nLab equivariant K-theory of projective G-space

Idea

For $G$ an abelian compact Lie group, the equivariant K-theory ring of projective G-spaces over a direct sum of complex 1-dimensinal linear representations is (2) the quotient ring of the polynomial ring in the tautological equivariant line bundle $L$ by the ideals generated by virtual differences $1 - 1_{{}_{V_i}} L$ between its external tensor product with each of these 1d representations and the trivial line bundle; see Prop. below.

This is the generalization to equivariant K-theory of the formula

(1)$$K\big( \mathbb{C}P^n \big) \;\simeq\; \mathbb{Z}\big[ L \big] \big/ \big( 1 - L \big)^{n+1}$$

(from the fundamental product theorem in topological K-theory) for the complex topological K-theory ring of complex projective space, where $L$ is the class of the tautological line bundle and $1- L$ the “Bott element”.

In generalization of how (1) exhibits complex orientation in topological complex K-theory, so the equivariant version (2) exhibits equivariant complex orientation of equivariant complex K-theory.

Statement

Proposition

(equivariant K-theory of projective G-space)

For $G$ an abelian compact Lie group, let

$$\underset{i}{\oplus} \, \mathbf{1}_{V_i} \;\; \in \;\; G Representations_{\mathbb{C}}^{fin}$$

be a finite-dimensional direct sum of complex 1-dimensional linear representations.

The $G$-equivariant K-theory ring $K_G(-)$ of the corresponding projective G-space $P(-)$ is the following quotient ring of the polynomial ring in one variable $L$ over the complex representation ring $R(G)$ of $G$:

(2)$$K_G \Big( P \big( \underset{i}{\oplus} \, \mathbf{1}_{V_i} \big) \Big) \;\; \simeq \;\; R(G) \big[ L \big] \big/ \underset{i}{\prod} \big( 1 - 1_{{}_{V_i}} L \big) \,,$$

where

• $L \,=\, \big[ \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} }\big]$ is the K-theory class of the tautological equivariant line bundle on the given projective G-space;

• $1_{{}_{V_i}} L \;=\; \big[ \mathbf{1}_{V_i} \boxtimes \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} } \big]$ is the class of its external tensor product of equivariant vector bundles with the given linear representation.

(Greenlees 01, p. 248 (24 of 39))

Applications

Complex orientation of equivariant complex K-theory

Corollary

(equivariant complex orientation of equivariant K-theory)

For $G$ an abelian compact Lie group and $\mathbf{1}_V \,\in\, G Representations_{\mathbb{C}}^{fin}$ a complex 1-dimensional linear representation, the corresponding representation sphere is the projective G-space $S^{\mathbf{1}_V} \,\simeq\, P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$ (this Prop.) and so, by Prop. ,

$$\begin{aligned} \widetilde K_G \big( S^{\mathbf{1}_V} \big) & \simeq\, K_G \big( P( \mathbf{1}_V \oplus \mathbf{1} ) ; \, \underset{ \simeq \, pt }{ \underbrace{ P( \mathbf{1} ) } } \big) \\ & \simeq ker \Big( R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \longrightarrow \underset{ \simeq \, R(G) }{ \underbrace{ R(G)\big[L\big] \big/ (1 - L) } } \Big) \\ & \simeq (1 - L) \cdot R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \end{aligned}$$

is generated by the Bott element $(1 - L)$ over $P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$. By the nature of the tautological equivariant line bundle, this Bott element is the restriction of that on infinite complex projective G-space $P\big(\mathcal{U}_G\big)$. The latter is thereby exhibited as an complex orientation in equivariant complex K-theory.

(Greenlees 01, p. 248 (24 of 39))

Related concepts

• fundamental product theorem in topological K-theory

• equivariant complex oriented cohomology theory

References

• John Greenlees, Section 10 of: Equivariant formal group laws and complex oriented cohomology theories, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 (euclid:hha/1139840255)