Sheets, Jordan classes and induced orbits in the exotic and enhanced modules
Filippo Ambrosio, Giovanna Carnovale, Francesco Esposito, Neil Saunders, Lewis Topley
TL;DR
The paper develops a Jordan-theoretic framework for two polar representations—the enhanced module for GL_n and the exotic module for Sp_{2n}—to extend the classical sheets/Jordan-class theory from the adjoint representation to these settings. It proves that V // G and Ṽ // G are affine, with irreducible, normal complete-intersection fibers, and that the inclusions V ⊆ Ṽ induce a poset isomorphism doubling orbit dimensions. A canonical Jordan–Kac–Vinberg decomposition is established, and enhanced/exotic Jordan classes are defined and classified; their closures are described via induction from Levi subgroups, mirroring Lusztig–Spaltenstein induction and Borho-type criteria. The authors show transitivity of induction and provide explicit combinatorial rules relating induced orbits to bipartitions, yielding a unified description of sheets for both enhanced and exotic modules. These results generalize the adjoint-space theory to symmetric pairs and polar representations, with potential implications for geometric realisations of Hecke algebras and related representation-theoretic structures.
Abstract
Kato developed an exotic Deligne-Langlands correspondence using a geometric model for the multiparameter affine Hecke algebra of type C, based on his exotic nilpotent cone. Achar-Henderson and Springer showed that this exotic nilpotent is intimately related to another, apparently simpler variety called the enhanced nilpotent cone. Each of these is defined as the Hilbert nullcone of a polar module, the exotic Sp(2n)-module and the enhanced GL(n)-module, respectively. In this paper we conduct a detailed study of the geometry of these two modules, by introducing the Jordan stratification, simultaneously generalising classical results on the adjoint representation as well as the symmetric space associated to (gl(2n), sp(2n)). One of the key tools we develop is the theory of induced orbits in the enhanced and exotic nilpotent cones, following the work of Lusztig-Spaltenstein. Our main application is a classification of sheets in these modules, inspired by a theorem of Borho.