Nonabelian Jacobian of smooth projective surfaces and representation theory

TL;DR

This paper explores the representation theory of a nonabelian Jacobian for smooth projective surfaces, linking geometric configurations to Lie algebra structures and representation categories, and relating it to Langlands duality.

Contribution

It introduces a nonabelian Jacobian with associated Lie algebraic structures, connecting geometry, representation theory, and Langlands duality in a novel framework.

Findings

Determines the sheaf of reductive Lie algebras associated to the nonabelian Jacobian.

Relates the Lie algebraic properties to configurations of points on the surface.

Establishes a connection between the nonabelian Jacobian and Langlands dual groups.

Abstract

The paper studies representation theoretic aspects of a nonabelian version of the Jacobian for a smooth complex projective surface $X$ introduced in [R1]. The sheaf of reductive Lie algebras $\bf\calG$ associated to the nonabelian Jacobian is determined and its Lie algebraic properties are explicitly related to the geometry of configurations of points on $X$. In particular, it is shown that the subsheaf of centers of $\bf\calG$ determines a distinguished decomposition of configurations into the disjoint union of subconfigurations. Furthermore, it is shown how to use $sl_2$-subalgebras associated to certain nilpotent elements of $\bf\calG$ to write equations defining configurations of $X$ in appropriate projective spaces. The same nilpotent elements are used to establish a relation of the nonabelian Jacobian with such fundamental objects in the representation theory as nilpotent orbits, Springer resolution and Springer fibres of simple Lie algebras of type $A_n$, for appropriate values of $n$. This leads to a construction of distinguished collections of objects in the category of representations of symmetric groups as well as in the category of perverse sheaves on the appropriate Hilbert schemes of points of $X$. Hence two ways of categorifying the second Chern class of vector bundles of rank 2 on smooth projective surfaces. We also give a `loop' version of the above construction by relating the nonabelian Jacobian to the Infinite Grassmannians of simple Lie groups of type $SL_n(\bf C)$, for appropriate values of n. This gives, via the geometric version of the Satake isomorphism, a distinguished collection of irreducible representations of the Langlands dual groups thus indicating a relation of the nonabelian Jacobian to the Langlands duality for smooth projective surfaces.