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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.

GA316
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    For further information not in the introductory book, you might want to consult the follow-up, Representations of Semisimple Lie algebras in the BGG category $\mathcal{O}$ (it is available for free on his website). – Tobias Kildetoft Feb 08 '14 at 09:38

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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.

Matt E
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    I am reading Humphreys book. I roughly know these are the things which we have to use. Can you explain it how it works or any where can I find this formulas applied in particular examples? Humphreys has given only one example. – GA316 Feb 08 '14 at 06:54
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    @GA316: Dear GA, In his comment, Tobias Kildetoft suggests a more advanced book. Let me make a suggestion in the opposite direction: the introductory rep'n theory book by Fulton and Harris gives an excellent introduction to highest weight reps. of Lie algebras, with many examples worked in the text and exercises. Regards, – Matt E Feb 08 '14 at 13:30