Categorification of generic Su-Zhang character formula

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.

💡 Research Summary

The paper addresses a long‑standing problem in the representation theory of the general linear Lie superalgebra gl(m|n): how to categorify a finite‑sum Weyl‑type character formula for finite‑dimensional simple modules. Two character formulas have been known for gl(m|n). The first, due to Serganova and Brundan, expresses the character of a simple module as an infinite sum involving Kazhdan–Lusztig‑type polynomials evaluated at −1 and Kac modules. The second, the Kac–Wakimoto formula, gives a finite sum but contains a factorial factor 1/atyp(λ)!, which makes it unsuitable for a direct categorification except in the most singular (totally disconnected) cases. Su and Zhang later showed that, when the highest weight λ lies sufficiently far from the walls of the Weyl chambers—what the author calls g‑1‑generic—the factorial factor disappears and the character simplifies to

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