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The aim of this paper is to describe structural properties of spaces ofdiagonal rectangular harmonic polynomials in several sets (say $k$) of $n$variables, both as $GL_k$-modules and $S_n$-modules. We construct explicit suchmodules associated to any hook shape partitions. For the two sets of variablescase, we conjecture that the associated graded Frobenius characteristiccorresponds to the effect of the operator Nabla on the correspondinghook-indexed Schur function, up to a usual renormalization. We prove identitiesthat give indirect support to this conjecture, and show that its restriction toone set of variables holds. We further give indications on how the several setscontext gives a better understanding of questions regarding the structures ofthese modules and the links between them.

(GL_k x S_n)-modules and nabla of hook-indexed Schur functions