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Organized by: MIT

Organizing committee: R. Bezrukavnikov (MIT), P. Etingof (MIT), G. Lusztig (MIT), M. Nevins (Ottawa), P. Trapa (Utah)

Sponsors:

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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	Mini-break
11:00AM	Peter Trapa	Relationships between unitary representations of real and p-adic 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 p-adic 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 L-functions in terms of this group.
2:45PM	Mini-break
3:00PM	Jean-Loup Waldspurger	Stabilization of the twisted trace formula : the local theorem	Abstract
		In a work in progress, joint with Moeglin, we stabilize the twisted Arthur-Selberg trace formula, following the fundamental work of Arthur in the non-twisted 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	Jing-Song 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 Arthur-packets as well as the stable characters in endoscopy.
5:30PM-6:30PM	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 ``cocenter-representation 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 Harish-Chandra modules	Abstract Slides
		We show that the coefficients of Kazhdan-Lusztig-Vogan polynomials attached to certain blocks of Harish-Chandra 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 Kazhdan-Lusztig 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 Kazhdan-Lusztig 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 tau-invariant, 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	Mini-Break
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 Amitsur-Levitski 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 Amitsur-Levitski theorem with equations for sheets in a reductive Lie algebra and with recent results of Kostant-Wallach 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 L-functions).
11:45AM	Mini-break
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 "perverse-coherent 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 Mirković--Vilonen conjecture. Parts of this are joint work.
1:00PM	David Vogan, Famous Last Words