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differential cohomology
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For $G$ a suitable (compact) Lie group, the $G$-equivariant K-theory of the point is the representation ring of the group $G$:
Accordingly the construction of an index (push-forward to the point) in equivariant K-theory is a way of producing $G$-representations from equivariant vector bundles. Specifically with $K \hookrightarrow G$ a suitable subgroup for the push-forward from $K$-equivariant to $G$-equivariant K-theory/representations, this method is also called Dirac induction since it is analogous to the construction of induced representations.
Applied to equivariant complex line bundles on coadjoint orbits of $G$, Dirac induction is a K-theoretic formulation of the orbit method.
For $G$ a compact Lie group with Lie algebra $\mathfrak{g}^\ast$, the push-forward in compactly supported twisted $G$-equivariant K-theory to the point (the $G$-equivariant index) produces the Thom isomorphism
Moreover, for $i \colon \mathcal{O} \hookrightarrow \mathfrak{g}^\ast$ a regular coadjoint orbit, push-forward involves a twist $\sigma$ of the form
and
$i_!$ is surjective
$ind_{\mathcal{O}} = ind_{\mathfrak{g}^\ast} \circ i_!$.
This is (FHT II, (1.27), theorem 1.28). Related results are in (Hochs 12, section 2.2). For more background see at orbit method.
The idea of Dirac induction goes back to Raoul Bott‘s formulation in the 1960s of index theory in the equivariant context.
elliptic complexes. II_. Applications. Ann. of Math. (2), 88:451– 491, 1968.
A generalization to super Lie groups is discussed in
An inverse to Dirac induction, hence a construction of good equivariant vector bundles that push to a given representation, is discussed in
The analog of Dirac induction for K-theory replaced by elliptic cohomology is discussed in
Last revised on October 29, 2013 at 10:47:49. See the history of this page for a list of all contributions to it.