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?**