  For an irreducible complex reflection group $W$ of rank $n$ containing $N$
reflections, we put $g=2N/n$ and construct a $(g+1)^n$-dimensional irreducible
representation of the Cherednik algebra which is (as a vector space) a quotient
of the diagonal coinvariant ring of $W$. We propose that this representation of
the Cherednik algebra is the single largest representation bearing this
relationship to the diagonal coinvariant ring, and that further corrections to
this estimate of the dimension of the diagonal coinvariant ring by $(g+1)^n$
should be orders of magnitude smaller. A crucial ingredient in the construction
is the existence of a dot action of a certain product of symmetric groups
acting on the parameter space of the rational Cherednik algebra and leaving
invariant both the finite Hecke algebra and the spherical subalgebra; this fact
is a consequence of ideas of Berest-Chalykh on the relationship between the
Cherednik algebra and quasi-invariants.
