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I'm trying to reconcile the differences between the (algebraic) based loop group and the affine grassmannian. I once believed that I understood the relationship, but I just read a paper which has ...

Let $G$ be a compact simply-connected Lie group. Then one can look at the homology $H_*(\Omega G;\mathbb{Z})$ of the based-loop space $\Omega G$ in at least two different ways: (1) Via Bott-Samelson'...

$\newcommand{\Ad}{\operatorname{Ad}}$ In the study of (smooth/algebraic) based loop spaces of compact groups, one often uses a Grassmannian model to study the space. In particular, the Grassmannian ...