Ramified convolution and monodromy bases for symmetrizable Kac--Moody algebras

Abstract

We realize the positive enveloping algebra of a symmetrizable Kac--Moody algebra by convolution on a ramified preprojective species over $\mathbb{C}((t))$. The coefficients are alternating traces of tame monodromy on representation flag varieties. We prove the relative Serre relations modulo classes induced from proper finite extensions. Monodromy evaluations at compatible generic points of the central reductions yield an integral basis respecting the top and socle crystals. The basis is unchanged by scaling the symmetrizer, induces bases of integrable representation lattices, and agrees with Lusztig's semicanonical basis when the Cartan matrix is symmetric. We prove the acyclic convolution conjecture of Geiß--Leclerc--Schröer for all symmetrizable Cartan matrices and all positive integral symmetrizers. Finally, we give a counterexample to the Support Conjecture of Geiß--Leclerc--Schröer and compute a ramified lift for which monodromy restores the Serre relation.