Realizing a subgroup of a Lie group as a stabilizer subgroup 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). 
 A: 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
A: Though it's not so readily available online, there is an elementary textbook treatment of the basic theory here in Representations of Compact Lie Groups by Brocker and tom Dieck (GTM 98, Springer, 1985).
1) As they note early in the book, the isotropy group $H$ of a point $v$ must be closed in an arbitrary topological group $G$ acting continuously: here $H$ is the inverse image of $v$ under the (continuous) orbit map $g \mapsto g \cdot v$.
2) As a corollary of the Peter-Weyl theorem (using representative functions),
they derive easily in III, (4.6) for any compact Lie group $G$ (not necessarily semisimple): Every closed subgroup $H$ of $G$ appears as the isotropy group of an element of some $G$-module.
3) Unfortunately, since all of this theory is somewhat abstract, it doesn't seem to shed light on your question about finding an irreducible representation.    However, you do have complete reducibility here of all (necessariy finite-dimensional) representations, which are usually studied in the essentially equivalent (complexified) Lie algebra setting.    So if there is a counterexample it would probably be best located there.   It seems hard to compute directly with group elements and group representations, but for example Willem de Graaf has worked extensively with computer algorithms for both real and complex Lie algebras.
P.S. Though Dan Mostow (who just turned 90) was on my thesis committee, I don't think you need to get into his more general results involving group actions on manifolds.
