Date/Time : 09 Sep 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Emile Takahiro Okada, National University of Singapore

Abstract: In my talk last year I introduced new methods for computing the wavefront set of depth 0 representations using local wavefront sets. In this series of talks I will begin by briefly recalling the main results from that talk before I go on to outline new results and open questions that have come from these ideas. These will fall into roughly 3 categories: the structure theory of nilpotent orbits, new results on the wavefront set, and finally applications to Arthurs packets.

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Date/Time : 02 Sep 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Hengfei Lu, National University of Singapore

Abstract: Let F be a local field of characteristic zero. Let GL(n,F) acts on the space of n by n matrices by conjugation. Let P be a mirabolic subgroup with last row (0,0,…,0,1). Bernstein proved that the P-invariant distributions on n by n matrices space are also GL(n,F)-invariant when F is nonarchimedean. Aizenbud-Gourevitch prove it for the archimedean case. Let H=GL(p,F)xGL(n-p,F) be a subspace of GL(n,F). Chen-Sun prove that (GL(2n,F),GL(n,F)xGL(n,F)) is a Gelfand pair for any good character \chi of H. In a similar way, we can show that (GL(2n+1,F),GL(n,F)xGL(n+1,F)) is also a Gelfand pair for the good character. Furthermore, if F is nonarchimedean, we can prove that the P’\cap H-invariant linear functional on an H-distinguished irreducible smooth admissible representation \pi of GL(2n+1,F) is also H-invariant. Here H=GL(n,F)xGL(n+1,F) and P’ is the mirabolic subgroup of GL(2n+1,F). As one application, we can show that certain L-function has a pole at s=0 is a necessary condition such that a unitary and unramified principal series representation is distinguished by GL(n,F)xGL(n+1,F). At the end, we will talk about the situation when the character on the subgroup GL(n,F)xGL(n+1,F) is nontrivial.

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Date/Time : 26 Aug 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jonathan Gruber, National University of Singapore

Abstract: In this talk, we explain how every object of a highest weight category can be associated with a \emph{minimal tilting complex}.

Minimal tilting complexes are well-behaved with respect to direct sums and short exact sequences, and they capture some important properties of the corresponding objects in the highest weight category.
After establishing these general properties, we will discuss the problem of finding the multiplicities of indecomposable tilting objects appearing in the terms of minimal tilting complexes. We develop some general techniques for computing these multiplicities and show that in many specific cases (including the category $\mathcal{O}$ for a complex simple Lie algebra or a Kac-Moody Lie algebra), they are governed by Kazhdan-Lusztig polynomials.

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Date/Time : 19 Aug 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jonathan Gruber, National University of Singapore

Abstract:  Highest weight categories provide an axiomatic framework which encompasses many categories that naturally appear in Lie theoretic representation theory, including the BGG category $\mathcal{O}$ for complex simple Lie algebras, representations of quantum groups at roots of unity and rational representations of reductive algebraic groups.

In this talk, we give the definition of highest weight categories and establish some of their key properties.  Then we discuss the classification of \emph{tilting objects} in highest weight categories (due to C.M. Ringel) and explain how the subcategory of tilting objects in a given highest weight category $\mathcal{C}$ controls the homological properties of $\mathcal{C}$.

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Date/Time : 20 Jun 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Yap Jit Wu, Harvard University

Abstract: For a rational function on P^1, it is known that the set of periodic points of period n converges weakly to a measure supported on its Julia set, known as the equilibrium measure, as n tends to infinity. In the 1980s, Szpiro-Ullmo-Zhang proved an arithmetic version of this for abelian varieties, where one is allowed to take just the Galois orbit of a torsion point, instead of all torsion points of order n. Since then, much more general arithmetic equidistribution theorems have been proven, along with the discovery of various applications to number theory.

In this talk, we will explain how having a stronger notion called logarithmic equidistribution will allow us to prove finiteness of S-integral points in certain situations and state some results in this direction.

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Date/Time : 16 Jun 2022 11:00
Venue : S17 #04-06 (SR1)
Speaker : Dinesh Thakur, University of Rochester

Abstract: We will explain some basic congruences of number theory, questions, conjectures and results about them and connections with Zeta values.

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Date/Time : 09 Jun 2022 11:00
Venue : S17 #04-06 (SR1)
Speaker : Dinesh Thakur, University of Rochester

Abstract: We will explain some basic congruences of number theory, questions, conjectures and results about them and connections with arithmetic derivatives.

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Date/Time : 01 Jun 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Edmund Karasiewicz, University of Utah

Abstract: The Fourier coefficients of automorphic forms are important objects of study due to their connection to L-functions. In the adelic framework, constructions of L-functions involving Fourier coefficients (e.g. Langland-Shahidi and Rankin-Selberg methods) naturally lead to spherical Whittaker functions on p-adic groups. Thus we would like to study these spherical Whittaker functions to better understand L-functions. Casselman-Shalika determined a formula for the spherical Whittaker functions, and basic algebraic manipulations reveal that their formula can be succinctly expressed in terms of characters of the Langlands dual group. We will describe a new approach to the Casselman-Shalika formula that provides a conceptual explanation of the appearance of characters. This is joint work with Nadya Gurevich.

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Date/Time : 19 Apr 2022 15:00
Venue : via Zoom
Speaker : Xiaoyu Zhang, University of Duisburg-Essen.

Join Zoom Meeting
https://nus-sg.zoom.us/j/88995853527?pwd=MHZNbFVGTy93QVJEMGFtdEtLdGJGdz09

Meeting ID: 889 9585 3527
Passcode: 388541

Abstract: Theta correspondence is a powerful tool in the theory of automorphic representations. Many fundamental problems in the complex coefficient case have been resolved thanks to the work of many people and have led to very deep applications in Langlands program and other problems in number theory. In this talk, we consider theta correspondence of orthogonal-symplectic type. Given the integral structures of the spaces of automorphic forms, we will show that if an automorphic form on orthogonal group is non-zero mod p, so is its theta lift. The proof involves a comparison of the Bessel period of the theta lift and the toric period of the original automorphic form and is reduced to an equidistribution result of CM points on the orthogonal group. If time permits, we will also discuss some problems related to these results.

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