TITLE: Macdonald polynomials and Macdonald positivity
QUESTION [7 upvotes]: 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?

REPLY [6 votes]: As far as I know there is no positivity statement for $P_\lambda$ as a linear combination of characters (which are the analog of Schur functions in general type).  There is a very general statement that $P_\lambda$ expands positively as a sum of monomial symmetric functions; this is proved in a root-system uniform way by Ram and Yip in
http://arxiv.org/abs/0803.1146
The proof they give is for the "constant parameters" situation, but the technique works in general.  There is some inefficiency in their formula (in type A it can be "compressed" to a formula of Haglund, Haiman and Loehr in http://arxiv.org/abs/math/0409538) which is the subject of current work. 
Presumably character-positivity of $P_\lambda$ actually fails already in type A, and in general type there is no notion of plethystic substitution to save us!