Let $\frak{g}$ be a semisimple Lie algebra over the complex number, and $M$ a finite dimensional representation. For any dominant weight $\lambda$, denote $M^\lambda$ the isotypic component of $\lambda$. Is there an explicit formula for the canonical projection $M \to M^\lambda$?
As an analogous example, for a finite dimensional complex representation $(V,\rho)$ of a finite group $G$, the canonical projection to the $\chi$-isotypic component is
$$p_\chi = \frac{\dim(\chi)}{|G|}\int_{t\in G} \chi(t)^* \rho_t,$$
where $\chi$ is an irreducible character [1]. (I find this nontrivial.)
Reference
- Theorem 2.6.8, Linear representations of finite groups, J.P. Serre.