[Paper Review] Rational and trigonometric degeneration of the double affine Hecke algebra of type $A$

Takeshi Suzuki|ArXiv.org|Feb 25, 2005

Advanced Algebra and Geometry7 references6 citations

TL;DR

This paper establishes a fully faithful embedding of the category $\mathcal{O}$ for the rational Cherednik algebra of type $A_n$ into the category $\mathcal{O}$ for the degenerate double affine Hecke algebra (DAHA), via an algebra embedding and induction functor. The key result is that irreducible and standard modules are preserved under this functor, enabling transfer of representation-theoretic results, including a classification of irreducible modules in the semisimple subcategory $\mathcal{O}^{ss}$.

ABSTRACT

We study a connection between the representation theory of the rational Cherednik algebra of type $GL_n$ and the representation theory of the degenerate double affine Hecke algebra (the degenerate DAHA). We focus on an algebra embedding from the rational Cherednik algebra to the degenerate DAHA and investigate the induction functor through this embedding. We prove that this functor embeds the category ${\mathcal O}$ for the rational Cherednik algebra fully faithfully into the category ${\mathcal O}$ for the degenerate DAHA. We also study the full subcategory ${\mathcal O}^{ss}$ of ${\mathcal O}$ consisting of those modules which are semisimple with respect to the commutative subalgebra generated by Cherednik-Dunkl operators. A classification of all irreducible modules in ${\mathcal O}^{ss}$ for the rational Cherednik algebra is obtained from the corresponding result for the degenerate DAHA.

Motivation & Objective

• To establish a connection between the representation theories of the rational Cherednik algebra $\mathbf{H}_\kappa$ and the degenerate DAHA $\widetilde{\mathbf{H}}_\kappa$ of type $A_n$.
• To construct an algebra embedding from $\mathbf{H}_\kappa$ into $\widetilde{\mathbf{H}}_\kappa$ that extends to a localization isomorphism.
• To study the induction functor $\widetilde{\mathbf{H}}_\kappa \otimes_{\mathbf{H}_\kappa} (-)$ and its action on category $\mathcal{O}$.
• To classify irreducible modules in the semisimple subcategory $\mathcal{O}^{ss}$ for the rational Cherednik algebra using known results for the degenerate DAHA.

Proposed method

• An algebra embedding $\mathbf{H}_\kappa \to \widetilde{\mathbf{H}}_\kappa$ is constructed, extending to an isomorphism between a localization of $\mathbf{H}_\kappa$ and $\widetilde{\mathbf{H}}_\kappa$.
• The induction functor $\widetilde{\mathbf{H}}_\kappa \otimes_{\mathbf{H}_\kappa} (-)$ is used to map modules from $\mathbf{H}_\kappa$-mod to $\widetilde{\mathbf{H}}_\kappa$-mod.
• The functor is restricted to map $\mathcal{O}(\mathbf{H}_\kappa)$ fully faithfully into $\mathcal{O}(\widetilde{\mathbf{H}}_\kappa)$, preserving standard and irreducible modules.
• The semisimple subcategory $\mathcal{O}^{ss}$ is defined as modules semisimple with respect to the commutative subalgebra generated by Cherednik-Dunkl operators.
• The induction functor maps $\mathcal{O}^{ss}(\mathbf{H}_\kappa)$ into $\mathcal{O}^{ss}(\widetilde{\mathbf{H}}_\kappa)$, leveraging known classification for $\widetilde{\mathbf{H}}_\kappa$.
• The classification of irreducible modules in $\mathcal{O}^{ss}(\mathbf{H}_\kappa)$ is derived from the classification in $\mathcal{O}^{ss}(\widetilde{\mathbf{H}}_\kappa)$ via the induction functor.

Experimental results

Research questions

• RQ1 How can the representation theory of the rational Cherednik algebra be related to that of the degenerate DAHA?
• RQ2 Does the induction functor induced by the algebra embedding $\mathbf{H}_\kappa \to \widetilde{\mathbf{H}}_\kappa$ preserve the structure of category $\mathcal{O}$?
• RQ3 Can the classification of irreducible modules in $\mathcal{O}^{ss}$ for the degenerate DAHA be used to classify those for the rational Cherednik algebra?
• RQ4 What is the behavior of the induction functor on standard and irreducible modules in $\mathcal{O}$?
• RQ5 How do the multiplicities of irreducible modules in standard modules compare across $\mathbf{H}_\kappa$, $\widetilde{\mathbf{H}}_\kappa$, and the original double affine Hecke algebra?

Key findings

• The induction functor $\widetilde{\mathbf{H}}_\kappa \otimes_{\mathbf{H}_\kappa} (-)$ embeds $\mathcal{O}(\mathbf{H}_\kappa)$ fully faithfully into $\mathcal{O}(\widetilde{\mathbf{H}}_\kappa)$.
• The functor preserves standard modules: it maps a standard module of $\mathbf{H}_\kappa$ to a standard module of $\widetilde{\mathbf{H}}_\kappa$.
• The functor preserves irreducibility: it maps an irreducible module of $\mathbf{H}_\kappa$ to an irreducible module of $\widetilde{\mathbf{H}}_\kappa$.
• The multiplicity of an irreducible module in the composition series of a standard module is preserved under the functor, and thus equal across $\mathbf{H}_\kappa$, $\widetilde{\mathbf{H}}_\kappa$, and the original double affine Hecke algebra.
• The subcategory $\mathcal{O}^{ss}(\mathbf{H}_\kappa)$ is mapped fully faithfully into $\mathcal{O}^{ss}(\widetilde{\mathbf{H}}_\kappa)$ by the induction functor.
• For $\kappa \in \mathbb{Z}_{>0}$, the irreducible modules in $\mathcal{O}^{ss}(\mathbf{H}_\kappa)$ are in one-to-one correspondence with partitions $\lambda \in \bigsqcup_{m=1}^n \Lambda^+_{\kappa}(m,n)$, where $\Lambda^+_{\kappa}(m,n) = \{\lambda \in \Lambda^+(m,n) \mid \kappa - m - \lambda_1 + \lambda_m \in \mathbb{Z}_{\geq 0}\}$.