An isotypic (maybe reducible) representation V of GL(V) may be represented by its highest weight subspace HW(V). We have dim HW(V) equal to the multiplicity of the irreducible representation inside V and we can identify GL(V) morphisms between two such components with (linear) morphisms between highest weight spaces.
I would like to ask if there is a similar theory for other (e.g. finite) groups. More precisely, is there a nice way to choose a subspace of a representation with the two properties above.
(Of course one can do it naively by choosing explicitely a decomposition. Then choose one vector in the irreducible component and the corresponding vectors (unique up to scalar by Schur lemma) in other components - the vector space spanned by all these vectors will have the desired property. However, I would like to know if there is a better way - an analogue of highest weight vectors. Of course some choices have to be made, but maybe they are more natural then choosing a decomposition and a vector inside, as this is in fact choosing a basis, not just a vector subspace)