An irreducible representation $(\pi,V_\pi)$ of a compact group $G$ is called spherical with respect to the pair $(G,K)$, $K$ is closed subgroup of $G$, if $V_\pi$ has a non-zero vector invariant by $K$, or equivalently, the trivial representation $1_K$ of $K$ appears in the decomposition of $\pi|_K$.

The pair $(G,K)$, $G=SO(2n)$, $K=SO(2n-1)$, has spherical representations $\pi_k$ with highest weights $k\varepsilon_1$ for $k\geq0$ (under the standard choice of root system and positivity, i.e. $\varepsilon_1-\varepsilon_2,\dots,\varepsilon_{n-1}-\varepsilon_n, \varepsilon_{n-1}+\varepsilon_n$ are the simple roots). We have the same situation in the case $G=SO(2n+1)$, $K=SO(2n)$.

The pair $G=Sp(n)$ and $K=Sp(n-1)\times Sp(1)$ has spherical representations $\pi_k$ with highest weights $k(\varepsilon_1+\varepsilon_2)$ for $k\geq0$. Here, the simple roots are $\varepsilon_1-\varepsilon_2,\dots,\varepsilon_{n-1}-\varepsilon_n, 2\varepsilon_{n}$.

Is there $K\subset G=Sp(n)$ (closed) such that the spherical representations have highest weights $k\varepsilon_1$ for $k\geq0$?.