The K-theory of the conjugation action

In 1999, Brylinski and Zhang computed the complex equivariant K-theory of the conjugation self-action of a compact, connected Lie group with torsion-free fundamental group. In this note we show it is possible to do so in under a page.

Brylinski and Zhang [2] showed that if G is a compact, connected Lie group with torsion-free fundamental group, then the equivariant K-theory of its conjugation action G Ad is isomorphic to the ring Ω * RG/Z of Grothendieck differentials on the complex representation ring RG of G. Their proof uses results on holomorphic differentials on complex manifolds, a reduction to the case G is a torus, and some algebraic geometry.We show a more concrete and arguably more natural expression for the ring K * G (G Ad ) can be obtained rapidly using only Hodgkin's Künneth spectral sequence [5], in the same manner they already use it, and elementary algebraic considerations.We then show this purely algebraic isomorphism admits a satisfying geometric interpretation, and remark finally that this geometric version gives back Brylinski and Zhang's description in terms of Grothendieck differentials at no added cost.

Theorem (Brylinski-Zhang [2, Thm. 3.2]).
Let G be a compact, connected Lie group with torsionfree fundamental group.Then K * G (G Ad ) is isomorphic to RG ⊗ K * G as an RG-algebra.Under this identification, the forgetful map f : K * G (G Ad ) −→ K * G becomes reduction with respect to the augmentation ideal IG of RG.
Here the two structure maps RG ⊗ RG −→ RG are the both the multiplication of RG.Recall [5,Prop. 11.1] that under our hypotheses, RG is the tensor product of a polynomial ring on generators y i ∈ IG and a Laurent polynomial ring on generators t j ∈ 1 + IG.Let P be the free abelian group on generators q i and w j and let γ : P −→ IG be the linear map taking q i to y i and w j to t j − 1.Then an (RG) ⊗ 2 -module resolution of RG is given by RG ⊗ ΛP ⊗ RG, with differential the derivation vanishing on RG ⊗ Z ⊗ RG and sending 1 To compute the Tor, apply − ⊗ RG ⊗ RG RG to this resolution to obtain the cdga RG ⊗ ΛP with 0 differential, RG in bidegree (0, 0), and P in bidegree (−1, 0).
The spectral sequence collapses because the differentials d r for r ≥ 2 send all generators into the right half-plane.Since π 1 G is torsion-free and X = G is locally contractible of finite covering dimension, the spectral sequence strongly converges to the intended target.Hence RG ⊗ ΛP is the graded algebra associated to a filtration (F p ) p≤0 of K * G (G Ad ) with F 0 ∼ = RG and F −1 /F 0 ∼ = RG ⊗ P .Since RG and ΛP are both free abelian, there is no additive extension problem, so K * G (G Ad ) is also free abelian as a group.Let z k be elements in F −1 lifting 1 ⊗ q i and 1 ⊗ w j under the isomorphism F −1 /F 0 ∼ = RG ⊗ P .Then the z k anticommute with each other because they lie in K 1 G (G Ad ) and square to 0 since K * G (G Ad ) contains no 2-torsion, and by induction, they generate K * G (G Ad ) as an RG-algebra, so To see the forgetful map f is as claimed, note that forgetting the (G × 1)-action on G bi induces a map to the spectral sequence Tor RG (Z, Z) =⇒ K * G, which again collapses by lacunary considerations.Computing Tor RG (Z, Z) ∼ = ΛP with the resolution ΛP ⊗ RG of Z shows the map Remark.We can be completely explicit about the exterior generators.As observed by Hodgkin [4, Thm.A], the injection U(n) U := lim − − → U(n) induces an additive map β : RG −→ K 1 G descending to a group isomorphism between the module IG/(IG) 2 of indecomposables of RG and the module P K * G of primitives of the exterior Hopf algebra K * G ∼ = ΛP K * G.In particular, a set of generators is given by β(λ i ) = β(λ i − dim λ i ) for λ i lifts in G of the fundamental representations of the commutator subgroup G and β(t j ) = β(t j − 1) for The map β in fact factors as f • β Ad for a map β Ad : RG −→ K * G (G Ad ), already giving surjectivity of f since P K * G generates K * G = ΛP K * G as a ring.Atiyah [1, Lem. 2, pf.] described β Ad and hence β geometrically: given a representation ρ : G −→ U(n), we can build a representative E of β(ρ) via the clutching construction, taking with two trivial bundles CG × C n over the cone CG on G and gluing them along G × C n via the relation (g , v) ∼ g , ρ(g )v to obtain a bundle over the suspension CG ∪ G CG.The action h • (g , v) = hg h −1 , ρ(h)v of G on G Ad × C n preserves this relation and so induces a G-action on E making it a G-equivariant bundle over the suspension of G Ad .
The RG-module structure on K * G (X ) is always given by σ The first paragraph of our proof is a variant of Brylinski-Zhang's §4 [2].Once the ring structure is determined as in our proof's second paragraph, replacing § §5-6, one knows their map φ : Ω * RG/Z −→ K * G (G Ad ) from the ring of Grothendieck differentials is an isomorphism as soon as one knows it is a well-defined RG-algebra map [2, Prop 3.1], for the class β Ad (ρ) ∈ K 1 G (G Ad ) from the previous remark is in fact the same as 1 Brylinski-Zhang's φ(d ρ), so that f • φ takes a basis {d ρ : ρ ∈ Q} of the free RG-module Ω 1 RG/Z to a Z-basis β(ρ) : ρ ∈ Q of P K * G.