•  On the spectral decomposition of affine Hecke algebras  (September 2003). An affine Hecke algebra $\H$ contains a large abelian subalgebra $\A$ spanned by Lusztig's basis elements $\theta_\lambda$, where $\lambda$ runs over the root lattice. The center $\Ze$ of $\H$ is the subalgebra of Weyl group invariant elements in $\A$. The trace of the affine Hecke algebra can be written as an integral of a rational $n$ form (with values in the linear dual of $\H$) over a certain cycle in the algebraic torus $T=spec(\A)$. We shall derive the Plancherel formula of the affine Hecke algebra by localization of this integral on a certain subset of $spec(\Ze)$.
•   Dunkl operators for complex reflection groups (Joint with Charles Dunkl, July 2001). Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameter family of  deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the rational Cherednik algebra'',and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups $G(m,p,N)$, the set of singular parameters in the parameter family of these structures is described explicitly, using the theory of nonsymmetric Jack polynomials.
•  On the category O for rational Cherednik algebras (joint with Victor Ginzburg, Nicolas Guay and Raphael Rouquier, 2002). We study the category O of representations of the rational Cherednik algebra A attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor, from O to the category of H-modules, where H is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between O/O_tor, the quotient of O by the subcategory of A-modules supported on the discriminant and the category of finite-dimensional H-modules. The standard A-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of cells'', provided W is a Weyl group and the Hecke algebra H has equal parameters. We prove that the category O is equivalent to the module category over a finite dimensional algebra, a generalized "q-Schur algebra" associated to W.