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} $\mathcal{S}({\square }^r)$ of powers of the Laplacian $\square$ 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 $\mathcal{S}({\square }^r)\cong \mathcal{D}(X)$ is isomorphic to a primitive factor ring of $U(\mathfrak{g})$. Interestingly, if (and only if) $n$ is even with $2r\ge n$ then both $\mathcal{D}(X)$ and its natural module $\mathcal{O}(X)$ have a finite dimensional factor. These results all have real analogues, with $\square$ replaced by the d'Alembertian on the pseudo-Euclidean space ${\mathbb{R}}^{p,q}$ and $\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$.

This is the final corrected version of the paper. To appear in Compositio Mathematica

References in corpus (1)

• Higher order singletons, partially massless fields and their boundary values in the ambient approach

Cited by in corpus (3)

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• Generalized symmetries of remarkable (1+2)-dimensional Fokker-Planck equation