Representation Theory, Quantum Groups, and Canonical Bases

本提案的主要主题是研究位于李群，量子群，簇代数和非交换代数几何的表示理论的十字路口的区域。本文提出了一种基于量子簇代数和几何晶体的研究Lusztig正则基和Kashiwara晶体基的新方法。本研究所得的新信息将应用于计算约化群表示的对称幂的多重性，以及构造新的全正变异。本研究的结果将用于解决在约化代数群和Cherednik代数的离散子群表示中出现的问题，以及用于解释和阐述相关的组合和几何结构，包括晶体碱的“几何提升”，作为理解局部朗兰兹对应的新工具。表征理论是现代数学中发展最活跃的领域之一。它对数学的其他领域产生了巨大的影响，并在其他自然科学中有许多应用。正则基和晶体基的概念对于表示理论具有重要意义：仅仅建立这些基的存在性就有助于解决经典的枚举问题，如计算不可约表示的复数或分解不可约表示的张量积。作者发现的一类新的正则基有望解决对称幂表示分解的老问题。因此，任何关于规范基或晶体基的信息都将对表征理论非常有益。在G. Lusztig和他的后续著作中揭示了纯离散正则基和晶体基的代数-几何对应物：完全正变异、几何晶体和簇代数。理解这些结构之间的关系是规范基础的主要优先事项之一。这种关系被证明是研究朗兰兹对应的一个有用的工具——朗兰兹对应是20世纪数学中代数和几何之间最神秘和最鼓舞人心的对应。

The main theme of this proposal is to investigate the area lying at the crossroads of the representation theory of Lie groups, quantum groups, cluster algebras, and noncommutative algebraic geometry. A new approach to the study of Lusztig's canonical bases and Kashiwara's crystal bases is proposed, based on quantum cluster algebras and geometric crystals as developed in recent papers of the proposer. New information resulting from this study will be applied to computing the multiplicities for the symmetric powers of representations of reductive groups, and constructing new totally positive varieties. The results of this study will be applied for solving problems emerging in the representations of discrete subgroups of reductive algebraic groups and Cherednik algebras as well as for explication and elaboration of related combinatorial and geometric structures including the ``geometric lifting'' of crystal bases as a new tool in understanding the local Langlands correspondence. Representation theory is one of the most dynamically developing fields of modern Mathematics. It has a large impact in other fields of Mathematics and numerous applications in other Natural Sciences. The concepts of canonical and crystal bases are of great importance for the representation theory: a mere establishing of existence of such bases has helped in solving classical enumeration problems like computing multiplicities of irreducible representations or decomposing tensor products of irreducible representations. A new class of canonical bases discovered by the proposer is expected to settle an old problem of decomposing symmetric powers of representations. Therefore, any information on canonical or crystal bases would be very beneficial for the representation theory. It has been revealed in the works of G. Lusztig and the subsequent works of the proposer that there are algebro-geometric counterparts for the purely discrete canonical and crystal bases: totally positive varieties, geometric crystals, and cluster algebras. Understanding the relationship between these structures underlying the canonical bases is one of main priorities of this proposal. This relationship has proved to be a useful tool in the study of Langlands correspondence -- the most mysterious and inspiring correspondence between Algebra and Geometry of the 20th century Mathematics.