June 1-5, 2015
Yale University, New Haven, Connecticut

# Program

All talks will be in Davies Auditorium, Dunham Lab, 10 Hillhouse Avenue

## Tuesday

 9:30-10:30 Wee Teck Gan On the Howe duality conjecture Abstract: The theory of local theta correspondence was initiated by Howe in two influential papers written in the 1970’s. A fundamental conjecture in this subject is the Howe duality conjecture. I will discuss a brief history of this conjecture, leading up to its proof by Minguez (for GL(N)) and by Takeda and myself (for classical groups). I will also discuss joint work with Savin on the analogous conjecture in the context of exceptional theta correspondence. 11:00-12:00 S. Kudla Some degenerate Whittaker functions for Sp_n(R) Abstract: In a classic paper, Goodman and Wallach showed that Whittaker functionals can be obtained from conical functionals by applying certain differential operators of infinite order. Matumoto subsequently proved the existence of such operators in great generality. In the case of an abelian unipotent radical, after passage to the Fourier transform, a Goodman-Wallach operator can be expressed in terms of multiplication by an analytic function. For the group G= SL_2(R), Goodman and Wallach gave an explicit formula for this function as an IJ-Bessel function. In this talk, I will describe a generalization of their computation to the case of degenerate principal series for the symplectic group Sp_n(R) of rank n. The result involves Bessel functions and confluent hypergeometric functions of a matrix argument as defined in the classic paper of Herz. (This is part of a joint project with J. Bruinier and J. Funke) 1:45-2:45 C. Moeglin Arthur's and Adams-Johnson's packets, joint work with N. Arancibia and D. Renard. Abstract: In this talk I will explain how to deduce from Johnson and Adams-Johnson's work the description of A-packets for the unitary representations having cohomology (it's a local result on real fields). And this is a joint work with N. Arancibia and D. Renard. At the end of the talk, I will suggest how the Howe duality and the coherent continuation could be used to describe the general A-packets. Unfortunatly, the results are only under very special hypotheses. 3:00-4:00 T. Kobayashi Analysis on non-Riemannian locally symmetric spaces --- an application of invariant theory Abstract: By using invariant thoery for multiplicity-free actions, we construct joint eigenfunctions on non-Riemannian locally symmetric spaces modelled on real forms of spherical varieties with overgroups. This is a joint work with F. Kassel. 4:00-4:30 Tea/Coffee 4:30-5:30 Y. Sakellaridis On the Schwartz space of a smooth semi-algebraic stack. Abstract: There are many instances in the field of automorphic forms where nice classification results do not exist for a single group or space, but they do exist if one considers some pure inner forms'' at the same time. It has been suggested by J. Bernstein that this phenomenon should be explained by replacing the spaces and groups by appropriate quotient stacks. In this talk I will explain how to extend the notions of Schwartz functions (or rather measures) from smooth, semi-algebraic ("Nash") manifolds, to the corresponding category of stacks. I will also describe an approach to orbital integrals (in this setting considered as "evaluation maps" on global Schwartz spaces) which does not use truncation.

## Wednesday

 9:30-10:30 B. Sun Conservation relations for local theta correspondence Abstract: In the theory of local theta correspondence, a fundamental question is the non-vanishing and the key principle governing non-vanishing is a conjecture of Kudla and Rallis from the mid 90’s, known as the conservation relations. I will discuss my recent work (joint with Chen-Bo Zhu) which establishes this conjecture in full generality. 11:00-12:00 J-L. Waldspurger On the fundamental lemma for all functions in the Hecke algebra Abstract: Ngo Bao Chau has proved the fundamental lemma of the theory of endoscopy for the unit of the Hecke algebra. Using a global method due to Clozel, Hales has proved that, for ordinary endoscopy, this implies the same lemma for all the functions in the Hecke algebra. In a joint work with Lemaire and Moeglin, we give a new proof of the result of Hales, which works well in the case of twisted endoscopy. I will explain this proof, which is local and based on an idea of Arthur 1:45-2:45 D. Ciubotaru Formal degrees of unipotent discrete series representations of semisimple p-adic groups Abstract: The formal degree is a fundamental invariant of a discrete series representation. With an appropriate normalization, it appears in the Harish-Chandra-Howe local character expansion as the coefficient of the Fourier transform of the nilpotent orbital integral for the zero nilpotent orbit. For discrete series with unipotent cuspidal support, a formula for formal degrees, conjectured by Hiraga-Ichino-Ikeda, was recently verified by Opdam. For split exceptional groups, this formula was previously known from the work of Reeder. I will present a different interpretation of the formal degrees of unipotent discrete series in terms of the nonabelian Fourier transform (introduced by Lusztig in the character theory of finite groups of Lie type) and certain invariants arising in the elliptic theory of the affine Weyl group, and explain how this relates to the conjectures of Lusztig on almost characters of p-adic groups. The talk is based on joint work with Eric Opdam. 3:00-4:00 D. Barbasch Slides Unipotent representations and the Theta-correspondence. Abstract: Unipotent representations for real groups are the building blocks of the unitary dual and residual spectrum. For classical groups large classes of such representations are related by the Theta correspondence. In this talk I will review some older known, and some new results about this relationship. 4:00-4:30 Tea/Coffee 4:30-5:30 M. Reeder Some new supercuspidal representations Abstract: I will discuss recent progress on the construction of epipelagic supercuspidal representations of split p-adic groups, via Geometric Invariant Theory. Thanks to recent work of Fintzen and Romano, the method is now uniform in p and gives new supercuspidal representations when p is small.

## Thursday

 9:30-10:30 D. Vogan Slides Branching to maximal compact subgroups Abstract: Suppose $G$ is a real reductive Lie group and $K$ is a maximal compact subgroup. The set $\widehat K$ of irreducible representations of $K$ is (more or less) a cone inside a lattice (in a finite-dimensional vector space). An irreducible representation $\pi$ of $G$ defines (by restriction to $K$) a multiplicity function $m_\pi$ on $\widehat K$. There are many beautiful examples due to Roger Howe and his collaborators in which understanding the geometry of the function $m_\pi$ (for example, the geometry of the set where it is nonzero) leads to deeper understanding of the representation $\pi$ (for example, how it can be reducible). I will recall some of the general things that are known about the functions $m_\pi$, and some of the many interesting open problems that remain. 11:00-12:00 F. Knop Reductive group actions Abstract: Let G be a connected reductive group defined over a field k of characteristic zero. We are going to present a structure theory for G-actions which closely resembles the theory of Borel, Tits, Satake for connected reductive groups. In particular, we define the notion of an anisotropic kernel of a G-action which serves as a black box and a reduced root system. Both together capture many features of the action. The theory is most powerful for k-spherical varieties where, by definition, a minimal parabolic k-subgroup acts with an open orbit. In particular, we construct wonderful compactifications for them. Joint work with Bernhard Krötz. 2:00-3:00 N. Wallach Slides A GIT dynamical system Abstract: Consider the dynamical system on $M_{n}(\mathbb{C})$: $\dot{X}=-[[X,X^{\ast}],X].$ Let $\phi_{t}$ be the corresponding local one parameter group of diffeomorphisms. On $M_{n}(\mathbb{C})$ we put the Hilbert-Schmidt inner product. One can show by elementary means, using the estimate $\left\Vert \lbrack X,X^{\ast}]\right\Vert ^{2}\leq\left\Vert \lbrack\lbrack X,X^{\ast}],X]\right\Vert \left\Vert X\right\Vert,$ that $\phi_{t}$ is defined for $t>0$ and that $\lim_{t\rightarrow+\infty}[\phi_{t}(X),\phi_{t}(X)^{\ast}]=0.$ Let $U(n)$ act on $M_{n}(\mathbb{C})$ by conjugation then $\phi_{t}$ induces an action on $M_{n}(\mathbb{C})/U(n)$ and $\lim_{t\rightarrow+\infty}\phi_{t}(U(n)\cdot X)=U(n)\cdot Y$ with $Y$ normal and conjugate relative to $GL(n,\mathbb{C})$ to the semi-simple part of the Jordan decomposition of $X$ . The convergence is uniform if $X$ in a compactum. Less elementary is an immediate implication of a result of Lojasiewicz that implies that there exist $1>\varepsilon>0$ and $C>0$ (depending on $n)$ such that $\left\Vert \lbrack X,X^{\ast}]\right\Vert ^{2}\leq C\left\Vert [[X,X^{\ast }],X]\right\Vert ^{1+\varepsilon}\left\Vert X\right\Vert ^{1-3\varepsilon}$ and this implies $\lim_{t\rightarrow+\infty}\phi_{t}(X)=Y$ with a unique choice of $Y$ uniformly for $X$ in a compactum. This result is a special case of a result of Neeman. The purpose of the lecture is to give some generalizations that yield applications to algebraic geometry, matrix theory and geometric invariant theory of the the amazing work of Lojasiewicz. 3:00-3:30 Tea/Coffee 3:30-4:30 J. Arthur Problems Beyond Endoscopy Abstract: There has been progress in recent years on the conjectural theory of endoscopy. It has led to a classification of automorphic representations for quasisplit classical groups, and has placed the theory for general classical groups within reach. These results include a proof of Langlands' principle of fuctoriality in several important cases. However, the general principle of functoriality is still very much unresolved. Langlands has put forward a possible strategy for attacking the general problem, which he calls Beyond Endoscopy. It entails a study of the trace formula that is quite new, and involves profound questions in analytic number theory, as well as other questions in arithmetic and representation theory. We shall discuss the general goals, and describe a few of the problems that arise. 6:30 Banquet

## Friday

 9:30-10:30 Soo-Teck Lee Slides Branching algebras for classical groups Abstract: Let G be a complex classical group and let H be a symmetric subgroup of G. Using classical invariant theory, we can construct an algebra whose structure encodes the branching rule from G to H. We call this algebra a branching algebra for (G,H). In this talk, I will give a survey on some of the works done by Roger Howe and his collaborators on branching algebras. 11:00-12:00 A. Moy A computation with Bernstein projectors for p-adic SL(2). Abstract: For the p-adic group $G=SL(2)$, we present results of the computations of the sums of the Bernstein projectors of a given depth. Motivation for the computations is based on a conversation with Roger Howe in August 2013. The computations are elementary, but they provide a expansion of the delta distribution $\delta_{1}$ into an infinite sum of $G$-invariant locally integrable essentially compact distributions supported on the set of topologically unipotent set. When these distributions are transferred, by the exponential map, to the Lie algebra, they give $G$-invariant distributions supported on the set of topologically nilpotent elements, whose Fourier Transforms turn out to be characteristic functions of very natural $G$-domains. The computations in particular rely on the SL(2) discrete series character tables computed by Sally-Shalika in 1968. This new phenomenon for general rank has also been independently noticed in certain unpublished recent work of Bezrukavnikov, Kazhdan, and Varshavsky. 1:45-2:45 D. Jiang On Automorphic Analogue of the HighestWeight Module Theory Abstract: In the Open Problems in Honor of W. Schmid, 2013, Arthur gave Problem No. 5: The endoscopic classification via stable trace formula provides cer- tain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. If this is considered as an automorphic analogue of the classical Weyl character formula, then, as asked by W. Schmid in 2012, what will be the automorphic analogue of the classical highest weight 3 module theory? that is, to construct modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters. In this talk, we propose a general framework on how to construct modules of cuspidal automorphic representations of general classical groups in terms of their global Arthur parameters, and report on the recent progress with Lei Zhang for the case when the global Arthur parameters are generic. This extends the automrphic descent of Ginzburg-Rallis-Soudry to great generality. One of the main ideas in the theory is to consider how Fourier coecients can be used to determine automorphic forms, where Roger Howe made fundamental contributions to the theory of automorphic forms. If time is permitted, we will also talk about some applications of the theory, including the global Gan-Gross-Prasad conjecture. 3:00-4:00 J. Adams Slides Theta Correspondence for Dummies Abstract: This is joint work with Dipendra Prasad and Gordan Savin. The theta correspondence concerns quotients of the oscillator representation omega restricted to a dual pair (G,G'). If pi is an irreducible representation of G the representation Theta(pi)=Hom_G(omega,pi) of G' has a unique irreducible quotient theta(pi). The theta, or Howe, correspondence is the bijection pi->theta(pi). Instead of Hom it is natural to consider Ext_G(omega,pi) and the Euler-Poincare characteristic EP_G(omega,pi). Doing so makes many of the subtleties involved in the theta correspondence disappear. Results for EP are both simpler to state and to prove, and some naive expectations for the theta correspondence turn out to be true for the EP version. This gives a new perspective on the theta correspondence itself, including the structure of Theta(pi).