Following [C23], [C28], and the previous chapter, we will study DAHA, its representations, and the q–Fourier transform. The basic representations of this chapter are those in Laurent polynomials, Laurent polynomials multiplied by the Gaussian, and in various spaces of delta functions. We also consider induced, semisimple, spherical, and finite dimensional representations in this chapter. The q–Fourier transform generalizes the Harish-Chandra spherical transform, its p–adic counterpart due to Macdonald–Matsumoto, and the Fourier transforms for the Heisenberg and Weyl algebras. The applications of this transform to Verlinde algebras, Gauss–Selberg integrals and sums, Dedekind–Macdonald-type η-type identities, and the diagonal coinvariants will be discussed.
There are two major directions in the q–Fourier analysis, compact and noncompact, generalizing the corresponding parts of the harmonic analysis on symmetric spaces. The compact direction is based on the imaginary integration and its variant, the constant term functional. The representation of DAHA in Laurent polynonials, called the polynomial representation, and in Laurent polynomials multiplied by the Gaussian, a counetrpart of the Schwartz space, are the key here. The theory of Fourier transforms of these spaces includes the Macdonald conjectures (now the theorems), the Mehta–Macdonald formula, and has other applications of combinatorial nature.
Compact case. Concerning the Macdonald conjecture and the Mehta–Macdonald formula, we refer to [C16, C19, C20, C21, M8]. They will not be proven in the chapter, as well as their Jackson counterparts. They were justified using the shift operator.