Math 600D: representations of GL(2, Q

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.

References:

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:

E. Hewitt and K. Ross "Abstract Harmonic Analysis", Vol. 1. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 115. Springer-Verlag, Berlin-New York, 1979.
A. Weil "L'integration dans les groupes topologiques et ses applications. (French)" [This book has been republished by the author at Princeton, N. J., 1941.] Actual. Sci. Ind., no. 869. Hermann et Cie., Paris, 1940. 158 pp. (Our library has a copy).
W. Rudin "Functional analysis" (contains my favourite proof of existence of Haar measure for compact groups).

Representation theory:

Representations of reductive p-adic groups (by Fiona Murnaghan)
Other very nice notes just appeared on Fiona Murnaghan's website. There are now some notes on topological groups (see Chapter 5 ), and the basics of the representation theory of finite groups (see Chapter 2 ); the other chapters are very useful as well.
Notes by Karl Rumelhart for Joseph Bernsteins's course on representation theory of p-adic groups.
W. Casselman's notes .
J. Cartier's article in the Corvallis proceedings .

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)

NO CLASS ON THURSDAY MARCH 19 !!

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).