# nLab Dirac induction

Contents

cohomology

### Theorems

#### Representation theory

representation theory

geometric representation theory

# Contents

## 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.

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