Representations of infinite-dimensional Lie algebras and related topics

主要研究人员：Edward Frenkel提案编号：0303529机构：加州大学伯克利分校摘要：无限维李代数及相关主题的表示主要研究内容包括：仿射Kac-Moody代数的局部朗兰兹对应；顶点代数与量子群；曲线上丛的模堆上微分算子层的上同调。在传统的局部朗兰兹通信中，人们希望用局部域的伽罗瓦群和朗兰兹对偶群来描述局部非阿基米德域上的约化群的光滑表示，例如有限域上的形式洛朗级数的域。当我们用复数域代替有限域时，自然会得到循环群和循环李代数以及后者的中心扩张，即仿射Kac-Moody代数。主要研究人员希望用与对偶群相关的几何数据来描述仿射Kac-Moody代数上的Harish-Chandra模的范畴。更具体地说，主要研究人员试图证明仿射代数上某一模范畴的派生范畴等价于朗兰兹对偶李代数的幂零元的Springer纤维上拟凝聚层范畴的派生范畴。此外，主要研究人员还提出了任意单李代数的W-代数到顶点代数的扩张，它通过顶点代数自同构来承载对偶群的作用。他试图证明这个顶点代数的表示范畴等价于与Langland对偶李代数相关的量子群的表示范畴。最后，他打算计算仿射Kac-Moody代数上的真空表示的上同调，并将其与代数曲线上的丛的模叠上的微分算子束的上同调联系起来。在过去的30年里，在将数论、自同构表示和代数几何中看似不相关的结构联系在一起的朗兰兹程序的发展中做了大量的工作。首席研究人员预计，在仿射Kac-Moody代数和更一般的顶点代数的新背景下揭示朗兰兹对偶模式将显著增强我们对朗兰兹对应的理解，这一对应至今仍是一个谜。特别是，朗兰兹对应在这种情况下被提升到类别的水平，因此人们可以看到比以前可能的更精细的结构。希望这一建议的跨学科性质将有助于通过将仿射Kac-Moody代数和顶点代数的表示理论与朗兰兹计划相联系来促进对它们的发现和理解，同时通过引入几何的新见解来刺激朗兰兹计划的发展。

Principal Investigator: Edward Frenkel Proposal Number: 0303529Institution: University of California-BerkeleyAbstract: Representations of infinite-dimensional Lie algebras and related topicsThe principal investigator proposes to conduct research in the following areas: local Langlands correspondence for affine Kac--Moody algebras; vertex algebras and quantum groups; cohomology of the sheaves of differential operators on the moduli stacks of bundles on curves. In the traditional local Langlands correspondence one wishes to describe smooth representations of a reductive group over a local non-archimedian field, such as the field of formal Laurent power series over a finite field, in terms of the Galois group of the local field and the Langlands dual group. When we replace the finite field by the complex field, we are naturally led to loop groups and loop Lie algebras and the central extensions of the latter, i.e., the affine Kac-Moody algebras. The principal investigator wishes to describe the categories of Harish-Chandra modules over an affine Kac-Moody algebra in terms of geometric data associated to the dual group. More specifically, the principal investigator intends to prove that the derived category of a certain category of modules over an affine algebra is equivalent to the derived category of the category of quasicoherent sheaves on the Springer fiber of a nilpotent element of the Langlands dual Lie algebra. In addition, the principal investigator proposes the construction of an extension of the W-algebra associated to an arbitrary simple Lie algebra to a vertex algebra, which carries an action of the dual group by vertex algebra automorphisms. He intends to prove that the category of representations of this vertex algebra is equivalent to the category of representations of a quantum group associated to the Langland dual Lie algebra. Finally, he intends to compute the cohomology of the vacuum representation over an affine Kac-Moody algebra and to relate it to the cohomology of the sheaf of differential operators on the moduli stack of bundles on an algebraic curve.A lot of effort has been made over the last thirty years in the development of the Langlands Program which ties together seemingly unrelated structures in number theory, automorphic representations and algebraic geometry. The principal investigator expects that uncovering the Langlands duality patterns in the new setting of affine Kac-Moody algebras and more general vertex algebra will significantly enhance our understanding of the Langlands correspondence, which to this day remains a mystery. In particular, the Langlands correspondence is elevated in this case to the level of categories and therefore one can see a much finer structure than was previously possible. It is hoped that the interdisciplinary nature of this proposal will serve to advance discovery and understanding of representation theory of affine Kac-Moody algebras and vertex algebras by relating them to the Langlands Program, and at the same time will stimulate the development of the Langlands Program by bringing in new insights from geometry.