Let $G$ be a reductive algebraic group over some algebraically closed field $k$. Recall that, given an algebraic variety $X$ with an action of $G$, it is said that this action is linearizable if there exists a line bundle $\pi: L \to X$ with a fiberwise linear action of $G$ on $L$ for which $\pi$ is equivariant.

As a particular case, we can take $X=G$ and consider the action of $G$ on itself by conjugation. Then, my questions are:

**Is this conjugation action linearizable?****If the conjugation action is linearlizable, under what conditions can the line bundle $L$ be taken ample?**

In particular, I am concerned with the case $G = SL(2,\mathbb{C})$. In that case, we know that $X // G = \mathbb{A}^1$. As mentioned in this paper of Doebeli, if the GIT quotient is zero-dimensional, the existence of linearlization follows from Luna's slice theorem. However, if the quotient is $1$-dimensional, as in the case of $G=SL(2,\mathbb{C})$, the problem is harder and, in general, the result is false.

Anycase, it seems like, using the results and definitions of that paper, it could be possible to define a linear model for this action satisfying conditions of proposition 2, and thus obtaining that this conjugation action is linearlizable. However, I belive that this procedure too complicated and there must exists a general argument to show that these conjugation actions are always linearizable, but I can't find it. Furthermore, even using that paper, we can't assure that such line bundle is ample.

Thank you so much in advance!