Representation theory of the rational Cherednik algebras of type Z/lZ via microlocal analysis

TL;DR

This paper uses microlocal analysis to construct and study modules of rational Cherednik algebras of type Z/lZ, revealing new geometric structures and connections to microlocal systems with regular singularities.

Contribution

It introduces a microlocalization approach to explicitly construct irreducible and standard modules for these algebras, advancing the understanding of their representation theory.

Findings

Construction of irreducible modules via microlocalization

Identification of sheaves of microlocal systems for holonomic systems

Establishment of links between algebraic and geometric structures

Abstract

Based on the methods developed in [Kashiwara-Rouquier], we consider microlocalization of the rational Cherednik algebra of type $\Z/l\Z$. Our goal is to construct the irreducible modules and standard modules of the rational Cherednik algebra by using the microlocalization. As a consequence, we obtain the sheaves of microlocal system corresponding to holonomic systems with regular singularities.