Exercise 2, § 4

Take $G =\mathbf{S}\mathbf{O}(n,\mathbf{R})$, with $n\geq 3$; put $n= 2l+ 1$ if $n$ is odd, and $n=$ $2l$ if $n$ is even. The algebra $\mathfrak{g}_{\mathbf{C}}$ can be identified with $\mathfrak{o}(n,\mathbf{C})$; we use the notations of Chap. VIII, §13, nos. 2 and 4. Denote by $(f_i)_{1\leq i\leq n}$ the canonical basis of $\mathbf{R}^n$. Put $e_j = \frac{1}{\sqrt{2}}(f_{2j-1}+if_{2j})$ and $e_{-j} = \frac{1}{\sqrt{2}}(f_{2j-1}-if_{2j})$ for $1\leq j\leq l$, and $e_0 = i\sqrt{2}f_{2l+1}$ if $n$ is odd; choose the Witt basis of $\mathbf{C}^n$ given by $e_1, . . . , e_l, e_{-l}, . . . , e_{-1}$ if $n$ is even (resp. $e_1, . . . , e_l, e_0, e_{-l}, . . . , e_{-1}$ if $n$ is odd).

a) Let $H_i$ be the subspace of $\mathbf{R}^n$ generated by $f_{2i-1}$ and $f_{2i}(1\leq i\leq l)$; show that the subgroup of G consisting of the elements $g$ such that $g(H_i)\subset H_i$ and det($g|H_i$) $= 1$ for $1\leq i\leq l$ is a maximal torus T of G, and that $L(T)_{(\mathbf{C})}=\mathfrak{h}$. b) Identify X(T) with a subgroup of $\mathfrak{h}^*$ by means of $\delta$. Show that the linear forms $\varepsilon_i$ belong to X(T); if $t\in T$ and if the restriction of $t$ to $H_j$ is a rotation through an angle $\theta_j$, then $\varepsilon_j(t) =e^{i\theta_j}$. The weights $\varpi_1, . . . , \varpi_{l-2}$; $2\varpi_{l-1},2\varpi_l, \varpi_{l-1}\pm \varpi_l$ belong to X(T). If $n$ is odd, $\varpi_{l-1}$ belongs to X(T). c) Let $\widetilde{G} =$ Spin$(n,\mathbf{R})$ and let $\varphi : \widetilde{G}\rightarrow G$ be the canonical covering. For

$\theta \theta = (\theta_1, . . . , \theta_l)\in \mathbf{R}^l$, put $t(\theta \theta ) =\prod_{i=1}^l$ (cos$\theta_i-f_{2i-1}f_{2i}$ sin $\theta_i$)$\in \widetilde{G}$. Show that the set of the $t(\theta \theta )$ for $\theta \theta \in \mathbf{R}^l$ is a maximal torus $\widetilde{T}$ of $\widetilde{G}$, such that $\varphi ( \widetilde{T}) = T$. If $X( \widetilde{T})$ is identified with a subgroup of $\mathfrak{h}^*$, we have $\varepsilon_j(t(\theta \theta )) =e^{2i\theta_j}$.

d) Show that the weights $\varpi_{l-1}$ and $\varpi_l$ belong to $X( \widetilde{T})$; deduce that Spin$(n,\mathbf{R})$ is simply-connected (cf. §3, Exerc. 5).