Spinoriality of orthogonal representations of reductive groups

Abstract

Let $ G$ be a connected reductive group over a field $ F$ of characteristic 0, and $ \varphi : G \to \operatorname {SO}(V)$ an orthogonal representation over $ F$. We give criteria to determine when $ \varphi $ lifts to the double cover $ \operatorname {Spin}(V)$.