Representation Theory Seminar (UofC)

University of Chicago Representation Theory Seminar

Speakers for spring quarter 2009
The seminar runs on Tuesdays, 3:00-4:20, in Eckhart 207

Benedict Gross (Harvard University): March 31
Title: Values of Artin L-functions at negative integers, and explicit computations with the trace formula
Abstract: Euler introduced the zeta function into number theory, and proved a number
of beautiful results on its values at negative integers, using ingenious techniques to sum
divergent series. I will review the modern theory, which uses analytic continuation to
define these values, and state what is known about their rationality and integrality. I will
then use the trace formula to relate these values to the problem of counting the number
of automorphic representations for a simple group G over a global field k, having
prescribed local behavior at a finite number of places of k.

Gross will also give a talk at Northwestern University on Monday, March 30.
The title and abstract for that talk are as follows.

Title: Restriction problems for classical groups
Abstract: I will discuss several restriction problems in the representation theory of classical
groups over local fields, including restriction of irreducible representations from U(n) to U(n-1).
This work, which is joint with W-T Gan and D Prasad, attempts to predict the multiplicities
in the restriction from number-theoretic data of the Langlands parameters.
Qëndrim Gashi (Max Planck Institute for Mathematics): April 7
Title: A converse to Mazur's Inequality
Abstract: Given an isocrystal N and a lattice M in it, Mazur proved that the Hodge vector
of M lies above the Newton vector of N. The converse to Mazur's Inequality is the assertion
that given a vector $v$ that lies above the Newton vector of N (and satisfies certain obvious
conditions), there exists a lattice M whose Hodge vector is equal to $v$. These statements
can be viewed as statements for the group GL_n and it is known that they can be formulated
for other reductive groups. We prove the (generalized) converse to Mazur's Inequality for
split and quasi-split groups; this was previously known for split classical groups.
Sug Woo Shin (University of Chicago): April 28
Title: Cohomology of Rapoport-Zink spaces of EL-type
Abstract: Rapoport-Zink spaces are moduli spaces of p-divisible groups with additional
structure and may be thought of as local analogues of Shimura varieties. Their l-adic
cohomology spaces are expected to realize the local Langlands correspondence, and as
such played an important role in Harris-Taylor's proof of the local Langlands conjecture.
We will review the known results and introduce some further questions.
Tom Haines (University of Maryland): May 5
Title: Shimura varieties with $\Gamma_1(p)$-level structure via Hecke algebra isomorphisms
Abstract: The Langlands-Kottwitz approach to the determination of the local zeta function
of Shimura varieties at primes of good reduction consists in expressing the number of points
modulo p as a sum of products of a certain volume term, an orbital integral away from p,
and a twisted orbital integral at p. It has been known for some time how to generalize this
method to certain bad reduction cases coming from parahoric level structure. In this talk,
I will explain how to generalize the method further to some deeper level situations.
This is joint work with Michael Rapoport.
Hugh Thomas (University of New Brunswick): May 12
Title: Faithfulness of Artin group actions on derived categories
Abstract: Inspired by homological mirror symmetry, Seidel and Thomas constructed Artin
group actions on derived categories of coherent sheaves of various varieties and proved
faithfulness of such actions for Artin groups of type A. I will discuss joint work with
Christopher Brav giving some faithfulness results for Artin group actions of types D and E.
Teruyoshi Yoshida (Harvard University): May 19
Title: On arithmetic geometry of Lubin-Tate spaces
Abstract: Lubin-Tate spaces are an example of local symmetric spaces for p-adic
groups (Rapoport-Zink spaces), whose etale cohomology groups realize local
Langlands correspondence for GL(n) (proven via global theory of Shimura
varieties). We discuss its moduli-theoretic stratifications and resolutions, and their
connection with the Deligne-Lusztig theory in the tame level case.
Akaki Tikaradze (University of Toledo): May 26
Title: Center of infinitesimal Hecke algebras
Abstract: For a reductive Lie algebra and its finite dimensional representation, one
can define a family of associative algebras called infinitesimal Hecke algebras (the
definition is due to Etingof, Gan, Ginzburg). This definition is similar to that of
widely studied symplectic reflection algebras. In this talk I will be concerned with
computing the center of an infinitesimal Hecke algebra and its applications to
representation theory both in characteristic 0 and in positive characteristic. Particular
attention will be paid to infinitesimal Cherednik algebras of GL_n, for which I will
discuss an analog of the BGG category O and its spectral decomposition.
Mitya Boyarchenko (University of Chicago): June 2
Title: Representations of unipotent groups over local fields and Gutkin's conjecture
Abstract: Let U be the group of unipotent upper-triangular matrices of size n over a
finite field with q elements (n and q are fixed). In 1966 J. Thompson conjectured
that the dimension of every complex irreducible representation of U is a power of q.

A much more general assertion was formulated in 1973 by E. Gutkin. Let F be
either a finite field or a local field (of arbitrary characteristic), and let A be a finite
dimensional associative nilpotent algebra over F. The set 1+A of formal expressions
1+x, where x is in A, is a topological group with the operation (1+x)(1+y)=1+(x+y+xy)
and with the topology induced by the natural topology on F. Gutkin conjectured that
every unitary irreducible representation of 1+A is induced from a 1-dimensional
character of a closed subgroup of the form 1+B, where B is an F-subalgebra of A.

The first proof of Thompson's conjecture was given by I.M. Isaacs in 1995, while
the first proof of Gutkin's conjecture in the case where F is finite was given by
Z. Halasi in 2004. In my talk I will report on a recent proof of Gutkin's conjecture
in full generality. I will also discuss some related results on smooth (complex)
representations of groups of the form G(F), where F is a local field of arbitrary
characteristic and G is a unipotent algebraic group over F.

Speakers for winter quarter 2009
The seminar runs on Tuesdays, 3:00-4:20, in Eckhart 207

Steven Spallone (The University of Oklahoma): January 20
Title: Residues of Intertwining Operators for SO(6)
Abstract: Let E be a quadratic extension of a p-adic field F, and chi a quasicharacter
of the norm-one elements of E. Also fix a self-dual supercuspidal representation pi of GL(2).
Let Pi be the parabolic induction to SO(6) of the tensor product of chi and pi. The question
of irreducibility of Pi comes down to computing a residue of a certain integral over a unipotent
subgroup of SO(6). I will demonstrate how to convert this problem into a pairing of orbital
integrals, and discuss the solution from there. In the case for which pi has nontrivial central
character, the pairing can be interpreted in terms of lifting from twisted endoscopy.
This is joint work with Freydoon Shahidi.
Mitya Boyarchenko (University of Chicago): February 3
Title: Character sheaves on unipotent groups over finite fields
Abstract: I will give an overview the results that have been proved so far about character
sheaves on unipotent groups, and their relation to representations of finite groups of the
form G_0(F_q), where G_0 is a unipotent algebraic group over a finite field F_q. The talk
is based on joint work with Vladimir Drinfeld.
Sandeep Varma (Purdue University): February 10
Title: On twisted endoscopy and the generic packet conjecture
Abstract: download PDF
Loren Spice (University of Michigan): February 17
Title: Supercuspidal characters and applications
Abstract: In 2000, J.-K. Yu described a construction of supercuspidal characters of general
reductive p-adic groups that generalises the classical construction of Howe and Moy (and is
often exhaustive, by J.-L. Kim). Together with J. Adler, we computed the characters of
many of these representations using Harish-Chandra's integral formula.

By analogy with the case of real groups, character formulae for p-adic groups are expected
to have many applications in harmonic analysis. One important application arises in the
study of the local Langlands correspondence, where one wants to be able to prove that
certain combinations of characters are (loosely speaking) invariant under geometric
conjugacy. We discuss joint results in this direction with S. DeBacker.
Ben Webster (MIT): February 24
Title: Representation theory and a strange duality for symplectic varieties
Abstract: In recent work with Braden, Licata and Proudfoot, we showed that certain algebras
constructed from hyperplane arrangements whose module categories have a number of nice
properties which are surprisingly reminiscent of the BGG category O; in particular, they are
Koszul, and Koszul duality corresponds to a well known combinatorial duality. I'll explain
why we think properties are connected to a geometric origin for both these categories, and
how this suggests an underlying duality between pairs of symplectic varieties.
David Nadler (Northwestern University): March 3
Title: Homotopical algebra of character sheaves
Abstract: I will describe joint work with David Ben-Zvi explaining how Lusztig's character
sheaves fit into 3d topological field theory.
Brian Smithling (University of Toronto): March 10
Title: Local models for even orthogonal groups
Abstract: Local models are schemes, defined in terms of linear algebra, that were introduced
by Rapoport and Zink to study the \'etale-local structure of certain integral models of PEL
Shimura varieties over $p$-adic fields. A basic requirement for these models, or equivalently
for the local models, is that they be flat. When working with local models for split $GO_{2n}$,
Genestier observed that the most naive definition of the local model does not yield a flat scheme.
In a recent preprint, Pappas and Rapoport introduced a new condition to the moduli problem
defining the local models and conjectured that the models representing this strengthened moduli
problem are flat. I will report on the proof of a weaker form of their conjecture, namely that
their new models are topologically flat.

Speakers for autumn quarter 2008
The seminar normally runs on Tuesdays, 3:00-4:20, in Eckhart 207

Ramin Takloo-Bighash (UIC): October 14
Title: Gross-Prasad for GSp(4)
Abstract: The conjectures of Gross and Prasad predict precise branching laws for representations of orthogonal
groups when restricted to orthogonal subgroups locally and globally. In this talk I will explain a recent work verifying
certain special cases with interesting arithmetic applications. This is joint work with Dipendra Prasad.
David Goldberg (Purdue University): October 28
Title: On dual R-groups for the classical groups
Zhiwei Yun (Princeton): November 5, 1:30pm-2:50pm, Ryerson 358
Note that the time and location are different, and the talk is on a Wednesday
Title: A global analogue of Springer representations
Abstract: Classical Springer representations originated from the study of representations of groups over finite fields.
I will consider "global Springer representations" coming from groups over a global function field of a curve. These
are affine Weyl group actions on the cohomology of Hitchin fibers with parabolic structures (in general non-semisimple
representations). The construction uses Hecke correspondences between Hitchin spaces and works sheaf-theoretically.
I will talk about an example in SL(2) in detail. Time permitting, I will also mention what happens when one compares
the situations of global Springer actions for two groups which are Langlands dual to each other.
Travis Schedler (MIT): November 11
Title: The numbers game
Abstract: No, this isn't about elections or statistics. Mozes's game of numbers, which originated in a 1986 IMO problem,
is a combinatorial game played on a graph whose vertices are labeled by numbers. It recently arose in Qendrim Gashi's
proof of the Kottwitz-Rapoport conjecture.

In this talk, I will explain the solution of Gashi's variant of the game, the "numbers game with a cutoff", for any Dynkin
or extended Dynkin diagram. I will also discuss several results about the extended Dynkin case, linking configurations
of the game to the Weyl chambers appearing in the unit cube of the coweight representation. Playing the game (or
triangulating the cube) leads to a graded poset, of cubic degree in the number of vertices, whose Hilbert polynomial is
closely related to the exponents of the affine Weyl group.

This is joint work with Q. Gashi, and benefited from discussions with D. Speyer.

Dinner with Travis will be held on Monday, November 10.
We will meet for dinner right after Etingof's talk in the geometric Langlands seminar.
Alexei Oblomkov (Princeton): November 18
Title: Quantum cohomology of Hilbert scheme of points of ADE resolution and loop algebras
Abstract: Let X be a resolution of the ADE singularity C^2/\Gamma. Together with D. Maulik we computed
the operators of divisor multiplication in the ring of quantum equivariant cohomology of Hilb_n(X). The answer
is given in terms of the loop algebra of the corresponding type and the structure of the formulas is reminiscent
of the Casimir operators. Conjecturally, these operators generate the whole ring of quantum cohomology. In my
talk I will mostly discuss the case of A_1 singularity. All necessary geometric definitions unfamiliar to the
audience will be reminded.
Apoorva Khare (UC Riverside): November 25
Title: Representations of infinitesimal Hecke algebras
Abstract: We study families of infinite-dimensional algebras that are similar to semisimple Lie algebras
as well as symplectic reflection algebras. Infinitesimal Hecke algebras are deformations of semidirect
product Lie algebras, and we study two families over sl(2) and gl(2). Both of them have a triangular
decomposition and a nontrivial center, which allows us to define and study the BGG Category O over
them - including a (central character) block decomposition, and an analog of Duflo's Theorem about
primitive ideals. We then discuss certain related setups.

We conclude with the undeformed case of semidirect product Lie algebras, and of graded modules over
them; this is motivated by the study of Kirillov-Reshetikhin modules, using truncated current Lie algebras.
Wee Liang Gan (UC Riverside): December 2
Title: Harish-Chandra homomorphisms and symplectic reflection algebras for wreath products
Abstract: I will speak on a natural construction of the spherical subalgebra of a symplectic reflection algebra
assoicated to a wreath product in terms of quantum Hamiltonian reduction of an algebra of differential operators
on a representation space of an affine Dynkin quiver. This is a joint work with Etingof, Ginzburg, and Oblomkov.

Speakers for spring quarter 2008
The seminar runs on Tuesdays, 3:00-4:20, in Eckhart 207

Freydoon Shahidi: April 29

Title: Complexity of group actions, Bessel functions and root numbers
Abstract: Root numbers are expected to be stable under twists by highly
ramified characters. On the other hand there is now an expression for a
general class of local coefficients, objects which define root numbers
from our method inductively, in which a generalized Bessel function
naturally appears. The expression resembles that of a Mellin transform of
this Bessel function and is precisely so in the cases needed for functoriality
where stability was established using this expression. In this talk we will
consider some cases where the expressions is in fact a classical Mellin
transform as the cases of functoriality for classical groups. The integrals
are studied by means certain group actions which turn out to have zero
complexity, in the sense of Luna and Vust, in the cases functoriality. The
case of exterior square root numbers for GL(n) has a higher complexity,
pointing to the difficulty in proving their stability. We will discuss these
issues and the geometry involved in this problem which may be of
interest on its own.

Hugh Thomas: May 6

Title: (Higher) cluster categories
Abstract: Cluster categories are categories coming from the representation theory of
associative algebras, which model the structure of cluster algebras. I will speak about
the definition of the cluster categories of Buan-Marsh-Reinecke-Reiten-Todorov, and
about how to extract information about cluster algebras from them. I will also talk
about the higher cluster categories, which model the combinatorics of the generalized
clusters of Fomin-Reading, and about the hints they provide towards the possibility of
higher cluster algebras.

Stephen DeBacker: May 13

Title: On generic representations in certain L-packets
Abstract: This is joint with Mark Reeder. A conjecture of F. Shahidi
predicts that each tempered L-packet should contain exactly one "generic"
representation. Recently some "L-packets" have been constructed that
consist entirely of (very cuspidal) supercuspidal representations attached
to "unramified elliptic tori". I will discuss how to go about determining
which of the representations in these L-packets is generic. The answer was
a bit surprising to us.

Rafael Bocklandt: May 20

Title: Toric orders, Calabi-Yau Algebras and Dimer Models
Abstract: Dimer models were introduced by physicists to give a nice
combinatorial description of certain Calabi Yau algebras. In this talk we introduce
the notion of a toric order as a noncommutative generalization of a toric
variety and we show that if we impose the Calabi Yau condition on a toric order implies
naturally that it comes from a dimer model.

Fiona Murnaghan: May 27

Title: Parametrization of tame supercuspidal representations
Abstract: Let G be a connected reductive p-adic group that
splits over a tamely ramified extension. We describe a parametrization
of the equivalence classes of tame supercuspidal representations of G.
Certain of these equivalence classes are parametrized by
sets of G-conjugates of quasicharacters of elliptic maximal tori
in G. We will also discuss the particular form of the parametrization
for self-contragredient tame supercuspidal representations.
If time permits, we will discuss how features of the parametrization
are reflected in properties of the characters of the representations.

Ivan Mirkovic: June 3

Title: Beilinson-Bernstein localization theory
for reductive Lie algebras in positive characteristic
Abstract: Representation theory of (reductive split) Lie algebras (or groups)
over fields of positive characteristic traditionally exhibits certain non-standard
complexities. The localization of representations to D-modules on flag varieties
explains these as complexities of the geometry of Springer fibers. This allows
one to verify Lusztig's conjectural description of the numerical structure of
modular representation theories. The main interest in this subject comes from
analogies with representation theory of affine Lie algebras at critical level.
This is a joint work with Bezrukavnikov, Rumynin, Milburn and Arkhipov.