Galleries, Hall-Littlewood polynomials and structure constants of the   spherical Hecke algebra
  We provide a combinatorial description of the coefficients appearing in the expansion of Hall-Littlewood polynomials in terms of monomial symmetric functions. We also give a Littlewood-Richardson rule for Hall-Littlewood polynomials. For proving this we use galleries to calculate Satake coefficients and structure constants of spherical Hecke algebras with arbitrary parameters. 
Christoph Schwer