Flag manifolds over semifields

Abstract

In this paper, we develop the theory of flag manifolds over a semifield for any Kac–Moody root datum. We show that a flag manifold over a semifield admits a natural action of the monoid over that semifield associated with the Kac–Moody datum and admits a cellular decomposition. This extends the previous work of Lusztig, Postnikov, Rietsch, and others on the totally nonnegative flag manifolds (of finite type) and the work of Lusztig, Speyer, Williams on the tropical flag manifolds (of finite type). As an important consequence, we prove a conjecture of Lusztig on the duality of a totally nonnegative flag manifold of finite type.