The Harder–Narasimhan stratification of the moduli stack of $G$-bundles via Drinfeld’s compactifications

Abstract

We use Drinfeld’s relative compactifications ${\overline{\mathop {{\mathrm{Bun}}}\nolimits }}_P$ and the Tannakian viewpoint on principal bundles to construct the Harder–Narasimhan stratification of the moduli stack $\mathop {{\mathrm{Bun}}}\nolimits _G$ of $G$-bundles on an algebraic curve in arbitrary characteristic, generalizing the stratification for $G=\mathop {\mathrm{GL}}\nolimits _n$ due to Harder and Narasimhan to the case of an arbitrary reductive group $G$. To establish the stratification on the set-theoretic level, we exploit a Tannakian interpretation of the Bruhat decomposition and give a new and purely geometric proof of the existence and uniqueness of the canonical reduction in arbitrary characteristic. We furthermore provide a Tannakian interpretation of the canonical reduction in characteristic $0$ which allows to study its behavior in families. The substack structures on the strata are defined directly in terms of Drinfeld’s compactifications ${\overline{\mathop {{\mathrm{Bun}}}\nolimits }}_P$, which we generalize to the case where the derived group of $G$ is not necessarily simply connected. Using ${\overline{\mathop {{\mathrm{Bun}}}\nolimits }}_P$, we establish various properties of the stratification, including finer information about the structure of the individual strata and a simple description of the strata closures.