[Paper Review] Semi-infinite cohomology and Kazhdan-Lusztig equivalence at positive level

Chia-Cheng Liu|arXiv (Cornell University)|Jul 4, 2018

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper establishes a bridge between semi-infinite cohomology of positive level affine Lie algebra modules and quantum group cohomology via a newly defined positive level Kazhdan-Lusztig functor, using Arkhipov-Gaitsgory duality. The key result shows that semi-infinite cohomology factors through this functor and quantum group cohomology, offering a new pathway to generalize the Kazhdan-Lusztig equivalence beyond negative levels and supporting the quantum local geometric Langlands correspondence.

ABSTRACT

A positive level Kazhdan-Lusztig functor is defined using Arkhipov-Gaitsgory duality for affine Lie algebras. The functor sends objects in the DG category of G(O)-equivariant positive level affine Lie algebra modules to objects in the DG category of modules over Lusztig's quantum group at a root of unity. We prove that the semi-infinite cohomology functor for positive level modules factors through the Kazhdan-Lusztig functor at positive level and the quantum group cohomology functor with respect to the positive part of Lusztig's quantum group.

Motivation & Objective

• To define a positive level Kazhdan-Lusztig functor using Arkhipov-Gaitsgory duality for affine Lie algebras.
• To establish a factorization of semi-infinite cohomology through this functor and quantum group cohomology.
• To provide a new approach to the Kazhdan-Lusztig tensor equivalence valid for arbitrary non-critical levels.
• To support the quantum local geometric Langlands correspondence by connecting Kac-Moody and quantum group categories at positive level.
• To extend the scope of the Kazhdan-Lusztig equivalence beyond negative levels, conjecturally to all non-critical levels.

Proposed method

• The paper constructs a positive level Kazhdan-Lusztig functor from $G(\mathcal{O})$-equivariant modules over the affine Lie algebra $\hat{\mathfrak{g}}_{\kappa}$ to modules over Lusztig’s quantum group $U_q(\mathfrak{g})$ at a root of unity.
• It uses Arkhipov-Gaitsgory duality to define the functor, leveraging the duality between Whittaker and Kac-Moody categories.
• The semi-infinite cohomology functor $\mathfrak{C}^{\frac{\infty}{2}}(\mathfrak{n}(\mathcal{K}), -)$ is shown to factor through the Kazhdan-Lusztig functor and the quantum group cohomology functor $\textup{C}^{\bullet}(U_q(\mathfrak{n}), -)$.
• The factorization is proven via a commutative diagram involving $\hat{\mathfrak{g}}_{\kappa}\textup{-mod}^{G(\mathcal{O})}$, $\hat{\mathfrak{t}}_{\kappa+\textup{shift}}\textup{-mod}^{T(\mathcal{O})}$, $U_q(\mathfrak{g})\textup{-mod}$, and $\textup{Rep}_q(T)$.
• The proof relies on compactly generated categories and the ind-completion of the full subcategory of Weyl modules $\mathbb{V}^{\kappa}_{\lambda}$ for dominant integral weights $\lambda$, using Lurie’s theory of ind-categories.
• It verifies the commutativity of the diagram by showing agreement on compact generators and morphisms, particularly using irreducibility of Weyl modules when $\kappa$ is irrational.

Experimental results

Research questions

• RQ1 How can the Kazhdan-Lusztig equivalence be generalized beyond negative levels to arbitrary non-critical levels?
• RQ2 Can semi-infinite cohomology of positive level affine Lie algebra modules be related to quantum group cohomology via a functorial construction?
• RQ3 Does Arkhipov-Gaitsgory duality at positive level yield a well-defined Kazhdan-Lusztig functor linking $G(\mathcal{O})$-equivariant modules to quantum group modules?
• RQ4 How does the factorization of semi-infinite cohomology through the Kazhdan-Lusztig functor and quantum group cohomology support the quantum local geometric Langlands correspondence?
• RQ5 What is the role of the Tate shift in the semi-infinite cohomology of positive level modules, and how does it affect the level shift in the quantum group side?

Key findings

• The semi-infinite cohomology $\mathfrak{C}^{\frac{\infty}{2}}(\mathfrak{n}(\mathcal{K}), M)^{\mu}$ of a $G(\mathcal{O})$-equivariant positive level module $M$ is isomorphic to the quantum group cohomology $\textup{C}^{\bullet}(U_q(\mathfrak{n}), \textup{KL}_G(M))^{\mu}$ for all weights $\mu$.
• The Kazhdan-Lusztig functor $\textup{KL}_G$ at positive level is constructed via Arkhipov-Gaitsgory duality, mapping $\hat{\mathfrak{g}}_{\kappa}\textup{-mod}^{G(\mathcal{O})}$ to $U_q(\mathfrak{g})\textup{-mod}$, with $\textup{KL}_G(\mathbb{V}^{\kappa}_{\lambda}) = \mathcal{V}_{\lambda}$.
• The commutativity of the diagram (32) is established by showing agreement of the two compositions on the full subcategory of compact generators, i.e., the Weyl modules $\mathbb{V}^{\kappa}_{\lambda}$, using the irreducibility of these modules when $\kappa$ is irrational.
• The isomorphism between the two functors on morphisms is verified by showing that both send the identity morphism on $\mathbb{V}^{\kappa}_{\lambda}$ to the identity on the corresponding cochain complex in $\textup{Rep}_q(T)$.
• The factorization of semi-infinite cohomology through the Kazhdan-Lusztig functor and quantum group cohomology provides a new, potentially simpler approach to the Kazhdan-Lusztig equivalence, valid for all non-critical levels.
• The result supports the quantum local geometric Langlands correspondence by providing a bridge from Kac-Moody categories to quantum group categories at positive level, with the critical level as a degenerate case.