Hecke algebras and symplectic reflection algebras

TL;DR

This paper surveys Hecke algebras, Kazhdan-Lusztig theory, and symplectic reflection algebras, focusing on their interconnections and potential to extend Kazhdan-Lusztig theory to complex reflection groups.

Contribution

It highlights the links between Hecke algebras and rational Cherednik algebras, aiming to generalize Kazhdan-Lusztig theory for complex reflection groups.

Findings

Connections between Hecke and Cherednik algebras are explored.

Potential for extending Kazhdan-Lusztig theory to complex reflection groups.

Emphasis on applications of these algebraic structures.

Abstract

The current article is a short survey on the theory of Hecke algebras, and in particular Kazhdan-Lusztig theory, and on the theory of symplectic reflection algebras, and in particular rational Cherednik algebras. The emphasis is on the connections between Hecke algebras and rational Cherednik algebras that could allow us to obtain a generalised Kazhdan-Lusztig theory, or at least its applications, for all complex reflection groups.