Galleries and q-analogs in combinatorial representation theory

Schur functions and their q-analogs constitute an interesting branch of combinatorial representation theory. For Schur functions one knows several combinatorial formulas regarding their expansion in terms of monomial symmetric functions, their structure constants and their branching coefficients. In this thesis we prove q-analogs of these formulas for Hall-Littlewood polynomials. We give combinatorial formulas for the expansion of Hall-Littlewood polynomials in terms of monomial symmetric functions, for their structure constants and their branching coefficients. Specializing these formulas we get new proofs for the formulas involving Schur functions. As a combinatorial tool we use the gallery model introduced by Gaussent and Littelmann and show its relation to the affine Hecke algebra. All assertions are then proven in the more general context of the Macdonald basis of the spherical Hecke algebra. We show a commutation formula in the affine Hecke algebra with which we obtain a Demazure character formula involving galleries. We give a geometric interpretation of Kostka numbers and Demazure multiplicities of a complex reductive algebraic group using the affine Grassmanian of its Langlands dual group. As a further application we prove some first results regarding the positivity of Kostka-Foulkes coefficients