Let J^[X; q, t] be the integral form of the Macdonald polynomial and set Hp [X; q, t] _ t^^ J^
X/O.—1/t); g, l/t], where 71 (11)= X!;(*—l)/^-This paper focusses on the linear operator V
defined by setting VH^= tn^ qn^^ H^. This operator occurs naturally in the study of the Garsia-
Haiman modules M^. 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 …

Let Jµ [X; q, t] be the integral form of the Macdonald polynomial and set Hµ [X; q, t]= tn (µ) Jµ
[X/(1− 1/t); q, 1/t], where n (µ)=∑ i (i− 1) µi. This paper focusses on the linear operator∇
defined by setting∇ Hµ= tn (µ) qn (µ′) Hµ. This operator occurs naturally in the study of the
Garsia-Haiman modules Mµ. 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 …