2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62299In the paper, we introduce and calculate difference Fourier transforms on representations of the double affine Hecke algebras in polynomilas, polynomials multiplied by the Gaussian, and various spaces of delta-functions including finite-dimensional ones, give a general description of the semisimple representations with a special consideration of the GL-case, and then gradually restrict ourselves with spherical, pseudo-unitary, and Fourier-invariant representations. The latter generalize the Verlinde algebras and lead to new Gauss-Selberg sums and Macdonald's eta-type identities.106 pgs, LaTeX, to appear in Inventiones Math. A latex variant with some changes in Intro and Ch.7Quantum AlgebraRepresentation Theorytext