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  •  Cuspidal hypergeometric functions  In the "non-compact" spectral theory of hypergeometric functions for root systems we can have discrete spectrum when the multiplicity parameters are negative. In this paper we give a classification of the discrete  spectrum, and we compute the L^2 norms of the corresponding cuspidal hypergeometric  functions.
  •  Harmonic analysis for affine Hecke algebras  (Joint with Gerrit Heckman). In this paper we study the "anti-spherical" spectral theory of Hecke algebras, and discuss several consequences for the computation of formal degrees of cuspidal unipotent  representations of  semi-simple p-adic groups. As an application we give the partition of the cuspidal unipotent representations for E_8 into L-packets.
  •   A generating function for the trace of the Iwahori-Hecke algebra (November 1999). The Iwahori-Hecke algebra has a canonical  trace. The trace is the evaluation at the identity element in the usual interpretation of the Iwahori-Hecke algebra as a sub-algebra of the convolution algebra of a p-adic semi-simple group. The Iwahori-Hecke algebra contains an important  commutative sub-algebra which was  described and studied by Bernstein, Zelevinski and Lusztig. In this paper we compute the generating function for the  value of the trace on a basis of the above commutative sub-algebra.
  •  On the spectral decomposition of affine Hecke algebras  (NEW VERSION of 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.
  • The Schwartz algebra of an affine Hecke algebra (joint with Patrick Delorme, 2003) For a general affine Hecke algebra H we study its Schwartz completion S. The main theorem is an exact description of the image of S under the Fourier isomorphism. An important ingredient in the proof of this result is the definition and computation of the constant terms of a coefficient of a generalized principal series representation. Finally we discuss some consequences of the main theorem for the theory of tempered representations of H.