Spherical Hall algebras of curves and Harder-Narasimhan stratas

TL;DR

This paper demonstrates that characteristic functions of Harder-Narasimhan strata in the moduli stack of vector bundles over a curve are contained within the spherical Hall algebra, and establishes a geometric analog involving intersection cohomology sheaves.

Contribution

It proves the inclusion of characteristic functions and intersection cohomology sheaves of Harder-Narasimhan strata in the spherical Hall algebra and its geometric analog.

Findings

Characteristic functions of Harder-Narasimhan strata belong to the spherical Hall algebra.

Intersection cohomology sheaves of these strata are simple Eisenstein sheaves.

Provides a geometric analog linking algebraic and sheaf-theoretic structures.

Abstract

Let X be any smooth projective curve defined over a finite field. We show that the characteristic functions of any Harder-Narasimhan strata S_a of Bun_{GL_n}X belongs to the spherical Hall algebra H_X^{sph} of X. We give a geometric analog of the above result: the intersection cohomology sheaf IC(S_a) belongs to the category of simple Eisenstein sheaves over Bun_{GL_n}X.