One of the slickest things to happen to topology was the proof of the Hopf invariant one using Adams operations in $K$-theory. The general idea is that the ring $K(X)$ admits operations $\psi^k$ that endows it more relations from which one can deduce restrictions on when a Hopf invariant one element can occur.

Similarly, the representation ring $R(G)$ also admits Adams operations. One may describe its effects on characters by $\Psi^k(\chi)(g) = \chi(g^k)$. Alternatively one can parallel the construction of the Adams operations in topology by defining the Adams operations as taking the logarithmic derivative of $$\lambda_t(V) := \Sigma_k [\Lambda^k V]t^k$$ (of course we extend this formula to virtual characters as well) and defining $\Psi^n(V)$ as $(-1)^n$ times the $n$-th coefficient of the logarithmic derivative.

Now, considering that my background in representation theory is minimal, my questions are:

- What is an application of the Adams operations in representation theory? Is there one that has the same flavor as the proof of the Hopf invariant one problem?
- We have the Atiyah-Segal completion theorem that gives isomorphism of rings $K(BG) \simeq \hat{R(G)}_I$ and there are Adams operations on both sides (I'd imagine that the Adams operations in group representations extends naturally to completion) - what is their relationship (if any)?