Frobenius–Schur indicators for semisimple Lie algebras

Abstract

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 ( 2 n ) , so ( 2 n + 1 ) and sp ( 2 n ) , this is the case for m greater or equal to 2 n − 1 , 4 n − 5 , 4 n − 3 and 2 n + 1 , respectively.