# Algebra homomorphisms are identified with evaluation at semisimple elements

For $$G$$ a semisimple simply connected linear algebraic group, $$R(G)$$ its representation ring, there is a canonical embedding of algebras $$R(G) \hookrightarrow \mathcal O(G)^G = \{\text{regular class functions on } G\}$$ obtained by mapping a representation to its character. Here, I understand a regular class function to mean a class function $$G \to \mathbb C$$ that is regular in the sense of a morphism of varieties. It is a 'well-known fact' that if $$G$$ is reductive then the induced map given by complexification of the left-hand side $$\mathbb C \otimes_{\mathbb Z} R(G) \to \mathcal O(G)^G$$ is an isomorphism.

Now any algebra homomorphism $$R(G) \to \mathbb C$$ can be identified with evaluation of a character $$z \in R(G)$$ at a semisimple element $$a \in G$$, where here I take semisimple to mean a diagonalisable element, i.e. up to conjugation lies in (some) maximal torus $$T \subset G$$. Why is this bijection true?

It is because for $$f$$ a class function on $$G$$ and for $$x \in G$$, $$f(x)=f(x_s)$$, where $$x=x_sx_u$$ is the semisimple-unipotent (Jordan-Chevalley multiplicative) decomposition. Thus really we identify maps $$R(G) \to \mathbb C$$ with evaluation at any element of $$G$$, but we may as well take its semisimple part (since there is a nice bijection of semisimple classes with elements of $$T//W$$).