Compactifications of reductive groups as moduli stacks of bundles

TL;DR

This paper introduces a new moduli stack of bundle chains that provides a unified framework for compactifying reductive groups, connecting toric geometry, moduli of bundles, and known compactifications.

Contribution

It constructs a canonical compactification of reductive groups via moduli of bundle chains, generalizing the wonderful compactification and linking to toric and curve moduli spaces.

Findings

Provides an equivariant toroidal compactification of reductive groups.

Shows all toric orbifolds can be obtained from this construction.

Establishes connections with Losev-Manin spaces and Kausz's compactification.

Abstract

Let G be a split reductive group. We introduce the moduli problem of "bundle chains" parametrizing framed principal G-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of G. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple G, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev-Manin's spaces of weighted pointed curves and with Kausz's compactification of GL(n).