Relative Langlands Program

Lichang ZHANG (Zhejiang University), Zheng WANG (Paris 13)

“I've looked at clouds from both sides now...”

Intro : This reading seminar explores the Relative Langlands Program through the geometry of spherical varieties and the harmonic analysis of automorphic periods. We will study how the geometry of a spherical variety governs representation-theoretic phenomena, especially those related to automorphic L-functions. Particular attention will be given to Gan--Gross--Prasad varieties for unitary groups and the Jacquet--Rallis relative trace formula. We will also investigate their p-adic and arithmetic refinements, especially the (possible) construction of p-adic L-functions and the study of arithmetic GGP cycles.

This is an online reading seminar. Please get in touch with us if you are interested !

Shamelessly borrowed from Disegni, who borrowed from Kandinsky

Main Reference :

Raphaël Beuzart-Plessis, Introduction to the Relative Langlands Program (2025).

(Possible) Topics

Spherical varieties: spherical roots, valuation cones, dual groups, and boundary degenerations.
Local harmonic analysis: distinction, Plancherel formulas, relative characters, and asymptotics.
Gan–Gross–Prasad: local and global conjectures, period integrals, and the Ichino–Ikeda formula.
Relative trace formulas: geometric and spectral expansions, orbital integrals, and comparison of trace formulas.
p-adic and arithmetic aspects: p-adic relative characters, p-adic L-functions, arithmetic periods, and p-adic heights.
Families: variation of relative periods and arithmetic constructions over p-adic eigenvarieties.

Further References

D. Disegni and W. Zhang, Gan–Gross–Prasad Cycles and Derivatives of p-adic L-functions (2026).
H. Jacquet and S. Rallis, On the Gross-Prasad Conjecture for Unitary Groups (2011).
D. Loeffler, Spherical Varieties and Norm Relations in Iwasawa Theory (2021).
Y. Liu, Bessel Periods and Selmer Groups over Ordinary Rankin–Selberg Eigenvariety (2024).
Y. Sakellaridis and A. Venkatesh, Periods and Harmonic Analysis on Spherical Varieties(2017).
W. Zhang, Periods, cycles, and $L$-functions: a relative trace formula approach (2017).
W. Zhang, High dimensional Gross--Zagier formula: a survey (2024).

Schedule : TBA

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