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:
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of a page from unpublished notes by Patrick Walls from a lecture by P. Kutzko at the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).