Representation theory of GL(2, Q_p) (Math 600D, Term II 2008/2009) Tuesday 10-11am, at Math Annex 1101, Thursday 10-12 at PIMS auditorium WMAX 100.
My office: Math 217.
  • Initial plans for the course
  • The main reference should have been Colin J. Bushnell and Guy Henniart, "The Local Langlands Conjecture for GL(2)", Springer Grundlehren series, 1996. Another reference is Chapter 3 of Bump's book.


    A very useful general reference that I didn't think of earlier, if you can find it, is:
    A.A. Kirillov, "Elements of the theory of representations." Grundlehren der Mathematischen Wissenschaften, Band 220. Springer-Verlag, Berlin-New York, 1976. xi+315 pp.
    (sadly, our library only has the original in Russian)

    About topological groups in general: Representation theory: Other related manuscripts on the web:

    Approximate detailed syllabus:

  • A growing list of problems. (The ones marked with a check mark have been discussed already. All the others are to be discussed on the next Problem Thursday).
  • January 6 - 22: Representations of GL(2, F_q), and some reminders about finite groups in general: Mackey's theorem, Principal series, Weil representation. The main reference was Bump, 3.1, skipping the proofs of irreducibility of the principal series, but including some exercises.
  • January 26 - February 10: Haar measure, Smooth and admissible representations. The action of the Hecke algebra; definition of the distribution character; Smooth and compact induction. The main reference: F. Murnaghan's notes, sections 4 and 5.
    Homological algebra proof of Frobenius reciprocity (this is an expanded version of a page from unpublished notes by Patrick Walls from a lecture by P. Kutzko at the University of Ottawa, January 2007).
  • February 12 - 19: Cartan, Iwasawa, and Bruhat decompositions of G, with the corresponding integration formulas. The modulus character of the standard Borel subgroup. Jacquet module.
  • February 26: Representations of Mirabolic subgroup. Reference: Sections 8.1 -- 8.3 of Bushnell and Henniart's book.
  • March 3 - 5: Some proofs for representation theory of the mirabolic subgroup. Kirillov and Whittaker models.
  • March 10 -12: Classification of the irreducible representations of GL(2)
  • March 17: Asymptotic behaviour of the functions in Kirillov models near 0. Definition of L-functions (Section 4.7 in Bump)
  • March 24-26: Other definitions of L-functions; spherical represntations.
  • April 1-3: Local functional equation, epsilon-factors
  • April 8-10 Automorphic forms, etc. (Note: there will be a class on April 10 instead of March 17).