Soon it will be time to explore some aspects of root systems and structure theory for reductive groups. Our goal is to understand everything in the classical setting over an algebraically closed field, and then explore reductive groups over arbitrary base schemes.

Before we do this, I will give a few examples for some of the technical machinery we shall rely on. In this post, we shall see how an algebraic group acting on a variety $ X$ and a function $ f\in k[X]$ gives rise to a representation of $ G$, and in the next post we shall see an example. I learnt the material in this section mainly from Jim Humphrey’s book “Linear Algebraic Groups”.

### Finite Dimensional, Infinite Dimensional

Our setting is an arbitrary algebraic group $ G$ over an algebraically closed field $ k$.

Let $ X$ be a variety over $ k$ on which $ G$ acts, so that $ G\times X\to X$ is a group action and a morphism of varieties. If $ g\in G$ then there is a translation algebra homomorphism $ \tau_g:k[X]\to k[X]$ defined by $ \tau_g(f)(y) = f(g^{-1}y)$. The inverse is there so that

$ [\tau_g(\tau_h(f))](y) = \tau_h(f)(g^{-1}y) = f(h^{-1}g^{-1}y) = f((gh)^{-1}y) = \tau_{gh}(y)$

In other words, $ G\to\mathrm{Aut}_{k}(k[X])$ is actually a group homomorphism. Now, $ k[X]$ is a $ k$-vector space and thus this gives a representation of $ G$, but it is infinite dimensional. How can we get finite dimensional representations?

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