2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107549This 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.Representation TheoryCombinatoricstext