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
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Be?linson-Bernstein localization?
For 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 by the ideals generated by virtual differences 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 is the class of the tautological line bundle and 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 an abelian compact Lie group, let
be a finite-dimensional direct sum of complex 1-dimensional linear representations.
The -equivariant K-theory ring of the corresponding projective G-space is the following quotient ring of the polynomial ring in one variable over the complex representation ring of :
where
is the K-theory class of the tautological equivariant line bundle on the given projective G-space;
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 an abelian compact Lie group and a complex 1-dimensional linear representation, the corresponding representation sphere is the projective G-space (this Prop.) and so, by Prop. ,
is generated by the Bott element over . By the nature of the tautological equivariant line bundle, this Bott element is the restriction of that on infinite complex projective G-space . The latter is thereby exhibited as an complex orientation in equivariant complex K-theory.
(Greenlees 01, p. 248 (24 of 39))
Last revised on November 12, 2020 at 13:45:28. See the history of this page for a list of all contributions to it.