Relèvements d’opérateurs différentiels sur les anneaux d’invariants. (Liftings of differential operators over rings of invariants).

*(French)*Zbl 0733.16009
Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honour of J. Dixmier, Paris/Fr. 1989, Prog. Math. 92, 449-470 (1990).

[For the entire collection see Zbl 0719.00018.]

Let \(G\to GL(V)\) be a finite dimensional representation of the reductive group G, and let V//G denote the affine variety whose ring of regular functions \({\mathcal O}(V//G)\) equals \({\mathcal O}(V)^ G\). The group G operates naturally on the ring \({\mathcal D}(V)\) of differential operators on V and maps each of the \({\mathcal O}(V)\)-modules \({\mathcal D}er^ m(V)\) of differential operators of order \(\leq m\) with zero constant term into itself. Then (V,G) is said to have the property of lifting to order m if the restriction morphism \(\phi\) : \({\mathcal D}er^ m(V)^ G\to {\mathcal D}er^ m(V//G)\) is surjective. If this holds for all m, then (V,G) is said to satisfy the lifting property, and this will imply the surjectivity of \(\phi\) : \({\mathcal D}(V)^ G\to {\mathcal D}(V//G)\). The author provides a sufficient condition for this to happen. In order to formulate his main result, let \({\mathcal A}(V//G)\) denote the ideal of \({\mathcal O}(V)^ G\) defining the complement of the principal stratum of V//G, let gr \(\phi\) :gr \({\mathcal D}(V)^ G\to gr {\mathcal D}(V//G)\) be the graded morphism associated with \(\phi\), and set \(R=gr {\mathcal D}(V)^ G/\ker (gr \phi)\). Then, if R is the intersection of all localizations of its height one primes, and if \({\mathcal A}(V//G)\) generates an ideal of height \(\geq 2\) in R, then (V,G) has the lifting property. Among other applications, this is used to treat the cases when G is finite or the group of nonzero complex numbers, since in both instances the structure of R is fairly well known.

Let \(G\to GL(V)\) be a finite dimensional representation of the reductive group G, and let V//G denote the affine variety whose ring of regular functions \({\mathcal O}(V//G)\) equals \({\mathcal O}(V)^ G\). The group G operates naturally on the ring \({\mathcal D}(V)\) of differential operators on V and maps each of the \({\mathcal O}(V)\)-modules \({\mathcal D}er^ m(V)\) of differential operators of order \(\leq m\) with zero constant term into itself. Then (V,G) is said to have the property of lifting to order m if the restriction morphism \(\phi\) : \({\mathcal D}er^ m(V)^ G\to {\mathcal D}er^ m(V//G)\) is surjective. If this holds for all m, then (V,G) is said to satisfy the lifting property, and this will imply the surjectivity of \(\phi\) : \({\mathcal D}(V)^ G\to {\mathcal D}(V//G)\). The author provides a sufficient condition for this to happen. In order to formulate his main result, let \({\mathcal A}(V//G)\) denote the ideal of \({\mathcal O}(V)^ G\) defining the complement of the principal stratum of V//G, let gr \(\phi\) :gr \({\mathcal D}(V)^ G\to gr {\mathcal D}(V//G)\) be the graded morphism associated with \(\phi\), and set \(R=gr {\mathcal D}(V)^ G/\ker (gr \phi)\). Then, if R is the intersection of all localizations of its height one primes, and if \({\mathcal A}(V//G)\) generates an ideal of height \(\geq 2\) in R, then (V,G) has the lifting property. Among other applications, this is used to treat the cases when G is finite or the group of nonzero complex numbers, since in both instances the structure of R is fairly well known.

Reviewer: G.Krause (Winnipeg)

##### MSC:

16S32 | Rings of differential operators (associative algebraic aspects) |

16W50 | Graded rings and modules (associative rings and algebras) |

20G05 | Representation theory for linear algebraic groups |

14A10 | Varieties and morphisms |

14L30 | Group actions on varieties or schemes (quotients) |

13N10 | Commutative rings of differential operators and their modules |