Reference request: Highest weight representations of Linear algebraic groups Where can I read about highest weight representations of linear algebraic groups? Where can I read about homogeneous polynomial algebras as representations of these groups, like $\Bbb C[x,y]_n$ for a representation of $\text{PGL}_2(\Bbb C)$?
I looked at a few references, but they all do representation of Lie algebras.
(Lecture notes, textbooks, video lectures. Any type of resource is acceptable.)
 A: A general remark: In this context, people sometimes talk about $\operatorname{Sym}^2 (\Bbb C^2)^\ast$ instead of $\Bbb C[x,y]_2$, because the two are the same as $\operatorname{GL}_2$-modules. More generally, if $W$ is a finite dimensional vector space and $G\subseteq\operatorname{GL}(W)$ an algebraic subgroup, you are interested in $V=\operatorname{Sym}^d W^\ast \cong \Bbb C[W]_d$ as a $G$-module.

The problem is that the situation can vary a lot depending on $G$. For $G=\operatorname{GL}(W)$ it is very easy, because $V$ is irreducible in this case. However, for arbitrary $G$, even if it is reductive, it is hard to say anything at all and there are even very particular special cases that are still not completely understood (Plethysm coefficients for example, just to throw in a Buzzword).
The following is a list of some books that might be less known and which I often refer to in this context, maybe you give all of them a look and see if they contain what you seek.


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*Derksen and Kemper - Computational Invariant Theory

*Kraft - Geometrische Methoden in der Invariantentheorie 
(It's in German, sorry. But the book is truly fantastic, so I have to mention it.)

*Claudio Procesi - Lie Groups - An Approach through Invariants and Representations

*Bernd Sturmfels - Algorithms in Invariant Theory

*Igor Dolgachev - Lectures on Invariant Theory

