Twisted loop groups and their affine flag varieties

TL;DR

This paper generalizes the theory of affine flag varieties to twisted loop groups over Laurent series fields, connecting to Kac-Moody algebras and applications in Shimura variety local models.

Contribution

It develops a new framework for affine flag varieties in the twisted case, extending existing theories to include Kac-Moody affine types and proposing a coherence conjecture.

Findings

Construction of affine flag varieties for twisted loop groups

Inclusion of Kac-Moody affine flag varieties

Application to local models of Shimura varieties

Abstract

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a "twisted case"; a consequence of our results is that our construction also includes the flag varieties for Kac-Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t)) and some applications to local models of Shimura varieties.