Date/Time : 11 Apr 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Dougal Davis, University of Melbourne

In a seminal paper from 2011, Kari Vilonen and Wilfried Schmid proposed that the deep theory of mixed Hodge modules can be applied to the old problem of determining the unitary representations of a real reductive Lie group. They conjectured that the unitarity of a representation is completely determined by a canonical filtration, the Hodge filtration, which arises naturally from the geometry of the complex flag variety. In this talk, I will explain a recent paper with Vilonen (arXiv:2309.13215) in which we prove this conjecture. A key role (in both the proof of the conjecture and in applications) is played by a Hodge-theoretic refinement of Beilinson-Bernstein localisation: we show that, for mixed Hodge modules on flag varieties, the Hodge filtration is always globally generated and has vanishing higher cohomology.

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Date/Time : 03 Apr 2024 16:30
Venue : S17 #04-06 (SR1)
Speaker : Xuanzhong Dai, RIMS, Kyoto University

It has been suspected for a long time that the Rankin-Cohen brackets of modular forms should be related to vertex operator algebras as initially suggested by W. Eholzer, Y. Manin and D. Zagier. Our attempt involves sheaf constructions known as the chiral de Rham complex, whose cohomology is linked to the Witten genus. For arbitrary congruence subgroup of modular group, the fractional linear transformation induces an action on the global section of chiral de Rham complex on the upper half plane. We consider the invariant sections under the action of the congruence subgroups with suitable cuspidal conditions. It is noteworthy that modular forms appear naturally on invariant sections.

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Date/Time : 21 Mar 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jinfeng Song, National University of Singapore

It is well-known that the Drinfeld–Jimbo quantum groups can be viewed as quantized coordinate algebras of the dual Poisson groups. A quantum symmetric pair consists of a quantum group and a coideal subalgebra, called an i-quantum group. In this talk, I will explain that i-quantum groups can be viewed as quantized coordinate algebras of Poisson homogeneous spaces of dual Poisson groups. The consruction fits into the general picture, called the quantum duality principal. The relation with cluster algebras and KZ equations will also be briefly discussed.

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Date/Time : 14 Mar 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Le Nhat Hoang, National University of Singapore

The Gan-Gross-Prasad (GGP) conjecture studies a family of restriction problems for classical groups and proposes precise answers to these problems using the local and global Langlands correspondences. It also has a twisted variant in the Fourier-Jacobi case, which is called the twisted Gan-Gross-Prasad conjecture. In this talk, I will report my progress on the local twisted GGP conjecture for tempered representations. The strategy is to adapt Waldspurger and Beuzart-Plessis’s method to develop a local relative trace formula as well as a twisted trace formula and compare their elliptic parts. Although the geometric sides of both trace formulae have not been developed, one can use a partial comparison and an instance of an elliptic L-packet to prove the statement. Moreover, I will propose a geometric multiplicity formula for the local twisted GGP and necessary ingredients to prove it.

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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 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 : 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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