Generalized Bruhat decompositions and infinite lattice
varieties: an introduction to Langlands duals in the theory of
loop and looplike spaces -- William Haboush

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If G is a reductive algebraic group over k then it is well
known that the triple consisting of a Borel subgroup, a Cartan
subgroup and a set of fundamental reflections constitutes a Tits
system (BN pair) and that the associated double coset
decomposition is the Bruhat decomposition.  I will explain how
the triple consisting of an Iwahori subgroup, a Cartan subgroup
and another set of reflections constitutes a Tits system and
how the corresponding double coset decomposition is indexed by
objects which are associated to the Langlands dual.  Since
these double cosets detemine cellular decompositions of the
loop Grassmannian, the classical Cartan-Chevalley-Demazure
computation of the cohomology rings, K-theory and Chow rings of
flag varieties suggests the program of Ginzburg, Drinfeld,
Bezrukavnikov and others.