Matsuki correspondence for the affine Grassmannian

Abstract

Let $\mathcal{K}$=ℂ((t)) be the field of formal Laurent series, and let $\mathcal{O}$=ℂ[[t]] be the ring of formal power series. In this paper we present a version of the Matsuki correspondence for the affine Grassmannian Gr=G($\mathcal{K}$)/G($\mathcal{O}$) of a connected reductive complex algebraic group G. Our main statement is an anti-isomorphism between the orbit posets of two subgroups of G($\mathcal{K}$) acting on Gr. The first subgroup is the polynomial loop group LG_ℝ of a real form G_ℝ of G; the second is the loop group K($\mathcal{K}$) 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.