Unique Functionals and Quantum Groups

这是一项介于数论、表象理论和理论物理之间的研究项目。研究人员和合作者最近的工作表明，在统计力学和相变研究的理论物理中出现的某些矩阵，称为R-矩阵，也出现在一类称为惠特克函数的函数的研究中，当这些函数被视为表示时，即在理论上，当考虑到固有的代数对称性时，该函数编码算术信息。这一发现开启了丰富的问题集合的大门，这些问题将在本项目中探索。预计在这个过程中，数论和数学物理之间的重要新联系将会得到发展。更详细地说，p-进群上的Whittaker函数是自同构形式的基本工具。关于这些群的亚辛覆盖上的Whittaker函数还不是很清楚，但已经计算了相应的交织算子上的矩阵。研究人员和合作者证明了GL(N)的n重矩阵覆盖的这些矩阵与一个量子群的R-矩阵一致，该R-矩阵是仿射Lie超代数gl(1/n)的量子化包络代数的Drinfield扭曲。Drinfield扭曲的效果是将高斯和引入R-矩阵，从而使其成为“数论”。这在很大程度上澄清了问题，但也提出了将在该项目中探索的新问题。另一个单独但相关的项目是正在进行的关于唯一模型和Hecke代数表示之间的联系的研究。

This is a research project at the interface of number theory, representation theory, and theoretical physics. Recent work of the investigator and collaborators has shown that certain matrices, known as R-matrices, that arise in theoretical physics in the study of statistical mechanics and phase transitions also arise in the study of a class of functions, known as Whittaker functions, that encode arithmetic information when these functions are viewed representation-theoretically, that is, when inherent algebraic symmetries are taken into account. This discovery opens the door to a rich collection of questions that will be explored in this project. It is anticipated that in the process, important new connections between number theory and mathematical physics will be developed.In more detail, Whittaker functions on p-adic groups are a fundamental tool in automorphic forms. Whittaker functions on metaplectic covers of these groups are less well-understood, but the corresponding matrices on intertwining operators have been computed. The investigator and collaborators have shown that these matrices for the n-fold mataplectic cover of GL(n) agree with the R-matrices for a quantum group, which is a Drinfield twist of the quantized enveloping algebra of the affinized Lie superalgebra gl(1/n). The effect of the Drinfield twisting is to introduce Gauss sums into the R-matrix, and therefore to make it "number-theoretic." This clarifies a great deal, but also presents new questions that will be explored in this project. A separate but related project that will be pursued is an on-going study of connections between unique models and representations of Hecke algebras.