# 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.

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What do you mean by "fix vectors in $\mathbb{C}$? If $\mathbb{C}$ denotes the trivial representation of the group then the answer is also trivial (namely, all of them). Is the group $G$ a complex Lie group (like $GL(n,\mathbb{C})$) or do you simply wish to consider complex representations (not necessarily on $\mathbb{C}$ - perhaps, you can specify whether they are supposed to be finite-dimensional). By the way, $U_n$ is not a complex Lie group, in fact, it is compact, and so it coincides with its own maximal compact subgroup $K.$ –  Victor Protsak May 14 '13 at 6:00
I understand the OP is interested in a classification of the unitary (or smooth, admissible) representation, which are irreducible and have a invariant vector under the maximal compact subgroup, or equivalently the trivial representation is contained in the restriction to it. I think in his context, he wants to consider either the $\mathbb{R}$- or $\mathbb{C}$-points of these classical algebraic group, $U(n)$ making no sense over $\mathbb{R}$, though, and having a trivial answer over $\mathbb{C}$. Similarly, for $SO(n)$ over $\mathbb{R}$. –  Marc Palm May 14 '13 at 6:50
@Marc @Victor thank you for your comments. I am indeed interested in what Marc managed to extract: namely, the classification of the unitary irreducible representations which have an invariant vector under the maximal compact subgroup. I am considering the $\mathbb{C}$-points of these classical algebraic groups, as you pointed out, $U(n)$ does not makes sense over $\mathbb{R}$. I have changed some language slightly and added some details to help with the reference request. Thanks again for your time. –  Josh Izzard May 14 '13 at 13:28

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.

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Note, that I don't know which one of the parabolically induced ones have irreducible subquotients or are unitarizabile, though. I am only saying a classification of the former gives pretty easily a classification of the latter. –  Marc Palm May 14 '13 at 16:17
I also would claim that the subquotients are never spherical, but I am not sure in the generality I have stated the results. –  Marc Palm May 14 '13 at 16:18
Thanks for the information Marc, this is helpful –  Josh Izzard May 14 '13 at 20:49

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.

No. What you state is simply the fact when $(G,K)$ is a Gelfand pair. The OP is search for a set of unitary representation, e.g., for $GL_2(\mathbb{C})$, it would be all unitary unramified continuous series representation and the $| \det |^s$ with $\Re s =1$. –  Marc Palm May 14 '13 at 15:01