Dominant characters form the representation-ring basis

Let $\mathcal{I}$ be the set of irreducible classes, indexed by dominant actual characters by [L1]. Every finite-dimensional representation is a finite direct sum of these by complete reducibility in [L1]. Its multiplicities are unique: unitarize the irreducibles by [L3], take the character of a decomposition using [L2], and integrate against the conjugate character of any fixed irreducible. Orthonormality returns exactly that irreducible's multiplicity. Thus the direct-sum monoid of representation classes is the monoid of finitely supported functions $\mathcal{I}\to {\mathbb{Z}}_{\ge 0}$. Its Grothendieck group is the free abelian group of finitely supported functions $\mathcal{I}\to \mathbb{Z}$: positive and negative parts realize each integer vector as a difference, and equality of differences is coordinatewise. This is $R(G)$ as an abelian group. Tensor product distributes over direct sums and therefore defines its ring multiplication. Additivity and multiplicativity of trace in [L2] give the character ring homomorphism.

Let $\pi$ be a finite-dimensional representation. After choosing the invariant inner product of [L3], the commuting unitary operators $\{\pi (t):t\in T\}$ have a common eigenbasis. Thus $V={⨁}_{\mu }{V}_{\mu },\pi (t)v=\mu (t)v(v\in V_{\mu }),$ where each eigenvalue function $\mu :T\to S^1$ is a continuous homomorphism, hence belongs to $X^{\ast }(T)$. It follows that ${\chi }_{\pi }{\mathrm{\mid }}_T={\sum }_{\mu }(\dim V_{\mu })\mu \in \mathbb{Z}[X^{\ast }(T)]\mathrm{.}$ If $n\in N_G(T)$, then $\pi (n)$ sends $V_{\mu }$ isomorphically to the weight space for the conjugate character $t\mapsto \mu (n^{-1}tn)$, so the multiplicities are permuted by $W$. Therefore the restricted character lies in $\mathbb{Z}[X^{\ast }(T){]}^W$.

The characters of inequivalent irreducible representations are orthonormal by [L2], hence linearly independent as class functions on $G$. If an integral linear combination of their restrictions to $T$ vanishes, the same combination vanishes on every element of $G$ by [L4], because characters are class functions. Its coefficients therefore vanish, so restriction gives an injective character map $R(G)\hookrightarrow \mathbb{Z}[X^{\ast }(T){]}^W$.

By [L1] the irreducible classes correspond bijectively to the dominant elements of $X^{\ast }(T)$, and step 2.1 makes their characters linearly independent; hence the dominant irreducible characters form the asserted $\mathbb{Z}$-basis, with tensor product corresponding to multiplication of characters and direct sum to addition.