Note that the printed journal has an unusual error: the citation atop the first page says
   Asian J. Math, Vol. 6, No. 3, pp. 363-420, September 1999
but this information is incorrect.  Please do not cite our article with this incorrect citation.  This is a different journal also from International Press, and the actual publication issue was December 1999.  The error was made at International Press, and will be corrected when the journal is made available online.

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Let J_\mu[X;q,t] be the integral form of the Macdonald polynomial and set H^~_\mu[X;q,t]=t^n(\mu)J_\mu[X/(1-1/t);q,1/t], where n(\mu)=\sum_i (i-1)\mu_i. This paper focusses on the linear operator \nabla defined by setting \nabla H^~_\mu=t^n(\mu)q ^n(\mu') H^~_\mu. This operator occurs naturally in the study of the Garsia-Haiman modules M_\mu. It was originally introduced by the first two authors to give elegant expressions to Frobenius characteristics of intersections of these modules (see [3]). However, it was soon discovered that it plays a powerful and ubiquitous role throughout the theory of Macdonald polynomials. Our main result here is a proof that \nabla acts integrally on symmetric functions. An important corollary of this result is the Schur integrality of the conjectured Frobenius characteristic of the Diagonal Harmonic polynomials [11]. Another curious aspect of \nabla is that it appears to encode a q,t-analogue of Lagrange inversion. In particular, its specialization at t=1 (or q=1) reduces to the q-analogue of Lagrange inversion studied by Andrews [1], Garsia [7] and Gessel [17]. We present here a number of positivity conjectures that have emerged in the few years since \nabla has been discovered. We also prove a number of identities in support of these conjectures and state some of the results that illustrate the power of \nabla within the Theory of Macdonald polynomials.