Conformal Blocks and Affine Grassmannian Associated to Parahoric Group Schemes

表示论是代数学的一个分支，研究对称性，特别是线性数学结构的对称性，使用可逆矩阵群。另一方面，代数几何是基于使用抽象的代数技术，主要是从交换代数，解决几何问题的零集的多变量多项式。该奖项所支持的研究是在表示论和代数几何的十字路口，使用表示论来解决几何问题，并使用几何方法来理解表示论。研究项目集中在两个主要主题上。首先，PI将发展曲线上一般parahoric群格式的共形块理论。主要研究Pappas-Rapoport猜想、消失猜想和扭曲共形块的Verlinde公式。这将导致应用于轨道共形场论和几何朗兰兹程序parahoric群计划。对于第二个主题，PI将发展扭曲仿射格拉斯曼几何和表示论之间的联系。 仿射Grassmannian的几何通过几何Satake对应与Kac-Moody理论、约化群及其基的表示理论有着密切的联系，并且在辛对偶中也起着重要的作用。沿着这一方向的研究将推进扭曲仿射格拉斯曼在Kac-Moody理论中的作用，分支几何Satake，辛对偶，以及与对称空间的Springer理论的联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.