Affine Grassmannian
In mathematics, the affine Grassmannian of an algebraic group G over a field k is an ind-scheme—a colimit of finite-dimensional schemes—which can be thought of as a flag variety for the loop group G(k((t))) and which describes the representation theory of the Langlands dual group LG through what is known as the geometric Satake correspondence. == Definition of Gr via functor of points == Let k be a field, and denote by k -Alg {\displaystyle k{\text{-Alg}}} and S e t {\displaystyle \mathrm {Set} } the category of commutative k-algebras and the category of sets respectively.