Affine Kac-Moody Lie algebras as torsors over the punctured line

Torsors, Reductive Group Schemes and Extended Affine Lie Algebras

We give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that we take draws heavily for the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows us to find a bridge between multiloop algebras and the work of J. Tits on reductive groups over complete local fields…

Local geometric Langlands correspondence and affine Kac-Moody algebras

By a local geometric Langlands correspondence for a complex reductive group G we understand a construction which assigns to a local system on the punctured disc for the Langlands dual group of G, a category equipped with an action of the formal loop group G((t)). We propose a conjectural description of these categories as categories of representations of the corresponding affine Kac-Moody algebra of critical level, and, in some cases, as categories of D-modules on the ind-schemes G((t))/K. We describe in detail these categories and interrelations between them and provide supporting evidence fo…

Local Geometric Langlands Correspondence: the Spherical Case

A module over an affine Kac--Moody algebra g^ is called spherical if the action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of the corresponding group G[[t]]. Consider the category of spherical g^-modules of critical level. In this paper we prove that this category is equivalent to the category of quasi-coherent sheaves on the ind-scheme of opers on the punctured disc which are unramified as local systems. This result is a categorical version of the well-known description of spherical vectors in representations of groups over local non-archimedian fields. It may be vie…

李代数表示中的中卷积

Middle convolution for Lie algebra representations

本文提出李代数的中卷积函子，将其视为Long-Moody函子的推广，并展示其在Fuchsian系统和logarithmic connections中的应用，最终建立holonomy李代数与局部系统间的Riemann-Hilbert对应。

正特征中的仿射 Harish-Chandra 中心

Affine Harish-Chandra center in positive characteristic

本文在正特征下将非临界水平 Kac-Moody 顶点代数与包络代数的 Harish-Chandra 中心等同于 Langlands 对偶群联络模空间上的函数代数，揭示了其非平凡且非特征零约化的新现象。