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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Date/Time : 02 Dec 2020 09:00
Venue : via Zoom
Speaker : Yu Zhao, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/85482725152?pwd=em9iM2ZlQk5EZEsyQ3ZIUm9ZVVN3Zz09 Meeting ID: 854 8272 5152 Password: 638381 The quantum toroidal algebra is the affinization of the quantum Heisenberg algebra. Schiffmann-Vasserot, Feigin-Tsymbaliuk and Negut studied the quantum toroidal algebra action on the Grothendieck group of Hilbert schemes of points on surfaces, which generalized the action by Nakajima and Grojnowski in cohomology. In this talk, we will categorify the above quantum toroidal algebra action. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities.

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Date/Time : 18 Nov 2020 09:00
Venue : via Zoom
Speaker : Pablo Alvarez, Institute for Advanced Study

Join Zoom Meeting https://nus-sg.zoom.us/j/89214308777?pwd=VmZBVmZaamF5Z1h5VnU1MERaZ2ZVQT09 Meeting ID: 892 1430 8777 Password: 369675 Abstract:  The question of the center of the quantum group at a generic parameter was understood by Drinfeld in 1989, but the center of this algebra when you specialize at a root of unity remained an open question. By work of Lachowska, Qi and Bezrukavnikov we had some description of the center of the small quantum group. In joint work with Bezrukavnikov, Shen and Vasserot, we prove a more concrete description of the center of the small quantum group, in terms of the cohomology of a certain affine Springer fiber. In joint work with Kivinen and Losev, we link certain questions of the center in Type A with the Hilbert scheme of points on C2.

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Date/Time : 12 Nov 2020 10:00
Venue : via Zoom
Speaker : Chengze Duan, University of Maryland, college park

Join Zoom Meeting https://nus-sg.zoom.us/j/85957375797?pwd=clE4UTk1WDF0VGMzdU5wVldIMnl2UT09 Meeting ID: 859 5737 5797 Password: 456131 Abstract:  n Vinberg’s study of characterization of reductive algebraic monoids and Vinberg monoids, he introduced the notion of perfect submonoids of dominant weights. Given a reductive algebraic monoids, we shall see a perfect submonoids of dominant weights of the group of units. Meanwhile, the perfect submonoids of dominant weights can also appears naturally from the tensor product decomposition in representation theory. We will see some explicit descriptions of the perfect submonoids of dominant weights and compare it with Vinberg’s results on reductive monoids.

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Date/Time : 05 Nov 2020 10:00
Venue : via Zoom
Speaker : Rahul Dalal, University of California, Berkeley

Join Zoom Meeting https://nus-sg.zoom.us/j/94719446250?pwd=bzRVL1Z6blh1RUViMTUzVnFFblllUT09 Meeting ID: 947 1944 6250 Password: 022183 Abstract:  Since Automorphic representations for general groups are very difficult to study individually, they are often studied in families instead. The Arthur-Selberg trace formula lends itself very naturally to answering questions about averages of various parameters of the local components of automorphic representations in so-called harmonic families. In their 2016 work, Shin and Templier realized that, in the special case of representations with discrete series at infinity, the trace formula simplified dramatically enough to compute statistics with good error bounds. These bounds were good enough for applications: an averaged Sato-Tate law analogous to Sato-Tate for families of elliptic curves and computations of the specific random-matrix statistics followed by low-lying zeros of L-functions in the family. Shin-Templier’s work used the invariant trace formula which disallowed families that distinguish representations with infinite component in the same L-packet. However, which member of this L-packet a representation might correspond to determines some important characteristics–whether the representation is holomorphic or quaternionic for example. I will discuss how methods related to the stable trace formula can remove this restriction. The key idea is applying a certain “hyperendoscopy” formulation of stabilization used first by Ferrari, though many technical difficulties come up.

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Date/Time : 29 Oct 2020 09:00
Venue : via Zoom
Speaker : Charlotte Chan, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/91964791213?pwd=ZFlvbVdySWNlL2NpNzJXSnlSOUtxQT09 Meeting ID: 919 6479 1213 Password: 006362 Abstract:  Deligne–Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. These give rise to so-called “depth-zero” supercuspidal representations of p-adic groups. In this talk, we discuss geometric constructions of positive depth supercuspidal representations and the implications of such realizations towards the Langlands program. This is partially based on joint work with Alexander Ivanov and joint work with Masao Oi.

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Date/Time : 22 Oct 2020 10:00
Venue : via Zoom
Speaker : Jonathan Wang, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/92431908138?pwd=NGs4eWRENWRERWlJRUZIUDdyVFVOUT09 Meeting ID: 924 3190 8138 Password: 865981 Abstract:  Part of the aim of the relative Langlands program is to produce new integral representations of automorphic L-functions in order to study their analytic properties. The program is still very much in progress, but generalized Ichino-Ikeda formulas of Sakellaridis-Venkatesh relate the global problem to certain problems in local harmonic analysis on spherical varieties. I will explain how we can use techniques from geometric Langlands to compute “new” integrals which give special values of unramified local L-functions over a local function field. Our results also give new integral representations of L-functions (in a right half plane) over global function fields when the integral “unfolds”. This is joint work with Yiannis Sakellaridis.

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