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
Sadly, I seem to have inadvertantly deleted the schedule, problem sets, and participant list!
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.
Here are David's own lecture notes:
Ben-Zvi Lecture 1
Ben-Zvi Lecture 2
Ben-Zvi Lecture 3
Ben-Zvi Lecture 4
Ben-Zvi Lecture 5
Ben-Zvi Lecture 6
Ben-Zvi Lecture 7
In addition, you can find a number of lecture notes on
Rahbar Virk's web page.
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.