Simplicity criterion for fully reflectable spherical modules

Prove that for any base Σ of the restricted root system ∆ of a supersymmetric pair (g,k) arising from a symmetrizable Kac–Moody superalgebra g, and any fully reflectable weight λ ∈ P+Σ (meaning that for every base Σ′ ⊆ ∆ there exists λΣ′ ∈ P+Σ′ such that VΣ(λ) ≅ VΣ′(λΣ′)), the highest weight module VΣ(λ) is simple if and only if for every base Σ′ ⊆ ∆ and each non-isotropic root β ∈ Σ′ with m(β) := −sdim(gβ) − sdim(g2β) ∈ Z≥0, the half-evaluation λΣ′(hβ)/2 does not belong to the integer set {m(β)+1, …, 2m(β)}.

Background

The paper studies spherical weights and modules for supersymmetric pairs (g,k) associated with symmetrizable Kac–Moody superalgebras. For a chosen base Σ of the restricted root system ∆, the authors construct highest weight modules VΣ(λ) that admit nonzero k-coinvariants and analyze their behavior under singular reflections, reducing many questions to rank-one cases.

A central classification problem (Question C) asks which finite-dimensional simple g-modules have nonzero k-coinvariants. Complications arise from singular, non-isotropic roots, where reflections can produce critical weights that obstruct simplicity. Drawing on rank-one analyses (including Casimir eigenvalue comparisons), the authors formulate a conjectural numerical criterion that aims to characterize simplicity for “fully reflectable” spherical weights uniformly across all bases, thereby advancing the classification of simple spherical modules.