Wild orbits and generalised singularity modules: stratifications and quantisation

We study truncated gauge-orbits through principal parts of irregular-singular connection germs, in the untwisted/unramified setting: for any connected complex reductive structure group $G$, in the general multilevel case. In particular, we compute the stabilisers of the formal normal forms using filtrations of Levi root systems, showing that they are connected. When the residue is semisimple we then stratify the space of orbits by the conjugacy class of the stabilisers, i.e., by quotients of root-valuation strata; the dense stratum corresponds to the generic setting of isomonodromic deformations, \`a la Jimbo--Miwa--Ueno. Then we adapt a result of Alekseev--Lachowska to deformation-quantise nongeneric orbits. The $\ast$-product involves affine-Lie-algebra modules, extending: (i) the parabolic Verma modules (in the case of regular singularities); and (ii) the `singularity' modules of F.--R. (in the case of generic irregular singularities). They contain Whittaker vectors for the Gaiotto--Teschner/Bonelli--Maruyoshi--Tanzini Virasoro pairs in irregular Liouville conformal field theory, and they provide all the quotients obtained by leaving the aforementioned dense strata. We also construct Shapovalov forms for the corresponding representations of truncated-current Lie algebras, which enter into the category $\mathcal O$ of Chaffe--Topley; and we state a sharp irreducibility criterion. Finally, we use these representations to construct vector bundles of genus-zero vacua/covacua, equipped with flat connections \`a la Knizhnik--Zamolodchikov/Reshetikhin.