Return to Question

The goal of to give a uniform approach to the Russian proofs of Lusztig's conjectures using higher representation theory (extending [Elias-Williamson]'s work on the Kazhdan-Lusztig conjecture). Instead of using Soergel's approach (using bimodules), the goal is to rely exclusively on Bernstein-Frenkel-Khovanov's theory (higher representation theory). Here are some questions; any references would be appreciated.

(A) Kazhdan-Lusztig conjecture (proven by Beilinson-Bernstein; reproven by [Elias-Williamson] in greater generality).

(A.1) 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 $k$).

(A.2) 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).

(Q.1) 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? The goal is to construct a theory, with the input data being a Dynkin diagram (i.e. a simple Lie algebra).

(Q.2) In types B/C/D, the correct set-up is "quantum symmetric pairs", following the paper Bao-Shan-Wang-Webster. What obstacles does one encounter when constructing an analogue of [Losev-Webster] for $\mathfrak{g}=\mathfrak{sp}_4$? Rouquier's theory of tensor product categorifications is also relevant (and hasn't been published yet).

(B) A new approach to the Kazhdan-Lusztig-Vogan conjectures, for Harish-Chandra modules (now a theorem).

(A.1) Recent work of Bezrukavnikov-Vilonen (https://arxiv.org/abs/1510.08343) is a first step in this direction; there's more to be done.

(A.2) The categorical set-up (following [Bernstein-Frenkel-Khovanov]) is an easy exercise. This is work in progress with P.Trapa (sl_2 categorification).

(Q.3) Generalizing this, and proving the KLV conjectures, will take a lot more work.

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).

The goal of to give a uniform approach to the Russian proofs of Lusztig's conjectures using higher representation theory (extending [Elias-Williamson]'s work on the Kazhdan-Lusztig conjecture). Instead of using Soergel's approach (using bimodules), the goal is to rely exclusively on Bernstein-Frenkel-Khovanov's theory (higher representation theory). Here are some questions; any references would be appreciated.

(A) Kazhdan-Lusztig conjecture (proven by Beilinson-Bernstein; reproven by [Elias-Williamson] in greater generality).

(A.1) 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 $k$).

(A.2) 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).

(Q.1) 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? The goal is to construct a theory, with the input data being a Dynkin diagram (i.e. a simple Lie algebra).

(Q.2) In types B/C/D, the correct set-up is "quantum symmetric pairs", following the paper Bao-Shan-Wang-Webster. What obstacles does one encounter when constructing an analogue of [Losev-Webster] for $\mathfrak{g}=\mathfrak{sp}_4$? Rouquier's theory of tensor product categorifications is also relevant (and hasn't been published yet).

(B) Lusztig's conjectures for algebraic groups in positive characteristic (proven by Andersen-Jantzen-Soergel, Kashiwara-Tanisaki, Kazhdan-Lusztig). New proof given in [Arkhipov-Bezrukavnikov-Ginzburg.

(A.1) Following [Chuang-Rouquier], on pg 58 on [Williamson-Riche], (https://arxiv.org/abs/1512.08296), the author construct a categorical action of affine $sl_p$ using blocks of $\textbf{SL}_n$-representations in positive characteristic. Giving a new proof requires the following.

(Q.1) Show that that the combinatorics of Lusztig's conjectures (affine KL polynomials) matches up with the combinatorics of the affine $sl_p$ action.

(Q.2) Extend [Losev-Webster], and construct a theory of tensor product categorifications for this datum. Prove a uniqueness + existence theorem, and complete the proof.

(Q.3) Can Bezrukavnikov-Mirkovic's techniques be adapted to this setting (i.e. algebraic groups, instead of Lie algebras)? See the introduction of https://arxiv.org/abs/1001.2562 for more details.

(A.4) What goes wrong in small characteristic? Can [Williamson]'s results in that direction be reproved using these techniques?

(C) A new approach to the Kazhdan-Lusztig-Vogan conjectures, for Harish-Chandra modules (now a theorem).

(A.1) Recent work of Bezrukavnikov-Vilonen (https://arxiv.org/abs/1510.08343) is a first step in this direction; there's more to be done.

(A.2) The categorical set-up (following [Bernstein-Frenkel-Khovanov]) is an easy exercise. This is work in progress with P.Trapa (sl_2 categorification).

(Q.3) Generalizing this, and proving the KLV conjectures, will take a lot more work.

The goal of to give a uniform approach to the Russian proofs of Lusztig's conjectures using higher representation theory (extending [Elias-Williamson]'s work on the Kazhdan-Lusztig conjecture). Instead of using Soergel's approach (using bimodules), the goal is to rely exclusively on Bernstein-Frenkel-Khovanov's theory (higher representation theory). Here are some questions; any references would be appreciated.

(A) Kazhdan-Lusztig conjecture (proven by Beilinson-Bernstein; reproven by [Elias-Williamson] in greater generality).

(A.1) 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 $k$).

(A.2) 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).

(Q.1) 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? The goal is to construct a theory, with the input data being a Dynkin diagram (i.e. a simple Lie algebra).

(Q.2) In types B/C/D, the correct set-up is "quantum symmetric pairs", following the paper Bao-Shan-Wang-Webster. What obstacles does one encounter when constructing an analogue of [Losev-Webster] for $\mathfrak{g}=\mathfrak{sp}_4$? Rouquier's theory of tensor product categorifications is also relevant (and hasn't been published yet).

(B) A new approach to the Kazhdan-Lusztig-Vogan conjectures, for Harish-Chandra modules (now a theorem).

(A.1) Recent work of Bezrukavnikov-Vilonen (https://arxiv.org/abs/1510.08343) is a first step in this direction; there's more to be done.

(A.2) The categorical set-up (following [Bernstein-Frenkel-Khovanov]) is an easy exercise. This is work in progress with P.Trapa (sl_2 categorification).

(Q.3) Generalizing this, and proving the KLV conjectures, will take a lot more work.

(A.2) Show that that the combinatorics of Lusztig's conjectures (affine KL polynomials) matches up with the combinatorics in the affine $sl_p$ action.

(A.3) Extend [Losev-Webster], and construct a theory of tensor product categorifications for this datum. Prove a uniqueness + existence theorem, and complete the proof.

(A.4) What goes wrong in small characteristic? Can [Williamson]'s results in that direction be reproved using these techniques?

(Q.1) Show that that the combinatorics of Lusztig's conjectures (affine KL polynomials) matches up with the combinatorics of the affine $sl_p$ action.

(Q.2) Extend [Losev-Webster], and construct a theory of tensor product categorifications for this datum. Prove a uniqueness + existence theorem, and complete the proof.

(Q.3) Can Bezrukavnikov-Mirkovic's techniques be adapted to this setting (i.e. algebraic groups, instead of Lie algebras)? See the introduction of https://arxiv.org/abs/1001.2562 for more details.

(A.4) What goes wrong in small characteristic? Can [Williamson]'s results in that direction be reproved using these techniques?

(C) A new approach to the Kazhdan-Lusztig-Vogan conjectures, for Harish-Chandra modules (now a theorem).

(A.1) Recent work of Bezrukavnikov-Vilonen (https://arxiv.org/abs/1510.08343) is a first step in this direction; there's more to be done.

(A.2) The categorical set-up (following [Bernstein-Frenkel-Khovanov]) is an easy exercise. This is work in progress with P.Trapa (sl_2 categorification).

(Q.3) Generalizing this, and proving the KLV conjectures, will take a lot more work.