  We use a Chern Simons Landau-Ginzburg (CSLG) framework related to hierarchies
of composite bosons to describe 2D harmonically trapped fast rotating Bose
gases in Fractional Quantum Hall Effect (FQHE) states. The predicted values for
$\nu$ (ratio of particle to vortex numbers) are $\nu$$=$${{p}\over{q}}$ ($p$,
$q$ are any integers) with even product $pq$, including numerically favored
values previously found and predicting a richer set of values. We show that
those values can be understood from a bosonic analog of the law of the
corresponding states relevant to the electronic FQHE. A tentative global phase
diagram for the bosonic system for $\nu$$<$1 is also proposed.
