1 1

We have

∀ y+ ∈ p+

y+ e+ = 0,

and hence

(§p+ )+ e+ = 0.

In other words

§p+ e+

consists of all scalar multiples of e+ .

Since

§p’ — §p+ ’ C(p), w’ — w+ ’ w’ w+

is a linear bijection, we see that

C(p)e+ = §p’ e+ .

This means that the left ideal generated by e+ in C(p) has dimension 2m , and

hence must be isomorphic as a left C(p) module to S. In particular it is a

minimal left ideal.

9.3. THE SPIN REPRESENTATIONS. 157

Let e’ , . . . e’ be a basis of p’ and for any subset J = {i1 , . . . , ij }, i1 <

m

1

i2 · · · < ij of {1, . . . , m} let

eJ := e’ § · · · § e’ = e’ · · · e’ .

’ i1 ij i1 ij

Then the elements

eJ e+

’

form a basis of this model of S as J ranges over all subsets of {1, . . . , m}.

For example, suppose that we have a commutative Lie algebra h acting

on p as in¬nitesimal isometries, so as to preserve each p± , that the e+ are i

’

weight vectors corresponding to weights βi and that the ei form the dual basis,

corresponding to the negative of these weights ’βi . Then it follows from (9.14)

that the image, ν(h) ∈ §2 (p) ‚ C(p) of an element h ∈ h is given by

1 1

βi (h)e+ § e’ = βi (h)(1 ’ e’ e+ ).

ν(h) = i i ii

2 2

Thus

ν(h)e+ = ρp (h)e+ (9.19)

where

1

(β1 + · · · + βm ).

ρp := (9.20)

2

For a subset J of {1, . . . , m} let us set

βJ := βj .

j∈J

Then we have

[ν(h), eJ ] = ’βJ (h)eJ

’ ’

and so

ν(h)(eJ e+ ) = [ν(h), eJ ]e+ + eJ ν(h)e+ = (ρp (h) ’ βJ (h))eJ e+ .

’ ’ ’ ’

So if we denote the action of ν(h) on S± by Spin± ν(h) and the action of ν(h)

on S = S+ • S’ by Spin ν(h) we have proved that

The eJ e+ are weight vectors of Spin ν with weights ρp ’ βJ . (9.21)

’

It follows from (9.21) that the di¬erence of the characters of Spin+ ν and Spin’ ν

is given by

1 1

chSpin+ ν ’ chSpin’ ν = e( βj ) ’ e(’ βj ) (1 ’ e(’βj )) .

= e(ρp )

2 2

j j

(9.22)

There are two special cases which are of particular importance: First, this

applies to the case where we take h to be a Cartan subalgebra of o(p) = o(C2k )

158CHAPTER 9. CLIFFORD ALGEBRAS AND SPIN REPRESENTATIONS.

itself, say the diagonal matrices in the block decomposition of o(p) given by the

decomposition

C2k = Ck • Ck

into two isotropic subspaces. In this case the βi is just the i-th diagonal entry

and (9.22) yields the standard formula for the di¬erence of the characters of the

spin representations of the even orthogonal algebras.

A second very important case is where we take h to be the Cartan subalgebra

of a semi-simple Lie algebra g, and take

p := n+ • n’

relative to a choice of positive roots. Then the βj are just the positive roots,

and we see that the right hand side of (9.22) is just the Weyl denominator, the

denominator occurring in the Weyl character formula. This means that we can

write the Weyl character formula as

(’1)w e(w • »)

ch(Irr(») — S+ ) ’ ch(Irr(») — S’ ) =

w∈W

where

w • » := w(» + ρ).

If we let Uµ denote the one dimensional module for h given by the weight µ we

can drop the characters from the preceding equation and simply write the Weyl

character formula as an equation in virtual representations of h:

(’1)w Uw•» .

Irr(») — S+ ’ Irr(») — S’ = (9.23)

w∈W

The reader can now go back to the preceding chapter and to Theorem 16

where this version of the Weyl character formula has been generalized from the

Cartan subalgebra to the case of a reductive subalgebra of equal rank. In the

next chapter we shall see the meaning of this generalization in terms of the

Kostant Dirac operator.

9.3.2 The odd dimensional case.

Since every odd dimensional space with a non-singular bilinear form can be

written as a sum of a one dimensional space and an even dimensional space (both

non-degenerate), we need only look at the Cli¬ord algebra of a one dimensional

space with a basis element x such that (x, x) = 1 (since we are over the complex

numbers). This Cli¬ord algebra is two dimensional, spanned by 1 and x with

x2 = 1, the element x being odd. This algebra clearly has itself as a canonical

module under left multiplication and is irreducible as a Z/2Z module. We may

call this the spin representation of Cli¬ord algebra of a one dimensional space.

Under the even part of the Cli¬ord algebra (i.e. under the scalars) it splits

into two isomorphic (one dimensional) spaces corresponding to the basis 1, x of

9.3. THE SPIN REPRESENTATIONS. 159

the Cli¬ord algebra. Relative to this basis 1, x we have the left multiplication

representation given by

1 0 0 1

1’ x’

, .

0 1 1 0

Let us use C(C) to denote the Cli¬ord algebra of the one dimensional or-

thogonal vector space just described, and S(C) its canonical module. Then

if

q=p•C

is an orthogonal decomposition of an odd dimensional vector space into a direct

sum of an even dimensional space and a one dimensional space (both non-

degenerate) we have