Intersection pairings for higher laminations

Abstract

One can realize higher laminations as positive configurations of points in the affine building [7]. The duality pairings of Fock and Goncharov [1] give pairings between higher laminations for two Langlands dual groups G and G∨. These pairings are a generalization of the intersection pairing between measured laminations on a topological surface. We give a geometric interpretation of these intersection pairings in a wide variety of cases. In particular, we show that they can be computed as the minimal weighted length of a network in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [9]. We also suggest the next steps toward giving geometric interpretations of intersection pairings in general. The key tools are linearized versions of well-known classical results from combinatorics, like Hall's marriage lemma, König's theorem, and the Kuhn-Munkres algorithm, which are interesting in themselves. Show more