Higher symmetries of powers of the Laplacian and rings of differential operators

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 \(\mathfrak{so}(p+1,q+1)\).