Matsuki correspondence for the affine Grassmannian

Abstract

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 $L{G}_R$ 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.

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

References in corpus (3)

• Perverse sheaves on a Loop group and Langlands' duality
• Perverse Sheaves on Real Loop Grassmannians
• Hyperbolic localization of intersection cohomology