# Comptes Rendus
Mathématique

Jeffrey D. Carlson

The K-theory of the conjugation action

Volume 359, issue 7 (2021), p. 795-796

Published online: 2 September 2021

https://doi.org/10.5802/crmath.235

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The spectral sequence collapses because the differentials $d_r$ for $r \ge 2$ send all generators into the right half-plane. Since $\pi_1 G$ is torsion-free and $X = G$ is locally contractible of finite covering dimension, the spectral sequence strongly converges to the intended target. Hence $RG \otimes \Lambda P$ is the graded algebra associated to a filtration $(F_p)_{p \le 0}$ of $K_G^*(G^{\text{Ad}})$ with $F_0 \cong RG$ and $F_{-1}/F_0 \cong RG \otimes P$. Since $RG$ and $\Lambda P$ are both free abelian, there is no additive extension problem, so $K_G^*(G^{\text{Ad}})$ is also free abelian as a group. Let $\tilde{z}_k$ be elements in $F_{-1}$ lifting $1 \otimes q_i$ and $1 \otimes w_j$ under the isomorphism $F_{-1}/F_0 \cong RG \otimes P$. Then the $\tilde{z}_k$ anticommute with each other because they lie in $K_G^1(G^{\text{Ad}})$ and square to 0 since $K_G^*(G^{\text{Ad}})$ contains no 2-torsion, and by induction, they generate $K_G^*(G^{\text{Ad}})$ as an $RG$-algebra, so $K_G^*(G^{\text{Ad}}) = RG \otimes \Lambda[\tilde{z}_k]$.

To see the forgetful map $f$ is as claimed, note that forgetting the $(G \times 1)$-action on $G^{\text{bi}}$ induces a map to the spectral sequence $\text{Tor}_{RG}(\mathbb{Z}, \mathbb{Z}) \implies K^*G$, which again collapses by lacunary considerations. Computing $\text{Tor}_{RG}(\mathbb{Z}, \mathbb{Z}) \cong \Lambda P$ with the resolution $\Lambda P \otimes RG$ of $\mathbb{Z}$ shows the map $E_2(f): RG \otimes \Lambda P \to \Lambda P$ is reduction modulo $IG$ and $\Lambda P \cong K^*G$. $\square$

**Remark.** We can be completely explicit about the exterior generators. As observed by Hodgkin [4, Thm. A], the injection $U(n) \to U := \lim_{\leftarrow} U(n)$ induces an additive map $\beta: RG \to K^1G$ descending to a group isomorphism between the module $IG/(IG)^2$ of indecomposables of $RG$ and the module $PK^*G$ of primitives of the exterior Hopf algebra $K^*G \cong \Lambda PK^*G$. In particular, a set of generators is given by $\beta(\lambda_i) = \beta(\lambda_i - \dim \lambda_i)$ for $\lambda_i$ lifts in $G$ of the fundamental representations of the commutator subgroup $G'$ and $\beta(t_j) = \beta(t_j - 1)$ for $t_j: G \to G/G' \to U(1)$ circular coordinate functions of the torus $G^{\text{ab}} = G/G' \cong (S^1)^{\text{rk}G - \text{rk}G'}$. Let $Q = \{\lambda_i, t_j\}_{i,j}$.

The map $\beta$ in fact factors as $f \circ \beta^{\text{Ad}}$ for a map $\beta^{\text{Ad}}: RG \to K_G^*(G^{\text{Ad}})$, already giving surjectivity of $f$ since $PK^*G$ generates $K^*G = \Lambda PK^*G$ as a ring. Atiyah [1, Lem. 2, pf.] described $\beta^{\text{Ad}}$ and hence $\beta$ geometrically: given a representation $\rho: G \to U(n)$, we can build a representative $E$ of $\beta(\rho)$ via the clutching construction, taking with two trivial bundles $CG \times \mathbb{C}^n$ over the cone $CG$ on $G$ and gluing them along $G \times \mathbb{C}^n$ via the relation $(g, v) \sim (g, \rho(g)v)$ to obtain a bundle over the suspension $CG \cup_G CG$. The action $h \cdot (g, v) = (hgh^{-1}, \rho(h)v)$ of $G$ on $G^{\text{Ad}} \times \mathbb{C}^n$ preserves this relation and so induces a $G$-action on $E$ making it a $G$-equivariant bundle over the suspension of $G^{\text{Ad}}$.

The $RG$-module structure on $K_G^*(X)$ is always given by $\sigma \cdot [E] = [\alpha(\sigma) \otimes E]$, where if $\sigma: G \to \text{Aut } V$ is a representation, then $\alpha(\sigma)$ is the trivial bundle $X \times V$ equipped with the diagonal $G$-action. Thus $\sigma \otimes \prod_k \beta(\rho_k) \to \alpha(\sigma) \cdot \prod_k \beta^{\text{Ad}}(\rho_k)$ for $\rho_k \in Q$ gives an explicit isomorphism $RG \otimes K^*G \xrightarrow{\sim} K_G^*(G^{\text{Ad}})$.

**Remark.** 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 $\phi: \Omega_{RG/Z}^* \to K_G^*(G^{\text{Ad}})$ 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 $\beta^{\text{Ad}}(\rho) \in K_G^1(G^{\text{Ad}})$ from the previous remark is in fact the same as<fnref n="1" /> Brylinski–Zhang's $\phi(d\rho)$, so that $f \circ \phi$ takes a basis $\{d\rho : \rho \in Q\}$ of the free $RG$-module $\Omega_{RG/Z}^1$ to a $\mathbb{Z}$-basis $\{\beta(\rho) : \rho \in Q\}$ of $PK^*G$.

## References

[1] M. F. Atiyah, “On the K-theory of compact Lie groups”, *Topology* **4** (1965), no. 1, p. 95-99.

[2] J.-L. Brylinski, B. Zhang, “Equivariant K-theory of compact connected Lie groups”, *K-Theory* **20** (2000), no. 1, p. 23-36.

[3] C.-K. Fok, “The Real K-theory of compact Lie groups”, *SIGMA, Symmetry Integrability Geom. Methods Appl.* **10** (2014), article no. 022 (26 pages).

[4] L. Hodgkin, “On the K-theory of Lie groups”, *Topology* **6** (1967), no. 1, p. 1-36.

[5] —, “The equivariant Künneth theorem in K-theory”, in *Topics in K-theory*, Lecture Notes in Mathematics, vol. 496, Springer, 1975, p. 1-101.

<footnote n="1">¹ the corrected version of—see Fok [3, Rmk. 2.8.1]</footnote>