SPHERICAL HALL ALGEBRAS OF CURVES AND HARDER-NARASIMHAN STRATAS

Abstract

We show that the characteristic function <TEX>$1S_{\underline{\alpha}}$</TEX> of any Harder-Narasimhan strata <TEX>$S{\underline{\alpha}}\;{\subset}\;Coh_X^{\alpha}$</TEX> belongs to the spherical Hall algebra <TEX>$H_X^{sph}$</TEX> of a smooth projective curve X (defined over a finite field <TEX>$\mathbb{F}_q$</TEX>). We prove a similar result in the geometric setting: the intersection cohomology complex IC(<TEX>${\underline{S}_{\underline{\alpha}}$</TEX>) of any Harder-Narasimhan strata <TEX>${\underline{S}}{\underline{\alpha}}\;{\subset}\;{\underline{Coh}}_X^{\underline{\alpha}}$</TEX> belongs to the category <TEX>$Q_X$</TEX> of spherical Eisenstein sheaves of X. We show by a simple example how a complete description of all spherical Eisenstein sheaves would necessarily involve the Brill-Noether stratas of <TEX>${\underline{Coh}}_X^{\underline{\alpha}}$</TEX>.