SPHERICAL HALL ALGEBRAS OF CURVES AND HARDER-NARASIMHAN STRATAS

Title & Authors
SPHERICAL HALL ALGEBRAS OF CURVES AND HARDER-NARASIMHAN STRATAS
Schiffmann, Olivier;

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