Applications of Noetherian Ring Theory

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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.