Exercise 18, § 1

Assume that $k=\mathbf{R},\mathbf{C}$ or a field complete for a discrete valuation and with residue field of characteristic $p\not= 0$ (for example a finite extension of the $p$-adic field $\mathbf{Q}_p$).

a) Let $\mathfrak{n}$ be a nilpotent Lie algebra, N the Lie group obtained by giving $\mathfrak{n}$ the Hausdorff law (Chap. III, §9, no. 5), and $\rho :\mathfrak{n}\rightarrow \mathfrak{g}\mathfrak{l}(E)$ a linear representation of $\mathfrak{n}$ on a finite dimensional vector space E. Assume that $\rho (x)$ is nilpotent for all $x\in \mathfrak{n}$, and put $\pi (x) =$ exp($\rho (x)$). Show that $\pi$ is the only homomorphism of Lie groups $\varphi : N\rightarrow \mathbf{G}\mathbf{L}(E)$ such that $L(\varphi ) =\rho$.

(When $k=\mathbf{R}$ or $\mathbf{C}$, use the fact that N is connected. When $k$ is ultrametric, show that the eigenvalues of $\varphi (x),x\in N$, are equal to 1; for this, prove first of all that, if $k'$ is a finite extension of $k$, and if $(\lambda_1, . . . , \lambda_n, . . .)$ is a sequence of elements of $k'$ such that $\lambda_n=\lambda^p_{n+1}$ for all $n$ and $\lambda_1= 1$, then $\lambda_n= 1$ for all $n.$)

b) Let $\rho :\mathfrak{s}\rightarrow \mathfrak{g}\mathfrak{l}(E)$ be a finite dimensional linear representation of $\mathfrak{s}$, and let $\pi$ be the homomorphism from $\mathbf{S}\mathbf{L}(2, k)$ to $\mathbf{G}\mathbf{L}(E)$ compatible with $\rho$ (no. 4). Show that $\pi$ is the unique homomorphism of Lie groups $\varphi :\mathbf{S}\mathbf{L}(2, k)\rightarrow$ $\mathbf{G}\mathbf{L}(E)$ such that $L(\varphi ) =\rho$. (Use $a$) to prove that $\pi$ and $\rho$ coincide on the exp($n$), with $n$ nilpotent in $\mathfrak{s}$, and remark that $\mathbf{S}\mathbf{L}(2, k)$ is generated by the exp($n$)$.)$