nLab
Dirac induction

Context

Cohomology

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Special and general types

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Special notions

Variants

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  • differential cohomology

Extra structure

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Operations

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Theorems

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Representation theory

Ingredients

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Definitions

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Geometric representation theory

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Theorems

Contents

Idea

For GG a suitable (compact) Lie group, the GG-equivariant K-theory of the point is the representation ring of the group GG:

K G(*)Rep(G). K_G(\ast) \simeq Rep(G) \,.

Accordingly the construction of an index (push-forward to the point) in equivariant K-theory is a way of producing GG-representations from equivariant vector bundles. Specifically with KGK \hookrightarrow G a suitable subgroup for the push-forward from KK-equivariant to GG-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 GG, Dirac induction is a K-theoretic formulation of the orbit method.

Properties

Relation to the orbit method

Proposition

For GG a compact Lie group with Lie algebra 𝔤 *\mathfrak{g}^\ast, the push-forward in compactly supported twisted GG-equivariant K-theory to the point (the GG-equivariant index) produces the Thom isomorphism

ind 𝔤 *:K G σ+dimG(𝔤 *) cptK G 0(*)Rep(G). ind_{\mathfrak{g}^\ast} \;\colon\; K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt} \stackrel{\simeq}{\to} K_G^0(\ast) \simeq Rep(G) \,.

Moreover, for i:𝒪𝔤 *i \colon \mathcal{O} \hookrightarrow \mathfrak{g}^\ast a regular coadjoint orbit, push-forward involves a twist σ\sigma of the form

Rep(G)K G 0(*)ind 𝒪K G σ(𝒪)+dim(𝒪)(𝒪)i !K G σ+dimG(𝔤 *) cpt Rep(G) \simeq K_G^0(\ast) \stackrel{ind_{\mathcal{O}}}{\leftarrow} K_G^{\sigma(\mathcal{O}) + dim(\mathcal{O})}(\mathcal{O}) \stackrel{i_!}{\to} K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt}

and

  1. i !i_! is surjective

  2. ind 𝒪=ind 𝔤 *i !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.

References

The idea of Dirac induction goes back to Raoul Bott‘s formulation in the 1960s of index theory in the equivariant context.

  • Raoul Bott, The index theorem for homogeneous differential operators, In: Differential and combinatorial topology (A symposium in honor of Marston Morse), 1965, Princeton Univ. Press, Princeton, NJ, 167–186
  • Michael Atiyah, Raoul Bott, A Lefschetz fixed point formula for

    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.