Erdal Emsiz
Pontificia Universidad Catolica de Chile
Discrete harmonic analysis on a Weyl alcove.
Abstract:
I will speak about recent work on a unitary representation
of the affine Hecke algebra given by discrete difference-reflection
operators acting in a Hilbert space of complex functions on the
weight lattice of a reduced crystallographic root system.
I will indicate why the action of the center under this
representation is diagonal on the basis of Macdonald spherical
functions (also referred to as generalized Hall-Littlewood
polynomials associated with root systems). I will furthermore
discuss a periodic counterpart of the above mentioned model
that is related to a representation of the double affine Hecke
algebra at critical level $q=1$ in terms of difference-reflection
operators. We use this representation to construct an explicit
integrable discrete Laplacian on the Weyl alcove corresponding
to an element in the center. The Bethe Ansatz method is employed
to show that our discrete Laplacian and its commuting integrals
are diagonalized by a finite-dimensional basis of periodic
Macdonald spherical functions. This is joint work in progress
with J. F. van Diejen.