Deligne-Lusztig Theory

Workshop on Algebra and Representation Theory, Held on Oregonian Grounds

13 August - 17 August 2018
University of Oregon
Eugene, OR

If you believe in the classification of finite simple groups, you know that every one of them falls into one of the following families:

  • the cyclic groups of prime order (boring),
  • the alternating groups,
  • the simple groups of Lie type (such as PSL(n,q) or PSp(2n,q)),
  • the 26 sporadic groups.

    This means that most of the finite (almost) simple groups have an underlying geometric nature, since they come from reductive groups over finite fields (called finite reductive groups). In 1976 Deligne and Lusztig [DL] introduced a family of algebraic varieties — now referred to as "Deligne-Lusztig" varieties — whose cohomology allows to construct all the irreducible representations of these groups. Shortly afterwards, in a series of papers culminating by the book [L2], Lusztig gave a complete classification of these representations in the case where the reductive group has connected center.

    The purpose of these lectures is to introduce you to the beautiful geometry of Deligne--Lusztig varieties. We shall not follow the usual textbooks [C] and [DM] on character theory of finite reductive groups, but we will rather aim at understanding a specific example of such a variety, the Coxeter variety [L1]. Building on the examples of the general linear group GL(n,q) and the symplectic group Sp(4,q), we will illustrate many general conjectures relating the cohomology of Deligne--Lusztig varieties, complex reflection groups, braid groups and their Hecke algebras.

    (For the references, see here.)

    A more thorough teaser with an overview of the daily topics can be found here.

    The workshop will be led by Olivier Dudas, and will consist of a combination of lectures and problem sessions. Jay Taylor will be on hand as our secondary lecturer and exercise helper.
    It will be organized by Ben Elias, under the watchful eye of Nicholas Proudfoot.

    Background reading

    If you plan to attend WARTHOG, please read this page which describes the background we expect you to acquire before the workshop begins. The link also contains many helpful references.

    Schedule of lectures

    A preliminary schedule can be found here and is subject to revision. All talks are in HEDCO 146.


    If you are interested in participating, email Ben Elias. Please include your school, advisor, and a brief description of your research interests. Please place [WARTHOG] in the subject line of your email.
    Funding for accommodations in Eugene (but not for travel) will be available to students and postdocs as long as space and funds remain. Let Ben know if you have any special needs or requests.

    Participant List

    If you are on this list then I am expecting you to come to the workshop, and you should expect (unless you have an asterisk by your name) that accommodations will be provided.

    Logistics for Participants

    General Schedule: There will be lectures and exercises Monday through Friday, from approximately 9AM to 5PM, with gaps between and a space for lunch. Wednesday is a half-day, with the afternoon being left open for an organized hiking trip to nearby Spencer Butte. Wednesday evening we will have a pizza party at a local park, sponsored by the workshop. On Monday, Tuesday, and Thursday evening there will be an optional, organized exercise session.

    Food: Starting at 830AM each day of the workshop, coffee and pastries will be catered to the lecture room. They take away the coffee at 1PM! Aside from this and the pizza party, participants will have to pay for their own food. Carson residence hall has a cafeteria with more substantial breakfast for purchase. There will be time to eat lunch at many local restaurants, and restaurant recommendations will be posted at a later date.

    Accommodations: Participants will be staying in a dormitory on campus. The cost of this will be covered by the workshop. Details will be emailed shortly before arrival.

    Accommodations and arrival/departure timing: It is assumed that participants will arrive in Eugene on Sunday before the workshop, and leave on either Friday night or Saturday after the workshop. Housing will be available for participants leaving on Sunday after the workshop as well, if necessary, but it should be requested. Participants arriving earlier or leaving later should contact me for possible options, but housing is not guaranteed.

    Accommodations and arrival: When you arrive, 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. When you depart, you will need to return your key to the Area Desk as well.

    Transportation costs: Unless you have been told otherwise, we will not be covering your transportation costs. If other options have not availed you, please email me (Ben Elias) and we can discuss options. (If we do end up covering your transportation, make sure to fly to EUG during the expected dates, or contact me to discuss price comparisons.)

    Flying to Eugene: You should fly to the Eugene airport, code EUG. There are direct flights connecting Eugene to Seattle, Portland, Salt Lake City, Denver, San Francisco, and Los Angeles. In my personal experience, if you fly through San Francisco then your flight will be delayed and you will miss your connection, so try to find another option if possible. It is also possible to fly to Portland, which is a little over a two hour drive from Eugene - there are also shuttles and a train which can get you from Portland to Eugene, but this is typically more complicated then just flying directly to Eugene.

    Airport to Campus: There is no public transportation from the airport, so you'll have to take a cab, which should take about twenty minutes and cost about $30. Cabs in Eugene are run by a host of private companies, some better than others. Avoid Oregon Taxi if you can: Muscle Arm Taxi, ECabs, Eugene City Hybrid Taxi (more expensive), GOTaxi, Emerald city cab, these have all been good to me. If they ask which route to take, just say to take 99. 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. A google group will be set up closer to the time of the conference to help coordinate this. Cabs back to the airport will be organized later in the conference.

    Getting around: During the conference you will be able to walk everywhere you need to go. If you are staying longer in Eugene or in Oregon for recreational purposes, it is easy to rent a bicycle, and car rentals (within the city, not from the airport) are very cheap.