A Paley–Wiener theorem for
the inverse spherical transform is proved for noncompact semisimple Lie groups G
which are either of rank one or with a complex structure. Let K be a fixed
maximal compact subgroup of G. For each K-bi-invariant function f in the
Schwartz space on G, consider the function f defined on a fixed Weyl chamber
a+ by f(H) := Δ(H)f(expH). Here Δ(H) :=∏α∈Σ+mα∕2,
where Σ+ is the set of positive restricted roots and mα is the multiplicity of the root
α. The K-bi-invariant functions f whose spherical transform has compact support
are identified as those for which f extends holomorphically and with a specific growth
to a certain subset of the complexification ac of a. The proof of the theorem
in the rank-one case relies on the explicit inversion formula for the Abel
transform.
