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Rings of differential operators on classical rings of invariants. (English) Zbl 0691.16019
Mem. Am. Math. Soc. 412, 117 p. (1989).
The algebra $${\mathcal D}(R)$$ of all differential operators on a commutative affine $${\mathbb{C}}$$-algebra R is known to be very well-behaved when R is the coordinate ring $${\mathcal O}(X)$$ of a nonsingular affine variety X - in particular, $${\mathcal D}(R)$$ is then a simple noetherian domain and an affine $${\mathbb{C}}$$-algebra of finite Gelfand-Kirillov dimension (see e.g. [J.-E. Björk, Rings of Differential Operators (1979; Zbl 0499.13009)] or [J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings (1987; Zbl 0644.16008)]). However, in the singular case these nice properties are often lost (see e.g. [I. N. Bernstein, I. M. Gel’fand and S. I. Gel’fand, Usp. Mat. Nauk 27, 185-190 (1972; Zbl 0253.58009)]), although $${\mathcal D}({\mathcal O}(X))$$ is always an affine noetherian $${\mathbb{C}}$$-algebra when X is an affine curve (see [J. L. Muhasky, Trans. Am. Math. Soc. 307, 705-723 (1988; Zbl 0668.16007)], and [S. P. Smith and J. T. Stafford, Proc. Lond. Math. Soc., III. Ser. 56, 229-259 (1988; Zbl 0672.14017)]). Moreover, $${\mathcal D}(R)$$ is known to be nice when R is the ring of invariants for a finite group of automorphisms of $${\mathcal O}({\mathbb{C}}^ n)$$ [T. Levasseur, in Lect. Notes Math. 867, 157-173 (1981; Zbl 0507.14012)], when R is the ring of invariants for the diagonal action of a torus on a polynomial ring [I. M. Musson, Trans. Am. Math. Soc. 303, 805-827 (1987; Zbl 0628.13019)], and when $$R={\mathcal O}(X)$$ for X a quadratic cone in $${\mathbb{C}}^ n$$ [T. Levasseur, S. P. Smith and J. T. Stafford, J. Algebra 116, 480-501 (1988; Zbl 0656.17009)].
In this paper, the authors prove that for several classical rings of invariants, $${\mathcal D}(R)$$ is a simple noetherian ring. They analyze the following four situations, taking R to be the ring of invariants for the standard action of the given group on the coordinate ring of the given variety: (A) GL(k,$${\mathbb{C}})$$ acting on $$M_{p,k}({\mathbb{C}})\times M_{k,q}({\mathbb{C}})$$, (B) O(k,$${\mathbb{C}})$$ acting on $$M_{k,n}({\mathbb{C}})$$, (C) Sp(2k,$${\mathbb{C}})$$ acting on $$M_{2k,n}({\mathbb{C}})$$, and (D) SO(k,$${\mathbb{C}})$$ acting on $$M_{n,k}({\mathbb{C}})$$. In each of the first three cases, R is isomorphic to the coordinate ring of an affine variety: (A) the variety of $$p\times q$$ matrices of rank at most k (which is singular if $$p\geq q>k)$$, (B) the variety of symmetric $$n\times n$$ matrices of rank at most k (singular if $$k<n)$$, (C) the variety of antisymmetric $$n\times n$$ matrices of rank at most 2k (singular if 2k$$\leq n-2).$$
In cases (A), (B), (C), $${\mathcal D}(R)$$ is shown to be a simple noetherian domain by proving that it is isomorphic to a suitable factor ring of an enveloping algebra U($${\mathfrak g})$$, where (A) $${\mathfrak g}={\mathfrak gl}(p+q)$$, (B) $${\mathfrak g}={\mathfrak sp}(2n)$$, (C) $${\mathfrak g}={\mathfrak so}(2n)$$. In case (D) (with $$k\leq n)$$, $${\mathcal D}(R)$$ is shown to be a simple noetherian ring finitely generated as a module over a simple factor ring of U($${\mathfrak sp}(2n)).$$
To support the line of approach, the paper contains much work on enveloping algebras of semisimple Lie algebras and on rings of $${\mathfrak k}$$-finite vectors. An appendix contains a proof for (a generalization of) Gabber’s Lemma: If $${\mathfrak g}$$ is a finite-dimensional Lie algebra, M is a finitely generated s-homogeneous (in the sense of GK-dimension) left U($${\mathfrak g})$$-module, and E is an essential extension of M, then the set of finitely generated U($${\mathfrak g})$$-modules $$M'$$ such that $$M\subseteq M'\subseteq E$$ and $$GK\dim (M'/M)\leq s-2$$ has a unique maximal element.
Reviewer: K.R.Goodearl

##### MSC:
 16P40 Noetherian rings and modules (associative rings and algebras) 14L30 Group actions on varieties or schemes (quotients) 17B35 Universal enveloping (super)algebras 16D30 Infinite-dimensional simple rings (except as in 16Kxx) 14M12 Determinantal varieties 16W20 Automorphisms and endomorphisms 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 17B20 Simple, semisimple, reductive (super)algebras 13N05 Modules of differentials 14L24 Geometric invariant theory 32C38 Sheaves of differential operators and their modules, $$D$$-modules
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