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

Abstract

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