 Speakers  Program  Directions  Contact  Photos  Participants 
Time  Speaker  Talk  

Monday, May 19  
8:30AM  Registration Opens  
9:30AM  Jeffrey Adams, Opening Remarks  
Benedict Gross  Newforms for odd orthogonal groups  Abstract Slides 

A compatible family of symplectic \(\ell\)adic Galois representations of dimension \(2n\) for a global field should correspond to a generic automorphic representation of the odd orthogonal group \(\mathrm{SO}_{2n+1}\). To make this conjecture more precise, we describe a distinguished line in this representation. At real and complex places, this uses the determination of the minimal \(K\)type.  
10:45AM  Minibreak  
11:00AM  Peter Trapa  Relationships between unitary representations of real and padic groups  Abstract Slides 
The dominant theme in David Vogan's work has been the classification of unitary representations of real reductive groups. I will explain how Vogan's work on unitary representation theory led him to important insights into the Local Langlands Correspondence. These insights, in turn, suggest intrinsic relationships between the unitary dual of split real and padic groups. I will give a new realization of those relationships in the case of \(\mathrm{GL}(n)\).  
12:00PM  Lunch Break  
1:45PM  James Arthur  On Langlands' automorphic Galois group and Weil's explicit formulas.  Abstract Slides 
The global theory of endoscopy is best formulated in terms of the hypothetical automorphic Galois group proposed by Langlands. We shall describe how to construct an explicit locally compact group, which is based on Langlands principle of functoriality, and which we conjecture is equal to the automorphic Galois group. If time permits, we shall then describe how the explicit formulas of Weil can be formulated for general automophic Lfunctions in terms of this group.  
2:45PM  Minibreak  
3:00PM  JeanLoup Waldspurger  Stabilization of the twisted trace formula : the local theorem  Abstract 
In a work in progress, joint with Moeglin, we stabilize the twisted ArthurSelberg trace formula, following the fundamental work of Arthur in the nontwisted case. The key point is a local theorem which describes the stabilization of weighted orbital integrals. I try to state this theorem and to explain the differences between the twisted and non twisted case.  
4:00PM  Coffee Break  
4:30PM  JingSong Huang  Elliptic representations, Dirac cohomology and endoscopy  Abstract Slides 
Elliptic representations for reductive groups are those whose distribution characters are not identically zero on the set of the regular elliptic elements and of fundamental importance for harmonic analysis. We show that the elliptic representations are closely related to the representations with nonzero Dirac cohomology. Furthermore, the determination of Dirac cohomology reveals the orthogonality relations and is useful for understanding the representations in Arthurpackets as well as the stable characters in endoscopy.  
5:30PM6:30PM  Wine and Cheese Reception  Lobby 13 at MIT  
Tuesday, May 20  
9:30AM  Diana Shelstad  Transfer results for real groups.  Abstract Slides 
We review and extend some theorems on endoscopic transfer for real reductive groups.  
10:30AM  Coffee Break  
11:00AM  George Lusztig  Conjugacy classes in a reductive group  Abstract 
Let \(G\) be a reductive connected algebraic group over an algebraically closed field. I will describe a partition of \(G\) into finitely many strata such that (1) the set of strata is indexed by a set which depends only on the Weyl group (not on the root system) and does not depend on the ground field; (2) each stratum is a union of conjugacy classes of the same dimension (which is again independent of the ground field).  
12:00PM  Lunch Break  
1:45PM  Toshiyuki Kobayashi  Branching problems of representations of real reductive Lie groups  Abstract 
Branching problems ask how irreducible representations \(\pi\) of groups
\(G\) "decompose" when restricted to subgroups \(G'\). For real reductive groups, branching problems include various important special cases, however, it is notorious that "infinite multiplicities" and "continuous spectra" may well happen in general even if \((G,G')\) are natural pairs such as symmetric pairs. By using analysis on (real) spherical varieties, we give a necessary and sufficient condition on the pair of reductive groups for the multiplicities to be always finite (and also to be uniformly bounded). Further, we discuss "discretely decomposable restrictions" which allows us to apply algebraic tools in branching problems. Some classification results will also be presented. If time permits, I will discuss some applications of branching laws of Zuckerman's derived functor modules to analysis on locally symmetric spaces with indefinite metric. 

2:45PM  Mini Break  
3:00PM  Jeffrey Adams  Galois and theta cohomology of real groups  Abstract Slides 
The real forms of a complex reductive group are parametrized by the Galois cohomology \(H^1(Gal,G_{ad})\). Cartan's classification in terms of the Cartan involution \(\theta\) amounts to computing \(H^1(\theta,G_{ad})\), where this means the cohomology of \(\mathbb{Z}_2\) acting by the holomorphic involution \(\theta\). Therefore \(H^1(Gal,G_{ad})=H^1(\theta,G_{ad})\). It turns out this is true for all \(G\) (not just the adjoint groups). This provides a convenient setting to state and prove many results relating the two pictures. We give some applications, including a computation of \(H^1(\Gamma,G)\) for all simply connected \(G\), and relation between "strong real forms" and conventional real forms.  
4:00PM  Coffee Break  
4:30PM  Michel Duflo  On Frobenius Lie subalgebras of simple Lie algebras.  Abstract 
Frobenius Lie algebras are Lie algebras which have an open coadjoint orbit. I will present results (obtained in collaboration with M. S. Khalgui, P. Torasso, and R. Yu) on the Frobenius Lie subalgebras of a complex simple Lie algebra \(S\) containing a Cartan subalgebra of \(S\).  
Wednesday, May 21  
9:30AM  Xuhua He  Cocenters and representations of affine Hecke algebras  Abstract
Slides 
It is known that the number of conjugacy classes of a finite group
equals the number of irreducible representations (over complex numbers). The conjugacy
classes of a finite group give a natural basis of the cocenter of its group algebra.
Thus the above equality can be refomulated as a duality between the cocenter of the
group algebra and the Grothendieck group of its finite dimensional representations. For affine Hecke algebras, the situtation is much more complicated. First, the cocenter of affine Hecke algebras is harder to understand than the cocenter of group algebras. Second, for an affine Hecke algebra, the dimension of its cocenter is countablly infinite and the number of irreducible representations is uncountablly infinite. However, the ``cocenterrepresentation duality'' is still valid. This is what I am going to explain in this talk. It is based joint works with S. Nie, and joint work with D. Ciubotaru. If time allows, I will also mention affine Hecke algebras at roots of unity and their ordinary and modular representations. 

10:30AM  Coffee Break  
11:00AM  W. Monty McGovern  Upper semicontinuity of KLV polynomials for certain blocks of HarishChandra modules  Abstract Slides 
We show that the coefficients of KazhdanLusztigVogan polynomials attached to certain blocks of HarishChandra modules satisfy a monotonicity property relative to the closure order on \(K\)orbits in the flag variety.  
12:00PM  Lunch Break  
1:30PM  Wolfgang Soergel  Graded versions of categories of representations and true motives  Abstract 
The triangulated categories of motivic sheaves recently constructed by Cisinski and Deglise allow a very smooth and transparent approach to the construction of graded versions of categories of representations. This is joint work with Matthias Wendt.  
2:30PM  Micro Break  
2:40PM  Geordie Williamson  Global and local Hodge theory of Soergel bimodules  Abstract 
Soergel bimodules are certain bimodules over polynomial rings which are defined for any Coxeter group. When the Coxeter group is a Weyl group they may be realised geometrically as equivariant intersection cohomology of Schubert varieties. I will describe a research program (joint with Ben Elias) giving algebraic proofs of these Hodge theoretic properties, independent of geometry. The "global" case gave a proof of the KazhdanLusztig positivity conjecture. I will focus on the "local" case, which is closely linked to the Jantzen filtration on Verma modules (via results of Soergel and Kübel).  
3:40PM  Extended coffee break (MIT faculty meeting)  
4:30PM  Meinolf Geck  Computing with left cells of type \(E_8\)  Abstract
Slides 
The talk is concerned with algorithmic aspects of the theory of KazhdanLusztig cells in finite Coxeter groups. We shall explain how the biggest challenge in this area, type \(E_8\), can be handled systematically and efficiently. A key role is played by Vogan's generalised tauinvariant, and generalisations thereof.  
6:00PM  Banquet at Royal East Restaurant (registration required)  
Thursday, May 22  
9:30AM  Stephen DeBacker  Nilpotent orbits revisited  Abstract 
We return to some questions about nilpotent and unipotent orbits for reductive \(p\)adic groups. These include questions about the existence of nilpotent orbits (as distributions), certain homogeneity results, and making an explicit parameterization of nilpotent orbits for certain exceptional groups.  
10:30AM  Coffee Break  
11:00AM  Wilfried Schmid  On the \(\mathfrak{n}\)cohomology of the limits of the discrete series  Abstract 
I shall describe an inductive procedure for calculating the \(\mathfrak{n}\)cohomology of the totally degenerate limits of the discrete series of a reductive Lie group \(G\). Here \(\mathfrak{n}\) denotes a maximal nilpotent Lie subalgebra of the complexified Lie algebra of \(G\), one that is normalized by a compact Cartan subgroup of \(G\). It turns out that this cohomology vanishes identically for most choices of \(\mathfrak{n}\); when it does not vanish, it is quite sparse. This is joint work, in part, with Dragan Milicic.  
12:00PM  Lunch Break  
1:45PM  Nolan Wallach  Gleason's theorem and unentangled orthonormal bases  Abstract Slides 
Gleason's theorem says that in a Hilbert space of dimension at least three the probability distributions that behave like mixed states relative to orthonormal bases are exactly the missed states. The unentangled Gleason theorem says that in a tensor product of Hilbert spaces with each factor of at least 3 dimensional the same result is true with the bases restricted to unentangled orthonormal bases. If any factor is of dimension 2 the result is false. This was seen by analyzing unentangled orthonormal bases of tensor products of Hilbert spaces with one 2 dimensional factor. In the important case of qubits (all factors two dimensional) Lebl and Shakeel and the speaker have done complete analysis of such bases and found that they have a beautiful combinatorial structure. This analysis should yield a more refined Gleason's theorem.  
2:45PM  MiniBreak  
3:00PM  Joseph Bernstein  Stacks in Representation Theory: What is a representation of an algebraic group?  Abstract
Slides 
In my talk I will discuss a new approach to the representation theory of
algebraic groups. In the usual approach one starts with an algebraic group \(\mathcal{G}\)
over some local (or finite) field \(F\), considers the group \(G=\mathcal{G}(F)\) of its
\(F\)points as a topological group and studies some category \(Rep(G)\) of continuous
representations of the group \(G\). I will argue that more correct objects to study are some kind of sheaves on the stack \(B\mathcal{G}\) corresponding to the group \(\mathcal{G}\). I will show that this point of view naturally requires us to change the formulation of some basic problems in Representation Theory. In particular this approach might explain the appearance of representations of all pure forms of a group \(G\) in Vogan’s formulation of Langlands’ correspondence. 

4:00PM  Coffee Break  
4:30PM  Bert Kostant  Generalized AmitsurLevitski Theorem and equations for sheets in a reductive complex Lie algebra.  Abstract Slides 
I connect an old result of mine on a Lie algebra generalization of the AmitsurLevitski theorem with equations for sheets in a reductive Lie algebra and with recent results of KostantWallach on the variety of singular elements in a reductive Lie algebra.  
Friday, May 23  
9:15AM  Dan Ciubotaru  On formal degrees of unipotent discrete series representations of semisimple \(p\)adic groups  Abstract 
I will present an interpretation (joint with E. Opdam) of the formal degrees of discrete series representations with unipotent cuspidal support in terms of the exotic Fourier transform (introduced by G. Lusztig in the character theory of finite groups of Lie type) and certain invariants arising in the elliptic theory of the affine Weyl group. I will also explain connections between these invariants, elliptic representations of the finite Weyl group, and genuine representations of the pin double cover of the finite Weyl group. The talk is based on joint works with X. He, E. Opdam, and P. Trapa.  
10:15AM  Coffee Break  
10:45AM  Akshay Venkatesh  Analytic number theory and harmonic analysis on semisimple Lie groups  Abstract 
I will describe some of the problems in the representation theory of semisimple Lie groups that arise from studying questions in analytic number theory (in particular, upper bounds for the size of Lfunctions).  
11:45AM  Minibreak  
12:00PM  Pramod Achar  Equivariant coherent sheaves on the nilpotent cone of a reductive algebraic group  Abstract Slides 
When I became David Vogan's student in 1998, he gave me a thesis problem about coherent sheaves on the nilpotent cone. Sixteen years later, I am still learning new things about this topic! In this talk, I will explain what "perversecoherent sheaves" on the nilpotent cone are, and how they are related to topics such as Koszul duality and parity sheaves; Springer theory and Kato's Kostka modules; and the geometric Satake equivalence and the MirkovićVilonen conjecture. Parts of this are joint work.  
1:00PM  David Vogan, Famous Last Words 