Journal of the
Korean Mathematical Society JKMS

We show that the characteristic function $\mathbf{1}_{S_{\underline{\alpha}}}$ of any Harder-Narasimhan strata $S_{\underline{\alpha}} \subset Coh^{\alpha}_X$ belongs to the spherical Hall algebra ${\boldsymbol H}^{sph}_X$ 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 $\underline{S}_{\underline{\alpha}} \subset \underline{Coh}^{\alpha}_{\overline{X}}$ belongs to the category $\mathcal{Q}_X$ 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 $\underline{Coh}^{\alpha}_{\overline{X}}$.

Keywords: Hall algebras, Harder-Narasimhan stratas, Eisenstein sheaves

MSC numbers: 17B37, 22E57