Unique Functionals and Quantum Groups

这是一个在数论，表示理论和理论物理界面的研究项目。研究者和合作者最近的工作表明，在理论物理学中统计力学和相变研究中出现的某些矩阵，称为r矩阵，也出现在一类函数的研究中，称为惠特克函数，当这些函数被视为理论上的表示时，也就是说，当固有的代数对称性被考虑在内时，它们编码算术信息。这一发现为本项目探索的一系列问题打开了大门。预计在这一过程中，数论和数学物理之间将发展出新的重要联系。更详细地说，p进群上的Whittaker函数是自同构形式中的一个基本工具。这些群的元复盖上的惠特克函数还不太清楚，但是已经计算出了交织算子上的相应矩阵。研究者和合作者证明了GL(n)的n重包络覆盖的这些矩阵与一个量子群的r -矩阵一致，该量子群是仿射李超代数GL（1/n）的量子化包络代数的德林菲尔德扭曲。德林菲尔德扭转的作用是将高斯和引入r矩阵，从而使其成为“数字理论”。这澄清了很多问题，但也提出了本项目将探索的新问题。另一个独立但相关的项目是正在进行的关于赫克代数的独特模型和表示之间联系的研究。

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.