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

摘要

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 e, we define a Z[q, q~(-1)]-module F_e 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 F_e 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 F_e by dominant integral weights makes F_e a useful combinatorial tool for determining decomposition numbers ofWeyl modules for quantum groups at roots of unity.