nLab Dirac induction

Idea

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

$$K_G(\ast) \simeq Rep(G) \,.$$

Accordingly the construction of an index (push-forward to the point) in equivariant K-theory is a way of producing $G$-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.

Properties

Relation to the orbit method

Proposition

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

$$ind_{\mathfrak{g}^\ast} \;\colon\; K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt} \stackrel{\simeq}{\to} K_G^0(\ast) \simeq Rep(G) \,.$$

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

$$Rep(G) \simeq K_G^0(\ast) \stackrel{ind_{\mathcal{O}}}{\leftarrow} K_G^{\sigma(\mathcal{O}) + dim(\mathcal{O})}(\mathcal{O}) \stackrel{i_!}{\to} K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt}$$

and

1. $i_!$ is surjective

2. $ind_{\mathcal{O}} = ind_{\mathfrak{g}^\ast} \circ i_!$.

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.

Related concepts

• Schubert calculus

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

• Gregory Landweber, Twisted representation rings and Dirac induction, J. Pure Appl. Algebra 206 (2006), no. 1-2, 21-54 (arXiv:math/0403524)

An inverse to Dirac induction, hence a construction of good equivariant vector bundles that push to a given representation, is discussed in

• Daniel Freed, Michael Hopkins, Constantin Teleman, part II, section 1 of Loop Groups and Twisted K-Theory

• Peter Hochs, section 2.2 of Quantisation of presymplectic manifolds, K-theory and group representations (arXiv:1211.0107)

The analog of Dirac induction for K-theory replaced by elliptic cohomology is discussed in

• Nora Ganter, The elliptic Weyl character formula (arXiv:1206.0528)