Let $V$ be a highest weight module with a max. vector $v^+ \in V_{\lambda}$. By Theorem 20.2 in Humphreys' book "Introduction to Lie algebras and representation theory" we get that
- $V$ is the direct sum of its weight spaces, i.e. $V= \bigoplus_\mu V_\mu$.
- The weights of $V$ are of the form $\mu = \lambda - \sum_i k_i \alpha_i$ for $k_i \in \mathbb{Z}_+$, i.e. $ \mu < \lambda$.
- For each $\mu \in H^*$ we have $\operatorname{dim} V_\mu < \infty $.
Does two imply that there are only finitely many weights? If yes, does one and three then imply that $V$ is finite dimensional as a finite direct sum of finite dimensional vector spaces?
I know there are infinite dimensional highest weight modules (Verma modules), so something cannot be right with my reasoning. Are there weights $\mu$ as in two with arbitrary big $k_i$ allowed? I think not, because $\mu < \lambda$ iff. $\lambda - \mu$ is a sum of positive roots.
The answer to the question is no.
Yes there are infinitely many weights and they are of the form $\mu = \lambda - \sum_i k_i \alpha_i$ with arbitrarily big $k_i \in \mathbb{Z}^+$ allowed and this is not a contradiction to $\mu < \lambda$, i.e. $\lambda - \mu$ being a sum (with positive coefficients) of positive roots.
Consider $\mu = \lambda - k \alpha$ for some $k \in \mathbb{Z}^+$ and $\alpha$ a simple root, i.e. positive root. Then $\lambda - \mu= \lambda - (\lambda - k \alpha) = k \alpha$, which is a sum of (with positive coefficients) of positive roots, so we have $\mu < \lambda$ for arbitrary big $k \in \mathbb{Z}^+$.
This shows we can have infinitely many weights.