A Fock space model for decomposition numbers for quantum groups at roots of unity

Abstract

In this paper we construct an abstract Fock space for general Lie types that serves as a generalization of the infinite wedge $q$-Fock space familiar in type A. Specifically, for each positive integer $\ell$, we define a $\mathbb{Z}[q,q^{-1}]$-module ${\mathcal{F}}_{\ell }$ with bar involution by specifying generators and straightening relations adapted from those appearing in the Kashiwara–Miwa–Stern formulation of the $q$-Fock space. By relating ${\mathcal{F}}_{\ell }$ to the corresponding affine Hecke algebra, we show that the abstract Fock space has standard and canonical bases for which the transition matrix produces parabolic affine Kazhdan–Lusztig polynomials. This property and the convenient combinatorial labeling of bases of ${\mathcal{F}}_{\ell }$ by dominant integral weights makes ${\mathcal{F}}_{\ell }$ a useful combinatorial tool for determining decomposition numbers of Weyl modules for quantum groups at roots of unity.