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