Categories of Lie algebra representations, primitive ideals, and geometry of homogeneous ind-spaces

这是李表示理论和相关几何领域的一个广泛的建议。它建立在作者的计划在这一领域的最新进展可追溯到20年。该提案的一个统一特征是研究作者及其合作者部分介绍的各种类别的表征。我们的第一个研究课题是半单复李代数g和任意sl（2）-子代数的（g，sl（2））-模范畴。我们计划研究几年前提出的一个重大猜想。我们还希望在研究范畴O的某些类似物g = sl（infty），o（infty），sp（infty）方面取得重要进展。后一个主题是整个申请的中心，因为它是博士学位。此外，我们正在启动一个计划，研究更一般的对角李代数的张量表示的张量类别，以及o（infty）和sp（infty）的张量表示类别的某些扩展。这是由于我们最近成功地使用sl（infty）的张量表示范畴来分类玻色子-费米子对应。另一个研究方向是U（o（infty））和U（sp（infty））的本原理想理论，我们建议通过使用最近为U（sl（infty））开发的方法并通过U（o（infty））和U（sp（infty））中理想格的一个引人注目的新同构来推进。最后，我们研究了G=SL（infty），O（infty），SP（infty）的齐性ind空间G/P的自同构群，其中P是G的分裂抛物子群，并对G的一般（非分裂）抛物子群P子集的齐性ind空间G/P展开了研究.

This is a broad proposal in the field of Lie representation theory and the related geometry. It builds upon recent advances in the author's program in this field going back for up to 20 years. A unifying feature of the proposal is the study of various categories of representations introduced in part by the author and his collaborators. Our first topic of study is the category of (g, sl(2))-modules for a semisimple complex Lie algebra g and any sl(2)-subalgebra. We plan to work on a major conjecture which we made a few years ago. We hope also to make essential progress in the study of certain analogues of the category O for g = sl(infty), o(infty), sp(infty). This latter topic is center to the entire application as it is the Ph. D. project which we propose to be funded.In addition we are launching a program to study tensor categories of tensor representations of more general diagonal Lie algebras, as well as certain extensions of the categories of tensor representations of o(infty) and sp(infty). This is motivated by our recent success in using the category of tensor representations of sl(infty) to categorify the boson-fermion correspondence. Another direction of study is the theory of primitive ideals of U(o(infty)) and U(sp(infty)), which we propose to advance by using methods developed recently for U(sl(infty)) and via a striking new isomorphism of the lattices of ideals in U(o(infty)) and U(sp(infty)). Finally, we propose to study the automorphism groups of the homogeneous ind-spaces G/P for G=SL(infty), O(infty), SP(infty) where P is a splitting parabolic subgroup of G, and to launch a study of the homogeneous ind-spaces G/P for general (nonsplitting) parabolic subgroups P subset of G.