[Paper Review] The combinatorics of category O for symmetrizable Kac-Moody algebras

Peter Fiebig|arXiv (Cornell University)|May 27, 2003

Algebraic structures and combinatorial models5 references4 citations

TL;DR

This paper establishes that the structure of non-critical blocks in category 𝒪 over symmetrizable Kac-Moody algebras is determined entirely by the combinatorics of the integral Weyl group and its action on weights. It proves that equivalent Coxeter systems with isomorphic actions on parameter spaces yield equivalent categories, thereby reducing the Kazhdan-Lusztig conjecture for non-integral blocks in finite and affine cases to the known integral case via categorical equivalence.

ABSTRACT

We show that the structure of blocks outside the critical hyperplanes of category O over any symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules by giving a combinatorial description of the projective objects. As an application we derive the Kazhdan-Lusztig conjecture for non-integral blocks from the integral case in finite and affine situations.

Motivation & Objective

• To generalize Soergel's categorical equivalence result from finite-dimensional semisimple Lie algebras to symmetrizable Kac-Moody algebras.
• To show that the structure of category 𝒪 outside critical hyperplanes depends only on the Weyl group and its action on weights.
• To establish a combinatorial description of projective objects in non-integral blocks via coinvariant algebras and translation functors.
• To reduce the Kazhdan-Lusztig conjecture for non-integral blocks in finite and affine types to the integral case using categorical equivalence.

Proposed method

• Use the coinvariant algebra $ C = C({ m W}( ext{L}), { m S}( ext{L})) $ associated to the Coxeter system $ ({ m W}( ext{L}), { m S}( ext{L})) $ as a key invariant for block structure.
• Construct a functor $ b{V} = { m Hom}(P(w_0. ext{L}), ullet) $ from category 𝒪 to $ C $-mod, which preserves Hom-spaces and intertwines translation functors.
• Prove that $ b{V} $ induces an equivalence $ { m O}_ ext{L} o C{ m -mod} $ when $ ext{L} $ is regular, using induction on the length of Weyl group elements.
• Use the tilting functor and truncation functors to extend the equivalence from positive level to negative level blocks.
• Apply the equivalence to reduce the Kazhdan-Lusztig conjecture for non-integral blocks to the integral case via isomorphism of Coxeter systems.
• Leverage known results on integral blocks (Beilinson-Bernstein, Brylinski-Kashiwara, Kashiwara-Tanisaki) to conclude the conjecture holds in non-integral cases.

Experimental results

Research questions

• RQ1 Does the structure of category 𝒪 for symmetrizable Kac-Moody algebras outside critical hyperplanes depend only on the Weyl group and its action on weights?
• RQ2 Can the Kazhdan-Lusztig conjecture for non-integral blocks in finite and affine Kac-Moody algebras be reduced to the integral case?
• RQ3 Is there a combinatorial description of projective objects in non-integral blocks using coinvariant algebras and translation functors?
• RQ4 Under what conditions does an isomorphism of Coxeter systems induce an equivalence of category 𝒪 blocks?
• RQ5 Can the tilting functor be used to extend categorical equivalences from positive to negative level blocks?

Key findings

• The category 𝒪 block $ { m O}_ ext{L} $ is equivalent to $ { m O}'_{ ext{L}'} $ whenever the associated Coxeter systems $ ({ m W}( ext{L}), { m S}( ext{L})) $ and $ ({ m W}'( ext{L}'), { m S}'( ext{L}')) $ are isomorphic and the action on weights is compatible.
• The projective objects in $ { m O}_ ext{L} $ are completely determined by the coinvariant algebra $ C $ and the action of the Weyl group, with $ { m Hom}(P(w. ext{L}), ullet) $ providing a full, faithful, and exact functor to $ C{ m -mod} $.
• For regular, finite or affine-type blocks outside critical hyperplanes, the Kazhdan-Lusztig polynomials $ P_{y,w} $ and $ Q_{w,y} $ govern the composition series of simple modules in Verma modules.
• The equivalence $ { m O}_ ext{L} o { m O}'_{ ext{L}'} $ preserves the classes of Verma and simple modules, so $ [M(x. ext{L}):L(y. ext{L})] = [M(x. ext{L}'):L(y. ext{L}')] $.
• The Kazhdan-Lusztig conjecture holds for non-integral blocks in finite and affine symmetrizable Kac-Moody algebras when the Weyl group is finite or affine and the block is regular.
• The result extends known proofs of the Kazhdan-Lusztig conjecture (e.g., by Kashiwara and Kashiwara-Tanisaki) from integral to non-integral blocks via categorical equivalence.