It is

well-known that the connected components of the $H$-fixed locus of the classifying space for equivariant K-theory is the representation ring $R(H)$

known that the possible twistings of equivariant K-theory over an $H$-fixed point include – in addition to the notorious “gerbe” – a complex line bundle with structure group $H^2(G; \mathbb{Z}) \simeq H^1\big(G; \mathrm{U}(1)\big) \,\subset \, \mathrm{U}(1)$ (aka “local system”).

But, in describing how this degree-1 twist actually acts on the classifying space, all authors I have seen (where “all” is no more than 2 or 3 groups, apparently) pass to the perspective of “delocalized” cohomology.

While the delocalized picture has some clear virtues, it does a fair bit of violence to the classifying picture of K-theory. In the latter picture, there is an evident guess for how the 1-twist acts: It *ought* to be the canonical operation of tensoring representations with group characters regarded as 1d reps:

Because what else can it be. But also because I think I have proven this now. (It follows from the observation mentioned in another thread, here.)

I keep wondering, though, if this has not been discussed elsewhere, before?

]]>a reference item to add to *twisted equivariant K-theory* once the edit-functionality is back:

- Alejandro Adem, José Cantarero, José Manuel Gómez,
*Twisted equivariant K-theory of compact Lie group actions with maximal rank isotropy*, J. Math. Phys.**59**113502 (2018) (arXiv:1709.00989, doi:10.1063/1.5036647)

For $G$ a finite group (at least),

it is known that classifying $G$-space for rational $G$-equivariant KU-theory assigns the rationalized complex representation rings

$G/H \;\mapsto\; \underset{k}{\prod} B^{2k} \; \mathbb{Q} \otimes R_{\mathbb{C}}(H)$and that the equivariant fundamental group of $B_G PU(\mathcal{H})$ assigns the character group

$G/H \;\mapsto\; Hom\big(H, U(1)\big) \,.$Since the character group has a canonical action on the representation ring, it *ought* to be the case for the classifying coefficient $G$-bundle for 3-twisted equivariant K-theory, that the action of the equivariant fundamental group of the base on the homotopy fibers is, at each stage $G/H$, the canonical action of the $H$-character group on the $H$-representation ring.

Is this proven anywhere?

]]>added pointer to:

- Jean-Louis Tu, Ping Xu, Camille Laurent-Gengoux,
*Twisted K-theory of differentiable stacks*, Annales Scientifiques de l’École Normale Supérieure Volume 37, Issue 6, November–December 2004, Pages 841-910 (arXiv:math/0306138, doi:10.1016/j.ansens.2004.10.002)

added pointer to:

- Christopher Dwyer,
*Twisted equivariant K-theory for proper actions of discrete groups*, K-Theory v. 38, n. 2 (January, 2008): 95-111 (arXiv:0710.2136, doi:10.1007/s10977-007-9016-z)

added pointer to:

- Valentin Zakharevich, Section 2.2 of:
*K-Theoretic Computation of the Verlinde Ring*, thesis 2018 (hdl:2152/67663, pdf)

added pointer to:

- Jose Cantarero,
*Equivariant K-theory, groupoids and proper actions*, Thesis 2009 (ubctheses:1.0068026, CantareroEquivariantKTheory.pdf:file)

added pointer to the general definition in terms of equivariant sections of equivariant bundles of equivariant classifying spaces for equivariant K-theory:

- Michael Atiyah, Graeme Segal, Section 6 of:
*Twisted K-theory*, Ukrainian Math. Bull.**1**, 3 (2004) (arXiv:math/0407054, journal page, published pdf)

added this pointer:

- Noe Barcenas, Jesus Espinoza, Bernardo Uribe, Mario Velasquez,
*Segal’s spectral sequence in twisted equivariant K-theory for proper and discrete actions*, Proceedings of the Edinburgh Mathematical Society Volume 61 Issue 1 (arXiv:1307.1003, doi:10.1017/S0013091517000281)

finally splitting this off, for ease of organizing references. Not much here yet…

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