Discrete harmonic analysis on a Weyl alcove
Seminar Room 1, Newton Institute
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.