahori structures, and diagrams of Hecke stacks, then uses a contracted semidirect product to lift neutral-block equivalences to the full categories. The main result is a canonical monoidal equivalence between ${\mathcal{H}}_{\mathrm{mon},\mathrm{aff}}$ and ${\mathcal{H}}_{\mathrm{mon},\mathrm{aff}}^{\flat}$ (and similarly for the unipotent variant) that is Frobenius-compatible and matches (co)standard objects. This provides a robust bridge between mixed and equal characteristic Langlands-type structures, recovering Satake-type equivalences and offering new tools for the study of local geometric Langlands in both characteristics.

Abstract

For a quasi-split tamely connected reductive group G over a p-adic field, we prove that its (monodromic) affine Hecke category is canonically equivalent to its equal characteristic counterpart as monoidal categories.

Theorems & Definitions (14)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 3.1
• Remark 3.2
• Theorem 3.3: Tao-Travkin
• Remark 3.4