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I may be too late to this particular party, but I thought it should be stated that the result is originally due to Leray, not Borel. For a timeline, due to Borel, of Leray's results in this vein, see …

Where can one find a table of generator–relator expressions for representation rings $R(G)$ of simple Lie groups $G$ and explicit maps between them? For example, given a maximal torus $T < G$ or a cov …

For $G = PSU(3)$, it is known that $\dim I(G;\mathbb Q) / I(G;\mathbb Q)^2 = 3$, while $H^{**}(BG;\mathbb Q)$ is obviously a power series ring in two indeterminates since $G$ has rank 2. This would wo …

Let $G$ be a compact, connected Lie group and $S$ a torus in $G$ not assumed maximal. Then conjugation in $G$ induces a faithful representation of $N = N_G(S)/Z_G(S)$ in the Lie algebra $\mathfrak s$ …

This is very late and you've no doubt learned this in the last five years, but for completeness, the relation is indeed that they are linked by completion and the Chern character, as suggested in one …