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Proof. The irreducibility of the corresponding GW module H follows from the
ergodicity of the Haar measure, as in (3.8).
It is straightforward to check that the functions ck satisfy (2.3) and (3.10), so,
by Theorem (3.9), the corresponding J must leave the real form HR invariant. Of
course, this can be deduced by direct calculation as well, using (2.5). HR must
be irreducible under C(Z), since any closed invariant subspace generates a closed
CC(Z)-invariant subspace in H.
Finally, suppose that the representation of C(Z) in HR could be extended to
one" CC(Z) (in HR itself). Denote by T the operation representing multiplication
of
by -1: T is an orthogonal complex structure in HR commuting with C(Z). Its
unique C-linear extension to all of H = HR •" iHR is unitary and commutes with
all the J. As we have already mentioned, this implies that T is given pointwise, by
an operator-valued measurable function k(x): (T f)(x) = k(x)f(x). In the present
case, k(x) is complex valued. Since k(x)2 = -1, we can write it as
k(x) = (x)i
for some measurable : X ’! {±1}. The condition for T to leave invariant the real
form HR and to commute with the representation J amount to, respectively,
(1 - x) = - (x) (x + ´k) = (x)
10 ESTHER GALINA, AROLDO KAPLAN AND LINDA SAAL
for almost all x and all k. The second equation implies that is actually constant on
each "-equivalence class. [According to [Go] ???], must then be constant almost
everywhere, contradicting the first equation. Hence, no such T can exist.
Representations with these parameters can be realized on the standard L2(T)
of complex-valued functions on the circle since, as a measure space, (X, µX ) is the
same as the interval (0, 1) -hence to the circle, equipped with the Lebesgue measure
and dimHx = 1. In this identification, translations in X do not correspond to rigid
rotations of T. However, the operation x ’! 1 - x in X (changing all the elements
of x) does correspond to x ’! 1 - x on (0, 1) which, on T ‚" C, becomes ordinary
complex conjugation. The operators ck can then be viewed as functions
ck : T ’! T,
Ü
satifying, of course, (2.3). According to Theorem 3.9, those that split over R can
be realized with
¯
L2(T)R = {f " L2(T) : f(t) = f(t)}.
as the invariant real form. The condition on the functions ck for this real form to
be invariant under the J s becomes
¯
(3.10) ck(t) = (-1)kck(t).
for t " T.
References
[BD] T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Springer-Verlag,
1985.
[BSZ] J. Baez, I. Segal and Z. Zhaou, Introduction of algebraic and Constructive Quantum
Field Theory, Princeton University Press, 1992.
[C] E. Cartan, Nombres complexes, Encyclopédie des Sciences Mathématiques 1 (1908),
329 468.
[G] V. Ya. Golodets, Classification of representations of the anti-commutation relations,
Russ. Math. Surv. 24 (1969), 1 63.
[GKS] E. Galina, A. Kaplan and L. Saal, Infinite dimensional quadratic forms admitting com-
position (to appear).
[GW] L. Gårding and A. Wightman, Representations of the anticommutation relations, Proc.
Natl. Acad. Sci. USA 40 (1954), 617 621.
CIEM - FAMAF, Universidad Nacional de ordoba, Ciudad Universitaria, (5000)


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