  This paper is motivated by several combinatorial actions of the affine Weyl
group of type $C_n$. Addressing a question of David Vogan, it was shown in an
earlier paper that these permutation representations are essentialy
multiplicity-free~\cite{arXiv:2009.13880}. However, the Gelfand trick, which
was indispensable in~\cite{arXiv:2009.13880} to prove this property for types
$C_n$ and $B_n$, is not applicable for other classical types. Here we present a
unified approach to fully answer the analogous question for all irreducible
affine Weyl groups. Given a finite Weyl group $W$ with maximal parabolic
subgroup $P\leq W$, there corresponds to it a reflection subgroup $H$ of the
affine Weyl group $\widetilde W$. It turns out that while the Gelfand property
of $P\leq W$ does not imply that of $H\leq \widetilde{W}$, but $Q=N_W(P)\leq W$
has the Gelfand property if and only if $K=QH\leq \widetilde{W}$ has. Finally,
for each irreducible type we describe when $(W,Q)$ is a Gelfand pair.
