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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 3rd 2020
• (edited Nov 3rd 2020)

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 5th 2020

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 9th 2021

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

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 9th 2021

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 10th 2021

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 10th 2021

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 15th 2021

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJan 10th 2022

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?

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJan 10th 2022
• (edited Jan 11th 2022)

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)