GEOMETRY OF FLAG VARIETIES AND REPRESENTATION THEORY

Shrawan Kumar的NSF项目摘要Shrawan Kumar打算继续在“李理论和几何”的一般领域工作。他已经提出了五个项目，需要不同和广泛的数学工具，从代数几何，经典和量子表示理论，以及拓扑学。简单地说，这些项目如下：为了试图理解Kostka-Macdonald系数，Garsia和Haiman提出了一个引人注目的猜想，称为n！猜想。这表明，2n个变量的某一多项式空间的维度正好是n！猜想的正确性将意味着Kostka-Macdonald多项式的系数是非负整数，这仍然是一个具有挑战性的开放问题。第一个项目的目的是(与来自奥胡斯的J·汤姆森教授一起)利用希尔伯特方案的几何来证明这一猜想。第二个项目涉及证明某些齐次紧复流形之间的非常数全纯映射的不存在性。在这个方向上，库马尔已经有了一个精确的猜测。Lusztig定义了量子化包络代数在单位根处的某种Frobenius态射，以及定义在所涉及的代数的正部分上的这种态射的某种‘分裂’。另一方面，定义在正特征域上的任何代数簇都有Frobenius态射。Mehta-Ramanathan引入了这类簇的Frobenius分裂的概念，并证明了与任何半单代数群相关的旗簇都允许Frobenius分裂。这就引出了一些关于旗帜簇几何的重要结果。现在Kumar-Littelmann已经获得了Frobenius态射和Frobenius分裂这两个(量子和几何)概念之间非常精确的联系，第三个项目涉及完成这项正在进行的工作。第四个项目涉及证明Kumar自己关于与具有齐次向量丛中系数的仿射Kac-Moody群相关的‘粗’旗簇的上同调的一个猜想。这一猜想的“直接”证明将导致与向量丛的模和Verlinde公式有关的几个重要结果的一致证明。第五个项目是一个长期持续的项目，预计将于今年夏末完成。它涉及到写一本书《Kac-Moody群，他们的旗帜变种和表示理论》，提交给Springer-Verlag数学系列的研究生课本。为了向一般科学界解释这些，库马尔项目背后的主要潜在主题是利用数学和数学物理中出现的问题的对称性，或者更专业地说，不变性。有几种自然现象表现出对称性，从水晶到我们生活的地球表面(后者当然在任何自转下都是不变的)。现在考虑一个数学或物理问题，比如研究一个物体的几何性质(从细胞到宇宙)，或者在数学和其他科学中解决一些在不同情况下出现的复杂方程。一种有效的方法是研究所考虑问题的所有对称性的集合(称为群)，并利用群的性质来阐明原始问题。自19世纪以来，这种方法已经成功地应用于各种问题，从而产生了一些开创性的工作。

ABSTRACT OF THE NSF PROJECT OF SHRAWAN KUMARShrawan Kumar intends to continue work in the general area of 'Lie Theory and Geometry'. He has proposed to work on five projects requiring diverse and extensive mathematical tools from Algebraic Geometry, Classical and Quantum Representation Theory, and Topology. Briefly described, these projects are as follows: In an attempt to understand the Kostka-Macdonald coefficients, Garsia and Haiman proposed a remarkable conjecture, known as the n! conjecture. This asserts that the dimension of a certain space of polynomials in 2n variables is exactly n!. The validity of the conjecture will imply that the coefficients of the Kostka-Macdonald polynomials are non-negative integers which remains an open challenging problem. The first project aims at proving this conjecture (jointly with Prof. J. Thomsen from Aarhus) by using the geometry of Hilbert schemes. The second project involves proving the non-existence of non-constant holomorphic maps between certain homogeneous compact complex manifolds. Kumar already has a precise conjecture in this direction. Lusztig has defined a certain Frobenius morphism for quantized enveloping algebras at roots of unity and also a certain 'splitting' of this morphism defined on the positive parts of the algebras involved. On the other hand, any algebraic variety defined over a field of positive characteristic admits a Frobenius morphism. Mehta-Ramanathan introduced the concept of Frobenius splitting of such a variety and showed that the flag varieties associated to any semisimple algebraic group do admit Frobenius splitting. This leads to some important results on the geometry of flag varieties. Now Kumar-Littelmann have obtained a very precise connection between these two (quantum and geometric) notions of Frobenius morphisms and Frobenius splittings and the third project involves completing this ongoing work. The fourth project involves proving a conjecture given by Kumar himself on the cohomology of 'thick' flag varieties associated to affine Kac-Moody groups with coefficients in homogeneous vector bundles. A 'direct' proof of this conjecture will lead to uniform proofs of several important results related to the Moduli of Vector Bundles and Verlinde Formula. The fifth is a long continuing project which is due for completion by the end of this summer. It involves writing the book "Kac-Moody Groups, their Flag Varieties and Representation Theory," to be submitted to the Graduate Texts in Mathematics series of Springer-Verlag.To explain these to a general scientific community, the main underlying theme behind Kumar's projects is to exploit 'symmetry' or in more technical terms 'invariance' in problems arising in mathematics and mathematical physics. Several natural phenomena exhibit symmetry, from crystals to the surface of the earth we live on (where the latter of course is invariant under any rotation). Now consider a mathematical or a physical problem, say studying the geometric properties of an object (from a cell to the universe), or solving some complex equations arising in diverse situations in mathematics and other sciences. An effective method has been to study the collection of all the symmetries of the problem under consideration (called a Group) and use the properties of the Group to shed light on the original problem. This method has been very successfully employed in a wide variety of problems leading to some pioneering works since the nineteenth century.