258 XVII DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

each z e G, TT~l (z) is the left coset (z, e)D, where D is the diagonal subgroup of G x G;
so we may identify G with (G x G)/D and TT with the canonical mapping

G xG->(G xG)/D.

Let v = 'pri(uo) A 'pr2(uo), which is the left-invariant 2«-form on G X G corresponding
to the Haar measure f$ = {$Q (x) ft0. For each differential 2«-form/u on G x G, where
/e ^(G x G), show that (/u)b =/bUo, where/b is the function on G (identified with

(G x G)/D) defined in Section 14.4, Problem 2, such that /b(z) = f f(zw,w) dflofyi).

J G

Deduce that, for each /z-current T on G, we have <r7r(T), /u> = <T,/V>>. If the
Dirac measure ee on G is identified with an /z-current, then *7r(fie) is identified with the
measure tr defined in (a).

6. REAL DISTRIBUTIONS. POSITIVE DISTRIBUTIONS

(17.6.1) Let X be a differential manifold. For each p the vector bundle
p
/\ T(X)* may be identified with a subbundle of the real vector bundle

( AT(X)*)(C), which is the direct sum A T(X)* © z y\ T(X)* (16.18.5). It
follows that fp(X) (resp. 4r)(X)) is equal to <?P,R(X) + i^fH(X)
(resp. ^(X) + /^(X)), where <?P|R(X) (resp. ^}R(X)) is the space of real
differential p-forms of class C°° (resp. Cr) on X. A p-current T on X is then
said to be real if its restriction to ^,,R(X) is real-valued, and a real p-current
is often identified with its restriction to <?P,R(X). As in (13.2), for each current
T the conjugate current T is defined; the real currents ^T = i(T + T) and
= (l/2f)(T — T) are called, respectively, the real and imaginary parts of T.

(17.6.2) Every positive distribution T on a differential manifold X (i.e., every
distribution T such that T(/) j> Qfor allf^Q in ^(X)) is a positive measure
onX.

We may assume that X is an open subset U of R", by virtue of (13.1.9).
Let K be a compact subset of U, and let h : U -> [0, 1] be a C°°-mapping with
compact support and equal to 1 on K. For each real-valued function
; K), we have

and therefore - ||/|| T(A) £ T(/) g ||/|| T(A). Hence, if now

with 7^/2 real-valued, we have |T(/)| g 2T(/z)||/||. This shows that T is a
distribution of order 0, hence a measure. Moreover, with the same notation as