We study the interplay between the minimal representations of the orthogonal Lie algebra
$\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$
and the 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 a highest-weight representation of
$\mathfrak{g}$
via the ring of differential operators
${\mathcal{D}}(X)$
on the singular scheme
$X=(\mathtt{F}^{r}=0)\subset \mathbb{C}^{n}$
, for
$\mathtt{F}=\sum _{j=1}^{n}X_{i}^{2}\in \mathbb{C}[X_{1},\ldots ,X_{n}]$
. In particular, we prove that
$U(\mathfrak{g})/K_{r}\cong \mathscr{S}(\Box ^{r})\cong {\mathcal{D}}(X)$
for a certain primitive ideal
$K_{r}$
. Interestingly, if (and only if)
$n$
is even with
$r\geqslant n/2$
, then both
$\mathscr{S}(\Box ^{r})$
and its natural module
${\mathcal{A}}=\mathbb{C}[\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n},\ldots ,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n}]/(\Box ^{r})$
have a finite-dimensional factor. The same holds for the
${\mathcal{D}}(X)$
-module
${\mathcal{O}}(X)$
. We also study higher-dimensional analogues
$M_{r}=\{x\in A:\Box ^{r}(x)=0\}$
of the module of harmonic elements in
$A=\mathbb{C}[X_{1},\ldots ,X_{n}]$
and of the space of ‘harmonic densities’. In both cases we obtain a minimal
$\mathfrak{g}$
-representation that is closely related to the
$\mathfrak{g}$
-modules
${\mathcal{O}}(X)$
and
${\mathcal{A}}$
. Essentially all these results have real analogues, with the Laplacian replaced by the d’Alembertian
$\Box _{p}$
on the pseudo-Euclidean space
$\mathbb{R}^{p,q}$
and with
$\mathfrak{g}$
replaced by the real Lie algebra
$\mathfrak{so}(p+1,q+1)$
.