Questions about categorification (& combinatorial simplification of the Russian approach to Lusztig's conjectures, in zero & positive characteristic) [closed]

Question

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 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 and Webster, 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).

Answer 1

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.

Answer 2

Concerning your other question (4), it's probably too early to say what will eventually happen with Lusztig's various conjectures in prime characteristic. By now there is a lot of literature, following Williamson's cautionary observations about Lusztig's Santa Cruz conjecture in 1979 modeled on the statement of the Kazhdan-Lusztig conjecture. The modular situation turns out to be even more complicated than expected. See for example the papers and preprints listed on Achar's and Williamson's homepages, such as here or here.

While the earlier work by Andersen-Jantzen-Soergel proved Lusztig's first conjecture for all "sufficiently large" primes $p$, the explicit bound on $p$ found afterward by Fiebig is extremely large. There remains a serious problem about "intermediate" primes, and no firm conjecture about what happens when $p$ is smaller than the Coxeter number. While the current work encourages hopes for a solution, the end result is likely to be far more complicated than the Kazhdan-Lusztig conjecture. It's definitely worthwhile to seek alternative approaches but probably premature to expect a simple answer involving just the Kazhdan-Lusztig polnomials for an affine Weyl group.