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Equivariant symbol calculus for differential operators acting on forms. (English) Zbl 1035.17034

From the introduction: Let \(D^k_{\lambda}(M)\) be the space of differential operators of order at most \(k\) acting on \(\lambda\)-densities over a smooth manifold \(M\). Let \(D_{\lambda}(M)\) be the filtered union of these spaces. Both \(D^k_{\lambda}(M)\) and \(D_{\lambda}(M)\) are in a natural way modules over the Lie algebra \(\text{Vect}(M)\) of vector fields over \(M\).
In [P. B. A. Lecomte and V. Yu. Ovsienko, Lett. Math. Phys. 49, 173–196 (1999; Zbl 0989.17015)], for \(M={\mathbb R}^n\), the space \(D_{\lambda}(M)\) was considered as a module over a subalgebra of \(\text{Vect} (M)\) made of infinitesimal projective transformations which is isomorphic to \(\text{sl}(n+1, {\mathbb R})\). It was showed that this module is isomorphic to the module of symbols \(\text{Pol}(T^{\ast}M)\) , which is the space of functions on \(T^{\ast}M\) which are polynomial along the fibres. Up to a natural normalization condition, the isomorphism from \(\text{Pol}(T^{\ast}M)\) to \(D_{\lambda}(M)\) is unique and named the projectively equivariant quantization map. Its inverse is the projectively equivariant symbol map. Explicit formulas for these isomorphisms are given in [Lecomte and Ovsienko (loc. cit.)] in terms of a divergence operator.
In the present paper, the authors present a first example of projectively equivariant symbol calculus for differential operators acting on tensor fields. Let \(D_p\) be the space of differential operators mapping \(p\)-forms into functions over a smooth manifold \(M\) and \(D_p^k\) be the subspace of operators with order at most \(k\). The corresponding symbol space \(S_p\) is made up of polynomial functions valued in contravariant antisymmetric tensor fields.
The authors show first that, for \(M={\mathbb R}^n\), there exists a unique (up to normalization) projectively equivariant symbol map from \(D_p\) to \(S_p\). They obtain an explicit formula in terms of the divergence operators and classical invariants of the space of symbols. Next, they use this formula to classify the \(\text{Vect}(M)\)-invariant maps from \(D_p^k\) to \(D_p^l\) over any manifold \(M\) such that \(\dim(M)>1\). Then they recover the results of [N. Poncin, Equivariant operators between spaces of differential operators from differential forms into functions, Preprint, Centre Universitaire de Luxembourg (2002)] for \(p<n-1\) and give the complete list of invariants for \(p=n-1\) and \(p=n\).

MSC:

17B66 Lie algebras of vector fields and related (super) algebras
53D55 Deformation quantization, star products
58J70 Invariance and symmetry properties for PDEs on manifolds

Citations:

Zbl 0989.17015
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