Frobenius–Schur indicators for semisimple Lie algebras

Let be a finite-dimensional complex semisimple Lie algebra, and let V be a finite-dimensional complex representation of . We give a closed formula for the mth Frobenius–Schur indicator, m>1, of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical complex Lie algebras , , and , this is the case for m greater or equal to 2n−1, 4n−5, 4n−3 and 2n+1, respectively.