Date/Time : 04 Nov 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Fang Xinyu, National University of Singapore

Abstract: A conjecture by Cohen and Lenstra (1984) predicts a distribution for the class group of a random quadratic field. This was one of the first motivating examples for the probability theory of random groups that arise in number theory. In this talk we will give a brief overview of the theory, and introduce some major results concerning the moment problem as well as universality. Then we will focus on some universality results for random abelian groups that arise as cokernels of random matrices, possibly with additional structure (such as being symmetric, Hermitian or alternating). We show that a method used to compute a formula for the distribution of cokernels of symmetric matrices also works in the case of Hermitian ones, when they are drawn from additive Haar measure.

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Date/Time : 14 Oct 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Le Nhat Hoang, National University of Singapore

Abstract: The local Gan-Gross-Prasad (GGP) conjecture studies a family of restriction problems for classical groups and proposes precised answers to these problems using the local Langlands correspondence. It also has a twisted variant in the Fourier-Jacobi case. In this talk, I will present my proof for the tempered part of the local twisted GGP conjecture for (GL(V),U(V)), which is obtained by adapting Waldspurger and Plessis’s method to construct a local trace formula and comparing its spectral and geometric sides. Moreover, I will talk about how the geometric multiplicity formula looks like in the remaining case, when we consider an another quadratic extension K/F and have the local twisted GGP conjecture for (U(V)(K),U(V)).

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Date/Time : 07 Oct 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Chung-Ru Lee, National University of Singapore

Abstract: One of the main application of the Relative Langlands Program is providing a description of distinguished representation, which itself has profuse interpretations. In this talk series I will introduce a method to study such problem through the comparison of relative trace formulae, where the theory of endoscopy plays a significant role. In particular, the study of representations on unitary groups that are distinguished by an orthogonal subgroup remains mysterious due to the presence of a type N-spherical root, which requires extra caution when dealt with.

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Date/Time : 30 Sep 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Chung-Ru Lee, National University of Singapore

Abstract: One of the main application of the Relative Langlands Program is providing a description of distinguished representation, which itself has profuse interpretations. In this talk series I will introduce a method to study such problem through the comparison of relative trace formulae, where the theory of endoscopy plays a significant role. In particular, the study of representations on unitary groups that are distinguished by an orthogonal subgroup remains mysterious due to the presence of a type N-spherical root, which requires extra caution when dealt with.

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Date/Time : 16 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 : 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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