  We introduce a representation of the double affine Hecke algebra at the
critical level q=1 in terms of difference-reflection operators and use it to
construct an explicit integrable discrete Laplacian on the Weyl alcove
corresponding to an element in the center. The Laplacian in question is to be
viewed as an integrable discretization of the conventional Laplace operator on
Euclidian space perturbed by a delta-potential supported on the reflection
hyperplanes of the affine Weyl group. 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.
