Let a linear finite-dimensional isotypic (i.e., decomposable into a direct sum of isomorphic irreducible subrepresentations) representation of a compact Lie group G in a finite-dimensional space V is given. How to describe all linear transformations of the space V, that commute with all elements of the image of this representation? This is probably known. I have my own assumption about this, but I want to know a specific link to the formulation and proof of a way to describe all such transformations

Let a linear finite-dimensional isotypic (i.e., decomposable into a direct sum of isomorphic irreducible subrepresentations) representation of a compact Lie group $G$ in a finite-dimensional space $V$ is given. How to describe all linear transformations of the space $V$, that commute with all elements of the image of this representation? This is probably known. I have my own assumption about this, but I want to know a specific link to the formulation and proof of a way to describe all such transformations