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If $V$ is an irreducible representation of a semi simple lie algebra having highest weight $\lambda$ then what will be the highest weight of the corresponding irreducible representation $V^*$ (Dual of $V$)?

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    It is $\lambda$ itself, see Humphrey's book "Representations of Semisimple Lie algebras in the BGG category O" section 3.22012-11-01
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    Actually, it is not always $\lambda$ itself. See the answer to this question here: http://mathoverflow.net/questions/111127/finding-highest-weight-of-dual-of-a-representation-of-a-semisimple-lie-algebra2013-05-10

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