Categorical Representation Theory

13 August - 17 August 2012
University of Oregon
Eugene, OR


Classical harmonic anaylsis describes the decomposition of spaces of functions under the action of symmetries. Geometric representation theory, in which vector spaces of functions are enhanced to categories of sheaves, calls for a new brand of "geometric" or categorical harmonic analysis. This workshop will explore the emerging theory of group actions on categories, combining tools from homotopy theory, motivations from topological field theory, and applications to classical representation theory of Lie groups. These applications require the injection of some machinery (such as D-modules, ∞-categories, and Hochschild homology) which will be reviewed as needed, but we'll emphasize intuitions and simple analogies that work surprisingly well.

We will discuss three settings for group actions. Finite groups provide a toy model in which we can identify the categorified analogues of basic themes in representation theory, such as Frobenius algebras, class functions, characters, induced representations, double cosets (or Hecke) algebras, and Morita equivalence. Topological field theory provides an invaluable organizing principle for these structures which we'll use throughout [7],[8],[10],[11],[12].

Our second setting is that of affine algebraic groups and their algebraic actions on derived categories (for example categories of quasicoherent sheaves on homogenous spaces). We'll see how all of the themes from the finite setting generalize smoothly to this setting, once some homotopical machinery is introduced [3].

The most challenging and rewarding setting is that of "locally constant" actions of algebraic groups, which are more closely analogous to smooth representations of p-adic groups. The two main classes of examples are categories of D-modules on homogeneous spaces and categories of representations of Lie algebras. The seminal Beilinson-Bernstein localization theorem relates the two, providing a powerful geometric tool to study questions in representation theory [1],[5],[9].

The symmetries of the Beilinson-Bernstein construction are provided by the finite Hecke category, a categorified analogue of the group algebra of the Weyl group. We will discuss the associated topological field theory and in particular the characters of Hecke representations, Lusztig's character sheaves. We will conclude with an application of this categorified character theory to Harish Chandra's classical theory of characters of infinite dimensional representations of Lie groups [2],[4],[6],[10].

[1] Beilinson and Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves.

[2] Beilinson, Ginzburg, and Soergel, Koszul duality patterns in representation theory.

[3] Ben-Zvi, Francis, and Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry.

[4] Ben-Zvi and Nadler, The character theory of a complex group.

[5] Ben-Zvi and Nadler, The symmetries of Beilinson-Bernstein localization (draft available soon).

[6] Ben-Zvi and Nadler, Geometric theory of Harish Chandra characters (in progress).

[7] Freed, Higher algebraic structures and quantization.

[8] Freed, Hopkins, Lurie, and Teleman, Topological Quantum Field Theories from Compact Lie Groups.

[9] Frenkel and Gaitsgory, Local geometric Langlands correspondence and affine Kac-Moody algebras.

[10] Lurie, On the classification of topological field theories.

[11] Müger, From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories.

[12] Ostrik, Module categories, weak Hopf algebras and modular invariants.


Here are a couple more preprints that some folks might find useful:

Ben-Zvi, Francis, Nadler, Morita equivalence for convolution categories.

Ben-Zvi, Nadler, Beilinson-Bernstein localization over the Harish-Chandra center.


The workshop will be led by David Ben-Zvi.



Program

Schedule (with notes by Pavel Safronov)

Syllabus

Participants



Lecture notes

Here are some lecture notes that were written specifically for this workshop.

Tanaka, An introduction to infinity-categories.

Ganev, Representations of finite groups.

Garcia Raboso, Stable infinity categories.



Problem sets

Day One

Day Two

Day Four



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.

At 19th Avenue and Agate Street you can find a couple of casual restaurants, a slightly fancier Italian restaurant, a bakery, and an ice cream shop; we will probably gather informally in this area in the evenings during the workshop. There are many more cheap restaurants on 13th Avenue just west of campus; Downtown Eugene is about a mile west of campus. The nearest grocery store is Market of Choice, on Orchard Street and Franklin Boulevard.



Funding

Accommodations will be provided for participants while there's still money left in the pot. To request a spot, please email Daniel Moseley with a brief description of your research interests.



This workshop is part of an annual series funded by an NSF CAREER grant. The 2010 workshop was led by Andre Henriques and the 2011 workshop was led by David Speyer.