Interpolating PBW Deformations for the Orthosymplectic Groups

Abstract

We propose to use interpolation categories to study PBW deformations, and demonstrate this idea for the orthosymplectic supergroups. Employing a combinatorial calculus based on pseudographs and partitions which we derive from a suitable Jacobi identity, we classify PBW deformations in (quotients of) Deligne's interpolation categories for the orthosymplectic groups. As special cases, our classification recovers families of infinitesimal Hecke algebras found by Etingof-Gan-Ginzburg (2005) for the symplectic groups and by Tsymbaliuk (2015) for the orthogonal groups together with their respective standard representations using completely different geometric methods. Our results can be viewed as an extension of these known results to the family of all orthosymplectic groups together with all of their fundamental representations, obtained by novel interpolation techniques for PBW deformations.