group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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
(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.
(equivariant K-theory of projective G-space)
For $G$ an abelian compact Lie group, let
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 $G$:
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))
(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. ,
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))