Abstract

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 in 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.

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