Conformal Blocks and Affine Grassmannian Associated to Parahoric Group Schemes

表示论是代数的一个分支，利用可逆矩阵群研究对称，特别是线性数学结构的对称。另一方面，代数几何基于抽象代数技巧的使用，主要来自交换代数，用于解决关于多变量多项式的零点集的几何问题。该NSF奖支持的研究处于表示论和代数几何的十字路口，用表示论解决几何问题，用几何方法理解表示论。研究项目围绕两个主要主题展开。在第一部分中，PI将为曲线上的一般类意群方案发展一个共形块理论。研究的重点是扭转共形块的Pappas-Rapoport猜想、消失猜想和Verlinde公式。这将在共形场理论和仿射群方案几何朗兰兹程序中得到应用。对于第二个主题，PI将发展扭曲仿射Grassman几何和表示理论之间的联系。仿射Grassmannians的几何可以通过几何Satake对应与Kac-Moody理论、约化群及其基的表示理论相联系，并在辛对偶中起着至关重要的作用。沿着这个方向的研究将推进扭曲仿射Grassmannians在Kac-Moody理论、分支几何Satake、辛对偶以及与对称空间的Springer理论的联系中的作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Representation theory is a branch of algebra studying symmetries, especially symmetries of linear mathematical structures, using groups of invertible matrices. On the other hand, algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometric problems about the sets of zeros of multivariable polynomials. The research supported by this NSF award is at the crossroads of representation theory and algebraic geometry, to use representation theory to solve geometric problems and to use geometric methods to understand representation theory.The research projects center on two main themes. In the first, the PI will develop a theory of conformal blocks for general parahoric group schemes over curves. The research will focus on Pappas-Rapoport conjecture, vanishing conjecture and Verlinde formula for twisted conformal blocks. This will lead to applications in orbifold conformal field theory and geometric Langlands program for parahoric group schemes. For the second theme, the PI will develop connections between the geometry of twisted affine Grassmannian and representation theory. The geometry of affine Grassmannians can be related to Kac-Moody theory, representation theory of reductive groups and their bases via geometric Satake correspondence, and it also plays crucial role in symplectic duality. The research along this direction will advance the role of twisted affine Grassmannians in Kac-Moody theory, ramified geometric Satake, symplectic duality, and a connection with Springer theory for symmetric spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.