# representation theory – 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?