For a Lie group $G$ with compact Lie subgroup $K$, we say that $(G,K)$ is a pair of Gelfand type if the representation $L^2(G/K)$ of $G$ is multiplicity free, that is, if it is a direct integral of distinct irreducible representations.

Can there exist a pair of dual representations in $L^2(G/K)$ for a Gelfand pair $(G,K)$?

Or do there exist non-self dual irreducible representations in $L^2(G/K)$?