Then K acts on F(G) — S or on L2 (G) — S and centralizes the action of diag r.

If U is a module for R, we may consider F(G) — S — U or L2 (G) — S — U ,

and K — 1 commutes with diag r — 1 and with the action ρ of R on U , i.e with

1 — 1 — ρ. If R is connected, this implies that K commutes with the diagonal

˜

action of R, the universal cover of R, on F — S — U or L2 (G) — S — U given by

k ’ r(k) — Spin(k) — ρ(k), k∈R

˜

where Spin : R ’ Spin(p) is the group homomorphism corresponding to the

Lie algebra homomorphism ν. If G/R is a spin manifold, the invariants under

this R action correspond to smooth or L2 sections of S — U where S is the

spin bundle of G/R and U is the vector bundle on G/R corresponding to U .

Thus K descends (by restriction) to a di¬erential operator ‚ on G/R and we

shall compute its G-index for irreducible U . The key result, due to Landweber,

asserts that if U belongs to a multiplet coming from an irreducible V of G, then

this index is, up to a sign, equal to V . If U does not belong to a multiplet, then

this index is zero. We begin with some preliminary results due to Bott.

10.7.1 The index of equivariant Fredholm maps.

Let E and F be Hilbert spaces which are unitary modules for the compact Lie

group G. Suppose that

E= En , F= Fn

n n

are completed direct sum decompositions into subspaces which are G-invariant

and ¬nite dimensional, and that

T :E’F

10.7. THE GEOMETRIC INDEX THEOREM. 179

is a Fredholm map (¬nite dimensional kernel and cokernel) such that

T (En ) ‚ Fn .

We write

IndexG T = Ker T ’ Coker T

as an element of R(G), the ring of virtual representations of G. Thus R(G)

is the space of ¬nite linear combinations » a» V» , a» ∈ Z as V» ranges over

the irreducible representations of G. (Here, and in what follows, we are regard-

ing any ¬nite dimensional representation of G as an element of R(G) by its

decomposition into irreducibles, and similarly the di¬erence of any two ¬nite

dimensional representations is an element of R(G).)

If we denote the restriction of T to En by Tn , then

IndexG T = IndexG Tn

where all but a ¬nite number of terms on the right vanish. For each n we have

the exact sequence

0 ’ Ker Tn ’ En ’ Fn ’ Coker Tn ’ 0.

Thus

IndexG Tn = En ’ Fn

as elements of R(G). Therefore we can write

(En ’ Fn )

IndexG T = (10.33)

in R(G), where all but a ¬nite number of terms on the right vanish. We shall

refer to this as Bott™s equation.

10.7.2 Induced representations and Bott™s theorem.

Let R be a closed subgroup of G. Given any R-action ρ on a vector space U ,

we consider the associated vector bundle G —R V over the homogeneous space

G/R. The sections of this bundle are then equivariant U -valued functions on G

satisfying s(gk) = ρ(k)’1 s(g) for all k ∈ R. Applying the Peter-Weyl theorem,

we can decompose the space of L2 maps from G to U into a sum over the

irreducible representations V» of G,

L2 (G) — U ∼ —

V» — V» — U,

=

»

with respect to the G — G — R action l — r — ρ. The R-equivariance condition

is equivalent to requiring that the functions be invariant under the diagonal

R-action k ’ r(k) — ρ(k). Restricting the Peter-Weyl decomposition above to

the R invariant subspace, we obtain

∼ R

—

L2 (G —R U ) — (V» — U )

» V»

=

(10.34)

∼

» V» — HomR (V» , U ).

=

180 CHAPTER 10. THE KOSTANT DIRAC OPERATOR

The Lie group G acts on the space of sections by l(g), the left action of G on

functions, which is preserved by this construction. The space L2 (G —H U ) is

thus an in¬nite dimensional representation of G.

The intertwining number of two representations gives us an inner product

V, W = dimC HomG (V, W )

G

on R(G), with respect to which the irreducible representations of G form an

ˆ

orthonormal basis. Taking the formal completion of R(G), we de¬ne R(G) to

be the space of possibly in¬nite formal sums » a» V» . The intertwining number

ˆ

then extends to a pairing R(G) — R(G) ’ Z.

If R is a subgroup of G, every representation of G automatically restricts

to a representation of R. This gives us a pullback map i— : R(G) ’ R(R),

corresponding to the inclusion i : R ’ G. The map U ’ L2 (G —H U ) dis-

cussed above assigns to each R-representation an induced in¬nite dimensional

G-representation. Expressed in terms of our representation ring notation, this

ˆ

induction map becomes the homomorphism i— : R(R) ’ R(G) given by

i— V» , U

i— U = R V» ,

»

the formal adjoint to the pullback i— . This is the content of the Frobenius

reciprocity theorem.

A homogeneous di¬erential operator on G/R is a di¬erential operator D :

“(E) ’ “(F) between two homogeneous vector bundles E and F that commutes

with the left action of G on sections. If the operator is elliptic, then its kernel and

cokernel are both ¬nite dimensional representations of G, and thus its G-index

is a virtual representation in R(G). In this case, the index takes a particularly

elegant form.

Theorem 18 (Bott) If D : “(G —H U0 ) ’ “(G —H U1 ) is an elliptic homoge-

neous di¬erential operator, then the G-equivariant index of D is given by

IndexG D = i— (U0 ’ U1 ),

ˆ

where i— (U0 ’ U1 ) is a ¬nite element in R(G), i.e. belongs to R(G).

In particular, note that the index of a homogeneous di¬erential operator

depends only on the vector bundles involved and not on the operator itself! To

prove the theorem, just use Bott™s formula (10.33), where the subscript n is

replaced by » labeling the the G-irreducibles.

10.7.3 Landweber™s index theorem.

Suppose that G is semi-simple and simply connected and R is a reductive sub-

group of maximal rank. Suppose further that G/R is a spin manifold, then we

can compose the spin representation S = S+ • S’ of Spin(p) with the lifted

10.7. THE GEOMETRIC INDEX THEOREM. 181

˜

map Spin : R ’ Spin(p) to obtain a homogeneous vector bundle, the spin bun-

dle S over G/R. For any representation of R on U the Kostant Dirac operator

descends to an operator

‚ U : “(S± — U) ’ “(S — U).