Séminaire Lotharingien de Combinatoire, 93B.37 (2025), 12 pp.

Kam Hung Tong

Orbits in the Affine Flag Variety of Type A

Abstract. It is a classical result that the set K\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 = G^θ is the fixed point subgroup of an involution of $G. In this work, 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, the symplectic group or product group over formal Laurent series. We construct explicit bijections between the double cosets K\G/B and certain twisted affine involutions or affine (p,q)-clans, which are affine involutions with plus or minus signs assigned to the fixed-points. This is the first combinatorial description of K-orbits in the affine flag variety of type A.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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