Categorical Representation Theory
13 August - 17 August 2012
University of Oregon
Eugene, OR
Syllabus:
A detailed syllabus and bibliography can be found here.
Abstract:
Classical harmonic analysis 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,
oo-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.
The workshop will be aimed at graduate students and postdocs, with most
of the talks given by the participants. We do not expect any of the
participants to be experts in all of the subjects that are represented
in this workshop. Rather, we hope to bring together participants
with diverse backgrounds, and to weave these backgrounds together into a coherent
picture through a combination of lectures and informal problem sessions.
The workshop will be led by
David Ben-Zvi.
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.