Exercise 1, § 4

Take G to be one of the groups $\mathbf{S}\mathbf{U}(n,\mathbf{C})$ or $\mathbf{U}(n,\mathbf{H})$; identify $\mathfrak{g}_{\mathbf{C}}$ with $\mathfrak{s}\mathfrak{l}(n,\mathbf{C})$ or $\mathfrak{s}\mathfrak{p}(2n,\mathbf{C})$, respectively, cf. §3, no. 4 and Exerc. 3. We use the notations of Chap. VIII, §13, nos. 1 and 3, with $k=\mathbf{C}$.

a) Show that the subgroup T of G consisting of the diagonal matrices with complex coefficients is a maximal torus, and that $L(T)_{(\mathbf{C})}=\mathfrak{h}$.

b) Identify X(T) with a subgroup of $\mathfrak{h}^*$ by means of the homomorphism $\delta$. Show that the linear forms $\varepsilon_i$ and the dominant weights $\varpi_j$ belong to X(T). If $t=$ diag($t_1, . . . , t_n$)$\in T$, then $\varepsilon_i(t) =t_i$ and $\varpi_j(t) =t_1. . . t_j$ for $1\leq i, j\leq n$. c) Deduce from b) another proof of the fact that the groups $\mathbf{S}\mathbf{U}(n,\mathbf{C})$ and $\mathbf{U}(n,\mathbf{H})$ are simply-connected (cf. §3, Exerc. 4 and 6).