Lie Groups and Representation Theory Archives for Academic Year 2012


The Real Chevalley Involution

When: Wed, September 5, 2012 - 2:00pm
Where: Math 1311
Speaker: Jeffrey Adams (University of Maryland)
Abstract: The Chevalley involution of an algebraic group takes g to a conjugate of its inverse, and therefore acts on algebraic representations by the contragredient. This is over an algebraically closed field. We consider whether there is such an involution over a local field. The answer is affirmative over R, and we give some applications to computing the orthogonal/symplectic (Frobenius-Schur) indicator.

Local Langlands over R done right

When: Wed, September 12, 2012 - 2:00pm
Where: Math 1311
Speaker: Jeffrey Adams (University of Maryland)
Abstract: I will give a self-contained description of the Local Langlands classification (in terms of L-groups) over R, and some properties of the classification. Tom Haines will give a talk next week about the p-adic case.

Local Langlands over R done right II

When: Wed, September 19, 2012 - 2:00pm
Where: Math 1311
Speaker: Jeffrey Adams (University of Maryland)
Abstract: I'll finish my description of the the local Langlands correspondence over R.

Geometry of Langlands parameters

When: Mon, September 24, 2012 - 2:00pm
Where: Math 1311
Speaker: Tom Haines (UMD) - www.math.umd.edu/~tjh
Abstract: The Bernstein variety is, roughly speaking, a variety structure on the set of all irreducible smooth representations of a p-adic group G. The set of all Langlands parameters for G also has a variety structure. I will describe these varieties and relate them using the local Langlands correspondence for G.

Geometry of Langlands parameters II

When: Wed, October 3, 2012 - 2:00pm
Where: Math 1311
Speaker: Tom Haines (UMD) - www.math.umd.edu/~tjh
Abstract: The Bernstein variety is, roughly speaking, a variety structure on the set of all irreducible smooth representations of a p-adic group G. The set of all Langlands parameters for G also has a variety structure. I will describe these varieties and relate them using the local Langlands correspondence for G.

New geometric structures in the local Langlands program

When: Mon, October 15, 2012 - 2:00pm
Where: Math 1311
Speaker: Mitya Boyarchenko (University of Michigan) - http://www.math.lsa.umich.edu/~mityab/
Abstract: The problem of explicitly constructing the local Langlands
correspondence for GL_n(K), where K is a p-adic field, contains as an
important special case the problem of constructing automorphic
induction (or "twisted parabolic induction") from certain
1-dimensional characters of L^* (where L is a given Galois extension
of K of degree n) to irreducible supercuspidal representations of
GL_n(K). Already in 1979 Lusztig proposed a very elegant, but still
conjectural, geometric construction of twisted parabolic induction for
unramified maximal tori in arbitrary reductive p-adic groups. An
analysis of Lusztig's construction and of the Lubin-Tate tower of K
leads to interesting new varieties that provide an analogue of
Deligne-Lusztig theory for certain families of unipotent groups over
finite fields. I will describe the known examples of this phenomenon
and their relationship to the local Langlands correspondence. Part of
the talk will be based on joint work with Jared Weinstein (Boston
University).

Realizing Smooth Representations

When: Wed, October 17, 2012 - 2:00pm
Where: Math 1311
Speaker: David Vogan (MIT)
Abstract: Gelfand's program of abstract harmonic analysis seeks to understand a
manifold X with an action of a Lie group G in four steps:

(1) understand all irreducible unitary representations of G;
(2) construct the Hilbert space L^2(X);
(3) decompose L^2(X) as a "direct integral" of irreducible unitary
representations.
(4) relate (hard) problems about X to L^2(X), and so (via (3)
to (easier) problems about irreducible unitary representations.

We are interested in two (very old) technical problems arising in
Gelfand's program. Suppose we wish to understand a G-invariant
differential operator Delta_X. The first problem is that
eigenfunctions of Delta_X may not be square-integrable, so
relating them to L^2(X) (step (4)) is subtle.

The second problem appears already in step (1), the construction of
irreducible unitary representations. Natural candidates for such
representations are simultaneous eigenspaces for G-invariant
differential operators on X; but if (as very often happens) the
eigenfunctions are not square-integrable, then it is not easy in this
way to find natural Hilbert space representations.

The general solution we propose is dualization: to replace
_kernels_ of differential operators acting on _distributions_ by
_cokernels_ of differential operators acting on _compactly
supported densities_.

I'll describe these problems and their resolution in some very
familiar examples related to SL(2,R) and one complex
variable.

Small unitary representations and graviton scattering (Joint work with Michael B. Green, Jorge Russo, and Pierre Vanhove)

When: Wed, October 24, 2012 - 2:00pm
Where: Math 1311
Speaker: Stephen Miller (Rutgers University)
Abstract: String theory posits correction terms for graviton scattering amplitudes at low energies. The main term in this expansion is classical, and corresponds to Einstein's general relativity. For toroidal compactifications we identify the next two terms as particular Eisenstein series. These are automorphic realizations of small "unipotent" representations, and as a result have very few nonvanishing Fourier coefficients. We use this to prove supersymmetry predictions regarding these amplitudes and describe exactly which Feynman diagrams contribute to them. Finally, the computational techniques developed can be used to establish the unitary of the unipotent representations involved, and of several others (confirming a conjecture of Arthur).

Lifting of the trivial representation

When: Mon, November 12, 2012 - 2:00pm
Where: Math 1311
Speaker: Wan-Yu Tsai (University of Maryland) -
Abstract: Let G be the real points of a simply laced, simply connected complex Lie group, and G~ be the nonlinear two-fold cover of G. We'll discuss a set of small genuine representations of G~, denoted by Lift(C), which can be obtained from the trivial representation of G by a lifting operator. The representations in Lift(C) can be characterized by the following properties: (a) the infinitesimal character is \rho/2; (b) they have maximal tau-invariant; (c) they have a particular associated variety \O. When G is split we will show that the representations in Lift(C) are parametrized by pairs (central character, real form of \O).

Integral structures in Steinberg representations and p-adic Langlands

When: Wed, December 5, 2012 - 2:00pm
Where: Math 3206 (Note change of room)
Speaker: Claus Sorensen (Princeton University) -
Abstract: As a vast generalization of quadratic reciprocity, class field theory describes all abelian extensions of a number field. Over Q, they are precisely those contained in cyclotomic fields.
However, there are a lot more non-abelian extensions, which arise naturally. The Langlands program attempts to systematize them, by relating Galois representations and automorphic forms; mathematical objects of rather disparate nature. We will illustrate the basic plot for GL(2) through the example of elliptic curves and modular forms - the context of Wiles' proof of Fermat's Last Theorem. The main goal of the talk will be to motivate a "p-adic" Langlands correspondence, which is at the forefront of contemporary number theory, but still only well-understood for GL(2) over Q_p. We will discuss, in some depth, the case of semistable elliptic curves, which provide the first non-trivial example. This leads naturally to a result we proved recently, which shows the existence of (many) integral structures in locally algebraic representations of "Steinberg" type, for any reductive group G (such as GL(n), symplectic, and orthogonal groups). As a result, there are a host of ways to p-adically complete the Steinberg representation (tensored with an algebraic representation). The ensuing Banach spaces should play a role in a (yet elusive) higher-dimensional p-adic Langlands correspondence. We hope to at least give some idea of the proof, which goes via automorphic representations and the trace formula. For the most part, the colloquium will be very low-key and widely accessible.

Affine Weyl groups, affine Hecke algebras, and affine Deligne-Lusztig varieties

When: Thu, December 13, 2012 - 3:30pm
Where: Math 3206
Speaker: Xuhua He (HKUST) -
Abstract: Using some nice combinatorial properties of affine Weyl groups, we establish a relation between representations of affine Hecke algebras and structure of affine Deligne-Lusztig varieties. More precisely, we have the "degree=dimension" theorem, which relates the degree of class polynomials of affine Hecke algebras and the dimension of affine Deligne-Lusztig varieties. As a consequence, we verify the Gortz-Haines-Kottwitz-Reuman conjecture on dimension of affine Deligne-Lusztig varieties.

Organizational Meeting

When: Wed, January 30, 2013 - 2:00pm
Where:


The oscillator representation

When: Wed, February 13, 2013 - 2:00pm
Where: Math 1311
Speaker: Jeffrey Adams (University of Maryland)
Abstract: This is an introductory talk, as part of our RIT covering the application of the oscillator representation to the representation theory of GL(2) over a p-adic field (following the book by Bushnell and Henniart)

The oscillator representation I (RIT)

When: Wed, February 27, 2013 - 2:00pm
Where: Math 1311
Speaker: Jon Cohen (Maryland) -

The oscillator representation II (RIT)

When: Wed, March 13, 2013 - 2:00pm
Where: 1311.0
Speaker: Jon Cohen (Maryland) - (rescheduled from the previous week)

The oscillator representation III (RIT)

When: Wed, March 27, 2013 - 2:00pm
Where: Math 1311
Speaker: Jeffrey Adams (Maryland) -
Abstract: This will be a talk bridging from Bump's chapter to Bushnell-Henniart.

The oscillator representation IV (RIT)

When: Wed, April 3, 2013 - 2:00pm
Where: 1311.0
Speaker: Ran Cui (Maryland) -
Abstract: We'll be starting on the material in the book by Bushnell and Henniart.

The oscillator representation V (RIT)

When: Wed, April 10, 2013 - 2:00pm
Where: Math 1311
Speaker: Jonathan Fernandes (Maryland) -

Heuristic notions of a Frobenius-Schur indicator for the unipotent characters of a finite Coxeter system

When: Wed, April 17, 2013 - 2:00pm
Where: Math 1311
Speaker: Eric Marberg (MIT) -
Abstract: To each finite, irreducible Coxeter system (W,S), Lusztig has
associated a set of "unipotent characters" Uch(W). When (W,S) is
crystallographic, Uch(W) arises from Lusztig's set of unipotent
representations of a corresponding finite reductive group, though for
non-crystallographic Coxeter groups the definition of Uch(W) is heuristic.
By construction, Uch(W) always contains as a subset the set Irr(W) of
complex irreducible characters of W. However, we typically view the
elements Uch(W) not as characters but simply as formal objects with a few
defining attributes. In this talk I will give Lusztig's definition of the
set Uch(W) for each finite Coxeter system, and then outline a surprising
way in which several different types of data attached to Uch(W) interact
with the irreducible multiplicities of a certain W-representation, to
define a notion of a "Frobenius-Schur indicator" for unipotent characters.


Oscillator Representation VI (RIT)

When: Wed, April 24, 2013 - 2:00pm
Where: Math 1311
Speaker: Wan-Yu Tsai (University of Maryland) -

Introduction to the Baum-Connes Conjecture

When: Wed, May 1, 2013 - 2:00pm
Where: Math 1311
Speaker: Jonathan Rosenberg (UMCP) - http://www.math.umd.edu/~jmr
Abstract: This will be the first of two talks on the Baum-Connes conjecture, explaining the origins of the conjecture and a few special cases.

Introduction to the Baum-Connes Conjecture, part 2

When: Wed, May 8, 2013 - 2:00pm
Where: Math 1311
Speaker: Jonathan Rosenberg (UMCP) - http://www.math.umd.edu/~jmr
Abstract: This will continue the talk from the previous week, with more emphasis on the p-adic case.

On Contractions of Lie algebra Representations and Direct Limits

When: Tue, May 14, 2013 - 2:00pm
Where: Math 1311 (Note: on Tuesday)
Speaker: Eyal Subag (Technion) -
Abstract: It is well known that classical mechanics can be regarded as a limit case of relativistic mechanics. At the early fifties Segal, Inonu, and Wigner showed in what sense the symmetry group of classical mechanics can be regarded as a limit of the symmetry group of relativistic mechanics. These kinds of limits are now known as contractions.
More precisely, contractions of Lie groups, Lie algebras and their representations is a formalism for obtaining a limit of these objects.
In my talk I will define contractions for Lie algebras and their representations, I will present some recent results including a new formalism for contraction of
Lie algebra representations which is based on the notion of direct limit. Prior acquaintance with the subject will not be assumed and many examples will be given.
This work was done while the speaker studied for his PhD under the supervision of Prof. Moshe Baruch, Prof. Ady Mann, and Prof. Joseph L. Birman.