logo Toward Harmonic Analysis on DAHA

There are three classical directions:

(1) the unitary dual (irreducible unitary representations),
(2) Fourier transforms (positivity, inverse transforms),
(3) decomposition of the standard unitary representations.

In the daha theory, (2,3) are different from those in the classical harmonic analysis on symmetric spaces and affine Hecke algebras. Fourier transform is defined for a given (irreducible) representaion. Recent developments indicate that (3) changes to:
(3') integral representations of the canonical traces.

We will discuss (2',3') in the case of A-one, for instance, the positivity of the kernel of the Fourier transform will be proven and (3') solved. The rational case will be considered, general results will be formulated, and a (direct) relation to the Heckman-Opdam approach in the AHA theory will be outlined.