Rings of differential operators on classical rings of invariants
Rings of differential operators on classical rings of invariants
复制标题
经典不变量环上的微分算子环
DOI：
10.1090/memo/0412
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发表时间：
1989
影响因子：
1.9
通讯作者：
J. T. Stafford
中科院分区：
文献类型：
--
作者：
T. Levasseur;J. T. Stafford
我们认为环的微分算子的经典环的不变量，在意义上的Weyl [我们]。因此，设X k是下列各种之一：（情况A）所有秩≤ k的复p × q矩阵;（情况B）所有秩≤ k的对称n × n矩阵;（情况C）所有秩≤ 2k的反对称n × n矩阵。本文证明了定义在正则函数环O（Xk）上的微分算子环D（Xk）= D（O（Xk））是一个简单的、非线性生成的Noether整环.进一步假设X k是奇异的（这是唯一有趣的情况）。证明了D（Xk）是包络代数U（g）的因子环.这里g = gl（p+ q），sp（2n）和so（2n）分别在情况A，B和C中。最后，令SO（k）以自然的方式作用在kn元复多项式环C[X]上。然后证明了D（C[X]）具有相似的令人愉快的结构，并且至少对k ≤ n，D（C [X]）是一个n-生成的U（sp（2n））-模. 1980年数学学科分类（1985年修订）13 N 05、14 L30、14 M12、17 B20、17 B35、16 A19、16 A33。
We consider rings of differential operators over the classical rings of invariants, in the sense of Weyl [We]. Thus, let X k be one of the following varieties: (CASE A) all complex p × q matrices of rank ≤ k ; (CASE B) all symmetric n × n matrices of rank ≤ k ; (CASE C) all antisymmetric n × n matrices of rank ≤ 2k . We prove that the ring of differential operators D(X k) = D(O(X k)) defined on the ring of regular functions O(X k) is a simple, finitely generated, Noetherian domain. Assume further that X k is singular (which is the only interesting case). Then the result is proved by showing that D(X k) is a factor ring of an enveloping algebra U(g) . Here g = gl(p+ q) , sp(2n) and so(2n) in the Cases A, B and C, respectively. Finally, let SO(k) act in the natural way on the ring C[X] of complex polynomials in kn variables. Then we prove that D(C[X]) has a similarly pleasant structure and, at least for k ≤ n , is a finitely generated U(sp(2n)) -module. 1980 Mathematics Subject Classification (1985 Revision) 13N05, 14L30, 14M12, 17B20, 17B35, 16A19, 16A33.