2 YURI BEREST AND PETER SAMUELSON

2. Lecture 1

2.1. Historical remarks. The theory of rational Cherednik algebras was his-

torically motivated by developments in two different areas: integrable systems and

multivariable special functions. In each of these areas, the classical representa-

tion theory of semisimple complex Lie algebras played a prominent role. On the

integrable systems side, one should single out the work of M. Olshanetsky and

A. Perelomov [OP83] who found a remarkable generalization of the Calogero-Moser

integrable systems for an arbitrary semisimple Lie algebra. On the special functions

side, the story began with the fundamental discovery of Dunkl operators [Dun89]

that transformed much of the present day harmonic analysis. In 1991, G. Heckman

[Hec91] noticed a relationship between the two constructions which is naturally

explained by the theory of rational Cherednik algebras. The theory itself was de-

veloped by P. Etingof and V. Ginzburg in their seminal paper [EG02b]. In fact,

this last paper introduced a more general class of algebras (the so-called symplectic

reflection algebras) and studied the representation theory of these algebras at the

‘quasi-classical’ (t = 0)

level1.

At the ‘quantum’ (t = 1) level, the representation

theory of the rational Cherednik algebras was developed in [BEG03a, BEG03b],

[DO03] and [GGOR03], following the insightful suggestion by E. Opdam and

R. Rouquier to model this theory parallel to the theory of universal enveloping

algebras of semisimple complex Lie algebras.

In this first lecture, we briefly review the results of [OP83] and [Dun89] in

their original setting and explain the link between these two papers discovered in

[Hec91]. Then, after giving a necessary background on complex reflection groups,

we introduce the Dunkl operators and sketch the proof of their commutativity

following [DO03]. In the next lecture, we define the rational Cherednik algebras

and show how Heckman’s observation can be reinterpreted in the language of these

algebras.

2.2. Calogero-Moser systems and the Dunkl operators.

2.2.1. The quantum Calogero-Moser system. Let h = Cn, and let hreg denote

the complement (in h) of the union of hyperplanes xi − xj = 0 for 1 ≤ i = j ≤ n.

Write C[hreg] for the ring of regular functions on hreg (i.e., the rational functions

on Cn with poles along xi − xj = 0). Consider the following differential operator

acting on C[hreg]

H =

n

i=1

∂

∂xi

2

− c(c + 1)

i=j

1

(xi − xj)2

,

where c is a complex parameter. This operator is called the quantum Calogero-

Moser Hamiltonian: it can be viewed as the Schr¨ odinger operator of the system of n

quantum particles on the line with pairwise interaction proportional to (xi

−xj)−2.

It turns out that H is part of a family of n partial differential operators Lj :

C[hreg]

→

C[hreg]

of the form

Lj :=

n

i=1

∂

∂xi

j

+ lower order terms ,

1As

yet another precursor of the theory of Cherednik algebras, one should mention the

beautiful paper [Wil98] which examines the link between the classical Calogero-Moser systems

and solutions of the (infinite-dimensional) Kadomtsev-Petviashvili integrable system.