This is a question about the proofs of Kazhdan-Lusztig's conjectures for category $\mathcal{O}$ using higher representation theory (avoiding Beilinson-Bernstein's geometric localization theory). 


Using [Bernstein-Frenkel-Khovanov](https://arxiv.org/abs/math/0002087) and generalizations (Sussan, Stroppel-Mazorchuk, etc), the Kazhdan-Lusztig conjectures in type A are equivalent to the following statement: the classes of the simple modules in their categorification correspond to a "dual canonical basis" in a tensor product representation of $\mathfrak{sl}\_k$ (for appropriately chosen $k$). 


Using [Losev-Webster](https://arxiv.org/abs/1303.1336) and [Webster](https://arxiv.org/abs/1309.3796), that statement can be deduced. They construct a theory of tensor product categorifications in type A (i.e. existence + uniqueness). 


Can [Losev-Webster] be simplified, so that the connection to Soergel's J.AMS paper (www.ams.org/jams/1990-03-02/S0894-0347-1990-1029692-5/) becomes clear? It would be interesting to construct a theory of tensor product categorifications, with the input data being a Dynkin diagram (i.e. a simple Lie algebra). 



Questions about categorification (& combinatorial simplification of the Russian approach to Lusztig's conjectures, in zero & positive characteristic)
This is not a truly independent proof. Both of the papers of mine above use the decomposition theorem for various collections of algebraic varieties, which essentially include the proof of the original KL conjecture as special cases. You should think of the techniques in those papers as a generalization of Soergel's JAMS paper, rather than as an independent approach to the same material.


