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## 2018李群与自守表示研讨会(四川 成都)

On central values of L-functions

10月27日 08:10—08:50  祥宇宾馆2楼祥庆厅

In this talk, we will discuss recent joint work with Qi Zhi on generalizing Baruch-Mao's formula from totally real fields to general number fields, which express the central values of automorphic L-function of some automorphic representation in terms of the Fourier coefficients of its Waldspurger-Shimura correspondence. Our approach is based on a relative trace formula by Jacquet. We will also discuss its applications.

Bessel identities over the complex field

10月27日  08:50—09:30  祥宇宾馆2楼祥庆厅

A fundamental class of special functions in the number theory and representation theory for $\mathrm{GL}_2 (\mathbb{R})$ is classical Bessel functions (over $\mathbb{R}_+$). Certain (integral) formulae for Bessel functions have deep representation theoretic interpretations. For example the formulae of Weber and Hardy on the Fourier transform of Bessel functions on $\mathbb{R}$ may be considered as a realization of the Waldspurger correspondence over $\mathbb{R}$. In this talk, we show how certain Bessel identities in the Waldspurger correspondence over $\mathbb{C}$ follow from the Weber-Hardy type formula over $\mathbb{C}$. This is a joint work with Jingsong Chai.

Regular supercuspidal representations and applications

10月27日  09:30—10:10  祥宇宾馆2楼祥庆厅

I will briefly review Kaletha's construction of regular supercuspidal representations, and discuss some of its applications.

On the local descent for p-adic unitary groups

10月27日10:30—11:10 祥宇宾馆2楼祥庆厅

Starting from the local Gan-Gross-Prasad conjecture, we explain the spectral decomposition of local descents at first occurrence for p-adic unitary groups via both Bessel and Fourier-Jacobi models, in terms of local Langlands parameters and local root numbers. This talk is based on part of a project joint with Dihua Jiang and Lei Zhang.

Non-vanishing of Archimedean Local Integrals of Friedberg-Jacquet

10月27日 11:10—11:50  祥宇宾馆2楼祥庆厅

In this talk, we will establish the non-vanishing property with explicitly constructed cohomological test vector for the archimedean local integral of the Friedberg-Jacquet global integral that represents the standard L-function for cuspidal automorphic representation \pi for GL_{2n} with non-zero Shalika period. This non-vanishing result was a non-vanishing hypothesis in many applications, including the work of A. Ash and D. Ginzburg on the construction of a p-adic L-function for GL_{2n}; the work of Ragharum on critical values of the standard L-function of GL_{2n}. This is a joint work with D. Jiang and F. Tian. It was started during Lin's visit at UMN.

Spin norm: combinatorics and representations

10月27日 14:00—14:40  祥宇宾馆2楼祥庆厅

We will introduce the following three preprints in 2017: arXiv:1702.01876 (joint with Jian Ding), arXiv:1707.01380 and arXiv:1708.00383. As a consequence, for a real reductive Lie group G(R) in the sense of [Adams, van Leeuwen, Trapa and D. Vogan, arXiv:1212.2192], we will report a finiteness theorem for the classification of all the irreducible unitary Harish-Chandra modules (up to equivalence) for G(R) with non-zero Dirac cohomology. In particular, when the rank of G(R) is small, we should be able to obtain the complete classification via atlas.

Unipotent representations and unitarity

10月27日 14:40—15:20   祥宇宾馆2楼祥庆厅

The classification of unitary representations of reductive Lie groups has been a longstanding problem in the field. For complex classical Lie groups, Barbasch gave explicitly the unitary spectrum using the notion of 'unipotent representations'. We will review how the these representations are defined, and explore how the constructions may be extended to the exceptional Lie groups.