Let $\frak{g}$ be a complex semisimple Lie algebra with root lattice $Q$ and positive weight space $P^+$. Let $\lambda, \mu \in Q \cap P^+$, with corresponding respective fin-dim irreducible representations $V_{\lambda}$ and $V_{\mu}$. Form the tensor product $V_{\mu} \otimes V_{\lambda}$ and then decompose it into irreducibles
$$
V_{\mu} \otimes V_{\lambda} \simeq V_{\gamma_1} \oplus \cdots \oplus V_{\gamma_m}.
$$
Will the dominant weights $\gamma_i$ still be contained in $Q \cap P^+$ or will they in general only be contained in $P^+$? If they are only contained in $P^+$, then what is a simple example that illustrates this?
Finally, if in general there exist such $\gamma$'s that live outside $Q \cap P^+$, can we categorize/classify the subset of all such dominant weights?
Best Answer
Just to summarize what was mentioned in the comments (1 2) and to have this question marked as answered, what follows is the explanation of why the answer is yes, the irreps that appear in the decomposition of $V_{\mu}\otimes V_{\lambda}$ will be indexed by root lattice elements.
First notice that every weight that appears in both $V_{\mu}$ and $V_{\lambda}$ must also belong to the root lattice (because we can get to every weight from the highest weight by going down along simple roots: this is a well-known fact about the structure of irreps for semisimple Lie algebras). Then note that $V_{\mu}\otimes V_{\lambda}$ is spanned by $u\otimes v$ where $u$ is a weight basis element of $V_{\mu}$ and $v$ a weight basis element of $V_{\lambda}$. But the weight of $u\otimes v$ is the sum of the weights of $u$ and of $v$. So all the $u\otimes v$ have weights that belongs to the root lattice. In particular, all highest weight vectors do too, hence all irreps that appear are indexed by root lattice elements.