From Character Sheaves to Deligne-Lusztig Representations via Categorical Traces

Generalizing Deligne-Lusztig Representations

• 💡 The talk aims to generalize Lusztig's result connecting character sheaves to Deligne-Lusztig representations.
• 🎯 This generalization utilizes categorical trace machinery as a core method.
• 🔑 The work builds upon Harish-Chandra induction, which is a functor independent of parabolic subgroups and maps class functions.

Geometric Interpretation and Challenges

• 🔬 Harish-Chandra induction has a geometric interpretation involving sheaves and the Grothendieck Springer functor.
• ⚡ The trace of Harish-Chandra corresponds to the trace of Frobenius on the Grothendieck Springer functor.
• ⚠️ Generalizing to "twisted Levi" subgroups with Deligne-Lusztig functors faces a challenge: the Grothendieck Springer functor does not generally lift to Frobenius equivariant objects.

Solution: Character Sheaves and Categorical Traces

• ✅ The solution involves restricting to character sheaves (or monadromic models), a specific subcategory.
• 🚀 Within this subcategory, the Grothendieck Springer functor lifts to equivariant objects, making the geometric interpretation valid.
• 🧠 The proof strategy constructs functors between infinity two categories (or monadromic models) and uses categorical traces to relate them.

Key Theorems and Proof Strategy

• 📌 A main theorem asserts the canonical independence of the Deligne-Lusztig functor from the choice of parabolic subgroup.
• 💡 This independence and the lifting property are corollaries of the higher-categorical construction.
• 🛠️ The "intertwining operator" between different parabolics becomes an isomorphism when restricted to monadromic sheaves, which is crucial for proving independence.
• ✨ The use of infinity two or three categories is essential for managing complex combinatorics and ensuring necessary functorial properties.