Say I have a Lie group $G$ and a closed subgroup $H$. Consider representations over the complex numbers $\mathbb{C}$.
We say that $(G,H)$ is a Gelfand pair if for any irrep $\rho$ of $G$, we have $\mathrm{dim} \mathrm{Hom}_H(\rho,\mathbb{C}) \leq 1$ (meaning that the restricted representation $\rho|_H$ decomposes into at most one copy of the trivial irrep).
However, suppose $(G,H)$ has this "Gelfand property" only for some irreps of $\rho$ of $G$ (but not for all of them). Has this been studied in the literature before?
I know that if you study this for irreps $\rho$ such that $\mathrm{dim} \mathrm{Hom}_H(\rho,\mathbb{C}) = 1$ then $\rho$ is called a spherical representation. However, I am actually more interested in those $\rho$ such that $\mathrm{dim} \mathrm{Hom}_H(\rho,\mathbb{C}) = 0$ (in other words, I am interested in those irreps $\rho$ such that the restricted representation $\rho|_H$ decomposes into zero copies of the trivial irrep, i.e., the trivial irrep has zero multiplicity).