Sheets, Jordan classes and induced orbits in the exotic and enhanced modules

TL;DR

This paper explores the geometry of exotic and enhanced nilpotent cones, introducing Jordan stratification and classifying sheets in these modules, advancing understanding of their orbit structures and representations.

Contribution

It introduces the theory of induced orbits in exotic and enhanced nilpotent cones, generalizing classical results and providing a classification of sheets in these modules.

Findings

Development of Jordan stratification for exotic and enhanced cones

Introduction of induced orbit theory in these modules

Classification of sheets in exotic and enhanced modules

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.