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 LGℝ of a real form Gℝ of G; the second is the loop group K($\mathcal{K}$) of the complexification K of a maximal compact subgroup Kc of Gℝ. The orbit poset itself turns out to be simple to describe.
Citation
David Nadler. "Matsuki correspondence for the affine Grassmannian." Duke Math. J. 124 (3) 421 - 457, 15 September 2004. https://doi.org/10.1215/S0012-7094-04-12431-5
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