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