The Lie group $G := SO(n, 1)$ acts on the hyperbolic space $\mathbb{H}^n$ by isometries. In particular, we have a representation of $G$ on $F := C^\infty(\mathbb{H}^n)$. I assume that $F$ is endowed with its standard Fréchet structure.
I am interested in classifying the finite dimensional representations that appear in $F$.
Let me denote by $\mathcal{H}_k$ the space of homogeneous (wave) harmonic polynomials on $\mathbb{R}^{n, 1}$ (Minkowski space). Each $\mathcal{H}_k$ is an irreducible finite dimensional representation of $G$. Let me identify $\mathcal{H}_k$ with the subspace of $F$ it induces by restriction on $\mathbb{H}^n$. The direct sum $\bigoplus_{k \in \mathbb{N}} \mathcal{H}_k$ is isomorphic as a $G$-space to the space of regular functions (in the sense of algebraic geometry) of $\mathbb{H}^n$. As a consequence, a simple argument using the Stone-Weierstrass theorem shows that $$ F = \overline{\bigoplus_{k \in \mathbb{N}} \mathcal{H}_k}. $$ How can one conclude from this that the $\mathcal{H}_k$ are the only finite dimensional irreducible subrepresentations of $F$?
In the compact setting this would be a simple consequence of the Peter-Weyl theorem but in the non-compact case I don't know how to do.
Thanks in advance for your answer!
-- Original question:
Assume F is a Fréchet space which is a representation of a simple Lie group G. Let V be a finite dimensional subrepresentation of G in F. Does there exist a closed subspace W of F which is stable for the action of G and such that $V \oplus W = F$?
