Rationality of quasi-elementary group actions

Question

I am learning the paper https://arxiv.org/pdf/1604.01005.pdf "Reductive group actions" by Knop and Krotz.

They defined that a linear k-algebraic group $H$ with unipotent radical $H_{u}$ is quasi-elementary if $H/H_{u}$ is connected and all k-rational elements are semisimple.

For a $k$-split solvable group $H$, we have the following theorem (theorem 15.11) from the book "Linear algebraic groups" by Borel: If $X$ over $k$ is a homogeneous $H$-variety, then $X$ is affine and $X(k)\neq \emptyset$.

I am wondering do we have a similar result for quasi-elementary groups actions? Since for a quasi-elementary group $H$, we have a factorization $H=MAN$, with $M$ anisotropic and $A$ a split torus, and $N$ its unipotent radical. I hope to reduce this problem to the anisotropic group action.

Any references will be appreciated!

Answer 1

Not all homogeneous varieties $X$ for quasi-elementary groups $H$ have a rational point. This is not even true for anistropic groups. An example would be the variety of Borel subgroups which has no rational points unless $H$ is quasi-split.

On the other hand, if $X(k)\ne\emptyset$ then $X$ is affine. That is Prop. 3.4 of our paper.