[Paper Review] Representation theory of the cyclotomic Cherednik algebra via the Dunkl-Opdam subalgebra

Ben Webster|arXiv (Cornell University)|Sep 18, 2016

Algebraic structures and combinatorial models26 references16 citations

TL;DR

This paper introduces an alternate presentation of the cyclotomic rational Cherednik algebra that explicitly incorporates the Dunkl-Opdam subalgebra, enabling a diagrammatic, local-relations approach. It provides a direct algebraic construction of the KZ functor, classifies simple Dunkl-Opdam modules, and establishes a natural isomorphism between the spherical Cherednik algebra and the Coulomb branch of a 3D gauge theory—offering a new framework for connecting Cherednik algebras to geometric representation theory and mathematical physics.

ABSTRACT

We give an alternate presentation of the cyclotomic rational Cherednik algebra, which has the useful feature of compatibility with the Opdam-Dunkl subalgebra. This presentation has a diagrammatic flavor, and it provides a simple explanation of several surprising facts about this algebra. It allows direct proof of the connection of category $\mathcal{O}$ to weighted KLR algebras, allows us to classify the simple Dunkl-Opdam modules over the Cherednik algebra and provides an algebraic construction of the KZ functor. Furthermore, one of prime motivations for considering this approach is to provide a better framework for connecting Cherednik algebras to Coulomb branches of 3-d gauge theories.

Motivation & Objective

• To provide a new, diagrammatically flavored presentation of the cyclotomic rational Cherednik algebra that explicitly contains the Dunkl-Opdam subalgebra.
• To classify simple modules over the Dunkl-Opdam subalgebra within the Cherednik algebra, analogous to Gelfand-Tsetlin modules in gl_n.
• To give a direct algebraic construction of the Knizhnik-Zamolodchikov (KZ) functor over any characteristic 0 field, bypassing analytic methods.
• To establish a natural isomorphism between the spherical Cherednik algebra and the Coulomb branch of a 3D N=4 gauge theory, clarifying geometric and physical connections.
• To lay a foundation for generalizing this framework to rational Galois orders and for studying Cherednik algebras in positive characteristic.

Proposed method

• Introduce a new presentation of the cyclotomic Cherednik algebra using generators and relations that make the Dunkl-Opdam subalgebra manifest, with relations involving parameters $k$, $h_r$, and roots of unity $\zeta$.
• Define the Dunkl-Opdam operators $u_i = y_i x_i + k \sum_{j>i} \sum_p t_i^p t_j^{-p} (ij) + p(t_i)$, which commute and generate a polynomial subalgebra.
• Use weight spaces for the Dunkl-Opdam subalgebra to analyze the representation theory of the Cherednik algebra, enabling a direct interpretation of category O and its relation to weighted KLR algebras.
• Construct the KZ functor as a direct sum of weight spaces for the Dunkl-Opdam subalgebra, valid over any characteristic 0 field.
• Establish an isomorphism between the spherical Cherednik algebra and the quantum Coulomb branch of a 3D gauge theory via an Iwahori-Hecke algebra realization.
• Use diagrammatic reasoning and Chern class computations to verify that the action of shift elements in $e'He'$ matches the Schubert cell classes in the Coulomb branch.

Experimental results

Research questions

• RQ1 How can the cyclotomic Cherednik algebra be re-presented to make the Dunkl-Opdam subalgebra manifest and compatible with its structure?
• RQ2 What is the classification of simple modules over the Dunkl-Opdam subalgebra in the cyclotomic Cherednik algebra?
• RQ3 Can the Knizhnik-Zamolodchikov functor be constructed algebraically and over arbitrary characteristic 0 fields using this new presentation?
• RQ4 Is there a natural isomorphism between the spherical Cherednik algebra and the Coulomb branch of a 3D gauge theory in this framework?
• RQ5 How does this presentation facilitate a geometric or diagrammatic understanding of the algebra’s representation theory and its connections to Hilbert schemes and coherent sheaves?

Key findings

• The new presentation of the cyclotomic Cherednik algebra makes the Dunkl-Opdam subalgebra manifest, with generators $u_i$ that commute and form a polynomial subalgebra.
• The KZ functor is realized as a direct sum of weight spaces for the Dunkl-Opdam subalgebra, providing a purely algebraic construction valid over any characteristic 0 field.
• The simple modules over the Dunkl-Opdam subalgebra are classified, and they correspond to Gelfand-Tsetlin-like modules in the Cherednik algebra setting.
• The isomorphism between the spherical Cherednik algebra and the Coulomb branch of a 3D gauge theory is shown to be natural and explicit in this presentation, resolving a key geometric conjecture.
• The action of shift elements $e'x_{\ell-1}^1 \sigma e'$ in $e'He'$ matches the Chern class of tautological bundles on the affine Grassmannian, verifying the isomorphism to the quantum Coulomb branch.
• The map from $e'He'$ to the quantum Coulomb branch $A'$ is surjective, as it hits the fundamental classes of all Schubert cells via the action of simple reflections.