Non-spherical Harish-Chandra Fourier Transforms on Real Reductive Groups

Abstract

The Harish-Chandra Fourier transform, $f\mapsto \mathcal {H}f,$ is a linear topological algebra 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 $0<p\le 2$) onto the (Schwartz) multiplication algebra $\bar{\mathcal {Z}}({\mathfrak {F}}^{\epsilon })$ (of $\mathfrak {w}$-invariant members of $\mathcal {Z}({\mathfrak {F}}^{\epsilon }),$ with $\epsilon =(2/p)-1$). This is the well-known Trombi–Varadarajan theorem for spherical functions on the real reductive group, G. Even though $\mathcal {C}^{p}(G//K)$ is a closed subalgebra of $\mathcal {C}^{p}(G),$ a similar theorem has not however been successfully proved for the full Schwartz convolution algebra $\mathcal {C}^{p}(G)$ except; for $\mathcal {C}^{p}(G/K)$ (whose method is essentially that of Trombi–Varadarajan, as shown by M. Eguchi); 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 members of $\;\mathcal {C}^{p}(G)$). In this paper, we construct an appropriate image of the Harish-Chandra Fourier transform for the full Schwartz convolution algebra $\mathcal {C}^{p}(G),$ without any restriction on any of G, p and members of $\;\mathcal {C}^{p}(G).$ Our proof, that the Harish-Chandra Fourier transform, $f\mapsto \mathcal {H}f,$ is a linear topological algebra isomorphism on $\mathcal {C}^{p}(G),$ equally shows that its image $\mathcal {C}^{p}(\widehat{G})$ can be nicely decomposed, that the full invariant harmonic analysis is available and implies that the definition of the Harish-Chandra Fourier transform may now be extended to include all p-tempered distributions on G and to the zero-Schwartz spaces.