Exercise 2, § 6

Let G be the universal covering of $\mathbf{SL}(2, \mathbf{R})$, $\pi$ the canonical morphism of G onto $\mathbf{SL}(2, \mathbf{R})$ and N its kernel. (a) Arguing as in Exercise 1, show that there exists an isomorphism of the real analytic manifold $\mathbf{U} \times \mathbf{R}^2$ onto the real analytic manifold $\mathbf{SL}(2, \mathbf{R})$. Deduce that N is isomorphic to $\mathbf{Z}$.

(b) If $\rho$ is an analytic linear representation of $G$ on a complex vector space, then $\mathrm{Ker}\,\rho \supset N$. (The representation $L(\rho)$ of $\mathfrak{sl}(2, \mathbf{R})$ defines, by complexification, a representation of $\mathfrak{sl}(2, \mathbf{C})$ and the latter is, by Exercise 1, of the form $L(\sigma)$, where $\sigma$ is an analytic linear representation of $\mathbf{SL}(2, \mathbf{C})$. Then $L(\sigma \circ \pi) = L(\rho)$ and hence $\sigma \circ \pi = \rho$.)