Let $G$ a compact semisimple Lie group, $H$ a subgroup of $G$. Is it always possible to find an irreducible representation $R$ of $G$ such that the stabilizer of an $x\in R$ is "locally isomorphic" to $H$? I am only interested in the case when $H$ is a continuous subgroup, and "locally isomorphic" means has the same Lie algebra. 


If the result is not true, what would be the simplest counterexample?
What about weaker results when $R$ is not required to be irreducible, or when stabilizers of a finite set and not just of a single element are allowed?


Let me give an example of what I have in mind. I am only interested in the case when both $G$ and $H$ are finite dimensional compact Lie groups. As a representative example let's take $G=SU(3)$ and two subgroups $H\_1=SU(2)$, $H\_2=SU(2)\times U(1)$ (all groups over $\mathbb{C}$). I can realize $H\_1$ as the stabilizer of $x=(0,0,1)^t$ in the fundamental representation of $SU(3)$. I can also realize $H\_2$ as the stabilizer of $x=diag(1,1,-2)$ in the adjoint representation of $SU(3)$. Can I always do this? What would be an algorithm to construct the representation given the set of generators of $G$ which generate $H$? 


Update: the answers below show that the answer is yes, if you allow reducible representations (which I don't mind). There remains a problem of how to construct $R$ concretely and simply, given the list of generators for $H$ inside $G$. Such constructions can be also extracted from the proofs below, however they do not look elementary (for me). 



Realizing a subgroup of a Lie group as a stabilizer subgroup
You need $H$ to be closed.
The Mostow-Palais theorem (1,2,3) then gives what you want -- with "equal" in place of "locally isomorphic", but with a possibly reducible representation. I'm not aware of conditions ensuring that the representation can be chosen irreducible.


(1): <http://en.wikipedia.org/wiki/Mostow-Palais_theorem>


(2): <http://books.google.com/books?id=oCO0xOzNLhAC&pg=PA373>


(3): <http://books.google.com/books?id=yqbocEpFdyQC&pg=PA104>


