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An analogue of the Gauss summation formula for hypergeometric functions related to root systems (Mathematisches Zeitschrift, 1993)
We prove a summation formula for the evaluation at the identity of the asymptotically free solutions of the hypergeometric system, normalized at infinity in the negative Weyl chamber.
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
(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 $spec(\Ze)$.
The central support of the Plancherel measure of an affine Hecke
algebra (July 2002)
We find an integral formula for the trace of an affine Hecke algebra.
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.