Bijection between irreducible modules in O(V_{-b}(g)) and elements of the subregular cell

Establish that irreducible modules L(Λ) in the category O(V_{-b}(g)) of the simple affine vertex algebra V_{-b}(g) at level −b, for integral highest weights Λ, are in bijection with the subset {w ∈ c_subreg | μ(w) = s_i and Λ_i(K) = b} of the subregular two-sided cell c_subreg of the affine Weyl group. Demonstrate that the bijection is given explicitly by the map w ↦ Λ = w^{-1} ∘ (−Λ_i). In particular, when the centralizer Z_e of a subregular nilpotent e is finite, show that these irreducible objects correspond to a specific right Lusztig cell in the affine Weyl group.

Background

The paper computes special values of parabolic affine inverse Kazhdan–Lusztig polynomials on the subregular cell and applies them to obtain explicit character formulas for representations of affine Lie algebras and simple vertex algebras V_k(g). Based on these computations, the authors propose a classification of irreducible objects in the category O(V_{-b}(g)) via the combinatorics of the subregular cell in the affine Weyl group.

Here, b is the maximal coefficient among the labels of the highest short coroot θ∨, μ(w) is the first simple reflection in the unique reduced decomposition of w ∈ c_subreg0, and K is the central element of the affine Lie algebra. The conjecture asserts a precise bijection from elements of the subregular cell to integral highest weights producing irreducible V_{-b}(g)-modules.

References

Conjecture\label{conj_label_irred_O_vertex} Irreducible objects L(\Lambda) \in \mathcal{O}(V_{-b}(\mathfrak{g})) with integral \Lambda are in bijection with the set {w \in c_{\mathrm{subreg}\,|\,\mu(w)=s_i\,\text{s.t.}\,\Lambda_i(K)=b}. The bijection is explicitly given by w \mapsto \Lambda=w{-1} \circ (-\Lambda_i). In particular, for \mathfrak{g} s.t. Z_{e} is finite, the irreducible objects as above are in bijection with a certain {right} Lusztig cell in the affine Weyl group.

— 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 Conjecture \ref{conj_label_irred_O_vertex}, Applications to representations of vertex algebras and Schur indices of 4D SCFT's (Introduction)