Adams Operations on $K$-theory and $R(G)$

Question

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:

1. 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?
2. 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)?

Answer 1

Adams operations played a key part in Borcherds's proof of the Conway-Norton Monstrous Moonshine Conjecture. The specific point is in section 8 of his paper (13th on the linked page), where we have a $\mathbb{Z} \times \mathbb{Z}$-graded Lie algebra $$E = \bigoplus_{m > 0, n \in \mathbb{Z}} E_{m,n}$$ equipped with an action of the Monster simple group, by homogeneous automorphisms. An important property of $E$ is that its Lie algebra homology ``lives on the boundary'', i.e., $H_0(E) \cong \mathbb{C}$, $H_1(E) \cong \bigoplus_{n \in \mathbb{Z}} E_{1,n}$, $H_2(E) \cong \bigoplus_{m > 0} E_{m,1}$, $H_i(E) = 0$ for $i \geq 3$. The Chevalley-Eilenberg isomorphism $H_*(E) \cong \bigwedge^* E$ of graded virtual Monster modules then expands to a K-theory identity:

$$\mathbb{C}p^{-1} + \sum_{m >0} E_{m,1} p^m - \sum_{n \geq -1} E_{1,n} q^n = p^{-1} {\bigwedge}^* \left(\bigoplus_{m>0,n\in \mathbb{Z}} E_{m,n} p^m q^n \right).$$

A second important property of $E$ is that as a Monster module, the isomorphism type of $E_{m,n}$ depends only on the product $mn$, i.e., we have a set of modules $\{V_k\}_{k \geq -1}$ such that $E_{m,n} \cong V_{mn}$ (I recently wrote about this in a blog post). By using properties of Adams operations, we find that for any element $g$ in the Monster we get the following character identity:

$$\sum_{m \geq -1} Tr(g|V_m) p^m - \sum_{n \geq -1} Tr(g|V_n) q^n = p^{-1} \exp \left(- \sum_{i>0} \sum_{m>0,n\in \mathbb{Z}} \frac{Tr(g^i|V_{mn}) p^{mi} q^{ni}}{i} \right).$$

Both identities have a left side that is a sum of power series that are pure in $p$ and $q$, while the right side has lots of mixed terms that necessarily cancel. The resulting relations constrain the graded characters $\sum Tr(g|V_n)q^n$ quite strongly, e.g., the vanishing of the $p^1 q^2$ term implies $V_4 \cong V_3 \oplus \wedge^2 V_1$, so the $\psi^2$ identity implies $$Tr(g|V_4) = Tr(g|V_3) + \frac{Tr(g|V_1)^2 - Tr(g^2|V_1)}{2}.$$

From these relations, Borcherds was able to show that the characters are Fourier expansions of modular functions that precisely match the list conjectured by Conway and Norton, by just checking the first few terms.

There have been some further developments following this application. The character identity has been reinterpreted in terms of equivariant Hecke operators by Charles Thomas and Nora Ganter (and some others who I am forgetting), with a view toward elliptic cohomology operations. Gerald Höhn applied this method to solve the Baby Monster case of the Generalized Moonshine conjecture, and I've used it in my own work for showing that ``well-behaved'' actions of groups on Lie algebras can be used to form Hauptmoduln.

Answer 2

Representation rings and complex $K$-theory are both naturally "Lambda rings", and there is a great deal of literature about this context. The lambda ring structure is determined by the formal series $\lambda_t(V)$, and lets you define Adams operations as you describe.

Question 2: the obvious map $R(G)\to K(BG)$ is induced by the operation which sends a representation $V$ to the Borel vector bundles $V\times_G EG\to BG$. This clearly takes exterior powers of representations to exterior powers of bundles, so it defines a map of Lambda rings.

The augmentation $R(G)\to R(e)=\mathbb{Z}$ is also a map of Lambda rings, so its kernel $I$ is preserved by the $\lambda$ operations (and thus by the Adams operations), so $\widehat{R(G)}_I$ is a Lambda ring too, and it formally must coincide with that on $K(BG)$.

I don't know anything about representation theory, so I have nothing to say for question (1), though I would be interested in an answer.