Lectures on Quasi-invariants of Coxeter Groups and the Cherednik Algebra

Introduction This paper arose from a series of three lectures given by the first author at Universitá di Roma “Tor Vergata” in January 2002, when the second author extended and improved her notes of these lectures. It contains an elementary introduction for non-specialists to the theory of quasi-invariants (but no original results). Our main object of study is the variety Xm of quasi-invariants for a finite Coxeter group. This very interesting singular algebraic variety arose in a work of O.Chalykh and A.Veselov about 10 years ago, as the spectral variety of the quantum Calogero-Moser system. We will see that despite being singular, this variety has very nice properties (Cohen-Macaulay, Gorenstein, simplicity of the ring of differential operators, explicitly given Hilbert series). It is interesting that although the definition of Xm is completely elementary, to understand the geometry of Xm it is helpful to use representation theory of the rational degeneration of Cherednik’s double affine Hecke algebra, and the theory of integrable systems. Thus, the study of Xm leads us to a junction of three subjects – integrable systems, representation theory, and algebraic geometry. The content of the paper is as follows. In Lecture 1 we define the ring of quasi-invariants for a Coxeter group, and discuss its elementary properties (with proofs), as well as deeper properties, such as Cohen-Macaulay, Gorenstein prop-