5. Parabolic induction and restriction functors for rational Cherednik
algebras
5.1. A geometric approach to rational Cherednik algebras. An important property
of the rational Cherednik algebra H1,c (G, h) is that it can be sheaed, as an algebra, over
h/G
4. The Macdonald-Mehta integral
4.1. Finite Coxeter groups and the Macdonald-Mehta integral. Let W be a nite
Coxeter group of rank r with real reection representation hR equipped with a Euclidean
W -invariant inner product (, ). Denote by h the complexica
3. The rational Cherednik algebra
3.1. Denition and examples. Above we have made essential use of the commutation
relations between operators x h , g G, and Da , a h. This makes it natural to consider
the algebra generated by these operators.
Denition 3.1
2. Classical and quantum Olshanetsky-Perelomov systems for finite
Coxeter groups
2.1. The rational quantum Calogero-Moser system. Consider the dierential operator
n
2
1
H=
c(c + 1)
.
2
xi
(xi xj )2
i=1
i=j
This is the quantum Hamiltonian for a system
1. Introduction
Double ane Hecke algebras, also called Cherednik algebras, were introduced by Chered
nik in 1993 as a tool in his proof of Macdonalds conjectures about orthogonal polynomials for
root systems. Since then, it has been realized that Cheredni