Category $\mathcal{O}$ and asymptotic characters

TL;DR

This paper introduces an asymptotic character map linking category O modules to rational functions, revealing connections between geometric multiplicities, KLR algebra modules, and canonical bases in representation theory.

Contribution

It defines a new asymptotic character map from category O to rational functions, relating geometric and algebraic structures in representation theory.

Findings

Asymptotic character computes equivariant multiplicities of characteristic cycles.

Formulas relate Mirković-Vilonen cycles to KLR algebra module characters.

Provides evidence for positivity of coefficients in basis change formulas.

Abstract

This paper defines an asymptotic character map which is a morphism from the Grothendieck group of category $\mathcal{O}$ of an integral filtered quantization to rational functions on the Lie algebra of a torus. We show that the asymptotic character of a module computes the equivariant multiplicity of its characteristic cycle. We then apply this construction to truncated shifted Yangians coming from simple, simply-laced Lie algebras and draw connections with characters of modules over KLR algebras using an equivalence of categories of arXiv:1806.07519. Our main theorem shows how this new formalism gives formulas relating equivariant multiplicities of Mirkovi\'{c}-Vilonen cycles and characters of modules over cyclotomic KLR algebras. We explain how this result provides evidence that the change-of-basis between Lusztig's dual canonical basis and the Mirkovi\'{c}-Vilonen basis of $\mathbb{C}[N]$ is computed by a characteristic cycle map whose domain is category $\mathcal{O}$ for truncated shifted Yangians, implying that the coefficients are non-negative integers.