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Published online by Cambridge University Press: 19 November 2021
The aim of this chapter is first to set some basic notation and preliminaries (to be used through the book) and then recall the definition of affine Kac--Moody Lie algebras and their basic representation theory and to study the associated groups and their flag varieties. We define the loop group G((t)) (without the central extension) and its various subgroups. Then, we study the associated infinite Grassmannian and prove that it is a reduced ind-variety when G is a semisimple group. We show that G((t)) is a reduced affine ind-scheme. We further study the central extensions of G((t)).
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