Applications of Noetherian Ring Theory

首先，考虑一个复李代数g，它有伴随群g、Cartan子代数h和Weyl群w。在这种情况下，Levasseur和Stafford教授已经证明了g上的g不变微分算子到h上的w不变微分算子的harsh - chandra同态是由tau(g)生成核的满射。其中tau表示G对G的伴随作用的微分。这在不变特征分布理论，G的表示理论和施普林格对应的推广中有许多应用。本文将研究这些结果的推广及其在更一般的g -表示V的g不变微分算子中的应用。特别是，Stafford教授打算研究G和对称空间上的不变微分算子。这些结果也与长期存在的关于g的通勤变化的问题有关（例如，它是否正常），斯塔福德教授将进一步研究这个问题。斯塔福德教授研究代数的一般领域，特别是研究乘法不可交换的代数。这样的代数出现在数学和科学的许多领域；例如，在量子群理论（它最终来自量子物理学）和微分方程理论中。“非交换诺埃尔代数”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.