Spherical Inversion on SL_{n}(R)
pp 309-324 |
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# General Inversion Theorems

## Abstract

The Fourier inversion formula is a standard fact of elementary analysis. Harish-Chandra developed an inversion for *K*-bi-invariant functions on a semisimple Lie group G, in other words he developed the theory of a spherical transform [Har 58a], [Har 58b], which is an integral transform, with a kernel called the spherical kernel. There are variations to the setting of this transform, involving various factors. Harish-Chandra’s general Plancherel inversion in the non-*K*-bi-invariant case is a lot more complicated, and highly non-abelian. The *K*-bi-invariant case turns out to be abelian. However, in this case, one can look at the inversion on various function spaces, ranging over *C* _{ c } ^{∞} , C_{ c }, Schwartz space, *L*^{1}*, L*^{2}, ad lib. Harish-Chandra dealt fundamentally with the Schwartz space, which he defined in the context of semisimple Lie groups, and *L*^{2}. Helgason pointed to the correspondence between *C* _{ c } ^{∞} and the Paley-Wiener space [Hel 66], complemented by Gangolli [Gan 71]. Then Rosenberg [Ros 77] gave a much simpler version of some parts of Harish inversion, using some lemmas from Helgason and Gangolli. Here we give this Rosenberg material, suitably axiomatized in a way which gives rise to a setting for pure inversion theorems in ordinary euclidean space, essentially along the lines of the similar elementary and standard theory of Fourier inversion. Roughly speaking, the situation is as follows.

## Keywords

Polar Decomposition Chapter VIII Schwartz Space Fourier Inversion Iwasawa Decomposition## Preview

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