Finding spherical representations of $GL(n, \mathbb{C})$. 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. 
 A: There is a paper by Kramer about pairs $(G, K)$ with $G$ connected Lie group and $K$ spherical in $G$ that is for all irreducible representations of $G$, the space of vectors fixed by $K$ is at most 1-dimensional.
If I recall correctly Kramer gives some propreties of spherical pairs and provides the full classifications of spherical pairs $(G, K)$ with $G$ compact and simple.
Here: link text
Is this what you was looking for?
A: In the case of $GL(n, \mathbb{C})$, it is known that every unitary, irreducible, infinite-dimensional representation (the others are one-dimensional and factor through the determinant) is given as induced representation $\pi$ from a minimal parabolic associated to the Levi $M(\mathbb{C})$ (being the group of diagonal matrices). This one is spherical iff the restriction $\pi$ to $M(\mathbb{C}) \cap U(n)$ is trivial.
The Mackey Induction Restriction formula plus the Iwasawa decomposition indicates that this is the same question for inducing the restriction of $\pi$ to $M(\mathbb{C}) \cap U(n)$ up to $U(n)$.
This is the case if and only if the restriction of $\pi$ to $M(\mathbb{C}) \cap U(n)$ is trivial by Frobenius reciprocity.
This strategy works more generally for all parabolically induced representation in real reductive groups. Then look possibly at the structure of their subquotients. It is a theorem of Casselman that for a real reductive Lie group all smooth, admissible reps are found as such subquotients of such parabolically induced representations.
