## New bases of some Hecke algebras via Soergel bimodules

**Advances in Math.** 228 (2011) 1043-1067.
I introduce a **new set of bases ** for Hecke algebras related to extra-large Coxeter groups, coming from the theory of Soergel bimodules. If my "Forking path conjecture" (see the paper "Gentle Introduction to Soergel bimodules" above) is correct, the same kind of bases would exist for the symmetric group. I believe that they have a deep meaning related to the Hecke category and the p-canonical basis.

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June 29, 2011