Math 263C: Topics in Representation Theory

 Spring 2024

MWF 10:30 AM - 11:20 AM, GESB150.
In this course, we will discuss recent developments in Langlands duality, arithmetic of automorphic representations and arithmetic geometry via generating series e.g. L-functions and theta series.
Instructor: Zhiyu Zhang.

Exercises:

Office Hour: by appointment.

Lecture Schedule
  • Lecture 1, April 1. Overview.
  • Lecture 2, April 3. Compact Lie groups and their representations, L^2-functions on spheres, Fock model of Weil representations, Plancherel formulas.
  • Lecture 3, April 5. Representations of p-adic reductive groups, Satake isomorphism, parabolic induction and Jacquet modules.
  • Lecture 4, April 8. Automorphic spectrum, unramfied spectrum and Langlands program. Why only care about cuspidal forms?
  • Lecture 5, April 10. Towards (regular) Langlands: symmetric spaces and Hodge theory over a base, flag varieties, period maps and Matsushima formula (dual BGG complex).
  • Lecture 6, April 12. Towards (regular) Langlands: motives, Fontaine-Mazur, Shimura varieties and inner forms.
  • Lecture 7, April 15. Automorphic L-functions: Rankin-Selberg for GL(2), inner forms and Arthur packets.
  • Lecture 8, April 17. Arithmetic L-functions: conjectures and orthogonal/symplectic motives.
  • Lecture 9, April 19. Towards (automorphic L-functions) Period integrals, relative Langlands.
  • Lecture 10, April 22. Trace formulas and local-global compatibility: orbital integrals, fundamental lemmas, transfers and applications.
  • Lecture 11, April 24. Towards arithmetic L-functions: automorphic realizations, cohomology of algebraic cycles and Gross-Zagier formulas.
  • Lecture 12, April 26. Towards local Langlands: affine flag varieties, local Shimura varieties (Rapoport-Zink spaces) and local-global compatibility.
  • Lecture 13, April 29. Points, mod p and p-adic geometry of local-global Shimura varieties. Why integral models?
  • Lecture 14, May 1. From cycles to numbers: Height pairings, winding numbers, K-theory and arakelov geometry.
  • Lecture 15, May 3. Arithmetic relative Langlands: cycles, modularity, arithmetic periods and applications.
  • Lecture 16, May 6. Local-global compatibility: arithmetic fundamental lemmas and transfers.
  • Lecture 17, May 8. Computation tools: caonical / quasi-canonical liftings, Gross-Keating, Dieudonne theory, deformation theory.
  • Lecture 18, May 10. Theta correspondences: modularity, Siegel-Weil, Rallis inner products.
  • Lecture 19, May 13. Kudla program and motivic theta correspondences, local arithmetic Siegel-Weil formulas (Kudla-Rapoport).
  • Lecture 20, May 15. Geometrization of relative Langlands I. Spherical varieties, Hamiltonian spaces and symplectic representations.
  • Lecture 21, May 17. Geometrization of relative Langlands II. local L^2 decomposition, finite multipilicity, local periods.
  • Lecture 22, May 20. Geometrization of relative Langlands III: Plancherel algebras.
  • Lecture 23, May 22. Geometrization of relative Langlands IV: Period sheaves and L-sheaves.
  • Lecture 24, May 24. Geometrization of relative Langlands V. Regularization.
  • May 27 (Memorial Day).
  • Lecture 25, May 29. Geometrization of relative Langlands VI.
  • Lecture 26, May 31. Geometrization of relative Langlands VII.
  • Lecture 27, June 3. Geometrization of relative Langlands VIII.
  • Last Lecture 28, June 5. Arithmetic conjectures and conclusion.

    First half topics:

  • Conjectures that we still do not know.
  • Representations of compact Lie groups and p-adic analogs.
  • Automorphic forms, generating series and Langlands program.
  • L-functions and the point view of motives (orthogonal / sympletic).
  • Methods of spectral traces, base change and Jacquet-Langlands.
  • Method of theta correspondence: modularity, Siegel--Weil formulas, Rallis inner product formulas.
  • (Categorical) trace methods and relative Langlands duality.
  • Endoscopic Fundamental Lemmas.
  • Explicit understanding of functions on geometric spaces e.g. spheres.
    Second half topics:
  • Hodge theory, period domains and superconnections.
  • Complex and p-adic geometry of cycles on moduli spaces (local models -> Shtukas over a family -> Shimura varieties).
  • A highlight of developments of p-adic Hodge and Bun_G with applications to Shimura varieties.
  • Geometric theta correspondence, Kudla--Milson lifts and Kudla program.
  • Story over function fields.
  • Connections between the analytic world and the arithmetic world e.g. Gross--Zagier formulas, arithmetic fundamental lemmas.
  • Applications to L-functions and arithmetic correspondences between motives e.g. symmetric power of elliptic curves.

    More references on relative Langlands:
  • Relative Langlands seminar, Fall 2023 .
  • Seminar on Relative Langlands Duality .
  • RTG SEMINAR ON RELATIVE LANGLANDS PROGRAM .
  • Seminar on Arithmetic Topological Field Theory and Relative Langlands Duality . Minhyong Kim.
  • Between electric-magnetic duality and the Langlands program , David Ben-Zvi.
  • Talk on Quantization and Duality for Spherical Varieties , David Ben-Zvi.
  • Talk on Electric-Magnetic Duality for Periods and L-functions , David Ben-Zvi.
  • Talk on Infinite sums of L-functions. , Akshay Venkatesh.
  • Talk on Periods and L-functions. , Yiannis Sakellaridis.
  • A relative local Langlands correspondence, D. Prasad.
  • Spherical varieties and Langlands duality, D. Gaitsgory, D. Nadler.
  • The Bessel-Plancherel theorem and applications, Raul Gomez.
  • Langlands' philosophy and Koszul duality, Wolfgang Soergel.
  • Lagrangian subvarieties of hyperspherical varieties, M. Finkelberg, V. Ginzburg, R. Travkin.
  • Equivariant Satake category and Kostant-Whittaker reduction, R. Bezrukavnikov, M. Finkelberg.
  • Intersection complexes and unramified L-factors, Y. Sakellaridis, J. Wang.
  • The Fundamental lemma for the Bessel and Novodvorsky subgroups of GSp(4), M. Furusawa, J. Shalika.
  • The Fundamental Lemma for the Shalika Subgroup of GL(4), S. Friedberg, H. Jacquet.
  • Some branching laws for symmetric spaces, B. Orsted, B. Speh.
  • Coulomb branches of noncotangent type, Alexander Braverman, Gurbir Dhillon, Michael Finkelberg, Sam Raskin, Roman Travkin.