Schur Weyl duality for sl_n representations

Question

Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl construction there corresponds to each partition of $d$ a representation of $GL(V)$. Taking $V=\mathbb{C}^n$ this in turn determines a representation of lie algebra $sl_n \mathbb{C}$. This is the approach adopted in Fulton and Harris.

My question is: What if we have an $sl_n\mathbb{C}$ structure on some space other than $\mathbb{C}^n$, do we have the same construction? To be precise, If we have such $sl_n$ representation $V$ then how can we decompose the tensor product $V^{\otimes d}$ into $sl_n$ irreps provided we have a $S_d$ irreducible decomposition.

Answer 1

Depends what you mean by "the same", for example:

For which representations $W$ we can find various reps as summands in $W^{\otimes n}$? A good idea of course is to look at faithful self-dual $W$.

For which representations $W$ does the centraliser of $SL_n$ in $End(W^{\otimes n})$ admit a "nice" description? (This is how Schur--Weyl duality works, - you know that the centraliser is $\mathbb{C}S_n$, and that's where Young diagrams come from.) This is much less obvious.

Or you mean something entirely different from these questions?