## ~ Abstracts ~

Transferring representations between finite reductive groups

Suppose $\tilde{G}$ is a connected reductive group over a finite field $k$, and $\Gamma$ is a finite group acting on $\tilde{G}$, preserving a Borel-torus pair. Then the connected part $G$ of the group of $\Gamma$-fixed points of $\tilde{G}$ is reductive, and there is a natural map from (packets of) representations of $G(k)$ to those of $\tilde{G}(k)$. I will discuss this map, its motivation in the study of $p$-adic base change, prospects for refining it, and a generalization: the pair of groups $(\tilde{G},G)$ must satisfy some axioms, but $G$ need not be a fixed-point subgroup of $\tilde{G}$, nor even a subgroup at all.

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On the local Langlands correspondence and Arthur conjecture for orthogonal groups
Hiraku Atobe, Kyoto University, Japan

I will state precisely the local Langlands correspondence for orthogonal groups established by Arthur. As an application, I will discuss on the local Gross-Prasad conjecture for orthogonal groups. Also, I will explain the Arthur multiplicity formula for even orthogonal groups. This is a joint work with Wee Teck Gan.

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A partition of the set of enhanced Langlands parameters of a reductive p-adic group
Anne Marie Aubert, Institut de Mathématiques de Jussieu, France

Let $F$ be a local non-archimedean field, and let $G$ be the group of $F$-points of a connected reductive algebraic group defined over $F$. We will first introduce a notion of cuspidality for the enhanced Langlands parameters of $G$ (that is, pairs formed by a Langlands parameter $\phi$ and an irreducible character $\eta$ of a certain component group attached to $\phi$). The cuspidal pairs $(\phi,\eta)$ are expected to correspond to the supercuspidal irreducible representations of $G$ via the local Langlands correspondence. It is proved in [3] that it is indeed the case for split classical groups. We will next describe a construction of a cuspidal support map for enhanced Langlands parameters, the key tool of which is an extension to disconnected complex Lie groups of the generalized Springer correspondence due to Lusztig (see [1]). We will used this map to obtain a partition of the set of enhanced Langlands parameters for $G$, which reflects the partition of the smooth dual of $G$ giving by the Bernstein Center. Finally, we will explain the expecte role of that partition towards a proof of the local Langlands correspondence in some cases [2]. It is joint work with Ahmed Moussaoui and Maarten Solleveld.

References:
[1] A.-M. Aubert, A. Moussaoui, M. Solleveld, Generalizations of the Springer correspondence and cuspidal Langlands parameters, arXiv:1511.05335.
[2] A.-M. Aubert, A. Moussaoui, M. Solleveld, Graded Hecke algebras for disconnected reductive groups, Work in progress.
[3] A. Moussaoui, Centre de Bernstein enrichi pour les groupes classiques, arXiv:1511.02521.

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Unipotent representations for complex groups
Dan Barbasch, Cornell University, USA

In this talk I will describe the unipotent representations for complex groups and how they fit as building blocks of the unitary dual of groups of classical type. For exceptional groups I will give a set of parameters which conjecturally platy the same role as in the classical cases.

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Tutorial on Hecke algebras
Dan Ciubotaru, University of Oxford, UK

The aim of these lectures is to offer an overview of harmonic analysis for affine Hecke algebras and the relations with smooth representations of reductive p-adic groups. I will explain first how affine Hecke algebras appear naturally in the world of representations of p-adic groups:
a) as Iwahori-Hecke algebras, and
b) as endomorphism algebras of certain projective generators.

I will show how certain important questions in harmonic analysis for p-adic groups can be translated to affine Hecke algebras: e.g., the Plancherel formula and the determination of the unitary dual. I will explain the classification and construction of discrete series modules for affine Hecke algebras with unequal parameters, the calculation of formal degrees, the density and trace Paley-Wiener theorems in this setting, and present some elements of the theory of Dirac operators for graded affine Hecke algebras.

The material for these lectures will be based on works by Bernstein, Kazhdan-Lusztig, Barbasch-Moy, Reeder, Opdam, Solleveld, Kato, Heiermann, Waldspurger. The parts from my own work that are relevant for these lectures are joint with Barbasch, He, Kato, Opdam, Trapa.

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On the explicit description of local theta correspondence for real reductive dual pairs
Xiang Fan, Peking University, China

In this talk, a brief survey is given on the problem to determine the local theta correspondence explicitly in terms of Langlands parameters for real reductive groups in dual pairs. The results achieved in the past three decades on this problem will be reviewed, together with some recent progress.

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On the Moy-Prasad filtration and supercuspidal representations
Jessica Fintzen, Harvard University, USA

Reeder and Yu gave recently a new construction of certain supercuspidal representations of p-adic reductive groups (called epipelagic representations). Their construction relies on the existence of stable vectors in the first Moy-Prasad filtration quotient under the action of a reductive quotient. We will explain these ingredients and present a theorem about the existence of such stable vectors for all primes p. This builds on a result of Reeder and Yu about the existence of stable vectors for large primes and generalizes the paper of the speaker and Romano, which treats the case of an absolutely simple split reductive group.

In addition, we will present a general set-up that allows us to compare the Moy-Prasad filtration representations for different primes p. This provides a tool to transfer results about the Moy-Prasad filtration from one prime to arbitrary primes and also yields a new description of the Moy-Prasad filtration representations.

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Applications of the Deligne-Kazhdan philosophy to the Langlands correspondence for split classical groups
Radhika Ganapathy, Tata Institute of Fundamental Research, India

The Deligne-Kazhdan theory loosely says that the complex representation theory of Galois groups and split reductive groups over a local field of characteristic p can be viewed as the limit, as the ramification index tends to infinity, of the representation theory of these groups over local fields of characteristic 0. In this talk, I will explain how this method can be used to prove the local Langlands correspondence for split classical groups over a local field of characteristic p (with some restrictions on the characteristic) using the corresponding result of Arthur in characteristic 0. This is joint work with Sandeep Varma.

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Recent applications of a couple of theorems on holonomic distributions
Dmitry Gourevitch, The Weizmann Institute of Science, Israel

First I will formulate the Bernstein-Kashiwara theorem about the finite dimensionality of the space of solutions of a holonomic system of PDEs. Then I will present the following recent applications of this result:

a) A bound for the dimension of the space of equivariant distributions on an algebraic G-manifold with finitely many orbits (where G is a real algebraic group).

b) Hausdorffness of the zero homology of the Lie algebra of G with coefficients in the space of Schwartz functions on such a manifold.

c) Equivalence of the Casselman comparison conjecture and the automatic continuity conjecture.

d) A version of the Kobayashi-Oshima-Krötz-Schlichtkrull bounds for multiplicities of irreducible representations.

If time permits I will also present our recent easy proof for meromorphic continuation of intertwining operators on relative principal series on reductive spherical spaces, using an old analytic continuation result of Bernstein.

The talk is based on joint works with A. Aizenbud, B. Krötz, A. Minchenko, G. Liu, S. Sahi and E. Sayag.

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Max Gurevich, The Weizmann Institute of Science, Israel

In the area of relative p-adic harmonic analysis there is much interest in a description of the representations of a reductive group G which can be embedded inside the space of smooth functions on a homogeneous space G/H.

A related question is whether such an embedding can be realized in a canonical form such as an H-integral over a (smooth) matrix coefficient. In a joint work with Omer Offen we treated the symmetric case, i.e., when H is the fixed point group of an involution. As part of the answer we provide a precise criterion for such integrability, which reduces in the group case to Casselman?s known square-integrability criterion.

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Non-Siegel degenerate Eisenstein series
Marcela Hanzer, University of Zagreb, Croatia

In my talk I will elaborate on the work on the degenerate Eisenstein series for symplectic groups I've done recently. First, I'll mention the results in the Siegel case--the location of poles was already known by the work of Kudla and Rallis (using theta correspondence), but we give the explicit description of the images. Then I will address the case of non-Siegel Eisenstein series, which was not dealt with in depth before, and, using combinatorial approach, describe the results.

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On the unramified spherical automorphic spectrum
Volker Heiermann, Université d'Aix-Marseille, France

For a split connected reductive group G defined over a number field F , we compute the part of the spherical automorphic spectrum which is supported by the cuspidal data containing (T, 1), where T is a maximal split torus and 1 is the trivial automorphic character. The proof uses residue distributions introduced by E. Opdam and G. Heckman in the study of graded affine Hecke algebras, and a result by M. Reeder on the weight spaces of the (anti)spherical discrete series representations of affine Hecke algebras. Both of these ingredients are of a purely local nature. For many special cases of reductive groups G similar results have been established by various authors. The main feature of the present proof is the fact that it is uniform and general.
Joint work with M. De Martino et E. Opdam.

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Small representations of finite groups
Roger Howe, Yale University, USA

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of values of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group.

In particular, the representations of small dimension tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of some of these facts. Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, this aspect remains obscure. This talk will discuss a conjectural method for constructing the small representations of finite classical groups. The method is closely related to the theory of theta series, and has an analog over local field.

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A homomorphism of Harish-Chandra for polar representations
Jing-Song Huang, Hong Kong University of Science and Technology, Hong Kong

A polar representation of a complex reductive algebraic group has a Cartan subspace just like an adjoint representation. We describe a homomorphism on invariant differential operators on a polar representation extending the Harish-Chandra homomorphism for an adjoint representation. Then we discuss extension of the works by Wallach, Hotta-Kashiwara and Levasseur-Stafford in connection with the Springer correspondence.

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The automorphic discrete spectrum of Mp(2n)
Atsushi Ichino, Kyoto University, Japan

In his 1973 paper, Shimura established a lifting from half-integral weight modular forms to integral weight modular forms. After that, Waldspurger studied this in the framework of automorphic representations and classified the automorphic discrete spectrum of the metaplectic group Mp(2), which is a nonlinear double cover of SL(2), in terms of that of PGL(2). We discuss a generalization of his classification to the metaplectic group Mp(2n) of higher rank. This is joint work with Wee Teck Gan.

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Simple epipelagic local Galois representations
Naoki Imai, University of Tokyo, Japan

We discuss a geometric construction of the n-dimensional irreducible representations of the Weil group of a non-archimedean local field, which we call simple epipelagic representations. We give also an explicit description of the local Langlands correspondences for GL(n) in the simple epipelagic representation cases.

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Automorphic L-functions and the Langlands functoriality
Dihua Jiang, University of Minnesota, USA

Automorphic L-functions are invariants attached to automorphic representations. The analytic properties of automorphic L-functions are expected to have impacts to the Langlands functoriality, which is one of the fundamental structures of automorphic representations. In this talk, we will review some known results about the relation between the poles of L-funcitons and the Langlands functoriality, and discuss with some details about the relation between the central value of L-functions and certain types of the Langlands functoriality, which is new and is based on my joint work with Lei Zhang.

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Doubling global constructions for tensor product L-functions
Eyal Kaplan, The Ohio State University, USA

In the first part of the talk I will present a joint work with Cai, Friedberg and Ginzburg.
In a series of constructions, we apply the "doubling method" from the theory of automorphic forms to covering groups. We obtain partial tensor product L-functions attached to generalized Shimura lifts, which may be defined in a natural way since at almost all places the representations are unramified principal series.

In the second part, I will describe a joint work with Jan Mollers. We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.

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Some computations of the GL(n) tensor product algebra
Sangjib Kim, Korea University, Korea

I will present some computational results regarding the 'GL(n) tensor product algebra' constructed and investigated by Howe, Jackson, Lee, Tan, and Willenbring, encoding the decomposition of tensor products of finite dimensional irreducible representations of the complex general linear group GL(n). This is a joint work with Donggyun Kim.

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Alpha-determinants and the Alon-Tarsi conjecture
Kazufumi Kimoto, University of the Ryukyus, Japan

Alpha-determinant of a square matrix $A$ is defined by replacing the sign of a permutation in the definition of $\det A$ to a certain power of a parameter conventionally denoted by $\alpha$. This interpolates the the ordinary determinant and the permanent. When we consider the natural action of the general linear Lie algebra $gl_n$ on the algebra of polynomials in the variables $x_{ij}$ ($1\le i,j\le n$), each of the determinant and permanent of the matrix $X=(x_{ij})$ generates an irreducible module. Sho Matsumoto and Masato Wakayama (2006) studied the cyclic $gl_n$-module generated by the alpha determinant and determine its irreducible decomposition explicitly. This result leads us to the study of the alpha-determinants with special parameters for which the structure of the cyclic module degenerates from the generic case; Using them, we can construct a polynomial function, which we call the wreath determinant, for particular type of rectangular matrices, say $n$ by $kn$, which is relative invariant under the natural left translation by $GL_n$. In the talk, we survey the representation-theoretic results on the alpha-determinants, the wreath determinants. We also explain the relation between the wreath determinants and the Alon-Tarsi conjecture on Latin squares.

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Symmetry breaking operators for real reductive groups
Toshiyuki Kobayashi, The University of Tokyo, Japan

Branching problems ask how irreducible representations of groups G decompose when restricted to subgroups G'.We present a program on branching problems, from abstract feature to concrete construction of symmetry breaking operators.

As an abstract feature, we provide a geometric criterion on the pair of reductive groups for the multiplicities of the branching laws to be always of uniformly bounded (or more weakly, to be finite) by using analysis on (real) spherical varieties.

As a concrete construction of symmetry breaking operators (SBOs), we explain an idea of the F-method in constructing differential SBOs. Finally, we discuss some classification results of all non-local and local SBOs by an example.

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Some aspects for representation theory of GLn over non-archimedean field
Erez Lapid, The Weizmann Institute of Science, Israel

This is a joint work with Alberto Mínguez.

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Recent development on global Gan-Gross-Prasad conjecture and arithmetic applications
Yifeng Liu, Northwestern University, USA

In this talk, we will summarize major recent development on the global aspect of the Gan-Gross-Prasad conjecture, which relates the central special values of L-functions and the automorphic periods. After that, we will survey its application in number theory and arithmetic geometry, some known and some speculated.

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Tits' construction of exceptional Lie algebras and Moy-Prasad filtrations
Hung Yean Loke, National University of Singapore

In this talk, I will first quickly review Tits' construction of exceptional Lie algebras using Jordan algebras and octonion algebras.

Next I will explain how this will lead to embeddings of buildings of exceptional groups.

Finally we recall Gan and Yu results in which they identify the building of split $F_4$ (resp. building of $G_2$) with certain set of lattice functions on the Jordan algebra (resp. octonion algebra). We will use these lattice functions and Tits' construction to obtain Moy-Prasad filtrations of the split Lie algebra of type $E_8$.

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K-type, moment map and local theta correspondence
Jiajun Ma, National University of Singapore

In this talk, I will present two joint works with Loke, both use moment maps as an essential tool. One studies the K-types of the (full) theta lifting in the Archimedean case, which leads to the correspondence of associated cycles; Another studies refined minimal K-types in the p-adic case, which leads to an explicit description of the correspondence between (tamely ramified) supercuspidal representations.

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First occurrence indices of tempered representations of metaplectic groups
Ivan Matic, Josip Juraj Strossmayer University Of Osijek, Croatia

We will discuss a determination of the first occurrence indices of tempered representations of metaplectic groups in terms of the first occurrence indices of their partial cuspidal support, in the metaplectic-odd-orthogonal reductive dual pair over a non-archimedean local field.

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Exceptional poles of some L factors for GL(n)
Nadir Matringe, Université de Poitiers, France

For F a nonarchimedean local field, some L factors of GL(n,F) (for example L factors of pairs) have an integral representation which naturally involves the mirabolic subgroup. Cogdell and Piatetski-Shapiro defined the notion of an exceptional pole for such factors.

For generic representations, the occurrence of such a pole at zero should characterise the existence of an appropriate local period on the representation. As a consequence, classifying distinguished representations in terms of their cuspidal support allows one to obtain the inductivity relation of such factors. We will explain this.

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On special Bessel periods and the Gross-Prasad conjecture for SO(2n+1) x SO(2)
Kazuki Morimoto, Kyoto University, Japan

I will talk about one particular case of the global Gross-Prasad conjecture. I will show that if an irreducible cuspidal automorphic representation $\pi$ of $SO(2n+1)$, which is generic at almost all places, admits special Bessel periods corresponding to a quadratic extension $E$, then $L(1 \slash 2, \pi) L(1 \slash 2, \pi \times \chi)$ does not vanish where $\chi$ is the quadratic character corresponding to the quadratic extension $E$. In particular, when $n=2$ and $\pi$ corresponds to a Siegel modular form for $Sp(4, \mathbb{Z})$, I will prove the equivalence between the non-vanishig of the special Bessel period and that of the corresponding central $L$-value. This is a joint work with Masaaki Furusawa.

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Gamma factor and converse theorem
Chufeng Nien, National Cheng Kung University, Taiwan

This talk is about gamma factor, converse theorem for cuspidal representations of GL(n) over finite fields and related problems.

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Conormal variety on exotic Grassmannian
Kyo Nishiyama, Aoyama Gakuin University, Japan

Let $V = \C^n$ be a complex vector space and $W = V \oplus V$ with the standard symplectic strucure, and consider the Grassmannian $\Gr_n(W)$ of $n$-dimensional spaces in $W$.

Let us put $X = \Gr_n(W) \times V$ with the diagonal action of $K = GL(V)$. We call $X$ an exotic Grassmannian (or we may call it "enhanced" Grassmannian instead). Note that $(G=Sp(W), K=GL(V))$ is a symmetric pair, and we are in the situation of double flag variety in the sense of Ochiai-N (2011).

Our goal is to consider a moment map for $K$-variety $X$; defining a conormal variety and an "exotic" nilpotent variety $\ExoticNullcone_X$; and then establishing a "Steinberg-Springer" theory for them (cf. Henderson-Trapa (2012)).

In the talk, I will describe the correspondence between the $K$-orbits in $X$ and exotic nilpotent orbits in $\ExoticNullcone_X$.
We can also play the same game for the "exotic" Lagrangian Grassmannian.

This is a joint work in progress with Lucas Fresse.

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Classifying $A_{\mathfrak{q}}(\lambda)$ modules by their Dirac cohomology
Pavle Pandzic, University of Zagreb, Croatia

We will briefly review the notions of Dirac cohomology and of $A_{\mathfrak{q}}(\lambda)$ modules, and recall a formula for the Dirac cohomology of an $A_{\mathfrak{q}}(\lambda)$ module. Then we will discuss to what extent an $A_{\mathfrak{q}}(\lambda)$ module is determined by its Dirac cohomology. This is joint work with Jing-Song Huang and David Vogan.

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The character and the wave front set correspondence in the stable range
Tomasz Przebinda, University of Oklahoma, USA

We relate the distribution characters and the wave front sets of unitary representation for real dual pairs of type I in the stable range.

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Exceptional Theta Correspondences for D_4
Gordan Savin, University of Utah, USA

We report on a recent progress, joint with Gan, on the exceptional theta correspondences for dual pairs involving a triality D_4.

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The local Jacquet-Langlands correspondence: Endo-classes and congruences
Vincent Sécherre, University of Versailles Saint-Quentin, France

Let F be a non-Archimedean locally compact field of residue characteristic a prime number p, and let G be an inner form of the general linear group GL(n,F). Attached to any complex discrete series representation of G, there is an invariant called is endo-class, arising when restricting the representation to certain pro-p-subgroups of G. It is conjectured that the local Jacquet-Langlands correspondence preserves endo-classes. We prove that two discrete series representations of G have the same endo-class if and only if they are related via a chain of congruence relations mod finitely many prime numbers different from p. Thanks to recent work with A. Minguez on the modular Jacquet-Langlands correspondence, this allows to reduce the aforementioned conjecture to the case where the discrete series representation of G and its transfer to GL(n,F) are both cuspidal. This is joint work with Shaun Stevens.

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Motives with Galois group of type G_2 - construction of Gross and Savin revisited
Sug Woo Shin, The University of California, Berkeley, USA

Serre asked whether there exists a motive (over Q) with Galois group G_2. Put it in another way, the question is to find (a compatible family of) ell-adic Galois representations whose image has Zariski closure G_2. This has been answered affirmatively since 2010 by Dettweiler and Reiter, Khare-Larsen-Savin, Yun, and Patrikis (including generalizations to exceptional groups other than G_2). In this talk I revisit the construction of Gross-Savin (which was conditional when proposed in 1998) which aims to realize such a motive in the cohomology of a Siegel modular variety of genus 3 via exceptional theta correspondence between G_2 and PGSp_6. Then I will explain that the construction is now unconditional due to my recent work with Arno Kret on the construction of GSpin(2n+1)-valued Galois representations in the cohomology of Siegel modular varieties.

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Kazhdan's orthogonality conjecture for real reductive groups
Binyong Sun, Chinese Academy of Sciences, China

We prove that Kazhdan's orthogonality conjecture holds for all real reductive groups G in Harish-Chandra's class, namely the Euler-Poincare pairing of two Harish-Chandra modules of G is equal to the elliptic pairing of the corresponding global characters. This is a joint work with Jing-Song Huang.

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Hecke algebra correspondences for the metaplectic group
Shuichiro Takeda, University of Missouri, USA

We define a certain Iwahori Hecke algebra on the metaplectic group Mp_{2n} over a non-archimedean local field by using a minimal type of the Weil representation, and show that this Hecke algebra is isomorphic to the Iwahori Hecek algebra on SO_{2n+1}. This isomorphism implies an equivalence of the Bernstein component containing the even Weil representation of Mp_{2n} and the Bernstein component containing the trivial representation of SO_{2n+1}. Our result generalizes an earlier result by Gan-Savin, which shows the same equivalence with the assumption that the residue field of the local field is odd. This is a joint work with A. Wood.

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Some genuine small representations of a nonlinear double cover and the model orbit

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 will 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. We will use the split real group of type D_{2n} as an example to exhibit the following further consequences: (1) all representations in Lift(C) are parametrized by pairs (central character, real form of \O); (2) The K-structure of the regular functions on a real form of \O is multiplicity free and matches up with the K-types of the small representations in Lift(C) attached to that real orbit.

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Pro-p Iwahori Hecke algebra, inverse Satake transform and change of weight in characteristic p
Marie-France Vigneras, Institut de Mathématiques de Jussieu, France

Let C be an algebraically closed field of positive characteristic p and let G be a reductive p-adic group. The classification of the admissible irreducible representations of G over C has been reduced to the classification of the supercuspidal ones by Abe, Henniart, Herzig and V. The proof uses the (injective) general Satake transform and the change of weight (proved using the pro-p Iwahori Hecke algebra of G). The image of the general Satake transform was an open question.

With N. Abe and Fl. Herzig, using the pro-p Iwahori Hecke algebra, we can now describe the image of the general Satake transform and give an explicit formula for its inverse. The explicit inverse Satake transform implies easily the change of weight.

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Langlands parameters and finite-dimensional representations
David Vogan, Massachusetts Institute of Technology, USA

A standard way to describe the representation theory of reductive groups over local fields is to say that compact groups are easy, so we should try to understand representations in terms of their restrictions to compact subgroups. I will argue that compact groups are in fact {\it not} that easy, and that we should try to understand their representations as restrictions of representations of noncompact reductive groups, and therefore in terms of arithmetic and Langlands dual groups. I'll explain how this looks for maximal compact subgroups of real reductive groups (which are more or less understood) and for finite Chevalley groups (which George Lusztig understands, but almost nobody else does).

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Waldspurger formula over function fields
Fu-Tsun Wei, Institute of Mathematics Academia Sinica, Taiwan

In this talk, I will present a function field version of the Waldspurger formula for the central critical value of Rankin-Selberg convolution L-functions. This formula says that the central critical value in question shows up in the ratio of "global period integral" to the product of "local period integrals". Applications of this formula will be discussed if time allows.

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L-groups for double covers of Chevalley-Steinberg groups
Martin Weissman, Yale-NUS College

I have recently proposed an L-group for Brylinski-Deligne covers of quasisplit reductive groups over local and global fields. In this talk, I will focus on the crucial case of double covers of quasisplit simply-connected absolutely simple groups (e.g., Sp(4), Spin(6,6), SU(2,1) ), with most examples over the real numbers. In this case, no familiarity with gerbes or with Brylinsnki-Deligne extensions is required. The L-group can be used for a choice-free parameterisation of discrete series representations over R, and of unramified representations over p-adic fields.

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Towards homotopy methods in representation theory
Tian An Wong, The City University of New York, USA

Recently J. Bernstein has suggested considering sheaves on algebraic stacks BG as a way of studying representations of all pure inner forms of G at once, over a local field. Motivated by Morel and Voevodsky's homotopy theory of schemes, I will indicate certain possibilities of extending Bernstein's suggestion, namely through the introductio of simplicial sheaves. One hopes that this will lead to a model category and other higher categorical structures, as in the case of Morel and Voevodsky.

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On the lifting of Hilbert cusp forms
Shunsuke Yamana, Kyoto University, Japan

Starting from a Hilbert cusp form, I will construct holomorphic cusp forms on certain Hermitian symmetric domains of higher degree.

In the case of Siegel cusp forms, this is a generalization of the Saito-Kurokawa lifting from degree two to higher degrees and the Ikeda lifting from the rational number field to totally real number fields.

This is a joint work with Tamotsu Ikeda.

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Tutorial on automorphic descent
Lei Zhang, National University of Singapore

These courses are intended to give a brief introduction of Arthur's endoscopic classification of automorphic representations of classical groups and explicit module constructions via automorphic descent. More precisely, the endoscopic classification parametrises those automorphic representations by the certain self-dual automorphic representations of general linear groups. On the other hand, in order to enhance our understanding of those representations, a natural question is how to construct concrete modules corresponding to Arthur's parameters. One idea is to use the structure of Fourier coefficients inspired by the idea of Roger Howe on the notion of rank and wave front set. By using certain Fourier coefficients of residual representations, the automorphic descent method of Ginzburg-Rallis-Soudry constructs generic cuspidal automorphic representations. To extend their method into great generality, Jiang and I formulate a general construction of cuspidal automorphic representations in generic Arthur packets.

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Shimura operators, Okounkov polynomials and their positivity
Genkai Zhang, Chalmers University of Technology, Sweden

Shimura defined a remarkable family of invariant differential operators on Hermitian symmetric spaces. These operators are formally nonnegative and form a basis for all invariant differential operators. We prove that the Harish-Chandra homomorphisms of the Shimura operators are the Okounkov polynomials. We determine the set of real parameters of spherical representations for which all eigenvalues of Shimura operators are nonnegative for the groups $SU(2, N)$. For rank one groups $SU(n, 1)$ this set is precisely the set of complementary series, and we prove that for $SU(2, N)$ it is a bigger set. This answers partly some question of Shimura.
Joint work with Siddhartha Sahi.

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