Exercise 7, § 7

Let $\tau : G\rightarrow \mathbf{S}\mathbf{O}(n,\mathbf{R})$ be a real unitary representation of G; denote by $\varphi :$ Spin$(n,\mathbf{R})\rightarrow \mathbf{S}\mathbf{O}(n,\mathbf{R})$ the canonical double covering. Then $\tau$ is said to be spinorial if there exists a morphism $\widetilde{\tau}: G\rightarrow$ Spin$(n,\mathbf{R})$ such that $\varphi \circ \widetilde{\tau}=\tau$.

a) Let $\Sigma$ be a subset of $P(T, \tau )$ such that $\Sigma \cup (-\Sigma ) = P(T, \tau )$ and $\Sigma \cap (-\Sigma ) =\emptyset$; denote by $\omega_{\Sigma}$ the sum of the elements of $\Sigma$. The class $\varpi$ of $\omega_{\Sigma}$ in $X(T)/2X(T)$ is independent of the choice of $\Sigma$. Prove that $\tau$ is spinorial if and only if $\varpi = 0$. b) Prove that $\rho \in X(T)$ if and only if the adjoint representation is spinorial.