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Several combinatorial actions of the affine Weyl group of type$\widetilde{C}_{n}$ on triangulations, trees, words and permutations arecompared. Addressing a question of David Vogan, we show that, modulo a naturalinvolution, these permutation representations are multiplicity-free. The proofuses a general construction of Gelfand subgroups in the affine Weyl groups oftypes $\widetilde{C}_{n}$ and $\widetilde{B}_n$.

Flip actions and Gelfand pairs for affine Weyl groups