Theorem 2

Let $\rho$ be a linear representation of $\mathfrak{s}\mathfrak{l}(2, k)$ on a finite dimensional vector space E.

(i) There exists a unique linear representation $\pi$ of $\mathbf{S}\mathbf{L}(2, k)$ on E that is compatible with $\rho$.

(ii) A vector subspace F of E is stable under $\pi$ if and only if it is stable under $\rho$.

(iii) Let $x\in E$. Then $\pi (s)x=x$ for all $s\in \mathbf{S}\mathbf{L}(2, k)$ if and only if $x$ is invariant under $\rho ($that is, $\rho (a)x= 0$ for all $a\in \mathfrak{s}\mathfrak{l}(2, k))$.

The existence of $\pi$ follows from the preceding and Th. 1. On the other hand, we know that the group $\mathbf{S}\mathbf{L}(2, k)$ is generated by the elements of the form

$e^+=e$ =

0 1 $t$ 1

where $t\in k($Algebra, Chap. III, §8, no. 9, Prop. 17). This proves the uniqueness of $\pi$.

Assertions (ii) and (iii) follow from what we have said, together with Chap. VII, §3, no. 1, Lemma 1 (i). Q.E.D.

Every finite dimensional $\mathfrak{s}\mathfrak{l}(2, k$)-module therefore has a unique $\mathbf{S}\mathbf{L}(2, k)$-module structure, which is said to be associated to its $\mathfrak{s}\mathfrak{l}(2, k$)-module structure.