  We prove Hardy-type inequalities for a fractional Dunkl--Hermite operator
which incidentally give Hardy inequalities for the fractional harmonic
oscillator as well. The idea is to use $h$-harmonic expansions to reduce the
problem in the Dunkl--Hermite context to the Laguerre setting. Then, we push
forward a technique based on a non-local ground representation, initially
developed by R. L. Frank, E. H. Lieb and R. Seiringer in the Euclidean setting,
to get a Hardy inequality for the fractional-type Laguerre operator. The
above-mentioned method is shown to be adaptable to an abstract setting,
whenever there is a "good" spectral theorem and an integral representation for
the fractional operators involved.
