Associated variety of V_{-b}(g) via BVLS duality

Determine whether the associated variety Ass(V_{-b}(g)) of the simple affine vertex algebra V_{-b}(g) at level −b equals the closure of the Barbasch–Vogan–Lusztig–Spaltenstein dual of the subregular nilpotent orbit, i.e., Ass(V_{-b}(g)) = \overline{D(\mathbb{O}_{subreg})}, where D denotes BVLS duality.

Background

The authors relate their subregular-cell character computations to geometric expectations about associated varieties of vertex algebras. Under the finiteness of Z_e, they state a conjectural identification of Ass(V_{-b}(g)) with the BVLS dual of the subregular orbit, suggesting a deep link between Lusztig cells in \widehat{W} and associated varieties of V_{-b}(g).

This conjecture complements the preceding classification conjecture and aligns with known results in certain types (referenced by Arakawa–Moreau and Arakawa–Dai–Fasquel–Li–Moreau).

References

From now on, assume that Z_e is finite. The following conjecture was communicated to us by Peng Shan. Conjecture\label{conj_ass_BVLS} Let D\colon {\mathcal{N}{\mathfrak{g}/G} \rightarrow {\mathcal{N}{\mathfrak{g}\vee}/G\vee} be the Barbasch-Vogan-Lusztig-Spaltenstein duality. Then \operatorname{Ass}(V_{-b}(\mathfrak{g}))=\overline{D(\mathbb{O}_{\mathrm{subreg})}.

— 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_ass_BVLS}, Applications to representations of vertex algebras and Schur indices of 4D SCFT's (Introduction)