Twisted loop groups and their affine flag varieties

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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.