interesting to compare it with the following very simple criterion for the extension of a representation in the case of affine algebraic groups without the hypothesis of normality (this subject will be treated in more detail and precision in Section \[subsection:obsgeo\]).

First we need to introduce some definitions.

Let $H \subseteq G$ be a closed inclusion of affine algebraic groups.

1.  A character $\chi:H \to \Bbbk$ is said to be extendible to $G$ if there is a polynomial function $f \in \Bbbk[G]$ such that $f(1)=1$ and for all $x \in H$, $x \cdot f =\chi(x)f$ –or equivalently, for all $y \in G$, $f(yx)=f(y)\chi(x)$.

2.  If $M=(M,\cdot)$ is a rational $H$–module, and $\chi$ is a character of $H$, we call $M_\chi$ the rational $H$–module $(M,\cdot_\chi)$ where $\cdot_\chi $ is defined on $M$ as $x
          \cdot_\chi m=\chi(x)(x\cdot m)$ for all $m \in M$. Clearly $M_\chi= M \otimes \Bbbk_\chi$ where $\Bbbk_\chi$ is the one dimensional $H$–module associated to the character $\chi$.

Next theorem guarantees that for affine algebraic groups, every representation can be extended “up to the twist by an extendible character”.

[@kn:hmbb Theorem 1][@kn:nosotros2 Theorem 8.2.3] \[theo:obse1\] If $H \subseteq G$ is a closed inclusion of affine algebraic groups for any rational 