Seminar on representation theory
and reductive groups - Fall 2006

The ground rule of the seminar is that all talks should be comprehensible to graduate students. Not necessarily with details of proofs, but references should be adequate.

For those graduate students interested in obtaining credit for this seminar, it is Mathematics 620A, section 101.

  • Meetings: Tuesdays
  • 2:30 - 3:30 in Math Annex 1118
  • 3:30 - Math Annex 1102


  • September 19 - Bill Casselman - The L-group.

    Written after my lecture: I had meant to talk about the L-group, and in fact to cover essentailly what is in the relevant section of my notes on spherical functions (mentioned later), but wound up talking about more elementary material.

  • September 26 - Hesam - Finite groups of Lie type.

    He writes in e-mail, "When I first learned about linear algebraic groups one of the hard parts was groups over a field which is not alg. closed. I think a good exposition of what happens to groups (or their root data) over finite fields is very useful in that regard. Also later if we do the same for p-adic groups it is nice to compare the results."

  • October 3 - Patrick Walls on the ring of adeles

  • October 10 - Julia Gordon on unramified principal series, mostly SL(2).

  • October 17 - Michael LeBlanc, `The Bruhat decomposition'.

  • October 24 - Hesaam - reductive groups over finite fields II

  • October 31 - Patrick Walls - Adeles II

  • November 7 - Me - on the construction of a group from its root datum.

    By the way, "data" is plural, "datum" singular. "These data", not "this data." So is "agenda": "He has hidden agenda," not "a hidden agenda." "Agenda" is literally "things to be acted upon." But even so, the use of "root datum" for an array of things is unfortunate.

  • November 14 - Michael LeBlanc - local zeta functions. Thsi will go over questions raised in Patrick's last talk.
  • November 21 - Julia Gordon - TBA


Good for browsing to choose topics.

Structure of p-adic groups

  • Robert Steinberg, Lectures on Chevalley groups. The .pdf file has been compressed with gzip, and meant to be downloaded before opening.

    This classic was originally a set of lecture notes published by the mathematics department of Yale University. It is posted here with Steinberg's generous permission, but copyright remains with him. A copy taken from here must be for personal use only.

  • Ian Macdonald - Spherical functions on groups of p-adic type, Madras, 1971.
  • Jacques Tits - Reductive groups over local fields, in Proceedings of Symposia in Pure Mathematics 33.

    Nearly all the proceedings of the Corvallis conference are available on line at

  • Kenneth Brown - Buildings.
  • Arjeh Cohen, Scott Murray, and Don Taylor - Computing in groups of Lie type.
  • Arjeh Cohen and Scott Murray, Algorithm for Lang's theorem

Representation theory

Weyl groups and root systems

  • James E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press.
  • James E. Humphreys, Introduction to Lie algebras and representation theory, Springer.
  • Nicholas Bourbaki, Lie groups and Lie algebras-Chapters IV, V, VI, Masson.
  • H. S. M. Coxeter, Regular polytopes.

    There are several editions, interestingly different. Contains valuable geometric interpretations of things others don't deal with, particularly the role of Coxeter elements.