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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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Thanks @Julian. –  Rekha Biswal Nov 1 '12 at 10:10
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It is $\lambda$ itself, see Humphrey's book "Representations of Semisimple Lie algebras in the BGG category O" section 3.2 –  Aaron Nov 1 '12 at 13:24
    
Actually, it is not always $\lambda$ itself. See the answer to this question here: mathoverflow.net/questions/111127/… –  Alistair Savage May 10 '13 at 21:26
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