Restricted Representations of Loop Groups and the Categorical Trace Conjecture

Introduction to Categorical Representations

• 💡 The talk introduces a framework for categorical representations of loop groups in the l-adic context.
• 🧠 It distinguishes between a global story (older, reminder/motivation) and a local story (more speculative).
• 📌 The core idea involves refining the Grothendieck dictionary of sheaves to functions using a categorical trace.

Global Story: Automorphic Sheaves and Langlands

• 🎯 The discussion focuses on automorphic sheaves on the moduli stack of G-bundles on a curve.
• ✅ A key theorem states that the trace of the Frobenius endomorphism on sheaves with nilpotent singular support is isomorphic to automorphic functions.
• 🔑 This connects to the geometric Langlands conjecture, proposing an equivalence between automorphic sheaves and quasi-coherent sheaves on a stack of local systems.

Defining Categorical Actions and Restricted Categories

• ⚠️ Naive definitions of categorical actions for groups are problematic due to the failure of the categorical unit formula.
• 🛠️ The framework of DJCAT ALGOME (Drinfeld's framework) is introduced to address these issues, allowing for a robust definition of categories with group actions.
• 🧐 Restricted categories are defined by imposing conditions on their invariants, aiming to trim down the enlarged DJCAT ALGOME framework to match the Langlands dual side.

Loop Groups and the Trace Conjecture

• 🔬 For loop groups, the definition of restricted categories involves concepts like "depth" and Mo-Prasad theory, as refined by JKU and Fininsson.
• 📊 The talk presents a conjecture that the trace of the Frobenius endomorphism on restricted loop group categories recovers sheaves on "iso-crystals."
• 🚀 This conjecture, combined with local geometric Langlands, implies the equivalence of the Fargues-Scholze category with the arithmetic version of local systems.