Date/Time : 07 Apr 2021 14:00
Venue : via Zoom
Speaker : Shu-Yen Pan, National Tsing Hua University

Join Zoom Meeting hhttps://nus-sg.zoom.us/j/82917531325?pwd=QzVzVVdNbUZhOHlwc09MRG5nWldudz09 Meeting ID: 829 1753 1325 Password: 555209 Let $(G,G’)$ be a reductive dual pair inside a finite symplectic group. By restricting the Weil representation to the dual pair, there exists a relation (called the finite Howe correspondence) between the irreducible representations of the two groups $G,G’$. In this talk, we would like to discuss some progress on the understanding of the correspondence by using Lusztig’s classification on the representations of finite classical groups. In particular, we will focus on the following three subjects: 1. the decomposition of the uniform projection of the Weil character. 2. the commutativity between the Howe correspondence and the Lusztig correspondence. 3. the description of the Howe correspondence on unipotent characters in terms of the symbols by Lusztig.

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Date/Time : 01 Apr 2021 09:00
Venue : via Zoom
Speaker : Jayce Getz, Duke University

Join Zoom Meeting https://nus-sg.zoom.us/j/87075061185?pwd=QnNmQzNQMGxudFZ0aEIwalFTV0ZCdz09 Meeting ID: 870 7506 1185 Password: 568903 Braverman and Kazhdan proposed a conjecture, later refined by Ngo and partially set in the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. In this talk I will describe explicit formulae for the Fourier transform on Braverman-Kazhdan spaces (also known as pre-flag varieties) and for certain spherical varieties that are built out of triples of quadratic spaces. Though my primary motivation comes from automorphic representation theory, these formulae should be of interest more broadly. In particular, they allow for many techniques of analytic number theory to be brought from the setting of vector spaces to the setting of spherical varieties.

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Date/Time : 23 Mar 2021 16:30
Venue : via Zoom
Speaker : Jack Thorne, University of Cambridge

Join Zoom Meeting https://nus-sg.zoom.us/j/85360804573?pwd=WUZCVlQ5NU5kTW5TbWdMRmtWb09YZz09 Meeting ID: 853 6080 4573 Password: 645892 Abstract: Langlands functoriality predicts the transfer of automorphic representations along maps of L-groups. In particular, the symmetric power representation Symm^{n-1} of GL(2) should give rise to a lifting from automorphic representations of GL(2) to automorphic representations of GL(n). I will discuss joint work with James Newton proving the existence of all symmetric power lifts for those automorphic representations of GL(2, A_Q) associated to holomorphic modular forms.

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Date/Time : 17 Mar 2021 15:00
Venue : via Zoom
Speaker : Gaetan Chenevier, Université Paris-Sud

Join Zoom Meeting https://nus-sg.zoom.us/j/89926187213?pwd=eGZ5M2gybnMvZWhCVGJHdnRSYWJqUT09 Meeting ID: 899 2618 7213 Password: 003341 Abstract: I will explain a method allowing to classify the isometry classes of unimodular integral lattices in all dimensions <= 28. This method has computational aspects, but also theoretical ones, involving automorphic representations of orthogonal groups. The talk should (hopefully) be accessible to a general number theorists audience.

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Date/Time : 09 Mar 2021 14:00
Venue : via Zoom
Speaker : Shilin Yu, Xiamen University

Join Zoom Meeting https://nus-sg.zoom.us/j/83991095732?pwd=bG9Ramlwc1NGajVDUjZGQitSNUE3UT09 Meeting ID: 839 9109 5732 Password: 858523 Abstract:  The “coadjoint orbit method” philosophy of Kirillov and Kostant suggests that there is a close connection between the unitary dual of a Lie group G, which is the set of all irreducible unitary representations of G, and the set of coadjoint orbits in the dual of its Lie algebra. Roughly, one expects that the representations arise as “quantizations” of certain vector bundles on the coadjoint orbits, which are naturally symplectic manifolds. When the Lie group is a noncompact real semisimple group, both the unitary dual and the orbit method are far from being fully understood, yet a lot of deep and interesting mathematics have been developed to study this case. Vogan suggested the candidates for the vector bundles on the coadjoint orbits to be quantized, which he called admissible vector bundles. In this talk, we show that these admissible vector bundles indeed correspond to representations, with some exceptions. The tools used here are deformation quantization of symplectic varieties and their Lagrangian subvarieties. The talk is based on a joint paper with Conan Leung and an ongoing joint work with Ivan Losev.

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Date/Time : 03 Mar 2021 15:00
Venue : via Zoom
Speaker : Eyal Kaplan, Bar Ilan University

Join Zoom Meeting https://nus-sg.zoom.us/j/89277873740?pwd=M09uNUdjMVVDT21CcHkrOUVaUi9Idz09 Meeting ID: 892 7787 3740 Password: 469660 Abstract: In recent works, the doubling method of Piatetski-Shapiro and Rallis from the 80s has been generalized in several aspects with interesting applications, including a result on global functoriality, and progress in the study of L-functions and local factors for representations of covering groups.  In this talk I will describe the new ingredients, which help complete the (global and local) picture: The multiplicity one result obtained in a joint work with Aizenbud and Gourevitch, and a new family of GL(c) x GL(k) doubling integrals. Parts of the talk are also based on a collaboration with Cai, Friedberg and Ginzburg.

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Date/Time : 24 Feb 2021 14:00
Venue : via Zoom
Speaker : Chen Rui, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/86900644508?pwd=S0FUdkI2RzhGTWJwV1R1Y1F6SmpBUT09 Meeting ID: 869 0064 4508 Password: 895425 Abstract:   This is a sequel to Jialiang’s talk, we shall consider some local counter part of that talk. We describe some behavior of local Arthur packets under the stable range theta correspondence using Moeglin’s explicit construction of local Arthur packets for quasi-split classical groups. In some sense our results could be regarded as giving a proof of Adams conjecture in the stable range case.

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Date/Time : 17 Feb 2021 14:00
Venue : via Zoom
Speaker : Zou Jialiang, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/83550826419?pwd=aUtXWDA1eU1naDVUT1hIU0E0QW1iQT09 Meeting ID: 835 5082 6419 Password: 223252 Abstract:  Based on a result of J-S Li on global theta lifts in the stage range, Gan-Ichino established a multiplicity formula for the tempered part of discrete spectra for $MP_2n$ by transporting the multiplicity formula of quasi-split special orthogonal groups. In this talk, I will describe how to use this idea to obtain the same results for non quasi-split even orthogonal and unitary groups. Moreover, we can also obtain a multiplicity formula for the full discrete spectra of anisotropic unitary and even orthogonal groups.

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Date/Time : 10 Feb 2021 14:00
Venue : via Zoom
Speaker : Allan Francis Merino, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/88282572726?pwd=a1QrdVFHSUt0WGxDZER4Uy9za3FmQT09 Meeting ID: 882 8257 2726 Password: 353941 Abstract:  For every irreducible reductive dual pair (G, G’) in Sp(W), R. Howe proved the existence of an isomorphism between the spaces R(G) and R(G’), where R(G) is the set of infinitesimal equivalence classes of irreducible admissible representations of \tilde{G} (preimage of G in the metaplectic group) which can be realized as a quotient of the metaplectic representation. All the representations appearing in the correspondence have a distribution character, and characters are analytic objects completely identifying the irreducible representations. In particular, one natural question is to understand the transfer of characters in the theta correspondence (or Howe’s duality). In 2000, T. Przebinda introduced the Cauchy-Harish-Chandra integral and conjectured that the transfer of characters should be obtained via this map. This conjecture is known to be true if G is compact and was proven by Przebinda for unitary representations in the stable range case. In general, it has been established that the Cauchy-Harish-Chandra integral sends the \tilde{G}-invariant eigendistributions on \tilde{G} into the \tilde{G’}-invariant eigendistributions on \tilde{G’} as long as rk(G) \leq rk(G’). In my talk, after recalling carefully the construction of the Cauchy-Harish-Chandra integral and stating Przebinda’s conjecture, I am going to explain, by using results of A. Paul, how to prove this conjecture for the pair (G,G’) = (U(p, q), U(r, s)), with p+q = r+s, starting with a discrete series representation of \tilde{G}. At the end of my talk, I will discuss some auxiliary results I got in my paper and an ongoing project on transfer of characters for (G, G’), with rk(G) \leq rk(G’), starting from a discrete series representation.

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