Spinorial Representations of Orthogonal Groups

Let G be a real compact Lie group, such that G = G 0 C 2 , with G 0 simple.Here G 0 is the connected component of G containing the identity and C 2 is the cyclic group of order 2 .We give criteria for whether an orthogonal representation : G O(V ) lifts to Pin(V ) in terms of the highest weights of and also in terms of character values.From these criteria we compute the first and second Stiefel-Whitney classes of the representations of the orthogonal groups.