Date/Time : 07 Mar 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Ziyang Gao, Leibniz University Hannover

The goal of this talk is to propose a possible framework to study quadratic relations among holomorphic periods of CM abelian varieties. We define, for each Shimura variety and a CM point, a bi-$\overline{\mathbb{Q}}$-structure on the tangent space. Then we explain that in the case of the Siegel moduli variety, the numbers comparing the two $\overline{\mathbb{Q}}$-structures are precisely the products of the holomorphic periods of the CM abelian varieties parametrized by the CM point (up to $2\pi i$). Next we propose a hyperbolic analytic subspace conjecture, which is the analogue of Wüstholz’s analytic subgroup theorem in this context, and explain why it implies the desired consequence on the quadratic relations among these holomorphic CM periods. This is joint work with Emmanuel Ullmo and (partially with) Andrei Yafaev.

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Date/Time : 27 Feb 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Shilin Yu, Xiamen University

Unipotent representations are believed to be the building blocks of unitary dual of semisimple/reductive Lie groups. In their study of special unipotent representations for complex semisimple groups, Barbasch and Vogan constructed a duality map between the nilpotent orbits of $G$ and that of its Langlands dual group $G^\vee$ (also discovered by Lusztig and Spaltenstein), which allows them to describe the unipotent ideals and representations of $G$ in terms of $G^\vee$. Later Sommers and Achar extended this duality map by considering pairs $(\mathbb{O}, C)$ consisting of a nilpotent orbit $\mathbb{O}$ and a conjugacy class $C$ in its Lusztig canonical quotient group.

In the joint work with Lucas Mason-Brown and Dmytro Matvieievskyi (arXiv:2309.14853), we have extended the Sommers’ duality map in another direction by considering covers of nilpotent orbits, which turns out to be equivalent to Achar’s duality map (in a nontrivial way). This extended duality map allows us to describe the (generalized) unipotent ideals and bimodules attached to nilpotent covers, proposed previously by Ivan Losev and my other two coauthors, in terms of the Langlands dual group.

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Date/Time : 22 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Bin Xu, Tsinghua University

The irreducible admissible representations of Arthur class are the local components of automorphic representations. They are conjectured to be parametrized by the Arthur parameters, and the set of irreducible representations associated with a single Arthur parameter is called an Arthur packet. For quasisplit symplectic and orthogonal groups, the Arthur packets have been determined by Arthur, and their structure in the p-adic case can be understood in a very complicated way by the works of Moeglin, Atobe and myself. In this talk, we would like to introduce some conjectures on their structure in the real case, which are motivated by the results in the p-adic case. For simplicity we will only consider the symplectic groups. This is an ongoing project with Taiwang Deng and Chang Huang.

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Date/Time : 19 Feb 2024 16:00
Venue : S17 #04-06 (SR1)
Speaker : Max Gurevich, Technion - Israel Institute of Technology

Pursuing local foundations for Arthur theory, Adams-Barbasch-Vogan have introduced the concept of a weak Arthur packet. Those are sets of irreducible representations that share a common infinitesimal character specified by a choice of a dual unipotent conjugacy class, and have the smallest possible Gelfand-Kirillov dimension among those.

In a separate context, the p-adic setting of groups of classical type shows that Arthur theory behaves well with respect to ramification properties. Namely, an Arthur-type spherical representation determines the unique Arthur packet which contains it – a consequence of the extended Shahidi conjecture.

In a joint work with Emile Okada, we unveil a ‘weak’ analogy to this phenomenon: Roughly, a weak Arthur packet is characterized as the union of those Arthur packets that contain a weakly spherical representation.

Along the way we explicate how geometry of adjoint nilpotent orbits governs the composition of weak Arthur packets, while exhibiting Springer theory as an effective tool for studying weak sphericity in p-adic groups.

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Date/Time : 19 Feb 2024 17:00
Venue : S17 #04-06 (SR1)
Speaker : Geo Kam-Fai Tam, Radboud University Nijmegen

Let $G$ be a classical group (orthogonal, symplectic, unitary) over a p-adic field $F$, and $\pi$ be a supercuspidal representation of $G(F)$. We propose a general strategy to express the Langlands parameter of $\pi$, requiring only that the residual characteristic $p$ to be odd, with proven examples by the speaker (partly joint with Corinne Blondel) and others. To briefly explain the strategy, we apply M{\oe}glin’s theory of Jordan blocks and Bushnell-Kutzko’s theory of covering types to determine the endoscopic lift of $\pi$ into the general linear group whose dual expresses the dual group of $G$ as a complex matrix group. The cuspidal support of this lifted representation is given explicitly in terms of the inducing type of $\pi$ defined by Stevens’ construction of supercuspidal representations using skew semi-simple strata. This strategy is completely on the representation side. On the Galois side, we provide an example of describing the Langlands parameters of epipelagic representations constructed by Reeder-Yu, generalizing the results obtained by Oi on simple supercuspidal representations constructed by Gross-Reeder.

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Date/Time : 15 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Yuanqing Cai, Kanazawa University

We discuss the trace formula for the n-fold covering group of SL_2. Along the way, we also revisit the work of Flicker on the trace formula for covering groups of GL(2).

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Date/Time : 08 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Rui Chen, IASM, Zhejiang University

Branching law is a classical topic in representation theory which concerns certain Hom functors. It is natural to consider Ext functors associated to those Hom functors. In this talk we will report some recent progress on the study of these Ext spaces, and show a conjecture of D. Prasad, which asserts that the Ext-analogue of GGP model vanishes for tempered representations. We will also discuss some applications.

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Date/Time : 01 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jeff York Ye, National University of Singapore

The decomposition of tensor products of finite-dimensional irreducible representations of simple Lie groups is a long-studied problem. I will discuss some known results describing the multiplicity using canonical bases for quantum groups and tropicalization. Then I will discuss the related saturation conjecture, which is proved in type A by Knutson and Tao and open for general types.

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Date/Time : 01 Feb 2024 10:00
Venue : S17 #04-06 (SR1)
Speaker : Wang Jialin, National University of Singapore

Fix a finite group G and an algebraically closed field F of characteristic p. For an FG-module M, the complexity of M is the rate of growth of a minimal projective resolution of M which is a cohomological invariant of M. The rank varieties introduced by Carlson are defined for modules for elementary abelian p-groups and can be extended to modules for G by looking at the restriction to elementary abelian subgroups of G. Moreover, the dimension of the rank variety gives the complexity of the module. In this talk, I will discuss some basic properties of rank varieties and complexities and then review some known results on complexities of some simple modules for symmetric groups.

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