## Ribbon tableaux, Hall{endash}Littlewood functions, quantum affine algebras, and unipotent varieties

We introduce a new family of symmetric functions, which are q analogs of products of Schur functions, defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation scr(F){sub q} of U{sub q}(sl{sub n}), and are related to Hall{endash}Littlewood functions via the geometry of flag varieties. We present a series of conjectures, and prove them in special cases. The essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of scr(F){sub q} using ribbon tableaux. {copyright} {ital 1997 American Institute of Physics.}