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476 PHILLIP A. GRIFFITHS
(4.5)
zr
o #(H, C) + H.
Ignoring momentarily questions of convergence, we integrate (4.5) over IP n*
and interchange an order of integration to obtain
where
71"
o #(H, C)dH +
77" tI
’ "IT lpn,
is the average of r//. We claim that
(4.6) n 0,
which will certainly prove (4.3).
Now (4.6) can be proved by direct computation, but here is an alternative
invariant-theoretic argument. We may consider as a current T on IW by
"On A a dH, a A zn l(Ipgt).
Since the unitary group U(n + 1) acts transitively on IP n* and satisfies
g* dH dH
g*’H-- TgH
for g U(n + 1), we deduce that T, is an invariant current. But it is well
known (and easily proved) that the invariant currents on any compact symmetric
space are just the harmonic forms, (4) and since the degree of is odd we
conclude that 0.
Finally, the justification of the interchange of limits follows by a standard
(and not particularly delicate) argument.
(b) Crofton’s formula for Schubert cycles. One generalization of (4.1) deals
with the intersection of Schubert cycles with an analytic subvariety V in the
Grassmannian G(n, N). Recall that a flag F in U is an increasing sequence of
subspaces
(0) W 0 C W C C WN_ C WN l_,N.
The unitary group U(N) acts transitively on the manifold F(N) of all flags. For
each flag F and sequence of integers a (al, ", an) the Schubert cycle Ea(F)
is defined by
(4.7) Xa(F) {T _. G(n, N)" dim T fq WN-n +i-a, >- i}.
Thus E(F) is the set of n-planes which fail to be in general position with respect
C