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).

ispractical. You start with $A$, the $\mathbf{R}$-algebra of real-valued polynomial functions on $G$; let $I$ be its ideal of functions vanishing on $H$. There exists a finite dimensional (say $d$-dimensional) $G$-invariant subspace $V$ of $A$ such that $I$ is generated by $W=V\cap I$. Then $H$ is precisely the stabilizer of $W$. This is close to what you require: to get $H$ as stabilizer of a line instead of a subspace, consider the exterior power $\Lambda^dV$. Then $H$ is the stabilizer of the line $\Lambda^dW$. $\endgroup$1more comment