Borel--Serre type bordifications and Canonical Pairs for Loop groups

TL;DR

This paper constructs a Borel--Serre type bordification for the symmetric space of a real loop group, analyzing its boundary, topology, and group actions, and introduces semi-stability concepts to understand its structure.

Contribution

It develops a loop group analogue of Borel--Serre bordification, studies its boundary and quotient, and introduces semi-stability for loop groups, extending classical theories to infinite-dimensional settings.

Findings

Boundary is homotopic to an affine, rational Tits building.

A loop analogue of an arithmetic group acts continuously on the bordification.

The quotient space is non-compact, related to the center of the loop group.

Abstract

To the symmetric space of the (positive half) of a real loop group, we attach a Borel--Serre type bordification and equip it with a Hausdorff topology. The attached boundary, indexed by certain rational parabolics of the loop group, is shown to be homotopic to an affine, rational Tits building. A loop analogue of an arithmetic group is also shown to act continuously on the bordification and its quotient by this action is studied using the reduction theory of H. Garland. While the quotient is no longer compact (as in the Borel--Serre construction from finite-dimensions) we relate the non-compactness to the center of the loop group. We also introduce a notion of semi-stability for loop groups, following works of Harder--Narasimhan, Behrend, and most recently Chaudouard, and use this to describe a partition of our loop symmetric space. This partition is then related to the rational bordification and its quotient.