  This paper makes precise the close connection between the affine Hecke
algebra, the path model, and the theory of crystals. Section 2 is a basic
pictorial exposition of Weyl groups and affine Weyl groups and Section 5 is an
exposition of the theory of (a) symmetric functions, (b) crystals and (c) the
path model. Sections 3 and 4 give an exposition of the affine Hecke algebra and
recent results regarding the combinatorics of spherical functions on p-adic
groups (Hall-Littlewood polynomials). The $q$-analogue of the theory of
crystals developed in Section 4 specializes to the path model version of the
``classical'' crystal theory. The connection to the affine Hecke algebra and
the approach to spherical functions for a $p$-adic group in Nelsen-Ram was made
concrete by C. Schwer who told me that ``the periodic Hecke module encodes the
positively folded galleries'' of Gaussent-Littelmann. This paper is a further
development of this point of view.
