Let $G$ be a connected, reductive quasi-split group over a field $k$, acting on an afffine $k$-variety $X$. Let $B = TU$ be a Borel subgroup of $G$ with maximal torus $T$ and unipotent radical $U$. Then $G$ acts on the $k$-algebra of global sections $\mathcal O_X(X)$ of $X$. If necessary, we can assume $k$ has characteristic zero, $k$ is algebraically closed, or $G$ is split.

I recently asked a question on here about what we can say about a fundamental domain for the quotient space of $X$ under the action of $U$ in the case where $X = \mathbb A_k^n$ and $G$ acts by linear automorphisms. Friedrich Knop gave a very helpful answer, and in order to better understand it and the references he provided, I wanted to ask about some basic terminology and results assumed there.

A *highest weight vector* for this action is a nonzero global section $f \in \mathcal O_X(X)$ with the property that $b.f = \chi(b)f$ for some $k$-rational character $\chi$ of $B$. The stabilizer of the line $kf$ through $f$ is then a parabolic $k$-subgroup of $G$ which contains $B$.

I haven't been able to find much about highest weight vectors defined in this sense. Searching online mostly returns references for the more traditional notion of highest weight vector, which is an element in the one dimensional weight space of a dominant integral weight of a finite dimensional representation of $G$.

What some good references to learn about actions of reductive groups on affine varieties and highest weight vectors in the above sense? Do highest weight vectors always exist? Is there any sense in which they are unique or have a finite multiplicity property?