Combinatorial flip actions and Gelfand pairs for affine Weyl groups.

Several combinatorial actions of the affine Weyl group of type C ˜ n on triangulations, trees, words and permutations are compared. Addressing a question of David Vogan, we show that, modulo a natural involution, these permutation representations are multiplicity-free. The proof uses a general construction of Gelfand subgroups in the affine Weyl groups of types C ˜ n and B ˜ n. [ABSTRACT FROM AUTHOR]