from $\k$ to $\overline{\k}$. Theorem \[proprappel\] has the following consequence.

$M\in\rep^\alpha(Q,\R)$ is absolutely indecomposable if and only if ${\rm End}(\overline{M})= I_{\overline{M}}\oplus\mathfrak{z}$ where $\mathfrak{z}=\mathfrak{z}(\overline{\k}):=\{(\lambda\, {\rm Id}_{\alpha(i)})_{i\in I}\,|\, \lambda\in\overline{\k}\}\simeq\overline{\k}$ and ${\rm Id}_{\alpha(i)}$ the identity endomorphism of $\overline{\R}{^{\alpha(i)}}$. \[coro\]

To speak of Jordan decomposition [@milne Theorem 9.18] we regard $\G_\alpha(\overline{\R})$ as an affine $\overline{\k}$-algebraic group and $\G_\alpha(\R)$ as the group of $\k$-points of $\G_\alpha(\overline{\R})$.

More precisely, we consider the affine $\k$-group scheme $\G_\alpha^{\R/\k}$ which represents the $\k$-group functor $$\Alg_\k\rightarrow\Gr,\hspace{1cm} \R'\mapsto \G_\alpha(\R\otimes_\k\R')$$ so that $\G_\alpha^{\R/\k}(\k)=\G_\alpha(\R)$ and $\G_\alpha^{\R/\k}(\overline{\k})=\G_\alpha(\overline{\R})$. This functor is indeed representable as $\R$ is finite-dimensional over $\k$ (see [@Neron Theorem 4]).

Let $\pi:\G_\alpha^{\R/\k}\rightarrow\G_\alpha$ be induced by the $\k$-algebra morphism $\R\rightarrow\k$ and denote by $\U_\alpha$ the kernel of $\pi$. Then $\G_\alpha^{\R/\k}=\U_\alpha\rtimes\G_\alpha$ and $\U_\alpha$ is a unipotent group.

For any $\k$-algebra $\R'$, we denote by $Z_\alpha(\R')\simeq\mathbb{G}_m(\R')$ the subgroup of $\G_\alpha(\R')$ of elements of the form $(\lambda\, {\rm Id}_{\alpha(i)})_{i\in I}$ for some $\lambda\in\R'{^\times}$ where ${\rm Id}_{\alpha(i)}$ denotes the identity of $\GL(\R'^{\alpha(i)})$. The set of semisimple elements of $Z_\alpha(\R)$ and $Z_\alpha(\overline{\R})$ are respectively $Z_\alpha(\k)$ and $Z_\alpha(\overline{\k})$.

Let $M\in\rep^\alpha(Q,\R)$. The following assertions are equivalent :

\(i) $M$ is absolutely indecomposable.

\(ii) The semisimple elements of ${\rm Stab}_{\G_\alpha(\overline{\R})}(\overline{M})$ all lie in $Z_\alpha(\overline{\k})$.

\(iii) The semisimple elements of ${\rm Stab}_{\G_\alpha(\R)}(M)$ 