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

TL;DR

This paper explores the relationship between minimal representations of orthogonal Lie algebras and the algebra of symmetries of powers of the Laplacian, revealing isomorphisms with differential operator rings and conditions for finite-dimensional modules.

Contribution

It establishes a connection between the algebra of symmetries of Laplacian powers and primitive factors of universal enveloping algebras of orthogonal Lie algebras, including real analogues.

Findings

 isomorphic to a primitive factor ring of U()

Finite-dimensional modules occur when n is even and 2r  n

Results extend to real pseudo-Euclidean spaces

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