Definition 3

The adjoint representation of the group $\mathbf{S}\mathbf{L}(2, k)$ is the linear representation Ad of $\mathbf{S}\mathbf{L}(2, k)$ on $\mathfrak{s}\mathfrak{l}(2, k)$ defined by

Ad($s$)$.a=sas^{-1}$

for all $a\in \mathfrak{s}\mathfrak{l}(2, k)$ and all $s\in \mathbf{S}\mathbf{L}(2, k)$. When $k$ is $\mathbf{R}$ or $\mathbf{C}$ or a complete non-discrete ultrametric field, we recover Def. 7 of Chap. III, §3, no. 12 (cf. loc. cit., Prop. 49).

By Chap. VII, §5, no. 4, Lemma 1 (i) and (ii), the adjoint representations of $\mathfrak{s}\mathfrak{l}(2, k)$ and $\mathbf{S}\mathbf{L}(2, k)$ are compatible. By Chap. VII, §3, no. 1, Remark 2, Ad($\mathbf{S}\mathbf{L}(2, k)$) $=$ Aut$_e(\mathfrak{s}\mathfrak{l}(2, k))$.