[Paper Review] Categorification of (induced) cell modules and the rough structure of generalized Verma modules

Volodymyr Mazorchuk, Catharina Stroppel|arXiv (Cornell University)|Feb 27, 2007

Algebraic structures and combinatorial models37 references4 citations

TL;DR

This paper presents a categorification of (induced) cell modules for Hecke algebras of finite Weyl groups, showing in type A that the categorification depends only on the isomorphism class of the cell module. The main contribution is a multiplicity formula for parabolically induced modules over type A Lie algebras, which determines the rough structure of generalized Verma modules and provides a positive answer to Kostant’s problem in many cases.

ABSTRACT

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type $A$ we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type $A$, which finally determines the so-called rough structure of generalized Verma modules. On the way we present several categorification results and give the positive answer to Kostant's problem from \cite{Jo} in many cases. We also give a general setup of decategorification, precategorification and categorification.

Motivation & Objective

• To develop a categorical framework for (right) cell modules and induced cell modules of Hecke algebras of finite Weyl groups.
• To establish a connection between representation theory of Lie algebras and Kazhdan-Lusztig combinatorics via categorification.
• To determine the rough structure of generalized Verma modules using multiplicity formulas derived from categorification.
• To provide a positive answer to Kostant’s problem in many cases, particularly in type A.
• To show that in type A, the categorification of cell modules is unique up to equivalence, depending only on the isomorphism class of the module.

Proposed method

• Construct a quotient category of a subcategory of category $\mathcal{O}$ associated to a parabolic subalgebra, stable under translation functors, to categorify cell modules.
• Use the Grothendieck group of the categorified category to recover the cell module as a module over the Hecke algebra.
• Employ a general setup of decategorification, precategorification, and categorification to formalize the categorical upgrade of cell modules.
• Apply parabolic induction via the category $\mathcal{O}\{\mathfrak{p},\mathscr{A}\}$ to relate different categorifications and induce module structures.
• Utilize the equivalence of cokernel categories $\mathscr{Y}_{\overline{N}}$ and $\mathscr{Y}_{L(y\cdot\mu)}$ to transfer multiplicity information between modules.
• Leverage Kazhdan-Lusztig combinatorics and the structure of standard and proper standard modules in highest weight categories to compute multiplicities.

Experimental results

Research questions

• RQ1 Can (induced) cell modules of Hecke algebras be categorified in a way that preserves their representation-theoretic structure?
• RQ2 In type A, is the categorification of a cell module unique up to equivalence, depending only on the isomorphism class of the module?
• RQ3 How can the rough structure of generalized Verma modules be determined using categorification and multiplicity formulas?
• RQ4 Does Kostant’s problem have a positive solution for generalized Verma modules in type A?
• RQ5 What is the relationship between the categorification of induced cell modules and the structure of parabolically induced modules in category $\mathcal{O}$?

Key findings

• In type A, the categorification of a cell module is unique up to equivalence: if two cell modules are isomorphic, their categorifications are equivalent via a functor that commutes with the Hecke algebra action.
• The paper establishes a multiplicity formula for the composition factors of generalized Verma modules $\Delta(\mathfrak{p}, V_X)$, showing that $[\Delta(\mathfrak{p}, V_X) : L(\mathfrak{p}, V_Y)] = [\Delta(\mathfrak{p}, V_{\hat{\xi}(X)}) : L(\mathfrak{p}, V_{\hat{\xi}(Y)})]$ via an equivalence of categories.
• The rough structure of generalized Verma modules is determined by reducing the problem to the standard category $\mathcal{O}$, where Kazhdan-Lusztig combinatorics apply.
• The irreducibility of a generalized Verma module $\Delta(\mathfrak{p}, L)$ is equivalent to the condition $w = \overline{w}$, where $(x,w)$ is the pair associated to the module in the block corresponding to the trivial central character.
• The paper gives a positive answer to Kostant’s problem in many cases, particularly for modules with trivial central character, and extends this to singular cases via translation functors.
• The equivalence between cokernel categories $\mathscr{Y}_{\overline{N}}$ and $\mathscr{Y}_{L(y\cdot\mu)}$ induces a bijection on simple modules and preserves the structure of proper standard objects, enabling multiplicity transfer.