Branching Laws
(11 - 31 Mar 2012)


~ Abstracts ~

 

Real Chevalley involutions and self-dual representations
Jeffrey Adams, University of Maryland, USA


The Chevalley involution of a complex group G takes any algebraic representation to its dual. I'll discuss a real analogue, which has the same property for infinite dimensional representations of a real group G(R). As an application I'll give conditions for every representation of G(R) to be self-dual, and compute the Frobenious Schur (symplectic/orthogonal) indicator for these representations.

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Some applications of representation theory to estimates of automorphic periods
Joseph Bernstein, Tel Aviv University, Israel


I will talk about new methods that use representation theory of real reductive groups to give some highly non-trivial estimates of periods of automorphic functions.

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The coherent cohomology of an algebraic group
Michel Brion, Université de Grenoble, France


The coordinate ring of any linear algebraic group G has a structure of Hopf algebra, which uniquely determines G. On the other hand, for an abelian variety G, Serre showed that the cohomology of G with coefficients in its structure sheaf is an exterior algebra on g = dim(G) primitive elements of degree 1. The talk will present a common generalization of these results to an arbitrary algebraic group G.

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Local theta correspondence and the Gross-Prasad conjecture
Wee Teck Gan, National University of Singapore


I will explain how the Gross-Prasad conjecture for orthogonal and symplectic/metaplectic groups are related via the local theta correspondence.

Given that the GP conjecture for orthogonal groups over p-adic fields are now known (a la Waldspurger and Moeglin-Waldspurger), I will explain what remains to be done to deduce the GP conjecture for the symplectic/metaplectic groups.

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The arithmetic of elliptic curves
Benedict Gross, Harvard University, USA


I will discuss a problem which has been central in number theory for several centuries - whether an elliptic curve in the plane has infinitely many rational solutions. This led to a precise conjecture by Birch and Swinnerton-Dyer in the 1960s, and to some partial progress in the 1980s. More recently, Manjul Bhargava has introduced a new method to study the average number of solutions. I will survey the progress that has been made in the field.

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The internal structure of parabolic and parahoric subgroups
Benedict Gross, Harvard University, USA


In the parabolic case, I will discuss the structure of the Levi quotient and its action on the abelian subquotients of the filtration of the unipotent radical. In the parahoric case, I will discuss the reductive quotient and its action on the abelian subquotients of the Moy-Prasad filtration of the pro-unipotent radical. In the first case, the representations have an open orbit. In the second case, they have a polynomial ring of invariants.

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Hibi algebras and iterated Pieri algebras
Roger Howe, Yale University, USA


In recent years, substantial progress has been made in invariant theory and representation theory by exploiting the idea of toric deformation to semigroup rings. In particular, the special class of semigroup rings constructed by T. Hibi have been found to play a significant role in answering a basic collection of representation-theoretic questions. This talk will focus on the appearance of Hibi rings in describing certain tensor products that generalize the classical Pieri Rule for GL_n.

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Dirac cohomology. K-characters and branching laws
Jing-Song Huang, Hong Kong University of Science and Technology, Hong Kong


Inspired by work of Enright and Willenbring (Ann. Math. 159), we prove a generalized Littlewood's restriction formula in terms of Dirac cohomology. Our approach is to use a character formula of irreducible unitary lowest weight modules instead of the Bernstein-Gelfand-Gelfand resolution, and the proof is much simpler. We also show that our branching formula is equivalent to the formula of Enright and Willenbring in terms of nilpotent Lie algebra cohomology. This follows from the close relationship between the Dirac cohomology and the corresponding nilpotent Lie algebra cohomology for unitary representations of semisimple Lie groups of Hermitian type, which was established by Huang, Pandizc and Renard. This is a joint work with Fuhai Zhu and Pavle Pandzic to appear in American Journal of Mathematics.

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Formal degrees and adjoint gamma factors (continued)
Atsuchi Ichino, Kyoto University, Japan


We give an update on the conjecture which relates the formal degree of a discrete series representation of a reductive group over a local field to a certain arithmetic invariant.

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Stabilization of the trace formula for the covering groups of SL(2) and its application to the theory of Kohnen plus space
Tamotsu Ikeda, Kyoto University, Japan


In this talk, we give a partial stabilization of the trace formula for the metaplectic covering of SL(2). As an application, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half integral weight.

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Path models for branching to symmetric subgroups
Allen Knutson, Cornell University, USA


Let K be a connected, symmetric subgroup of G, the principal example being G = K x K, and let lambda be a dominant G-weight. I define a polyhedral complex P associated to lambda, and a lattice structure on each face (that may become coarser on smaller faces), together with a linear projection to K's positive Weyl chamber.

Theorem 1. For several pairs (G,K), such as (KxK,K), (GL(2n),Sp(n)), and (SL(3),SO(3)), the multiplicity of the K-weight mu in the finite-dimensional representation V_lambda is the number of lattice points in P lying over mu.

Theorem 2. For every pair (G,K), the corresponding asymptotic statement holds (to leading order in N, where lambda,mu are replaced by N*lambda, N*mu).

The points in P correspond to certain piecewise-linear paths in G's positive Weyl chamber, which in the G = KxK case are closely related to Littelmann's path model for tensor product multiplicities.

Time permitting, I will describe the difficulties in extending these results to infinite-dimensional representations.

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Finite multiplicity theorems and real spherical varieties
Toshiyuki Kobayashi, University of Tokyo, Japan


I plan to discuss geometric conditions that control the multiplicities of irreducible representations of real reductive groups occurring in branching laws (restriction) and Plancherel formulas (induction).

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Ubiquity of Schubert varieties
Venkatraman Lakshmibai, Northeastern University, USA


We shall first discuss the classical Schubert varieties in the Grassmannian and the Flag Variety. We shall then show examples of several affine varieties related to Schubert varieties including examples from Classical Invariant Theory. We shall also present results for affine Schubert varieties in the infinite dimensional Flag Variety associated to Kac-Moody Lie alge.

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On a class of representations of GL(n) over a p-adic field
Erez Lapid, Hebrew University of Jerusalem, Israel


Some time ago Tadic gave a remarkable formula expressing a Speh representation as an alternating sum of standard modules. His proof was subsequently simplified by Chenevier - Renard. I will discuss a different approach which sharpens the statement and works for a wider class of representations. Applications will also be discussed.
Joint work with Alberto Minguez

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P-adic L-functions on unitary groups
Jian-Shu Li, Hong Kong University of Science and Technology, Hong Kong


This is a report of work in progress with Eischen, Harris and Skinner on the construction of certain p-adic L-functions on unitary groups, with emphasis on those parts related to branching laws.

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On a pairing of Goldberg-Shahidi for SO(6n)
Wen-Wei Li, Morningside Center of Mathematics, China


In their study of the tempered spectrum of quasisplit even orthogonal groups, Goldberg and Shahidi defined a pairing between supercuspidal matrix coefficients and conjectured that its regular term should be related to twisted endoscopic transfer. These predictions had not been verified except for SO(6), by Shahidi and Spallone. I will discuss the general case in this talk.

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Derived functor modules, dual pairs and U(g)^K actions
Jia Jun Ma, National University of Singapore


Applying the derived functors on highest weight modules constructs interesting representations. When the highest weight module is in a good range, its derived functor modules are well understood. We consider the singular case. In works of Enright at. el. (1985), Frajria (1991) and Wallach (1994), they established several criteria of irreducibility and unitarity and constructed families of examples for the singular case. Building on the work of Wallach and Zhu, I show these examples are related to theta lifts of characters. This is proved by comparing U(g)^K actions.

Motivated by by the above result and a joint work with Loke, I will also give examples beyond the case of theta lifts of characters, which suggests that derived functor construction is related to dual pair correspondence in a much more general setting.

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A Fuchsian differential equation with accessory parameters and Zuckemann's tensoring
Tomonori Moriyama, Osaka University, Japan


If we wish to write a "hyperbolic" Fourier expansion of Maass form of weight one, we encounter a second-order Fuchsian differential equation with five singularities, which seems difficult to analyze at a first glance. But we can get a solution of this equation expressed by Gauss's hypergeometric functions by virtue of an interesting relation among differential operators. This is worked out in the master thesis of M. Maeda (Osaka University, 2012). We also explain how the above-mentioned relation can be foreseen by using Zuckemann's tensoring technique.

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Multiple flag varieties and spherical actions
Kyo Nishiyama, Aoyama Gakuin University, Japan


It is well known that product of two partial flag varieties of a reductive group G have finitely many orbits under the diagonal action of G (the Bruhat decomposition), and Steinberg considered a conormal variety on it. Recently there are several works on triple flag varieties also.

In my talk, I will introduce a double flag variety for a symmetric pair (G, K), which generalize the triple flag varieties above. On this variety K acts diagonally. Sometimes it has finitely many orbits, and they are related to both the Bruhat decompositions and KGB decompositions. It is also related to spherical flag varieties, which can be classified in some sense. We will give an outline of the classification and discuss applications to representation theory/combinatorics.

The talk is based on the joint work with Xuhua He, Hiroyuki Ochiai and Yoshiki Oshima.

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Buildings by Morse
Gordan Savin, University of Utah, USA


I will apply Bestvina's PL Morse theory to study the affine building for SL(n). Using a PL Morse function I will construct a contraction of the building. Then, following an idea of Opdam and Solleveld, the contraction is used to give a simple proof of a result of Schneider and Stuhler on projective resolutions of smooth representations.

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Shalika models and cohomological test vectors
Binyong Sun, Chinese Academy of Sciences, China


Let $\pi$ be an irreducible admissible representation of $GL(2n)$ over a local field, with a Shalika model. For arithmetic applications, it is important to find explicit vectors in the Shalika model of $\pi$ whose zeta integral produces the standard L-function of $\pi$. We try to solve this problem when the local field is the field of real numbers, and the representation is essentially tempered and has non-zero cohomology.

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L-functions and theta correspondence
Shunsuke Yamana, Osaka City University, Japan


The doubling method of Piatetski-Shapiro and Rallis applies in the local situation to define local factors of representations of classical groups. On the one hand, the L-factor is defined as a g.c.d. of the local zeta integrals for all good sections. On the other hand, it is defined from the gamma factor by using the Langlands classification. In this talk I develop a theory of the zeta integral to prove that the two candidates of the L-factor agree. Applications include a characterization of nonvanishing of global theta liftings in terms of the analytic properties of the complete L-functions and the occurrence in the local theta correspondence.

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