Date/Time : 06 Oct 2021 15:00
Venue : via Zoom
Speaker : Jan Frahm, Aarhus University

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https://nus-sg.zoom.us/j/81603760900?pwd=d0lGRktVY0hJSTM5ODBCWmZmTVRrUT09

Meeting ID: 816 0376 0900
Passcode: 883638

Abstract:  ranching problems ask for the behaviour of the restriction of an irreducible representation of a group $G$ to a subgroup $H$. In the context of smooth representations of real reductive groups, this typically leads to the study of multiplicities with which an irreducible representation of $H$ occurs as a quotient of an irreducible representation of $G$. Here, both quantitative results such as multiplicity-one theorems and qualitative results such as the Gan-Gross-Prasad conjectures are of interest. In the context of unitary representations of real reductive groups, one can go a step further and explicitly decompose in irreducible representation of $G$ into a direct integral of irreducible representations of $H$. I will explain how branching laws for unitary representations are related to those in the smooth category, and how one can use an analytic continuation procedure along a principal series parameter to obtain explicit branching laws from certain Plancherel formulas for homogeneous spaces.

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Date/Time : 29 Sep 2021 14:00
Venue : via Zoom
Speaker : Shunsuke Yamana, Osaka City University

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https://nus-sg.zoom.us/j/86263979122?pwd=WXNuYUdnNXdhQkNOODJJNlB3MnRhQT09

Meeting ID: 862 6397 9122
Passcode: 630599

Abstract:  p-adic L-functions involve modified p-factors which measure the discrepancy between the p-adic and complex L-values in the interpolation formula. It is a puzzling fact that this factor can vanish at the central point. Then the p-adic L-function trivially vanish at the point, and such a zero is called an exceptional zero. The p-adic L-function of an elliptic curve has an exceptional zero if and only if it has split multiplicative reduction at p, and the precise relation between derivative of the p-adic L-function and the algebraic part of the complex L-value was conjectured by Mazur-Tate-Teitelbaum and proved by Greenberg-Stevens. 　 There have been many attempts to extend this result of Greenberg-Stevens to more general automorphic forms.

In this talk I will consider the exceptional zeros of the cyclotomic twisted triple product p-adic L-function associated to elliptic curves over rationals and a real quadratic field, and prove an identity between derivatives of the p-adic L-function and complex L-values. I will also consider exceptional zeros of a certain p-adic L-function of degree 6 associated with two rational elliptic curves.

This is a joint work with Ming-Lun Hsieh.

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Date/Time : 22 Sep 2021 14:00
Venue : via Zoom
Speaker : Hideya Watanabe, Osaka City University

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https://nus-sg.zoom.us/j/82636421526?pwd=WUhpZzJPMW1mWXJtRkhtZHRFeVBOUT09

Meeting ID: 826 3642 1526
Passcode: 189069

Abstract: In the representation theory of a quantum group $U$ of finite
type, a based module means a $U$-module with a canonical and crystal
basis. One of the most important properties of canonical bases is
cellularity. In this talk, I generalize the notion of based modules to the
representation theory of the i-quantum group of type AI, and explain that
the associated canonical bases are cellular. Taking the crystal limit of
the canonical bases, a new combinatorial model for irreducible modules
over the orthogonal group is obtained.

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Date/Time : 15 Sep 2021 14:00
Venue : via Zoom
Speaker : Kazuki Morimoto, Kobe University

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https://nus-sg.zoom.us/j/82991610093?pwd=SndoVzZzUmErbjhzV2FCZWZ3bHJ0QT09

Meeting ID: 829 9161 0093
Passcode: 131130

Abstract:  Lapid and Mao formulated a conjecture on Ichino-Ikea type formula of Whittaker periods for quasi-split reductive groups. In the unitary group case, they showed that their conjecture is reduced to a certain local identity. In my previous work, I showed this local identity for even unitary groups at non-split non-archimedean places. In this talk, I will talk about my ongoing work on this local identity at both non-archimedean and archimedean split places, If time permits, I will explain an application of this formula to Ichino-Ikea type formula of Bessel periods for (U(2n), U(1)).

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Date/Time : 10 Sep 2021 15:00
Venue : via Zoom
Speaker : Emile Takahiro Okada, University of Oxford

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https://nus-sg.zoom.us/j/82201157603?pwd=WktRRC9hbHFsaFR6VlF6amdGNk5tZz09

Meeting ID: 822 0115 7603
Passcode: 509310

Abstract:  The wavefront set is a powerful invariant which one can attach to representations of reductive groups over both global and local fields. Notably there is a non-trivial interplay between the wavefront set of an automorphic representation and that of its local factors. In this talk we will discuss recent results computing the wavefront set for all spherical Arthur representations in the p-adic case and the new techniques used in the computation.

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Date/Time : 01 Sep 2021 14:00
Venue : via Zoom
Speaker : Yongqi Feng, Shantou University

Join Zoom Meeting https://nus-sg.zoom.us/j/89953315552?pwd=bERiQUVLSEhPNlFsRWNIR21qWGZOQT09 Meeting ID: 899 5331 5552 Passcode: 911014 Let k be a non-archimedean local field, and let G be a connected reductive k-group which splits over an unramified extension of k. Hiraga, Ichino and Ikeda conjectured that the formal degree d(π) of a discrete series representation π of G(k) can be computed via the associated discrete (i.e. elliptic) local Langlands parameter (φπ, ρπ). The supercuspidal unipotent representations form a convenient class to test this conjecture, with known classification and formal degrees when G is simple and adjoint. We shall discuss, for supercuspidal unipotent representations of classical groups, the proof of the HII formal degree conjecture. This leads to a local Langlands correspondence for such representations, which agrees with the parametrization obtained by Morris and Lusztig when G is simple and adjoint. This is joint-work with Eric Opdam and Maarten Solleveld.

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Date/Time : 25 Aug 2021 14:00
Venue : via Zoom
Speaker : Korkeat Korkeathikhun, University of North Carolina, Chapel Hill

Join Zoom Meeting https://nus-sg.zoom.us/j/89816394951?pwd=OXRvc1pXY0kxSGhYRkloMnFkL0dSUT09 Meeting ID: 898 1639 4951 Passcode: 903999 Let $G$ be a simply-connected simple algebraic group, $\mathfrak{g}$ its Lie algebra. Let $\mathrm{Gr}_G$ be the affine Grassmannian of $G$, and $\mathrm{Gr}_0^-$ the opposite open Schubert cell in $\mathrm{Gr}_G$. Achar and Henderson defined a map $\pi:\mathrm{Gr}_0^- \to \mathfrak{g}$ and used it to exhibit the relation between Schubert varieties in $\mathrm{Gr}_G$ and nilpotent cone in $\mathfrak{g}$. In our work, we consider a twisted analouge. We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar-Henderson in the twisted setting. We also get some applications for the geometry and topology of the order 2 nilpotent varieties in certain classical symmetric spaces.

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Date/Time : 18 Aug 2021 14:00
Venue : via Zoom
Speaker : Jinfeng Song, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/82727387952?pwd=WEo0M0lsd0tETHdhZWZIRGJUaEhEUT09 Meeting ID: 827 2738 7952 Passcode: 557651 Any variety over a field of characteristic p>0 possesses a Frobenius endomorphism F, which fixes all the points and raises the function to p-th power. A variety X is called Frobenius split if the p-th power map splits as a morphism of sheaves of modules. The most important class of examples of varieties which are Frobenius split are flag varieties. This is firstly proved by Mehta and Ramanathan using geometric methods. Later, Kumar and Littelmann construct an alternative proof using the methods of quantum groups. In this talk, I plan to introduce Kumar and Littelmann’s method, and explain its potential generalisation to quantum symmetric pairs.

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Date/Time : 21 Apr 2021 14:00
Venue : via Zoom
Speaker : Ming-Lun Hsieh, Academia Sinica

Join Zoom Meeting https://nus-sg.zoom.us/j/82234939628?pwd=TVRlLzBmS1VsaXhVZ3k5OWtXVlZYdz09 Meeting ID: 822 3493 9628 Password: 838848 Abstract: Darmon, Dasgupta and Pollack in 2011 applied the Eisenstein congruence for Hilbert modular forms to prove the rank one Gross conjecture for Deligne-Ribet p-adic L-functions under some technical assumptions. These assumptions were later lifted by Ventullo. In this talk, we will explain how to apply their ideas in the setting of CM congruence and compute the first derivative of the Katz p-adic L-functions associated with ring class characters of imaginary quadratic fields at the triviall zero. We will present a precise first derivate formula of the Katz p-adic L-functions in terms of certain Gross p-adic regulator and p-adic logarithms of elliptic units. This proves a formula proposed in a recent work of Betina and Dimitrov. This talk is based on a joint work with Masataka Chida.

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