Categorification of generic Su-Zhang character formula
Shunsuke Hirota
TL;DR
The paper addresses the problem of categorifying a Weyl-type finite-sum character formula for GL(m|n) in the generic g_{-1}-generic region by constructing a BGG-type resolution built from narrow Verma modules connected through odd reflections between non-conjugate Borel subalgebras. The authors introduce narrow Verma modules N^{()}(λ), defined as the image of a Verma map between distinguished and antidistinguished Borels, and analyze their structure via restriction to the even subalgebra and the Kac functors. The main result is an exact sequence 0 → N^{()}(w_0·λ) → ⊕_{w : ℓ(w)=ℓ(w_0)−1} N^{()}(w·λ) → … → N^{()}(λ) → L^{()}(λ) → 0 for λ that are dominant integral and g_{-1}-generic, with the crucial character formula ch N^{()}(λ) = ch M^{()}(λ) / ∏_{β ∈ Γ_{w·λ}} (1 + e^{−β}). This yields a closed character formula for all g_{-1}-generic antidominant simple highest-weight modules and extends the classical BGG framework to Lie superalgebras in a generic region, linking to Su–Zhang and Kac–Wakimoto character formulas.
Abstract
For semisimple Lie algebras, the BGG resolution is often viewed as a categorification of the Weyl character formula. For general linear Lie superalgebras, Brundan--Stroppel constructed an infinite resolution of the so-called Kostant simple modules by Kac modules, but their construction does not directly generalize the classical BGG resolution. In this paper we construct, for weights lying outside a neighborhood of the walls of the Weyl chambers, a resolution that categorifies a known Weyl-type finite-sum character formula in the same spirit as the Kac--Wakimoto formula. Our resolution is built from images of canonical homomorphisms between Verma modules attached to non-conjugate Borel subalgebras related by odd reflections. In particular, the construction developed here does generalize the classical BGG resolution.