2024/10/25 by Kam Hung Tong, Tong, Kam Hung
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2410.19442
It is a classical result that the set K\backslash G /B is finite, where G is a reductive algebraic group over an algebraically closed field with characteristic not equal to two, B is a Borel subgroup of G, and K = Gθ is the fixed point subgroup of an involution of G. In this paper, we investigate the affine counterpart of the aforementioned set, where G is the general linear group over formal Laurent series, B is an Iwahori subgroup of G, and K is either the orthogonal group or the symplectic group over formal Laurent series. We construct explicit bijections between the double cosets K \backslash G/B and certain twisted affine involutions. This is the first combinatorial description of K-orbits in the affine flag variety of type A.