  Originally motivated by connections to integrable systems, two natural
subalgebras of the rational Cherednik algebra have been considered in the
literature. The first is the subalgebra generated by all degree zero elements
and the second is the Dunkl angular momentum subalgebra.
  In this article, we study the ring-theoretic and homological properties of
these algebras. Our approach is to realise them as rings of invariants under
the action of certain reductive subgroups of $\mathrm{SL_2}$. This allows us to
describe their centres. Moreover, we show that they are Auslander-Gorenstein
and Cohen-Macaulay and, at $t = 0$, give rise to prime PI-algebras whose
PI-degree we compute.
  Since the degree zero subalgebra can be realized as the ring of invariants
for the maximal torus $\mathrm{T} \subset \mathrm{SL_2}$ and the action of this
torus on the rational Cherednik algebra is Hamiltonian, we also consider its
(quantum) Hamiltonian reduction with respect to $\mathrm{T}$. At $t = 1$, this
is a filtered quantization of the quotient of the minimal nilpotent orbit
closure $\overline{\mathcal{O}}_{\min}$ in $\mathfrak{gl}(n)$ by the reflection
group $W$. At $t = 0$, we get a graded Poisson deformation of the symplectic
singularity $\overline{\mathcal{O}}_{\min}/W$.
