Harmonic analysis and non-symmetric Macdonald polynomials? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:26:17Z http://mathoverflow.net/feeds/question/119440 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119440/harmonic-analysis-and-non-symmetric-macdonald-polynomials Harmonic analysis and non-symmetric Macdonald polynomials? Alex K. 2013-01-21T04:01:02Z 2013-01-21T04:01:02Z <p>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 <strong>non-positive</strong> (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.</p> <p>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. </p> <p>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. </p> <p>Anyway, the questions I have are:</p> <p>(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?</p> <p>(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? </p>