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

Part of: Partial differential equations on manifolds; differential operators Lie algebras and Lie superalgebras Rings and algebras arising under various constructions Classical field theories Classical differential geometry

Published online by Cambridge University Press:  08 March 2017

T. Levasseur  and
J. T. Stafford
T. Levasseur
Affiliation:
Laboratoire de Mathématiques de Bretagne Atlantique, CNRS - UMR6205, Université de Brest, 29238 Brest cedex 3, France email Thierry.Levasseur@univ-brest.fr
J. T. Stafford
Affiliation:
School of Mathematics, Alan Turing Building, The University of Manchester, Oxford Road, Manchester M13 9PL, UK email Toby.Stafford@manchester.ac.uk
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Abstract

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

Keywords

higher symmetriesLaplacianrings of differential operatorsharmonic polynomialsprimitive idealsscalar singletonsconformal geometry

MSC classification

Primary: 16S32: Rings of differential operators 58J70: Invariance and symmetry properties 17B08: Coadjoint orbits; nilpotent varieties 17B81: Applications to physics 53A30: Conformal differential geometry 70S10: Symmetries and conservation laws
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 4 , April 2017 , pp. 678 - 716
Copyright
© The Authors 2017 

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