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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Date/Time : 03 Feb 2021 14:00
Venue : via Zoom
Speaker : Allan Francis Merino, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/86961364011?pwd=SU9Xa05IUW9kL1NVY21peGVUUTNHUT09 Meeting ID: 869 6136 4011 Password: 065002 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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Date/Time : 21 Jan 2021 09:00
Venue : via Zoom
Speaker : Chao Li, Columbia University

Join Zoom Meeting https://nus-sg.zoom.us/j/84339085667?pwd=ZDZvZkZPMEJiSW1HcDNNNnVIbHlSQT09 Meeting ID: 843 3908 5667 Password: 816448 Abstract: For certain automorphic representations π on unitary groups, we show that if L(s, π) vanishes to order one at the center s=1/2, then the associated π-localized Chow group of a unitary Shimura variety is nontrivial. This proves part of the Beilinson-Bloch conjecture for unitary Shimura varieties, which generalizes the BSD conjecture. Assuming the modularity of Kudla’s generating series of special cycles, we further prove a precise height formula for L'(1/2, π). This proves the conjectural arithmetic inner product formula, which generalizes the Gross-Zagier formula to Shimura varieties of higher dimension. We will motivate these conjectures and discuss some aspects of the proof. This is joint work with Yifeng Liu.

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Date/Time : 17 Dec 2020 09:00
Venue : via Zoom
Speaker : Yau Wing Li, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/87836014196?pwd=MGdrTmVWN1NuWHFCbFg3YTBjMEZsUT09 Meeting ID: 878 3601 4196 Password: 339627 Abstract: Affine Hecke categories are categorifications of Iwahori-Hecke algebras, which are essential in the classification of irreducible representations of loop group LG with Iwahori-fixed vectors. The affine Hecke category has a monodromic counterpart, which contains sheaves with prescribed monodromy under the left and right actions of the maximal torus. We show that the neutral block of this monoidal category is equivalent to the neutral block of the affine Hecke category (with trivial torus monodromy) for the endoscopic group H. It is consistent with the Langlands functoriality conjecture.

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