I am reading Macdonald's Symmetric Functions and Orthogonal Polynomials lecture notes, and have several related questions:

  1. In both the type A case (chapter 1) and the general irreducible root system case (chapter 2), he makes the assumption that $t=q^k$ for some positive integer $k$, and does not seem to give any indication how to prove the existence of Macdonald polynomials without this assumption. However, he still goes to the trouble on p27 to define the inner product used in the existence proof for general $q$ and $t$, and then specializes it to the $t=q^k$ case. In Symmetric Functions and Hall Polynomials I believe he shows existence without that assumption, but this is only the type $A$ case and I am interested in the general case. I would like to know either (a) how to prove existence for general root systems without this assumption, or (b) a good argument for why we only ever care about the Macdonald polynomials when $t=q^k$ and so don't need to do the full proof.
  2. (somewhat related) On p28 he shows that as $q \to 0$ the Macdonald polynomials reduce to what is basically the generalization of Hall-Littlewood polynomials to arbitrary root systems, i.e. $$P_\lambda = W_\lambda(t)^{-1} \sum_{w \in W} w\left(e^\lambda \prod_{\alpha \in R^+} \frac{1-te^{-\alpha}}{1-e^{-\alpha}}\right)$$ where $W$ is the Weyl group and $W_\lambda$ is the analogue of the type $A$ stabilizing factor. I don't understand how to prove this specialization by using the definition of Macdonald polynomials as orthogonal to the inner product $$\langle f, g \rangle = |W|^{-1} \int_T f \bar{g} \Delta$$ defined in (4.4) of the book (refer to the book for full terminology if this isn't notation you're used to--essentially we identify polynomials $f,g$ with functions on a torus and integrate over it), unless we assume $t=q^k$. In the case $t=q^k$ this integral is equal to the constant term of $f\bar{g}\Delta$ because $\Delta$ reduced to a Laurent polynomial, but without this assumption it does not and therefore to show that the specialization works we must show the vanishing of various integrals over rational functions on the torus which are not clearly zero.
  3. Are there any other good sources to complement this one for Macdonald polynomials of arbitrary root systems?


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