Kazhdan-Lusztig theory and Soergel bimodules

10 August - 14 August 2015
University of Oregon
Eugene, OR

The goal of this workshop will be to first get a solid handle on Soergel bimodules and the philosophy of algebraic categorification, and then understand the recent proof of the Soergel conjecture in the following paper:

The Hodge theory of Soergel bimodules, by Ben Elias and Geordie Williamson.

The workshop will be led by Ben Elias, and will consist of a combination of lectures and problem sessions.
It will be organized by Nicholas Proudfoot and Alexander Ellis.

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Teaser

(Bait) One of the groundbreaking achievements in modern representation theory is the Localization Theorem (Beilinson-Bernstein, Brylinski-Kashiwara), which gives an equivalence between

O: a certain category of representations of a complex semisimple Lie algebra, introduced by Bernstein-Gelfand-Gelfand, and

P: a certain category of (perverse) sheaves on the associated flag variety.

Both categories (in some sense) categorify the Hecke algebra of the associated Weyl group, and the classes of simple objects in the Grothendieck group give a distinguished basis known as the Kazhdan-Lusztig (KL) basis. Not only does this prove that the KL basis has a number of positivity properties, but it gives representation-theoretic and geometric interpretations of certain coefficients attached to this basis.

This will be the topic of the workshop...but not really.

(Switch) The KL basis has a purely algebraic definition, which makes sense in the Hecke algebra attached to any Coxeter group, not only Weyl groups. The same positivity properties were observed and conjectured for the KL basis in this generality, but unfortunately, general Coxeter groups have neither representation theory nor geometry to back them up. Enter Soergel, who introduced

B: a certain category of bimodules over the coordinate ring of the reflection representation.

This third category is algebraically defined, makes sense for any Coxeter group, and categorifies its Hecke algebra. For Weyl groups, it agrees with the additive subcategory of projective objects in O, or the additive subcategory of semisimple objects in P (this reflects the inherent self Koszul duality of O=P). Soergel conjectured that the indecomposable objects in B should descend to the KL basis, giving a proof of positivity, as well as giving an algebraic interpretation of the coefficients. This conjecture was recently proven, by adapting de Cataldo and Migliorini's proof of the Decomposition Theorem for perverse sheaves to this more general algebraic context.

Unlike categories O and P, category B is very accessible. Recently, diagrammatic tools have been introduced to make it computable and combinatorial. Our goal will be to understand Soergel bimodules, be able to compute with them, and to understand the fundamental arguments in the proof of the Soergel conjecture.

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Program

Here is a preliminary schedule of talks. All lectures will be held in 142 HEDCO. Pastries and coffee will be available in the lecture room every day beginning at 8:30. Ben has also compiled some helpful references.

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Practical information

Participants will be staying in the Carson residence hall on campus. When you arrive on Sunday, you need to check in at the Area Desk of the Living Learning Center, which is on 15th Avenue between University Street and Agate Street; there you will receive a key to your room. The confusing thing is that the building is disconnected (there is a South component and a North component). The component on 15th Avenue is the South component, but the component that contains the Area Desk is the North component (which is not bordered by any street at all). Also, if you arrive after 8pm, you will have to call this number: (541) 346-5686.

There is no public transportation from the airport, so you'll have to take a cab, which should take about half an hour and cost about $30. Since a lot of you will be arriving around the same time, you might consider trying to find each other at the airport and sharing a cab. To help coordinate this, you can use this page.

Here are a few suggestions for where to eat.

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History

This is the fifth of a series of summer workshops that have been funded by an NSF CAREER grant. You can read about the previous ones here:

2010: Operator Algebras and Conformal Field Theory (with André Henriques)
2011: Cluster Algebras and Lusztig's Semicanonical Basis (with David Speyer)
2012: Categorical Representation Theory (with David Ben-Zvi)
2013: Quantum/Affine Schubert Calculus (with Allen Knutson)

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Registration

If you are interested in participating, email me. Please include your school, advisor, and a brief description of your research interests. Funding for accommodations in Eugene (but not for travel) will be available to students and postdocs as long as space and funds remain.

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Participants

If you are on this list, then you can expect me to arrange and pay for your accommodations in Eugene for the duration of the workshop. I will contact each of you to confirm your participation later in the year.