Lie Groups, Lie Algebras and their Representations

(page last edited October 31, 2015)
The Lie Theory Workshop series of weekend conferences has been meeting several times a year since 1990.  Most years the NSF supports graduate students and postdoctoral participants, and the institution at which the workshop takes place also provides some support.  Most, but not all, of the workshops take place at West Coast universities.  Each workshop program is determined by the local organizers, and the workshop series program itself is organized by a committee currently consisting of Geoff Mason (University of California, Santa Cruz)), Susan  Montgomery (University of Southern California) and Joseph Wolf (University of California at Berkeley).

The purpose of the program is to communicate results and ideas rather than to deliver polished presentations. The program also serves to acquaint graduate students in this area with a range of researchers in Lie theory and representation theory.  The fall workshop is scheduled for November 7-8, a spring workshop is scheduled for April 16-17, 2016, and another spring workshop is scheduled for May 30 - June 4, 2016.  Here is the 2015-16 schedule as of September 14, 2015; some  earlier workshops are listed below that.

   Fall 2015.  Louisiana State University (LSU), Baton Rouge, November 7-8, 2015.  The local organizers are Pramod Achar (, Gestur Olafsson (, Daniel Sage (  and Milen Yakimov (  The emphasis will be on analytic and algebraic aspects of Lie theory.  The local web page for the conference is , and the program is

     Saturday November 7:
                   9:30-10:00 Coffee

                 10:00-11:00 Joseph Wolf (Berkeley)                                              Real Lie Groups and Complex Flag Manifolds
                                                                                                                        Abstract: Let G be a complex simple direct limit group.  Let G_R be a real form of G that corresponds to
                                                                                                                        an hermitian symmetric space.  I'll describe the corresponding bounded symmetric domain in the context
                                                                                                                        of the Borel embedding, Cayley transforms, and the Bergman-Shilov boundary.  Let Q be parabolic    
                                                                                                                        subgroup of G.  In finite dimensions this means that G/Q is a complex projective variety, or equivalently   
                                                                                                                        has a Kaehler metric invariant under a maximal compact subgroup of G.  Then I'll show just how the
                                                                                                                        bounded symmetric domains describe cycle spaces for open G_R orbits on G/Q.  These cycle spaces
                                                                                                                        include the complex bounded symmetric domains.  In finite dimensions they are tightly related to moduli
                                                                                                                        spaces for compact Kaehler manifolds and to representations of semisimple Lie groups; in infinite
                                                                                                                        dimensions there are more problems than answers.  Finally, time permitting, I'll indicate how some of this
                                                                                                                        goes over to real and to quaternionic bounded symmetric domains.

                 11:15-12:15 Raul Gomez (Cornell University)                            Invariant trilinear forms on induced representations of real rank one groups. (Joint work with B. Speh)
                                                                                                                   Abstract: Bernstein and Reznikov introduced a triple integral formula to describe a family of invariant trilinear
                                                                                                                   forms for induced representations of $PGL(2, \mathbb{R})$. However, they left open the question of
                                                                                                                   computing the full space of invariants. Using the definition of the Schwartz space of a Nash manifold,
                                                                                                                   together with some homological algebra, we will show how to describe the remaining trilinear forms. We will
                                                                                                                   then show how these results can be extended to induced representations of real rank one groups, refining in
                                                                                                                   this way some results previously obtained by Clerc, Kobayashi, {\O}rsted and Pevzner
                  2:00-  3:00 Ben Harris (Simon's Rock College)                         Regular Elliptic Discrete Spectra
                                                                                                                   Harmonic analysis on the unit circle involves writing a function as an infinite direct sum while harmonic analysis
                                                                                                                   on the real line involves writing a function as a continuous integral. The first decomposition is discrete while the
                                                                                                                   second is continuous. Philosophically, this is because the first group is compact and the second group is non-compact.
                                                                                                                   In general, we say that an element of a Lie group is elliptic if it is contained in a compact subgroup. In this talk, we will
                                                                                                                   survey old and new results linking elliptic elements with the existence of certain types of discrete spectra in abstract
                                                                                                                   harmonic analysis.

                  3:15-  4:15 Hadi Salmasian (University of Ottawa)                   The Capelli problem and spectrum of invariant differential operators
                                                                                                                    Abstract:  The Capelli identity is a mysterious result in classical invariant theory with a long history. It was
                                                                                                                    demystified by Roger Howe, who used it in an ingenious and elegant fashion in the modern theory of
                                                                                                                    representations of real reductive groups. In this talk, I will introduce  the Capelli identity, and exhibit the
                                                                                                                   relationship between an extension of this identity with certain polynomials which describe the spectrum of
                                                                                                                   invariant differential operators on symmetric superspaces. These polynomials are analogs of the Jack and
                                                                                                                    Knop-Sahi/Okounkov-Olshanski polynomials. This talk is based on a joint project with Siddhartha Sahi.   

                  4:30-  5:30 Tsao-Hsien Chen (Northwestern University)            Springer theory for symmetric spaces
                                                                                                                   Abstract: We consider the Springer correspondence in the case of symmetric spaces. In this setting various new
                                                                                                                   phenomena  occur which are not present in the classical Springer theory. For example, we obtain representations of
                                                                                                                   (the Tits extension) of the braid group rather than just Weyl group representations. These representations come from
                                                                                                                   cohomology of families of Hessenberg varieties. In the situations we consider the Hessenberg varieties can be interpreted
                                                                                                                   as classical objects in algebraic geometry: Jacobians and moduli spaces of vector bundles on curves with extra
                                                                                                                   structure, Fano varieties of k-planes in the intersection of two quadrics, etc. This is joint work with Kari Vilonen and Ting Xue.

     Sunday November 8:
                 8:30- 9.00 Coffee

                 9:00-10:00 Cheng-Chiang Tsai (MIT)                                        Stratification of affine Springer fibers
                                                                                                                   Abstract: We describe an inductive stratification for arbitrary affine Springer fibers. Each stratum is an iterated étale
                                                                                                                   locally trivial fiber bundle over a slight generalization of affine Springer fibers for smaller groups. If time permits, we will
                                                                                                                   describe some examples of the fibers of these fiber bundles, and discuss further expectations.

               10:10-11:10 Sean Rostami (University of Wisconsin)                  On Fixers of Stable Functionals
                                                                                                                  Abstract: The epipelagic representations of Reeder-Yu, a generalization of the "simple supercuspidals" of
                                                                                                                  Gross-Reeder, are certain low-depth supercuspidal representations of reductive algebraic groups G. From a
                                                                                                                  linear functional f (on a vector space V coming from a Moy-Prasad filtration) which is stable in the sense of
                                                                                                                  Geometric Invariant Theory, such a representation can be constructed. It is known that these representations
                                                                                                                  are compactly induced from the fixer in G of f and it is important to identify all the elements that belong to this
                                                                                                                  fixer in as explicit and familiar a way as possible. In the situation of Gross-Reeder, there is a uniform answer,
                                                                                                                  which I finished in summer 2015. In the more general case, qualitatively different answers can occur and
                                                                                                                 interesting objects (e.g. the Legendre symbol) appear, even when the context (i.e. G and V) is fixed. This work
                                                                                                                 is in-progress.

               11:30-12:30 Peter Fiebig (FAU Erlangen-Nürnberg)                       Intersection matrices and modular representation theory
                                                                                                                   Abstract:  We review the approach of Andersen, Jantzen and Soergel to Lusztig's conjecture on
                                                                                                                  the irreducible rational characters of reductive algebraic groups. Then we show that the problem of the
                                                                                                                  degeneration of categoric structures for ``small'' primes cumulates in the problem of understanding a certain
                                                                                                                  matrix constructed in terms of the root system.

               12:40-  1:40 Christoper Bremer (LSU)                                        

More schedule details will be posted here and on the local web page ( as the information becomes available.

Hotel information:  A block of rooms is reserved at the Staybridge Suites-Baton Rouge, 4001 Nicholson Drive, Baton Rouge 70808,  Phone 225-456-5430,
The negotiated reduced price is $97 + tax per night.  Workshop participants have to make their own reservations, but must let the hotel know that they are part of the workshop in order to obtain the reduced rate.   The code is LTW.

Spring 2016:  University of Washington, Seattle, April 16-17, 2016.  The emphasis will be Hopf algebras and actions.  James Zhang ( is the local organizer.

Spring 2016:: The Lie Theory Workshop will be in cooperation with the Second US-Mexico Conference on Representation theory, Categorification, and Noncommutative Algebra, University of Southern Caaifornia (USC), May 30 - June 4, 2016.  The organizers are Aaron Lauda (, Andrea Appel (, and Christof Geiss (


Spring 2015:  University of Alberta, Edmonton, May 16-17, 2015.  The local organizers were Arturo Pianzola ( and Terry Gannon (  The emphasis was on geometric methods in Lie theory.  Speakers were  Kai Behrend (UBC), Jon Brundan (Oregon), Thomas Creutzig (Alberta), David Evans (Cardiff),Johanna Hennig (UCSD) , Shashank Kanade (Rutgers)  and Johannes Walcher (McGill).
Winter 2015:  University of California, Riverside, February 14-15, 2015.  The local organizers were Jacob Greenstein (,  Vyjayanthi Chari ( and Carl Mautner.  The emphasis was on representation theory.  The list of speakers included Achar, Dobrovolska, Friedlander, Lauda, Loseu, Pevtsova, Rosso, Yun and Zhu.  The conference web site is

The Fall 2014 workshop took place at the  University of Washington (Seattle), November 15-16, 2014.  The local organizer was James Zhang and the emphasis will be on Hopf algebras and actions.

The Spring 2014 workshop took place at the University of Southern California (USC), May 19-22, 2014, local organizers were Aravind Asok, Robert Guralnick, Aaron Lauda and Susan Montgomery .  The conference was joint with a conference in celebration of the 70th birthday of Eric Friedlander.  The Friedlander conference web site is

The Winter 2014 workshop took place at Stanford University, February 1-2.  The local organizers were Apoorva Khare, Daniel Bump and Persi Diaconis. 
The Fall 2013 workshop took place at the University of California, Berkeley, October 12-13.  The local organizer was Joseph Wolf (

The Spring 2013 workshop took place at the University of Oregon, April 27-28.  The local organizer was Jon Brundan (  The emphasis was on tensor categories and categorification.

The Winter 2013 workshop in the program took place at the University of California San Diego (UCSD), January 19-20.  The local organizers were Alireza Salehi Golsefidy, Daniel Rogalski and Efim Zelmanov.  The emphasis was on Lie groups and related areas.

The Fall 2012 workshop took place at Louisiana State University, October 6-7.  The local organizers were Gestur Olafsson and Milen Yakimov.  The emphasis was on  Representation theory and analysis on homogeneous spaces.

The Spring 2012 workshop took place at the University of Southern California (USC), May 5-6.  The local organizers were Susan Montgomery and  Miodrag Iovanov.  The emphasis was on  Hopf algebras and related areas.

The Winter 2012 workshop took place at Stanford University.  The local organizers were Apoorva Khare, Daniel Bump and Anne Schilling.  The emphasis was on Quantum Groups and related areas.