Equivalences entre conjectures de Soergel

TL;DR

This paper investigates the relationships between different conjectures in Soergel's categorification of Hecke algebras, establishing equivalences under certain representations and simplifying cases over the real numbers.

Contribution

It proves that for certain representations, Soergel's conjecture over different categories are equivalent, especially highlighting the case over the real numbers with the geometric representation.

Findings

Equivalence of Soergel's conjecture for different representations V and V'.

Reduction of the conjecture to the geometric representation over the real numbers.

Establishment of conditions under which conjectures are equivalent.

Abstract

Soergel's category B_k(V) over a field k is defined from a Coxeter system (W,S) and a k-linear representation V of W. It's a categorification of the Hecke algebra of (W,S). In this article we prove that for some representations V and V' of W, Soergel's conjecture over B_k(V') is equivalent to that over B_k(V). In particular, when k=IR we can choose V' to be the geometric representation.