I am looking for literature that might contain the spherical representations of $GL(n, \mathbb{C})$. Here a spherical representation is an irreducible representation $\rho$ of $G$ on $\mathbb{C}$ such that $\rho_{K}$, for $K$ a maximal compact subgroup, fixes a vector in $\mathbb{C}$. I realize my question is similar to this one, however I am looking for the spherical representations that may involve $Sp_{2n}$, $U_n$ and $SO_n$ for $n$ odd and even. I apologize if I have erred with my vocabulary or if the question lacks sufficient detail for a meaningful reference; I will gladly supply more details if there is confusion.

The motivation for this query is an attempt to find out which maximal subgroups of $GL(n^2)$, for $n \in \mathbb{N}$, stabilize one-dimensional subspaces when the representation $GL(n^2) \to GL(V)$ for $V = \mathrm{Sym}^n \mathbb{C}^{n}$ is restricted to this maximal subgroup. One such subgroup that fixes a 1-dimensional subspace that has been found is $GL(n) \times GL(n)$ under the tensor product representation, which fixes $\wedge^n \mathbb{C}^n \times \wedge^n \mathbb{C}^n$ i.e., the determinant.

As a side note, another technique I have been using for examining whether certain maximal subgroups have invariant vectors is the restriction formula found in Fulton and Harris for restricting representations of $GL(n)$ to $O(n)$ and the branching rule involved with these representations.

nota complex Lie group, in fact, it is compact, and so it coincides with its own maximal compact subgroup $K.$ $\endgroup$