Representations of Quantum Groups, Special Functions, and Geometry

基里洛夫这项研究涉及到量子群的表示理论及其与组合学和数学物理的关系。首席调查员将致力于两个问题。第一个问题涉及Lusztig正则基的几何实现，它使用Scheck htman和Varchenko提出的量子群上的模与配置空间上的某个局部系统的同调的标识。这样的认识将揭示组合学和数学物理几何结构之间的一种新的深层联系。第二部分继续研究表示论中出现的特殊函数和相关的组合恒等式。特别是，主要的研究人员将致力于将Macdonald多项式的内积恒等式推广到仿射根系，并研究这种内积的进一步性质，例如正性及其与保形块空间上的共形场论中的内积的关系。本研究属于李代数及其相关对象表示论的一般领域，对数学和物理都具有重要意义。

KIRILLOV This research is concerned with the representation theory of quantum groups and its relation with combinatorics and mathematical physics. The principal investigator will work on two problems. The first problem concerns a geometric realization of Lusztig's canonical basis, using the identification of modules over a quantum group with the homology of a certain local system on the configuration space, which is due to Schechtman and Varchenko. Such a realization would reveal a new deep relation between combinatorics and the geometric structures of mathematical physics. The second continues the study of special functions appearing in representation theory and related combinatorial identities. In particular, the principal investigator will work on the generalization of the inner product identities for Macdonald's polynomials to affine root systems, and on on further properties of this inner product, such as positivity and its relation with the inner product on the space of conformal blocks in conformal field theory. This study is in the general area of representation theory of Lie algebras and related objects, and is important both for mathematics and physics.