Group actions on compact homogeneous spaces with an open orbit

In the talk, I will consider the questions of existence of an open orbit and of finiteness of orbit sets for various group actions on compact homogeneous spaces of reductive groups (such spaces are sometimes called generalized flag varieties). Let $G$ be a semisimple algebraic group, and let $P$ be a parabolic subgroup. We will discuss solutions of the following problems:
1. Suppose that $G$ does not contain simple components of type A. One has to find all numbers $n$ such that $G$ acts on the product of $n$ copies of $G/P$ (this product will be denoted by $(G/P)n$) with an open orbit.
2. Let $G$ be an arbitrary semisimple group. One has to find all numbers n such that the set of $G$-orbits on $(G/P)n$ is finite.
3. One has to classify actions of the commutative unipotent group of dimension $dim(G/P)$ on $G/P$ with an open orbit.
To understand the talk, listeners need to know basic notions of complex simple Lie algebra theory and algebraic group theory.