Return to Revisions

I think there must be some different ideas of what your question means, looking at the comments. I understood it to mean as in the decomposition of tensor powers of $\mathfrak{g}$-representation. In which case, the answer is a definite yes. For a start, the action of $\mathfrak{g}$ on any symmetric power of a representation preserves the degree. So in this way $S(\mathfrak{g}) = \bigoplus_{i=0}^\infty S^{i}(\mathfrak{g})$. Then the isotypic components in this sense just refers to the decomposition of these finite dimensional pieces into irreducibles which are automatically finite dimensional themselves. The explicit decompositions themselves can be quite hard to work out although there are formulae for these things.