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.