6 2. ACTIONS O F MATCHED PAIRS
P92(VxV*) = VxV*. Therefore, Ad(VU) sends 11^'^ onto
n^°2\
which proves
the second statement.
1. Cocycles and cohomology group
In the rest we will assume that G = N x H is a semi-direct product with
d=N and G2 = H. For n G TV, h G H we set
nh
=
hTxnh
(G N).
Note
(Uln2)h = n\n\, (nhl)h2 = nhlh2.
When (a, (3) is an action of the matched pair (TV, H), for each (h,n) e H x N one
finds a unitary u(h,ri) in satisfying
phanhph-i = Ad(u(h,ri))an.
(Also note that u(h, TV) is determined up to a scalar, which will be a source of a
coboundary.) For each fixed h G H, the above right side (and hence the left side)
is an action of TV. Thus, we find a scalar rjh(ni,n2) G T satisfying
u(h,nin2) = rjh(nun2)u(h,n1)ani(u(h,n2)).
Since
Ph1h2anhih2P(h1h2)-1 =
&hx
yPh2a(nhi)h2Ph-iJ
Ph-i
= /3hl (Ad(u(h2,nhl))anhl)(3h-i
-
Ad(phl(u(h2,nhl)))
phlanhl0h-i
= Ad (/3hl
(u(h2,nhl))u(hun))
an,
we similarly find a scalar (n(hi, h2) G T satisfying
ufah^n)* = Cn(fti, ^ 2 ) ^ 1 , n)*/?^ (iz(h2, n*11)*).
Note that for each hi,h2, the function n G T V —• Cn(^i, ^2) G T is an element in
U(£°°(TV)), the unitary group of£°°(N).
LEMMA
2.2. (i) For each h e H, we have nh G
Z2(TV,T),
(ii) We have ( G
Z2(H,JJ(£oc(N)))
with the natural H-action on £°°(N), that is,
(n(hih2, h3)Cn(h1,h2) = Cn(hi, h2h3)(nhl (ft2, h3).
(Hi) The two cocycles {rih}heH,C are related by
Vh1(nun2)rjh2(nll\n21)
=
Cnin2(hi,h2)
VhxhAni,^) Cn1(^l,^2)Cn2(^l,^2)'
PROOF. We compute
u(h, (nin2)n3) = r}h(nin2, n3)u(h, nin2)aniTl2 (u(h, n3))
= r)h(mn2, n3) (r)h(ni,n2)u(h, n{)ani (u(h, n2)) anin2(u(h, n3)),
u(h, ni(n2n3)) = nh(nun2n3)u(h, ni)ani (u(h, n2n3))
= f]h(nun2n3)u(h, rti)ani (rjh(n2, n3)u(h, n2)an2(u(h, n3))).
From the above computations we conclude 77^ G
Z2(N,
T).
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