Frobenius–Schur indicators for semisimple Lie algebras

Let g be a finite-dimensional complex semisimple Lie algebra, and let V be a finite-dimensional complex representation of g. 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 sl(n), so(2n), so(2n+1) and sp(2n), this is the case for m greater or equal to 2n−1, 4n−5, 4n−3 and 2n+1, respectively.