  We study the interplay between the minimal representations of the orthogonal
Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the \emph{algebra
of symmetries} $\mathscr{S}(\Box^r)$ of powers of the Laplacian $\Box$ on
$\mathbb{C}^{n}$. The connection is made through the construction of highest
weight representation of $\mathfrak{g}$ via the ring of differential operators
$\mathcal{D}(X)$ on the singular scheme $X=(F^r=0)\subset \mathbb{C}^n$, where
$F$ is the sum of squares.
  In particular we prove that $ \mathscr{S}(\Box^r)\cong \mathcal{D}(X)$ is
isomorphic to a primitive factor ring of $U(\mathfrak{g})$. Interestingly, if
(and only if) $n$ is even with $2r\geq n$ then both $\mathcal{D}(X)$ and its
natural module $\mathcal{O}(X)$ have a finite dimensional factor.
  These results all have real analogues, with $\Box$ replaced by the
d'Alembertian on the pseudo-Euclidean space $\mathbb{R}^{p,q}$ and
$\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$.
