28 NONLINEAR REPRESENTATION AND SPACES
which shows that
^2xj[a0j1V) = im^Tx^ ~ 2^2^ji{xjdt - xidj).
3 3 3l
Therefore
AD = 2A - m2 + ^2(xidj - Xjdi)2 + ] T ) X J ( A - m2)xi + im^xai. (2.21)
i3 i I
Using the fact that
^2(xidj - Xjdi)2 = ^^(djXiXidj - diXiXjdj), (2.22a)
i3 i,3
and that
y] XjAxj y] djXjXjdj, (2.22b)
we obtain
AD = —m2 + 2A m 2 |x| 2 + 3 V^SjXiXi^ \^diXiXjdj + imS^xiji. (2.23)
It follows from (2.23) that
(a, - A
D
a ) = m2\\a\\2L2 + 2 £ ||0,a||*a + m2 £ ||^a|| 2
2
(2.24)
+ 3^11x^^111 2 - H ^ ^ ^ a H ^ -my}Tf(a,i'yjxja)
i,3 i 3
m2(\\a\\l2
+ J2 WxMh) + 2 £
U*all£'
i i
i,3 3
As pointed out in the proof of Theorem 2.2, the unitary representation V in Ep is equivalent
to a unitary representation V in Exl2 by the isomorphism | V| p ~ i 0 / : MP®D M 1 / 2 © / } .
V is a direct sum: V = F M P 0 F D and V = VM'/2 0 F D . The generators of VM'/2 are
given by
ti£" ( / , / ) = (/, A / ) , (2.25)
$f'2
=
ft,
1 * 3,
Tflij XiOj ~r XjOi^
3
dC2
(/, /) = (*«/, E a^i/). i 3.
i = i
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