Prescribed spherical representations, symplectic group $Sp(n)$

Question

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 $k\geq0$?.

Answer 1

Though I can't provide an answer to the question, I can add a few more comments in community-wiki mode.

1) In Bourbaki's construction of root systems, the starting point is always an orthonormal basis $\varepsilon_i$ for a real euclidean space. Though the dimension of this space is the rank $n$ in types $B_n, C_n, D_n$, it's sometimes bigger for other types. Making the usual identifications, one can regard the $\varepsilon_i$ in those three classical types as a basis for $\mathfrak{h}^*$ after complexifying.

2) It's more customary to express highest weights of finite dimensional irreducible representations (of complex or compact simple groups, or the corresponding complex Lie algebras) in terms of fundamental dominant weights $\varpi_i$. For example, in type $D_n$ (or $B_n$) here, $\pi_k$ has highest weight $k\varpi_1$ (= $k\varepsilon_1$), where the first fundamental weight belongs to the natural representation of dimension $2n$ (or $2n+1$). In type $C_n$, on the other hand, the weight $\varepsilon_1 + \varepsilon_2 = \varpi_2$, whereas $\varepsilon_1 = \varpi_1$ (the natural $2n$-dimensional representation). Note that $\varpi_2$ is the highest weight of the big irreducible summand of the second exterior power of the natural representation, whose dimension is $[n(n-1)/2] -1$.

3) Also, it should be observed that the notion of "spherical representation" comes up frequently in the study of semisimple Lie groups. In the case of a real noncompact group $G$ the subgroup $K$ is usually taken to be a maximal compact subgroup. When $G$ itself is compact, $K$ might be any proper (say nontrivial) closed subgroup, as in this question. In that case there is some variation in the condition for a representation of $G$ to be called "spherical": the restriction to $K$ involves the trivial representation just once, or else at most once. I'm not sure about the history and motivation for these definitions.