Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero (I am particularly interested in the case $G=GL_n(K))$. Let $H$ be a closed subgroup of $G$. It is known (from e.g. the book of Humphreys on linear algebraic groups) that we have a scheme structure on the space $G/H$. Are there cases in which $H$ is not reductive and $G/H$ is affine? If for example $H=B$ is a Borel subgroup, then $G/H$ is projective. The same holds if we just know that $H$ contains some Borel subgroup. Is there any other choice of non-reductive $H$ for which the quotient space $G/H$ will have a structure of affine variety?
When $G$ is a non-reductive connected affine algebraic group with $\dim R(G) > 0$, $G$ can be embedded as the unit group of an irreducible affine algebraic monoid $M$. We also assume that $G$ is a nonnilpotent group. Most of time, $M$ has infinitely many minimal idempotents, all lying in the kernel of $M$ (two-sided semigroup-theoretic minimal ideal of $M$), $\ker(M)$. Conversely, all idempotents in $\ker(M)$ are minimal.
In this case, the one-sided, two-sided centralizers of an idempotent $e \in \ker(M)$ in $G$, $Z_G^l(e)$, $Z_G^r(e)$, $Z_G(e)$, all proper connected closed non-normal subgroups of $G$. Let $H$ be such a subgroup of $G$. Since $R_u(H) < R_u(G)$ (ref: W. Huang, Kernels, regularity, unipotent radicals in a linear algebraic monoids, Forum Math. 23(2011), 803-834.), by Grosshans LNM 1673, Theorem 7.1, the homogeneous space $G/H$ is affine.