Applications of Noetherian Ring Theory

斯塔福德，9801148首先，考虑一个约化复李代数g，它具有伴随群G，Cartan子代数h和Weyl群W。 在这种情况下，Levasseur和斯塔福德教授证明了g上G-不变微分算子到h上W-不变微分算子的Harish-Chandra同态是满射的，其核由tau（g）生成，其中tau表示G在g上的伴随作用的微分。 这有许多应用的理论不变的本征分布，代表理论的g和推广的施普林格对应。 这些结果的推广及其应用G-不变微分算子更一般的G-表示V将进行研究。 特别是，教授斯塔福德打算研究的不变微分算子的G和对称空间。 这些结果也与长期存在的问题有关的交换品种的g（例如，它是否是正常的），其中斯塔福德教授将进一步调查。教授斯塔福德工程在一般领域的代数，特别是在代数的研究乘法是不交换的。 这种代数出现在数学和科学的许多领域;例如，量子群理论（最终来自量子物理学）和微分方程理论。 “非交换诺特代数”的技术可以应用于研究这些不同领域的问题。

Stafford, 9801148Initially, one considers a reductive, complex Lie algebra g with adjoint group G, Cartan subalgebra h and Weyl group W. In this case, Professors Levasseur and Stafford have shown that the Harish-Chandra homomorphism from G-invariant differential operators on g to W-invariant differential operators on h is surjective with kernel generated by tau(g), where tau denotes the differential of of the adjoint action of G on g. This has numerous applications to the theory of invariant eigendistributions, to the representation theory of g and to generalizations of the Springer correspondence. Generalizations of these results and their applications to the G-invariant differential operators for more general G-representations V will be studied. In particular, Professor Stafford intends to study the invariant differential operators on G and on symmetric spaces. These results are also related to longstanding questions about the commuting variety of g (for example, whether it is normal), which Professor Stafford will investigate further.Professor Stafford works in the general area of algebra, specifically in the study of algebras for which multiplication is not commutative. Such algebras arise in many areas of mathematics and science; for example, in the theory of quantum groups (which, ultimately, come from quantum physics), and in the theory of differential equations. The techniques of ``noncommutative Noetherian algebra'' can be applied to study questions in these diverse areas.