Applications of removing the simplicial-cone and subgroup restrictions in the higher-dimensional embedding

Identify concrete mathematical applications that arise from extending the almost embedding μ^γ(T^*M) → Sh(M, Λ_0^γ) beyond the case where γ ⊂ ℝ^n is a simplicial closed polyhedral proper convex cone and any associated subgroup restrictions on G are imposed, specifically determining how such a generalization to arbitrary convex cones γ and more general subgroups G ⊂ ℝ^n yields new insights or results.

Background

In the higher-dimensional variant, the paper defines μ^γ(T^*M) using ℝ^n-equivariant sheaves with microsupport controlled by the polar cone γ^∨, and proves an almost embedding μ^γ(T^*M) ↪ Sh(M, Λ_0^γ) under the assumption that γ is simplicial. This result generalizes the one-dimensional Tamarkin-Novikov correspondence to certain multidimensional settings.

The author notes that, although it seems feasible to lift the simplicial assumption on γ (and potentially relax restrictions on the subgroup G), they currently do not know of any applications of such a broader generalization. The open issue is therefore not the existence of the generalization per se, but the absence of known use-cases or consequences that would motivate or exploit it.