2-representations of Soergel bimodules

Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer, Xiaoting Zhang

Published 2019-06-27Version 1

In this paper we study the graded 2-representation theory of Soergel bimodules for a finite Coxeter group. We establish a precise connection between the graded 2-representation theory of this non-semisimple 2-category and the 2-representation theory of the associated semisimple asymptotic bicategory. This allows us to formulate a conjectural classification of graded simple transitive 2-representations of Soergel bimodules, which we prove under certain assumptions. Along the way we also show several results and provide examples which are interesting in their own right, e.g. we show that Duflo involutions have a Frobenius structure (in a certain quotient) and give an example of a left cell for which the underlying algebra of the cell 2-representation is not symmetric.