# Harmonic analysis and non-symmetric Macdonald polynomials?

I have recently been reading a lot about Macdonald polynomials, the symmetric and the non-symmetric ones. One thing that strikes me is that the symmetric Macdonald polynomials admit a positive theory, while the measure employed by Cherednik is non-positive (actually not even real) and the non-symmetric Macdonald polynomials are only (bi-)orthogonal with respect to the Cherednik measure. The symmetric Macdonald polynomials are orthogonal with respect to original positive measure of Macdonald.

One of the limits of the Macdonald polynomials are the so-called Jacobi polynomials of Heckman and Opdam. Here the non-symmetric polynomials admit a positive theory, actually the measure for the symmetric and the non-symmetric polynomials is the same.

Another limit is the so-called p-adic limit, i.e. the $q\rightarrow 0$ limit of the Macdonald polynomials, leading to the Macdonald spherical functions. It seems that the non-symmetric Macdonald spherical functions are only bi-orthogonal with respect to the $p$-adic limit of the Cherednik measure, still a non-positive measure.

Anyway, the questions I have are:

(1) Can someone explain why the symmetric and the non-symmetric Jacobi polynomials of Heckman and Opdam admit a positive theory but the non-symmetric Macdonald polynomials not?

(2) Is there a way to modify the complex measure to a positive measure in a natural way so that the non-symmetric Macdonald polynomials/spherical functions are orthogonal with respect to the later measure?

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