Rational and trigonometric degeneration of the double affine Hecke algebra of type $A$

TL;DR

This paper explores the relationship between the rational Cherednik algebra of type $GL_n$ and the degenerate double affine Hecke algebra, establishing a faithful embedding of their category ${ m O}$ and classifying certain irreducible modules.

Contribution

It introduces an algebra embedding from the rational Cherednik algebra to the degenerate DAHA and fully characterizes the semisimple modules in the associated category ${ m O}$.

Findings

The embedding from the rational Cherednik algebra to the degenerate DAHA is fully faithful.

The classification of irreducible modules in the semisimple subcategory ${ m O}^{ss}$ is achieved.

The study reveals a deep connection between the representation theories of these algebras.

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.