We show that the characteristic function $1S_{underline{alpha}}$ of any Harder-Narasimhan strata $S{underline{alpha}};{subset};Coh_X^{alpha}$ belongs to the spherical Hall algebra $H_X^{sph}$ of a smooth projective curve X (defined over a finite field $mathbb{F}_q$). We prove a similar result in the geometric setting: the intersection cohomology complex IC(${underline{S}_{underline{alpha}}$) of any Harder-Narasimhan strata ${underline{S}}{underline{alpha}};{subset};{underline{Coh}}_X^{underline{alpha}}$ belongs to the category $Q_X$ 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 ${underline{Coh}}_X^{underline{alpha}}$.