Notes on Cells of Harish-chandra Modules and Special Unipotent Representations

Let GR denote the real points of a connected complex reductive algebraic group defined over R). (For the basic results on cells below, this class of groups is unnecessarily restrictive. But we shall need the assumption when we discuss unipotent representations below.) Let gR denote the Lie algebra of GR and write g for the complexification of gR. Let KR denote a maximal compact subgroup in G, and write K for its complexification. Let HCλ be the category of Harish-Chandra modules with infinitesimal character λ. Cells are designed to capture some of the information of tensoring Harish-Chandra modules with finite-dimensional modules. More precisely, given two objection X and Y in HCλ, write X > Y if there exists a finite-dimensional representation of G appearing the tensor algebra T (g) such that Y appears a subquotient of X⊗F . Write X ∼ Y if both X > Y and Y > X . Then ∼ is an equivalence relation.