Weights in O(V_k(g)) for general negative levels

Investigate whether, for arbitrary negative integers k in {−1, …, −b}, every integral highest weight Λ yielding L(Λ) ∈ O(V_k(g)) conforms to the classification pattern described in Lemma 2.11 (i.e., the explicit forms of Λ linked to the subregular cell conditions).

Background

Beyond the principal level k = −b considered in the main conjecture, the authors point out uncertainty about the structure of integral highest weights Λ that produce modules in O(V_k(g)) for other negative levels in the range −1 to −b.

This question asks whether the same subregular-cell-based classification persists for all such k, extending the reach of their character computations and geometric framework.

References

In Conjecture \ref{conj_label_irred_O_vertex} above we only consider simple vertex algebra V_{k}(\mathfrak{g}) for k=-b. One can consider arbitrary k \in {-1,\ldots,-b}. We do not know if it is natural to expect that every integral \Lambda such that L(\Lambda) \in \mathcal{O}(V_{k}(\mathfrak{g})) is as in Lemma \ref{lem:descr_poss_lambda} above.

— Affine Kazhdan-Lusztig polynomials on the subregular cell in non simply-laced Lie algebras: with an application to character formulae (with an appendix by Roman Bezrukavnikov, Vasily Krylov, and Kenta Suzuki) (2401.06605 - Krylov et al., 12 Jan 2024) in Remark following Conjecture \ref{conj_label_irred_O_vertex}, Applications to representations of vertex algebras and Schur indices of 4D SCFT's (Introduction)