Levi-Civita connection on G/R twisted by U, and has the same index by Bott™s

theorem. For the precise relation between this Dirac operator coming from

K and the dirac operator coming from the Levi- civita connection we refer to

Landweber™s thesis.)

The following theorem of Landweber gives an expression for the index of this

Kostant Dirac operator. In particular, if we consider G/T , where T is a maximal

torus (which is always a spin manifold), this theorem becomes a version of the

Borel-Weil-Bott theorem expressed in terms of spinors and the Dirac operator,

instead of in its customary form involving holomorphic sections and Dolbeault

cohomology.

Theorem 19 (Landweber) Let G/R be a spin manifold, and let Uµ be an

irreducible representation Uµ of R with highest weight µ. The G-equivariant

index of the Dirac operator ‚ U is the virtual G-representation

IndexG ‚ Uµ = (’1)dim p/2 (’1)w Vw(µ+ρH )’ρG (10.35)

if there exists an element w ∈ WG in the Weyl group of G such that the weight

w(µ + ρH ) ’ ρG is dominant for G. If no such w exists, then IndexG ‚ Uµ = 0.

Proof. For any irreducible representation V» of G with highest weight » we

have

(’1)w Uw•»

V» — (S+ ’ S’ ) =

w∈C

by [GKRS]. Hence

HomR (V» — (S+ ’ S’ ), Uµ ) = 0

if µ = w • » for some w ∈ C while

HomR (V» — (S+ ’ S’ ), Uµ ) = (’1)w

if µ = w • ». But, by (10.33) and Theorem 18 we have

—

V» — (V» — (S+ ’ S’ ) — Uµ )R

IndexG ‚ U =

»

HomR (V» — (S+ ’ S’ )— , Uµ ).

=

Now (S+ ’ S’ )— = S+ ’ S’ if dim p ∼ 0 mod(4) while (S+ ’ S’ )— = S’ ’ S+

=

∼ 2 mod(4). Hence

if dim p =

IndexG ‚ Uµ = (’1)dim p/2 HomR (V» — (S+ ’ S’ ), Uµ ). (10.36)

182 CHAPTER 10. THE KOSTANT DIRAC OPERATOR

The right hand side of (10.36) vanishes if µ does not belong to a multiplet, i.e

is not of the form

w • » = w(» + ρg ) ’ ρr

for some ». The condition w • » = µ can thus be written as

w’1 (µ + ρr ) ’ ρg = ».

If this equation does hold, then we get the formula in the theorem (with w

replaced by w’1 which has the same determinant). QED

In general, if G/R is not a spin manifold, then in order to obtain a similar

result we must instead consider the operator

r r

‚ Uµ : L2 (G) — (S± ) — Uµ ’ L2 (G) — (S ) — Uµ

viewed as an operator on G, restricted to the space of (S — Uµ )-valued functions

on G that are invariant under the diagonal r-action (Z) = diag(Z) + σ(Z),

where σ is the r-action on Uµ . Note that if S —Uµ is induced by a representation

of the Lie group R, then this operator descends to a well-de¬ned operator on

G/R as before. In general, the G-equivariant index of this operator ‚ Uµ is once

again given by (10.35). To prove this, we note that Bott™s identity (10.33) and

ˆ

his theorem continue to hold for the induction map i— : R(r) ’ R(g) using the

representation rings for the Lie algebras instead of the Lie groups. Working in

the Lie algebra context, we no longer need concern ourselves with the topological

obstructions occurring in the global Lie group picture. The rest of the proof of

Theorem 19 continues unchanged.

Chapter 11

The center of U (g).

The purpose of this chapter is to study the center of the universal enveloping

algebra of a semi-simple Lie algebra g. We have already made use of the (second

order) Casimir element.

11.1 The Harish-Chandra isomorphism.

Let us return to the situation and notation of Section 7.3. We have the monomial

basis

f11 · · · fm hj1 · · · hj ek1 · · · ekm

i im

m

1 1

of U (g), the decomposition

U (g) = U (h) • (U (g)n+ + n’ U (g))

and the projection

γ : U (g) ’ U (h)

onto the ¬rst factor of this decomposition. This projection restricts to a projec-

tion, also denoted by γ

γ : Z(g) ’ U (h).

The projection γ : Z(g) ’ U (h) is a bit awkward. However Harish-Chandra

showed that by making a slight modi¬cation in γ we get an isomorphism of

Z(g) onto the ring of Weyl group invariants of U (h) = S(h). Harish-Chandra™s

modi¬cation is as follows: As usual, let

1

ρ := ±.

2 ±>0

Recall that for each i, the re¬‚ection si sends ±i ’ ’±i and permutes the

remaining positive roots. Hence

si ρ = ρ ’ ±i

183

184 CHAPTER 11. THE CENTER OF U (G).

But by de¬nition,

si ρ = ρ ’ ρ, ±i ±i

and so

ρ, ±i = 1

for all i = 1, . . . , m. So

ρ = ω1 + · · · + ωm ,

i.e. ρ is the sum of the fundamental weights.

11.1.1 Statement

De¬ne

σ : h ’ U (h), σ(h) = h ’ ρ(h)1. (11.1)

This is a linear map from h to the commutative algebra U (h) and hence, by

the universal property of U (h), extends to an algebra homomorphism of U (h)

to itself which is clearly an automorphism. We will continue to denote this

automorphism by σ. Set

γH := σ —¦ γ.

Then Harish-Chandra™s theorem asserts that

γH : Z(g) ’ U (h)W

and is an isomorphism of algebras.

11.1.2 Example of sl(2).

To see what is going on let™s look at this simplest case. The Casimir of degree

two is

12

h + ef + f e,

2

as can be seen from the de¬nition. Or it can be checked directly that this

element is in the center. It is not written in our standard form which requires

that the f be on the left. But ef = f e + [e, f ] = f e + h. So the way of writing

this element in terms of the above basis is