is obtained by taking the $\G_\alpha(\R)$-invariant part of the $\G_\alpha(\R)$-equivariant isomorphism in Proposition \[proprep\] (i).

We now want to prove the analogue of the above corollary for absolutely indecomposable representations. Instead of taking the $\G_\alpha(\R)$-invariant part (i.e. the isotypical component of the trivial character) we will take the isotypical component of another linear character of $\G_\alpha(\R)$. This approach is very different from Kac’s proof [@kac] of independence of the orientation in the finite field case (i.e. when $\R=\k$). We assume now in the remainder of this section that $\R$ is a split local Frobenius $\k$-algebra with maximal ideal $\mathfrak{m}$.

If $M\in\rep^\alpha(Q,\R)$, then

$${\rm End}(M)=\left.\left\{(X_i)_{i\in I}\in\bigoplus_{i\in I}\gl(\R^{\alpha(i)})\,\right|\, M_{i\rightarrow j}X_i=X_j M_{i\rightarrow j}\text{ for all }i\rightarrow j\in E\right\}.$$ Note that

$${\rm End}(M)^\times={\rm Stab}_{\G_\alpha(\R)}(M).$$

In the following we put $\overline{\R}:=\R\otimes_\k\overline{\k}$. As $\overline{\R}$ is finite-dimensional over $\overline{\k}$, any $M'\in\rep^\alpha(Q,\overline{\R})$ is a finite-dimensional $\overline{\R}Q$-module and so we have the following well-known result [@Brion Theorem 1.15].

Let

$$M'\simeq\bigoplus_{i=1}^rm_i M'_i$$ be the decomposition of $M'\in\rep^\alpha(Q,\overline{\R})$ into indecomposables, then

$${\rm End}(M')= I_{M'}\oplus K,$$ with $I_{M'}$ a nilpotent ideal and $K$ a $\overline{\k}$-subalgebra of ${\rm End}(M')$ isomorphic to $\prod_{i=1}^r{\rm Mat}_{m_i}(\overline{\k})$. \[proprappel\]

If $M\in\rep^\alpha(Q,\R)$, we denote by $\overline{M}$ the representation of $Q$ over $\overline{\R}$ obtained from $M$ by extension of scalars 