Spinorial representations of Lie Groups

Abstract

We solve the question: which finite-dimensional irreducible orthogonal representations of connected reductive complex Lie groups lift to the spin group? We have found a criterion in terms of the highest weight of the representation, essentially a polynomial in the highest weight, whose value is even if and only if the corresponding representation lifts. The criterion is closely related to the Dynkin Index of the representation. We deduce that the highest weights of the lifting representations are periodic with a finite fundamental domain. Further, we calculate these periods explicitly for a few low-rank groups.