2 WILLIAM H. BARKER
We now describe the contents of the various sections of this paper. Sections 2 and 3
introduce general notation and the
Lp-
Schwartz spaces
CP(G).
Sections 4 through 8
consider the basic properties of the canonical matrix coefficients for both the principal
series and the discrete series of representations of SL(2, R). Most of the results in these
sections are already known, although locating some of them in the literature can be time
consuming. The major result of these sections is Theorem 8.1, a slight extension for
SL(2,R) of the discrete series matrix coefficient bound of Trombi and Varadarajan [27].
In Sections 9 through 11 we return to
CP(G),
considering the basic properties of the
Fourier transform T on this space and the natural decomposition of
CP(G)
into the
Lp
cusp forms space Cg(G) and the perturbated Lp wave packets space Cg(G). The space
CP(G)
= C*(G) x C^G),
which we will show to be the image of C
p
(G) under T , is defined in Section 9, and T is
shown to be a continuous injection of
CP(G)
into
CP(G)
(Theorem 9.6). From Arthur's
work [1] it is known that T is an isomorphism from C (G) onto C (G), with the inverse
denoted by S. Establishing the surjectivity of T from
CP(G)
onto
CP(G),
and the
continuity of S on
CP(G),
are the goals of the remaining portions of the paper.
Sections 12 through 14 undertake a detailed study of the Harish-Chandra expan-
sions for matrix coefficients of the principal series and discrete series for SL(2, R). To
determine the image of C (G) under the Fourier transform for any reductive group G,
only the first, or "constant", term needs to be considered. This remains true for CP(G)
when G = SL(2, R) and 1 p 2. However, for p 1, we need to make heavy use of
further terms in the Harish-Chandra expansions, the number of required terms increasing
as p decreases toward zero. Truncating the expansions at the required number of terms
yields asymptotic approximations to the matrix coefficients on the vector group A, and it
is these approximations that will be needed in the subsequent sections. The proofs of all
the major results in these sections depend on being able to explicitly write out the
differential equations that the matrix coefficients must satisfy. This is where we would
Previous Page Next Page