Page 157 - Curvature and Homology

4.4 HARMONIC FORMS 139

4.4. Harmonic forms on a compact semi-simple Lie group

In terms of the metric (4.3.4) on G the star operator may be defined
and we are then able to prove the following

Proposition 4.4.1. Let a be an invariant p-form on G. Then,
(i) da is invariant;
(ii) *a is invariant, and
(iii) if a = d/3, /3 is invariant.
Let X be an element of the Lie algebra L of G. Then, B(X)da =
dB(X)a = 0; B(X)*a = *B(X)a = 0 by formulae (3.7.7) and (3.7.11).
Hence, (i) and (ii) are established. By the decomposition theorem of 5 2.9
we may write a = d8Ga where G is the Green's operator (cf. II.B.4).
Since 6 = (- l)np+n+l*d* on p-forms we may put a = d*dy where y
is some (n - p)-form. Then, 0 = B(X)a = 9(X)d*dy = dB(X)*dy
= dd(X)dy = d*dB(X)y, from which 8dO(X)y = (- l)np+l*d*dO(X)y
= 0. Since (SdB(X)y, B(X)y) = (dB(+, dB(X)y) and B(X)dy = dB(+,
dy is invariant. Thus, from (ii), *dy is invariant. This completes
the proof of (iii).
Proposition 4.4.2. The harmonic forms on G are invariant.
This follows from lemma 4.3.1 and theorem 3.7.1.

Proposition 4.4.3. The invariant forms on G are harmonic.
Indeed, if /3 is an invariant p-form it is co-closed. For, by lemma 4.2.4,

Hence, by prop. 4.2.1, /3 is harmonic.
Note that prop. 4.4.1 is a trivial consequence of prop. 4.4.3.
Therefore, in order to find the harmonic forms /3 on a compact Lie
group G we need only solve the equations

where /3 = B4.. 'ral A ... A 'rap. The problem of determining the