Sur la cat\'egorie des bimodules de Soergel

TL;DR

This paper provides a combinatorial description of morphism spaces in the Soergel bimodule category associated with Coxeter systems, linking it to the principal block of category O, thus advancing understanding of their structure.

Contribution

It introduces a combinatorial framework for morphism spaces in Soergel bimodules and relates it to the morphisms in the principal block of category O.

Findings

Explicit combinatorial description of morphism spaces in B

Correspondence between morphisms in B and O_0-proj

Enhanced understanding of the structure of Soergel bimodules

Abstract

The Soergel category B of a Coxeter system (W,S) is a bimodule category over a polynomial algebra on which W acts. It's a categorification of the Hecke Algebra of (W,S). In this article we give a combinatorial description of morphism spaces in B. As a corollary, we give an analogous description of the morphisms in O_0-proj, where O_0 is the principal block of the BGG category O. ----- La cat\'egorie B de Soergel d'un syst\`eme de Coxeter (W,S) est une cat\'egorie de bimodules sur une alg\`ebre de polyn\^omes sur laquelle W agit. C'est une cat\'egorification de l'alg\`ebre de Hecke de (W,S). Dans cet article nous donnons une description combinatoire des espaces de morphismes dans B. En corollaire, on obtient une description analogue des morphismes dans O_0-proj, o\`u O_0 est le bloc principal de la cat\'egorie O de BGG.