But then

dim ¦(U (g))nC(g) = 2s+2N = dim C(g).

This implies that

C(g) = ¦(U (g))nC(g) = ¦(U (g))n(§b’ ), (9.28)

proving that C(g) is primary of type Vρ with multiplicity 2s+N as a represen-

tation of g under the left multiplication action of ¦(g).

This implies that any submodule for this action, in particular any left ideal

of C(g), is primary of type Vρ . Since we have realized the spin representation

of C(g) as a left ideal in C(g) we have proved the important

Theorem 17 Spin ad is primary of type Vρ .

One consequence of this theorem is the following:

Proposition 29 The weights of Vρ are

ρ ’ φJ (9.29)

where J ranges over subsets of the positive roots and

φJ = φJ

φ∈J

each occurring with multiplicity equal to the number of subsets J yielding the

same value of φJ .

Indeed, (9.21) gives the weights of Spin ad, but several of the βJ are equal

due to the trivial action of ad(h) on itself. However this contribution to the

multiplicity of each weight occurring in (9.21) is the same, and hence is equal

to the multiplicity of Vρ in Spin ad. So each weight vector of Vρ must be of the

form (9.29) each occurring with the multiplicity given in the proposition.

Chapter 10

The Kostant Dirac operator

Let p be a vector space with a non-degenerate symmetric bilinear form. We

have the Cli¬ord algebra C(p) and the identi¬cation of o(p) = §2 (p) inside

C(p).

10.1 Antisymmetric trilinear forms.

Let φ be an antisymmetric trilinear form on p. Then φ de¬nes an antisymmetric

map

b = bφ : p — p ’ p

by the formula

(b(y, y ), y ) = φ(y, y , y ) ∀ y, y , y ∈ p.

This bilinear map “leaves ( , ) invariant” in the sense that

(b(y, y ), y ) = (y, b(y , y )).

Conversely, any antisymmetric map b : p — p ’ p satisfying this condition

de¬nes an antisymmetric form φ. Finally either of these two objects de¬nes an

element v ∈ §3 p by

’2(v, y § y § y ) = (b(y, y ), y ) = φ(y, y , y ). (10.1)

We can write this relation in several alternative ways: Since

’2(v, y § y § y ) = ’2(ι(y )ι(y)v, y ) = 2(ι(y)ι(y )v, y )

we have

b(y, y ) = 2ι(y)ι(y )v. (10.2)

Also, ι(y)v ∈ §2 p and so is identi¬ed with an element of o(p) by commutator

in the Cliford algebra:

ad(ι(y)v)(y ) = [ι(y)v, y ] = ’2ι(y )ι(y)v

163

164 CHAPTER 10. THE KOSTANT DIRAC OPERATOR

so

ad(ι(y)v)(y ) = [ι(y)v, y ] = b(y, y ). (10.3)

10.2 Jacobi and Cli¬ord.

Given an antisymmetric bilinear map b : p — p ’ p we may de¬ne

Jac(b) : p — p — p ’ p

by

Jac(b)(y, y , y ) = b(b(y, y ), y ) + b(b(y , y ), y) + b(b(y , y), y )

so that the vanishing of Jac(b) is the usual Jacobi identity. It is easy to check

that Jac(b) is antisymmetric and that if b satis¬es (b(y, y ), y ) = (y, b(y , y ))

then the four form

’ (Jac(b)(y, y , y ), y )

y, y , y , y

is antisymmetric. We claim that if v ∈ §3 p as in the preceding subsection, then

1

ι(y )ι(y )ι(y)v 2 = Jac(b)(y, y , y ). (10.4)

2

To prove this observe that

ι(y)v 2 = (ι(y)v)v ’ v(ι(y)v)

ι(y )ι(y)v 2 = (ι(y )ι(y)v)v + (ι(y)v)(ι(y )v) ’ (ι(y )v)(ι(y)v) + v(ι(y )ι(y)v)

ι(y )ι(y )ι(y)v 2 = ’(ι(y )ι(y)v)ι(y )v + (ι(y )v)(ι(y )ι(y)v) + (ι(y )ι(y)v)ι(y )v

+(ι(y)v)(ι(y )ι(y )v ’ (ι(y )ι(y )v)(ι(y)v) ’ (ι(y )v)(ι(y )ι(y)v))

= [ι(y )v, ι(y )ι(y)v] + [ι(y )v, ι(y)ι(y )v] + [ι(y)v, ι(y )ι(y )v]

1

= Jac(b)(y, y , y )

2

by (10.2) and (10.3).

Equation (10.4) describes the degree four component of v 2 in terms of Jac(b).

We can be explicit about the degree zero component of v 2 . We claim that

n

1

2

[y ’

(v )0 = tr j b(yj , b(yj , y))], := (yj , yj ). (10.5)

j

24 j=1

Indeed, by (9.6) we know that (v 2 )0 = ’(v, v) and since yi § yj § yk , i < j < k

10.3. ORTHOGONAL EXTENSION OF A LIE ALGEBRA. 165

form an “orthonormal” basis of §3 p we have

±(v, yi § yj § yk )2 , ± =

’(v, v) =’ ijk

1¤i<j<k¤n

n,n,n

1

±(v, yi § yj § yk )2

=’

6

i=1,j=1,k=1

n,n,n

1

±(ι(yk )ι(yj )v, yi )2

=’

6

i=1,j=1,k=1

n,n,n

1

±(b(yj , yk ), yi )2

=’

24

i=1,j=1,k=1

n,n

1

=’ j k (b(yj , yk ), b(yj , yk ))

24

j=1,k=1

n,n

1