Categorification of generic Su-Zhang character formula

TL;DR

This paper develops a new categorification of a Weyl-type character formula for general linear Lie superalgebras, extending classical resolutions to weights outside Weyl chamber walls using Verma modules and odd reflections.

Contribution

It constructs a novel resolution that generalizes the classical BGG resolution for weights outside Weyl chamber walls in Lie superalgebras.

Findings

Constructs a resolution categorifying a Weyl-type character formula.

Uses images of canonical homomorphisms between Verma modules.

Generalizes classical BGG resolution to broader weights.

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.