[Paper Review] The Hodge theory of Soergel bimodules

Ben Elias, Geordie Williamson|arXiv (Cornell University)|Dec 4, 2012

Algebraic structures and combinatorial models4 citations

TL;DR

This paper proves Soergel's conjecture on the characters of indecomposable Soergel bimodules for arbitrary Coxeter systems by establishing the hard Lefschetz theorem and Hodge-Riemann bilinear relations in the algebraic setting of Soergel bimodules. The key result is the proof of positivity of Kazhdan-Lusztig polynomials and an algebraic derivation of the Kazhdan-Lusztig conjecture, completing Soergel's program via Hodge-theoretic methods.

ABSTRACT

We prove Soergel's conjecture on the characters of indecomposable Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.

Motivation & Objective

• To prove Soergel's conjecture on the character of indecomposable Soergel bimodules in arbitrary Coxeter systems.
• To establish the hard Lefschetz and Hodge-Riemann bilinear relations for Soergel bimodules in the algebraic setting.
• To provide an algebraic proof of the Kazhdan-Lusztig conjecture, independent of geometric realizations.
• To extend the Hodge-theoretic approach of de Cataldo and Migliorini to Soergel bimodules, using Lefschetz operators and perverse filtrations.
• To demonstrate that Kazhdan-Lusztig polynomials have positive coefficients for all Coxeter systems, resolving a long-standing conjecture.

Proposed method

• Adapts Hodge-theoretic techniques from algebraic geometry, particularly the hard Lefschetz and Hodge-Riemann relations, to the category of Soergel bimodules.
• Introduces a Lefschetz operator ρ on the Grothendieck group of Soergel bimodules, defined via Rouquier complexes and perverse filtrations.
• Uses the perverse filtration on complexes and minimal complexes to define a Hodge structure on the category of Soergel bimodules.
• Applies the Lefschetz operator to prove injectivity of ρ^k on graded components of bimodules, establishing hard Lefschetz.
• Employs the decomposition of bimodules into upward and downward components (B↑, B↓) to analyze the differential in the complex and verify injectivity.
• Relies on the compatibility of the Lefschetz operator with the differential and the induced form, ensuring that ρ^k(b) ≠ 0 when d(b) ≠ 0.

Experimental results

Research questions

• RQ1 Does the hard Lefschetz theorem hold for Soergel bimodules in arbitrary Coxeter systems, without geometric realization?
• RQ2 Can the Hodge-Riemann bilinear relations be established algebraically for Soergel bimodules?
• RQ3 Are Kazhdan-Lusztig polynomials with positive coefficients for all Coxeter systems, as conjectured by Kazhdan and Lusztig?
• RQ4 Can the Kazhdan-Lusztig conjecture be proven algebraically via Soergel bimodules, without relying on D-modules or the Riemann-Hilbert correspondence?
• RQ5 Is the Lefschetz operator ρ on the Grothendieck group of Soergel bimodules a genuine Lefschetz operator satisfying the required algebraic properties?

Key findings

• The hard Lefschetz theorem holds for Soergel bimodules in arbitrary Coxeter systems, with the Lefschetz operator ρ acting via left multiplication.
• The Hodge-Riemann bilinear relations are satisfied by the graded components of Soergel bimodules, ensuring the positivity of the bilinear form.
• Kazhdan-Lusztig polynomials have positive coefficients for all Coxeter systems, confirming a long-standing conjecture.
• The indecomposable Soergel bimodules correspond precisely to the Kazhdan-Lusztig basis in the Grothendieck group, proving Soergel's conjecture.
• The Lefschetz operator ρ induces an injective map ρ^k on the graded components of the bimodule, which is essential for the hard Lefschetz result.
• The decomposition of the complex F_x F_s into B↑, B↓, and B_x B_s allows for a reduction of the problem to verifying injectivity of the differential and compatibility with ρ.