with strict inequality unless » ’ w’1 (γ) = 0 = ρ ’ w’1 (ρ ’ φJ ), and this last

equality implies that J = Jw . QED

We have the spin representation Spin ν where ν : r ’ C(p). Call this

module S. Consider

V» — S

as a r module. Then, letting γ denote a weight of V» , we have

“(V» — S) = {µ = γ + ρp ’ φJ } (10.23)

where

1

¦+ := ¦+ /¦+ .

ρp = φ, p r

2

J∈¦+

p

In other words, ¦p are the roots of g which are not roots of r, or, put another

way, they are the weights of p considered as a r module. (Our equal rank

176 CHAPTER 10. THE KOSTANT DIRAC OPERATOR

assumption says that 0 does not occur as one of these weights.) For the weights

µ of V» — S the form (10.23) gives

J ‚ ∆+ .

µ + ρr = γ + ρ ’ φJ , p

So if we set Z = V» — S as a r module, (10.22) says that

(» + ρ, » + ρ) ≥ mZ .

But we may take J = … as one of our weights showing that

mZ = (» + ρg , » + ρg ). (10.24)

To determine “max (Z) as in Prop. 30 we again use Prop.31 and (10.23): A

µ = γ + ρp ’ φJ belongs to “max (Z) if and only if γ = w» and J = Jw . But

then

ρg ’ φJ = wρg .

Since ρg = ρr + ρp we see from the form (10.23) that

µ + ρr = w(» + ρg ) (10.25)

where w is unique, and

J w ‚ ¦+ .

p

We claim that this condition is the same as the condition w(D) ‚ Dr de¬ning

our cross-section, C. Indeed, w ∈ C if and only if (φ, wρg ) > 0, ∀ φ ∈ ¦+ . But

r

(φ, wρ) = (w’1 φ, ρ) > 0 if and only if φ ∈ w(¦+ ). Since Jw = w(’¦+ ) © ¦+ ,

we see that Jw ‚ ¦+ is equivalent to the condition w ∈ C.

p

Now for µ ∈ “max (Z) we have

µ = w(» + ρ) ’ ρr =: w • » (10.26)

where γ = w(») and so has multiplicity one in V» .

Furthermore, we claim that the weight ρp ’ φJw has multiplicity one in S.

Indeed, consider the representation

Zρr — S

of r. It has the weight ρ = ρr + ρp as a highest weight, and in fact, all of

the weights of Vρg occur among its weights. Hence, on dimensional grounds,

say from the Weyl character formula, we conclude that it coincides, as a rep-

resentation of r, with the restriction of the representation Vρg to r. But since

ρg ’ φJw = wρg has multiplicity one in Vρg , we conclude that ρp ’ φJw has

multiplicity one in S.

We have proved that each of the w • » have multiplicity one in V» — S with

corresponding weight vectors

zw•» := vw» — eJw e+ .

’

10.6. EIGENVALUES OF THE DIRAC OPERATOR. 177

So each of the submodules

Zw•» := U (r)zw•» (10.27)

occurs with multiplicity one in V» — S. The length of w ∈ C (in terms of

the simple re¬‚ections of W determined by ∆) is the number of positive roots

changed into negative roots, i.e. the cardinality of Jw . This cardinality is the

sign of det w and also determines whether eJ e+ belongs to S+ or to S’ . From

’

Prop.31 and equation (10.24) we know that the maximum eigenvalue of Casr

on V» — S is

(» + ρg , » + ρg ) ’ (ρr , ρr ).

Now K» ∈ End(V» — S) commutes with the action of r with

V » — S+ ’ V » — S’

K» :

V » — S’ ’ V » — S+ .

Furthermore, by (10.19), the kernel of K2 is the eigenspace of Casr correspond-

»

ing to the eigenvalue (» + ρ, » + ρ) ’ (ρr , ρr ). Thus

Ker( K2 ) = Zw•» .

»

w∈C

Each of these modules lies either in V — S+ or V — S’ , one or the other but not

both. Hence

Ker( K2 ) = Ker( K» )

»

and so

Ker( K» )|V»—S = Zw•» (10.28)

+

w∈C, det w=1

and

Ker( K» )|V»—S = Zw•» (10.29)

’

w∈C, det w=’1

Let

K± := Zw•» . (10.30)

w∈C, det w=±1

It follows from (10.28) that K» induces an injection of

(V» — S+ )/K+ ’ V — S’

which we can follow by the projection

V» — S’ ’ (V» — S’ )/K’ .

Hence K» induces a bijection

˜

K» : (V — S+ )/K+ ’ (V» — S’ )/K’ . (10.31)

178 CHAPTER 10. THE KOSTANT DIRAC OPERATOR

In short, we have proved that the sequence

0 ’ K + ’ V » — S+ ’ V » — S’ ’ K ’ ’ 0 (10.32)

is exact in a very precise sense, where the middle map is the Kostant Dirac

operator: each summand of K+ occurs exactly once in V» — S+ and similarly for

K’ . This gives a much more precise statement of Theorem 16 and a completely

di¬erent proof.

10.7 The geometric index theorem.

Let r be the representation of G on the space F(G) of smooth or on L2 (G) of

L2 functions on G coming from right multiplication. Thus