This question is about Macdonald's symmetric polynomials theory. Going through related papers and literature, it seems to me that the magical part of his theory lies in how the k-th weight function $\Delta_{q,k}$ is defined. Since once that is done, Macdonald's polynomials can be characterized by the orthogonality with respect to his k-th pairing $$ (f,g) = <f\bar{g}\,\Delta_{q,k}>_1 $$
I am trying to understand his motivation: how did he get the correct definition of $\Delta_{q,k}$? Well, I imagine the motivation was two-fold.
- Define the "classical" $\Delta_k$, which I believe was the work of Jack (please correct me if I am wrong).
- Find its q-analogue.
I am more interested in the first part. So, perhaps my questions should be "what was the motivation of Jack?".
I have heard somewhere else that the weight function does something with the radial weight function on a compact symmetric Riemannian manifold, which I am not familiar with. Could anyone point out any relations or references between the two? Thank you very much in advance.
