[Paper Review] Multi-fusion categories of Harish-Chandra bimodules

Victor Ostrik|arXiv (Cornell University)|Apr 25, 2014

Algebraic structures and combinatorial models29 references3 citations

TL;DR

This paper introduces multi-fusion categories arising as semisimple subquotients of the tensor category of Harish-Chandra bimodules over a complex semisimple Lie algebra, using associated varieties to define a filtration. It shows these categories, called cell categories, are equivalent to representation categories of finite groups and identifies them via Drinfeld center constructions and character sheaves, yielding new insights into Lusztig's classification and finite W-algebras.

ABSTRACT

We survey some results on tensor products of irreducible Harish-Chandra bimodules. It turns out that such tensor products are semisimple in suitable Serre quotient categories. We explain how to identify the resulting semisimple tensor categories and describe some applications to representation theory.

Motivation & Objective

• To understand the tensor category structure of Harish-Chandra bimodules, which are non-semisimple and thus difficult to classify directly.
• To define and study semisimple subquotients—called cell categories—via a filtration by associated varieties, simplifying the representation-theoretic structure.
• To identify these cell categories as multi-fusion categories equivalent to representation categories of finite groups, using categorical tools like the Drinfeld center.
• To apply the framework to finite W-algebras, recovering information about their finite-dimensional simple modules.
• To generalize the construction to arbitrary Coxeter groups and positive characteristic, linking to character sheaves and Soergel bimodules.

Proposed method

• Use the associated variety filtration on Harish-Chandra bimodules to define a graded category, whose associated graded pieces are the cell categories.
• Apply the theory of multi-fusion categories to classify the cell categories, showing they are equivalent to categories of twisted group representations, $\mathrm{Coh}_{\bar{A},\omega}(Y \times Y)$.
• Employ the Drinfeld center construction to relate the cell categories to unipotent character sheaves on the group $G$, via the functor $\Gamma = \pi_! f^*$ on the flag variety.
• Use truncated convolution and perverse sheaf techniques to realize the cell categories geometrically, enabling generalization to positive characteristic.
• Leverage results from Bezrukavnikov, Losev, and others on Whittaker categories and W-algebras to analyze the cell categories' structure.
• Apply the theory to finite W-algebras by realizing their representation categories as actions of the cell categories, recovering the number of finite-dimensional simple modules.

Experimental results

Research questions

• RQ1 How can the non-semisimple tensor category of Harish-Chandra bimodules be simplified via a filtration based on associated varieties?
• RQ2 What is the structure of the semisimple subquotients (cell categories) obtained from this filtration, and how can they be classified?
• RQ3 Can the cell categories be realized geometrically using perverse sheaves or character sheaves on the flag variety or the group $G$?
• RQ4 How do these categories relate to finite W-algebras and their finite-dimensional representations?
• RQ5 Can the framework be extended to arbitrary Coxeter groups or positive characteristic, and what new structures emerge?

Key findings

• The cell categories associated with nilpotent orbits in $\mathfrak{g}$ are multi-fusion categories, and they are equivalent to $\mathrm{Coh}_{\bar{A},\omega}(Y \times Y)$, the category of twisted sheaves on a finite set $Y \times Y$.
• The Drinfeld center of each cell category $\mathcal{J}_C$ is equivalent to the category of unipotent character sheaves $\mathcal{U}_C$ on $G$, establishing a bijection between simple objects in the center and $\mathcal{U}_C$.
• For finite W-algebras, the action of the cell category on Whittaker categories allows computation of the number of finite-dimensional simple modules, recovering known results.
• The construction generalizes to arbitrary Coxeter groups, with cell categories arising as $\mathrm{Coh}_Q(Y \times Y)$, though they may not be multi-fusion if $Q$ is infinite.
• In the dihedral case of order 10, a cell category contains a fusion subcategory with two simple objects and $X \otimes X = \mathbf{1} \oplus X$, showing it is not of the form $\mathrm{Coh}_{A,\omega}(Y \times Y)$.
• The identification of $\mathcal{U}_C$ with the simple objects of $\mathcal{Z}(\mathcal{J}_C)$ provides a new categorical approach to Lusztig’s classification of unipotent character sheaves.