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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?

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They are the same as the usual highest weight vectors but the $G$-representation is realized in the coordinate ring of some affine variety $X$. The usual argument goes if $\alpha:G \times X \to X$ is an algebraic action and $f \in \mathcal{O}_{X}$ then $\alpha^*(f) \in \mathcal{O}_G \otimes \mathcal{O}_X$ satisfies $\alpha^{*}(f)(g,x) = \sum_{i=1}^{n} \phi_i(g)f_i(x)$ hence $f$ generates a finite dimensional $G$-stable module spanned by the $f_i$. Thus the coordinate ring can be decomposed as a direct sum (using reductivity + char 0) of the irred. finite dimensional $G$-modules that appear in $\mathcal{O}_X$. By the usual highest weight theory these correspond to dominant weights $\lambda$ and as such there is an $f_{\lambda}$ generating a $G$-module $\cong V(\lambda)$. $f_{\lambda}$ spans the unique line stabilized by $B$ which acts on it through $\lambda$, hence it is a semi-invariant.

If $X$ is affine, normal and has an open Borel orbit (e.g. is spherical) then you can show multiplicity $\leq 1$ for the $\lambda$'s that appear basically imitating the argument for $G$ acting on itself. You can look up "algebraic Peter Weyl" or look at the end of Humphrey's LAG book and Timashev's Homogeneous Spaces and Equivariant Embeddings chapsters 1 and 5.

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  • $\begingroup$ Thanks. If $X = \mathbb A_k^n$ and $G$ acts irreducibly by linear isomorphisms, is there any connection between highest weight vectors in $X$ and those in $\mathcal O_X(X)$? $\endgroup$ – D_S Feb 4 at 16:26

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