Demazure operators for double cosets

TL;DR

This paper introduces Demazure operators for double cosets in Coxeter systems, forming a basis for morphism spaces in the nilCoxeter category and generalizing Frobenius extension theorems to support singular Soergel bimodules.

Contribution

It defines Demazure operators for double cosets, constructs the nilCoxeter category with generators and relations, and generalizes Frobenius extension theorems to this setting.

Findings

Demazure operators form a basis for morphism spaces.

The nilCoxeter category is presented by generators and relations.

A generalized Frobenius extension theorem ensures proper behavior of singular Soergel bimodules.

Abstract

For any Coxeter system, and any double coset for two standard parabolic subgroups, we introduce a Demazure operator. These operators form a basis for morphism spaces in a category we call the nilCoxeter category, and we also present this category by generators and relations. We prove a generalization to this context of Demazure's celebrated theorem on Frobenius extensions. This generalized theorem serves as a criterion for ensuring the proper behavior of singular Soergel bimodules.