Fourier Transform of Schwartz Algebras on Groups in the Harish-Chandra class

Abstract: It is well-known that the Harish-Chandra transform, $f\mapsto\mathcal{H}f,$ is a topological isomorphism of the spherical (Schwartz) convolution algebra $\mathcal{C}^{p}(G//K)$ (where $K$ is a maximal compact subgroup of any arbitrarily chosen group $G$ in the Harish-Chandra class and $0<p\leq2$) onto the (Schwartz) multiplication algebra $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon})$ (of $\mathfrak{w}-$invariant members of $\mathcal{Z}({\mathfrak{F}}^{\epsilon}),$ with $\epsilon=(2/p)-1$). The same cannot however be said of the full Schwartz convolution algebra $\mathcal{C}^{p}(G),$ except for few specific examples of groups (notably $G=SL(2,\mathbb{R})$) and for some notable values of $p$ (with restrictions on $G$ and/or on $\mathcal{C}^{p}(G)$). Nevertheless the full Harish-Chandra Plancherel formula on $G$ is known for all of $\mathcal{C}^{2}(G)=:\mathcal{C}(G).$ In order to then understand the structure of Harish-Chandra transform more clearly and to compute the image of $\mathcal{C}^{p}(G)$ under it (without any restriction) we derive an absolutely convergent series expansion (in terms of known functions) for the Harish-Chandra transform by an application of the full Plancherel formula on $G.$ This leads to a computation of the image of $\mathcal{C}(G)$ under the Harish-Chandra transform which may be seen as a concrete realization of Arthur's result and be easily extended to all of $\mathcal{C}^{p}(G)$ in much the same way as it is known in the work of Trombi and Varadarajan.