Let $k$ be an algebraically closed field of characteristic $p\geq0$. An affine algebraic group $G$ is an affine algebraic variety (a Zariski closed subset of $k^m$ for some $m$) such that multiplication and inversion map are morphisms of varieties. An example is the special linear group $SL_n(k)=SL_n$.
Let $G$ be an affine algebraic group and let $H\leq G$ be a closed subgroup. In general $\Omega=G/H$ is a variety (quasi-projective a priori, however if they are both reductive $\Omega$ is affine).
Assume $\dim\Omega>0$. Is $\Omega$ (non-) irreducible, in general? What about if we make stronger assumption on $G$ and $H$? For example, what happen if $G=GL_{2n}$ and $H=GL_n\wr S_2$; or $G=GL_n$ and $H=O_n$? In the first case $H$ is the subgroup that fixes a decomposition $V=V_1\oplus V_2$ of $V=k^{2n}$ where $\dim V_1=\dim V_2$; in the second case $H=O_n$ is the orthogonal group, i.e. the stabiliser of a non-degenerate quadratic form defined on $V=k^n$ ($n$ even if $p=2$).