Working Seminar on Nilpotent orbits and the orbit method

• We have a discord channel now! (Thanks, Kristaps). It has the pdf of Kirillov's book and might soon have recordings of the talks.

  Schedule of talks.

  All talks will be in Math 126 and on zoom, 1-2:30pm PDT. (Note: the days alternate between Mondays and Wednesdays for a while; pay close attention!)
  • Wednesday September 17. Mishty.
    Introduction to representation theory of locally compact groups: induced representations, Fell topology, etc.
  • Monday September 22. Sarah.
    Mackey theory. (unfortunately, recording started 1 hour late).
  • Wednesday October 1. Peilin.
    Introduction to symplectic geometry.
  • Monday October 6. Different location: PIMS, 4th floor! Raphael.
    The Heisenberg group.
  • Wednesday October 15. Kristaps.
    Introduction to Lie groups and Lie algebras.
  • Monday October 20. NO MEETING.
  • Monday October 27. Kin Ming, Orbit method for nilpotent groups.
  • Monday November 3. Julia, recap of structure theory and representation theory for complex and compact real Lie groups
  • Wednesday November 12. UBC BREAK. NO MEETING.
  • Monday November 17. Atonu, Chevalley map.
  • Monday November 24. 1. Mishty, Harish-Chandra homomorphism 2. Atonu, Borel-Weil Theorem; the orbit method for compact Lie groups.
  • Monday December 1. Ethan, Geometric quantization.
  
  
  This seminar will be a mix or talks at different levels aiming at developing our understanding the geometry of the nilpotent nilpotent orbits in a Lie algebra and its linear dual, and the role they play in representation theory of Lie groups and p-adic groups. There will be self-contained subsets of talks on different subjects, and students can choose which talks to attend. The condition for getting credit at UBC is giving at least one talk. The seminar will be in person in Math 126 and streamed live on zoom. We might also have some talks based at the University of Calgary via videoconferencing.
  Prerequisites: Nothing required but some familiarity with Lie theory, representation theory, and basic algebraic geometry could be helpful for some parts (and especially for motivation).
  

  Tentative list of topics (for now, in random order; and to be expanded)

  • The Heisenberg Lie algebra and its representations
  • Poisson manifolds and Lie groups
  • The orbit method for nilpotent complex Lie groups; the notion of quantization
  • Partial flag varieties
  • The moment map
  • Orbit method for compact Lie groups
  • Nilpotent orbits: Dynkin-Kostant classification
  • Richardson orbits and Richardson parabolics
  • Borel-Weil-Bott theorem

  References, links, etc:

  • A.A. Kirillov, "Lectures on the orbit method" see also the review by D. Vogan and his very helpful list of corrections
  • Collingwood and McGovern "Nilpotent orbits in semisimple Lie algebras" (available online at UBC library).
  • By the end of the seminar, it would be good to understand IHES lecture by Lucas Mason-Brown.