Endomorphism Algebras in Coxeter Categorifications and Harish-Chandra 2-Categories
Given the data of a Coxeter system (W,S), a Coxeter categorification is a 2-category in which the objects are subsets of S, the generating 1-morphisms categorify induction and restriction functors associated to parabolic subgroups, and the generating 2-morphisms impose certain coherence conditions and structural properties among the 1-morphisms. Of particular interest is the structure of the 2-homomorphism spaces of these 1-morphisms. Furthermore, given a connected, reductive, algebraic group G over an algebraically closed field k, a chosen Frobenius endomorphism F on G determines a parameter q in k, and the Weyl group of G gives rise to a Coxeter system. When this system is of rank 1, we construct by generators and relations an extension of the Coxeter categorification, independent of q, where the 2-homomorphism spaces are free modules of finite rank over the ring of Laurent polynomials with integer coefficients. An explicit description of the 2-homomorphism spaces between generating 1-morphisms is given, along with an algorithm lifting these descriptions to the 2-homomorphism spaces of arbitrary 1-morphisms. Then a nontrivial 2-functor from this 2-category is constructed into the 2-category of bimodules. Some conjectural constructions are given in the case that W has arbitrary finite rank, in particular a proposal for the endomorphism ring of the generating 1-morphism from the empty set to S that is an extension of an algebra introduced by Marin.