Let G = 5I(2,R), the group of real, 2 x 2 matrices of determinant one, K = SO(2),
the subgroup of rotations in the plane, and A the vector subgroup consisting of elements
which are positive on the main diagonal and zero elsewhere. For each 0 p ^ 2,
will denote the //-Schwartz space on G, defined in Section 3, and C (G) will denote the
zero-Schwartz space, defined as the intersection of all the //-Schwartz spaces.
The main result of this paper is the characterization of the images of Cp (G) and
C (G) under the operator-valued Fourier transform. The case p = 2 has a long history,
starting with Ehrenpreis and Mautner [11]. In their paper the image of a space similar
to C'(G) was characterized for G=PS£(2,R). Arthur, in [1], determined the image of
C (G) for any G of real rank one, while in [8,9] Eguchi considered the case where G has
just one conjugacy class of Cartan subgroups. The complete p = 2 result, for any reduc-
tive group, was established by Arthur in [2,3].
For values of p other than 2, Trombi [26] considered groups G of real rank one and
determined the image of CP(G:F), the subspace of CV(G) consisting of those functions
whose K-types fall within the finite set F. Kawazoe [20] considered the same problem,
but with a more restricted set of p values. Recently Trombi announced the solution of
the K-finite problem without the restriction to groups of real rank one. In our current
paper the restriction to finite K-type is removed for the group G = SL(2, R).
The basic organization of this paper comes from Trombi [26], with modifications to
remove the finite K-type restriction strongly influenced by the arguments in Arthur [1].
Much of the detailed structural information needed for SL(2, R) was derived from
material given by Milicic in [22]. The results concerning C (G) were developed in colla-
boration with Henrik Schlichtkrull.
Received by the editor March 23, 1987.
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