Matsuki correspondence for the affine Grassmannian

Details Matsuki correspondence for the affine Grassmannian Matsuki correspondence for the affine Grassmannian

2003-01-09

arxiv.org/abs/math/0301091v4

Mathematics

Algebraic Geometry

28 pages; final version, to appear in Duke Math J

Scientific paper

We present a version of the Matsuki correspondence for the affine Grassmannian $Gr=G(C((t)))/G(C[[t]])$ of a connected reductive complex algebraic group $G$. The main statement is an anti-isomorphism between the orbit posets of two subgroups of $G(C((t)))$ acting on $Gr$. The first is the polynomial loop group $LG_R$ of a real form $G_R$ of $G$; the second is the loop group $K(C((t)))$ of the complexification $K$ of a maximal compact subgroup $K_c$ of $G_R$. The orbit poset itself turns out to be simple to describe.

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