Geometric Methods in Representation Theory

Kumar The Pi(Shrawan Kumar)打算继续在‘李论和几何学’的一般领域工作。他提议开展四个项目。第一个项目旨在解决Kollar和Peskine在1988年提出的关于完全交叉口的下列问题。设C(T)是三维射影空间中的一族复杂光滑曲线族，使得一般成员是一个完全交。那么，特殊成员也是一个完全的交集吗？PI把这个问题转化为从三维仿射空间减去原点到SU(2)联络的瞬子模空间的等变态射不存在的问题，在四维欧几里得空间上模为基础规范等价的秩二丛。这种模空间已经得到了广泛的研究，PI希望利用这些结果来解决上述问题。设G是复数上的半单代数群，S是G的具有固定子群L的图自同构。第二个方案的目的是确定L的特征锥以及L表示的张量积分解。PI和Belkale提出了一些精确的猜想。一旦求解，它将给出非单酸群的饱和问题的最优解，前提是单酸群的饱和保持。第三个项目(与Arzu Boysal联合)旨在明确确定融合代数。设g是一个简单李代数。固定一个正整数k(称为电平)。设R(G)是g的表示环，R(g；k)(k级)是R(G)在环同态f下的商环。Gepner对特殊的线性李代数确定了它的核，随后由Bourdeau-Mlawer-Riggs-Schnitzer和Bouwkenert-Ridout对辛李代数确定了它。对于其他经典群和例外的G(2)型李代数，Pi和Boysal已经给出了f的核的显式猜想刻画。PI建议证明这一猜想描述，并为所有其他例外群找到f的核的类似描述。第四个项目旨在写一本关于弗林德公式及其完整证明和结果的书。E.Verlinde在1988年给出了共形块空间的维度的猜想公式。这个空间是有理共形场理论中的一个基本对象。在Wess-Zumino-Witten模型的特殊情况下，共形块空间允许解释为射影曲线上向量丛的模理论中产生的广义theta函数空间。这一解释为研究射影曲线上向量丛(更一般地，主G-丛)的模开辟了一个全新的视野。这些发展导致了Verlinde公式的证明以及各种应用，分散在文献中，没有包含这些的单一来源。这本拟出的书将填补文献上的这一空白。建议的项目强调了数学统一的主题，因为它们有望从几个数学领域获得想法，包括拓扑学、组合学、代数几何、表示论和数学物理。国际和平研究所建议的这些项目代表了一些当前感兴趣的非常困难和重要的问题，如果这些问题得到解决，也将有助于澄清现有已知案件的结果。此外，预计这些解决方案将在该地区产生大量活动。私家侦探在他提议使用的技术方面拥有相当多的专业知识。国际和平研究所提出的关于“弗林德公式、其完全证明和后果”的书将是关于这一主题的第一本书。预计它将成为研究生和专业数学家的基本资料来源，从而大大促进教与学。私人侦探早先写的两本书(其中一本是与布里昂先生合著的)已经成为关于这个主题的标准文本。PI已经成功地指导了六名博士生。

KumarThe PI (Shrawan Kumar) intends to continue work in the general area of `Lie Theory and Geometry.' He proposes to work on four projects. The first project aims at settling the following problem on complete intersections raised by Kollar and Peskine in 1988. Let C(t) be a family of complex smooth curves in the three dimensional projective space such that the general member is a complete intersection. Then, is the special member also a complete intersection? The PI has converted this problem into a problem of the nonexistence of equivariant morphisms from the three dimensional affine space minus the origin to the instanton moduli space ofSU(2) connections on a rank two bundle over the four dimensional Euclidean space modulo the based gauge equivalence. This moduli space has been extensively studied and the PI expects to use these results to solve the above problem. Let G be a semisimple algebraic group over the complex numbers and let s be a diagram automorphism of G with fixed subgroup L. The second project aims at determining the eigencone of L as well as the tensor product decomposition for the representations of L in terms of that of G. The PI and Belkale have formulated some precise conjectures to determine this. Once solved, it would give an optimal solution of the saturation problem for non-simplylaced groups provided the saturation holds for simplylaced groups. The third project (jointly with Arzu Boysal) aims at determining the fusion algebra explicitly. Let g be a simple Lie algebra. Fix a positive integer k (called the level). Let R(g) be the representation ring of g. The fusion ring R(g;k) (at level k) is a quotient ring of R(g) under a ring homomorphism f. Gepner determined its kernel for the special linear Lie algebra and it was subsequently determined by Bourdeau-Mlawer-Riggs-Schnitzer and Bouwknegt-Ridout for the symplectic Lie algebras. The PI and Boysal have come up with explicit conjectural description of the kernel of f for other classical groups as well as for the exceptional Lie algebra of type G(2). The PI proposes to prove this conjectural description and also find an analogous description of the kernel of f for all other exceptional groups. The fourth project aims at writing a book on ``Verlinde formula, its complete proof and consequences." E. Verlinde gave a conjectural formula in 1988 for the dimension of the space of conformal blocks. This space appears as a basic object in Rational Conformal Field Theory. In the special case of the Wess-Zumino-Witten model, the space of conformal blocks admits an interpretation as the space of generalized theta functions arising in the theory of moduli of vector bundles on projective curves. This interpretation opened a completely new horizon for the study of the moduli of vector bundles (more generally, principal G-bundles) on projective curves. These developments leading to the proof of the Verlinde formula as well as various applications are scattered through the literature and there is no single source containing these. The proposed book would fill this void in the literature.The proposed projects underline the theme of unity in mathematics as they are expected to derive ideas from several areas of mathematics including Topology, Combinatorics, Algebraic Geometry, Representation Theory and Mathematical Physics. These projects suggested by the PI represent some of the very difficult and important problems of current interest which, if solved, should also help clarify the results in the existing known cases. In addition, it is expected that the solutions will spawn a lot of activity in the area. The PI has considerable expertise in the techniques he is proposing to use. The PI's proposed book on ``Verlinde formula, its complete proof and consequences" would be the very first book on the subject. It is expected that it will serve as a basic source for graduate students and professional mathematicians alike thus substantially promoting teaching and learning. The two books written earlier by the PI (one coauthored with M. Brion) have become standard texts on the subject. The PI has successfully supervised six PhD students.