Invariants under tori of rings of differential operators and related topics.

*(English)*Zbl 0928.16019
Mem. Am. Math. Soc. 650, 85 p. (1998).

If \(G\) is a reductive algebraic group acting rationally on a smooth affine variety \(X\) then the ring of invariant differential operators \(D(X)^G\) is of considerable interest. It is generally believed that this ring has properties very similar to those of enveloping algebras of semisimple Lie algebras. The paper is concerned with the case when \(G\) is a torus and \(X=k^r\times(k^*)^s\). Denote \(n=r+s\), then \(D(X)=k[x_1,\dots,x_r,x^{\pm 1}_{r+1},\dots,x^{\pm 1}_n,\partial_1,\dots,\partial_n]\), where \(\partial_i=\tfrac\partial{\partial_i}\).

The paper gives a precise description of the primitive ideals of \(D(X)^G\), and the minimal primitive quotients of \(D(X)^G\) are of the following form. Let \({\mathfrak g}=\text{Lie}(G)\), then each character \(\chi\in{\mathfrak g}^*\) gives rise to a corresponding central quotient \(B^\chi=D(X)^G/({\mathfrak g}-\chi({\mathfrak g}))\), where \({\mathfrak g}-\chi({\mathfrak g})\) is the set of all \(v-\chi(v)\) with \(v\in{\mathfrak g}\). The paper gives a fairly exhaustive description of the properties of \(B^\chi\). Namely, 1) when \(B^\chi\) is simple; 2) when \(B^\chi\) has finite global dimension; 3) the various classical dimensions associated to \(B^\chi\): Krull-dimension, GK-dimension, injective dimension and homological dimension; 4) the lattice of primitive ideals in \(B^\chi\) and the corresponding primitive quotients; 5) the category of finite dimensional representations of \(B^\chi\). Also the authors prove that if \(G\) is a torus acting rationally on a smooth affine variety then \(D(X//G)\) is a simple ring.

The paper gives a precise description of the primitive ideals of \(D(X)^G\), and the minimal primitive quotients of \(D(X)^G\) are of the following form. Let \({\mathfrak g}=\text{Lie}(G)\), then each character \(\chi\in{\mathfrak g}^*\) gives rise to a corresponding central quotient \(B^\chi=D(X)^G/({\mathfrak g}-\chi({\mathfrak g}))\), where \({\mathfrak g}-\chi({\mathfrak g})\) is the set of all \(v-\chi(v)\) with \(v\in{\mathfrak g}\). The paper gives a fairly exhaustive description of the properties of \(B^\chi\). Namely, 1) when \(B^\chi\) is simple; 2) when \(B^\chi\) has finite global dimension; 3) the various classical dimensions associated to \(B^\chi\): Krull-dimension, GK-dimension, injective dimension and homological dimension; 4) the lattice of primitive ideals in \(B^\chi\) and the corresponding primitive quotients; 5) the category of finite dimensional representations of \(B^\chi\). Also the authors prove that if \(G\) is a torus acting rationally on a smooth affine variety then \(D(X//G)\) is a simple ring.

Reviewer: Victor Petrogradsky (Ulyanovsk)

##### MSC:

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

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

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16E10 | Homological dimension in associative algebras |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |

20G05 | Representation theory for linear algebraic groups |