Matsuki Correspondence for the Affine Grassmannian
نویسنده
چکیده
Let K = C((t)) be the field of formal Laurent series, and let 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(K)/G(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(K) acting on Gr. The first subgroup is the polynomial loop group LGR of a real form GR of G; the second is the loop group K(K) of the complexification K of a maximal compact subgroup Kc of GR. The orbit poset itself turns out to be simple to describe.
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