Let $\lambda \in H^*$ be the irreducible standard cyclic module $V(\lambda)$ of weight $\lambda$ of a semisimple Lie algebra $L$. What are all the possible ways to determine :
1) Which $V(\lambda)$ are finite dimensional?
2) Which weight $\mu$ occur in $V(\lambda)$ and with what multiplicity?
I am reading Humphreys Lie algebra book VI chapter Representation theory, He has given some formulas, but can any one tell me overview of them and which one among them is efficient one and how to use it to finding the answer for the above questions in the particular examples.
Is there any other methods which are not in Humphrey's book?
I am trying to understand and get a solution for the above two question, which are very important for my research.
Thanks in Advance.
The representation $V(\lambda)$ is finite dimensional iff $\lambda$ is dominant integral. This is the theory of highest weight representations of $L$.
To compute the dimensions of the weight spaces in $V(\lambda)$, say when $\lambda$is finite dimensional, there is the Weyl character formula and the Kostant multiplicity formula.