I will give an exposition of a celebrated series of papers by the three referenced authors. They proved that if G is
a semisimple algebraic group over a field of characteristic 0 and if
{I_λ : λ ∈ Λ} is a set of representatives of the isomorphism classes
of irreducible representations of G then the coordinate ring of G
is the direct sum k[G] = \bigoplus_{λ∈Λ} I_λ ⊗k I*_λ. The proof uses mostly
elementary notions such as the left and right translation actions
on k[G] and the notion of the matrix coefficients of an algebraic
representation.
