I would like to have some order in my head about different version of Macdonald polynomials and positivity statements about them. I understand the following:
1) There is a definition of Macdonald polynomials for any root system. These can be defined, for example as $W$-invariant polynomials on the torus $T$ of a semi-simple group $G$, which are orthogonal polynomials with respect to Macdonald scalar product and normalized in such a way that $$ P_{\lambda}(q,t,x)=e^{\lambda}+\text{lower order terms} $$ where $\lambda$ is a dominant weight and $x\in T$.
2) In type $A$ there is a notion of transformed Macdonald polynomials, which were extensively studied by Haiman. Haiman denotes them by $\tilde{H}_{\lambda}$
(here $\lambda$ is a partition, which can be thought of as a domonant weight of $GL(n)$); he proved the Macdonald positivity conjecture, which says that ${\tilde H}_{\lambda}(q,t,x)$
is a linear combination of Schur functions in $x$ whose coefficients are polynomials in $q$ and $t$ with non-negative integral coefficients. The definition of ${\widetilde H}_{\lambda}(q,t,x)$ appears for example on page 4 of http://math.berkeley.edu/~mhaiman/ftp/nfact/polygraph-jams.pdf
My questions are these:
a) What is the relation between $P_{\lambda}$ and ${\tilde H}_{\lambda}$? It is not clear to me from the definition.
b) Are there positivity statements for $P_{\lambda}$ itself? Or is there a version of the positivity conjecture for any root system?
