Exercise 14, § 13

a) Let G be the group associated to $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(n, k)$ by the procedure of §7, Exerc. 26. The natural $\mathfrak{g}$-module structure on $k^n$ gives rise to a homomorphism $\varphi : G\rightarrow \mathbf{G}\mathbf{L}(n, k)$. Use loc. cit. h) to prove that $\varphi$ is injective, and loc. cit. f) to prove that

Im($\varphi$ ) $=\mathbf{S}\mathbf{L}(n, k)$.

b) Let E be a finite dimensional $\mathfrak{s}\mathfrak{l}(n, k$)-module, and $\rho$ the corresponding representation of $\mathfrak{s}\mathfrak{l}(n, k)$. Show that there exists a unique representation $\pi :\mathbf{S}\mathbf{L}(n, k)\rightarrow \mathbf{G}\mathbf{L}(E)$ such that $\pi (e^x) =e^{\rho(x)}$ for every nilpotent element $x$ of $\mathfrak{s}\mathfrak{l}(n, k)$ (use $a$)$)$. We say that $\rho$ and $\pi$ are compatible. Generalize the results proved for $n= 2$ in §1, no. 4.

c) Assume that $k$ is $\mathbf{R},\mathbf{C}$ or a complete field for a discrete valuation with residue field of characteristic $\not= 0$. Show that $\rho$ and $\pi$ are compatible if and only if $\pi$ is a homomorphism of Lie groups such that $L(\pi ) =\rho$ (same method as in Exerc. $18b$) of §1).

d) Prove analogous results for $\mathfrak{s}\mathfrak{p}(2n, k)$ and $\mathbf{S}\mathbf{p}(2n, k)$.