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

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.