Bill Casselman's research page

• Original Notes on admissible representations of p-adic groups.
  • Introduction to admissible representations of p-adic groups
  These are the original notes written at the IAS in June, 1974, with a major correction suggested a couple of years later by Cartier and a few other minor ones. They were subsequently put into TEX under Paul Sally's direction at the University of Chicago. They contain a number of small mathematical errors. The most egregious is the formulation of the Bruhat-Tits KTK reduction.
• New edition. This will appear in fragments.
  • Analysis on profinite groups
  • Smooth representations
  • The Jacquet module
  • The asymptotic behaviour of matrix coefficients
  • Cuspidal representations
• On some results of Atkin and Lehner, Mathematische Annalen 201 (1973), 301-314 (link to the Göttingen archive)
• The restriction of a representation of GL(2,k) to GL(2,o), Mathematische Annalen 206 (1973), 311-318 (link to the Göttingen archive)
• The unramified principal series of p-adic groups I. Compositio Mathematica 40 (1980), 387-406 (link to NUMDAM)
• The unramified principal series of p-adic groups II (with Joe Shalika) Compositio Mathematica 41 (1980), 207-231 (link to NUMDAM)
• Notes on p-adic Whittaker functions
  This is a collection of notes and letters written by me to Freydoon Shahidi in the course of putting together our joint paper.
• On irreducibility of standard modules for generic representations (with Freydoon Shahidi), Annales scientifiques de l'École Normale Supérieure 31 (1998), 561--589.
• Jacquet modules and the asymptotic behaviour of matrix coefficients
  Based on a talk at the 2007 Purdue conference in honour of Freydoon Shahidi.
• The letters to Jim Arthur
  These are referred to in the previous paper.
• Langlands' Fundamental Lemma for SL(2)
  Started out as informal notes to accompany a talk at TIFR, February 2009. I have now broken it up into pieces, which are still under construction:
  • The origins of endoscopy in a paper of Erich Hecke
  • Notes on the Bruhat-Tits tree (incomplete)
• Notes on spherical functions
  Written to accompany a reprinting of Ian Macdonald's book that has not materialized.
• Notes on the Bruhat-Tits tree

Representations of real groups

• The asymptotic behaviour of matrix coefficients of admissible representations (with Dragan Milicic) Duke Mathematics Journal 49 (1982), 869-930.
• Bruhat filtrations and Whittaker vectors (with Henryk Hecht and Dragan Milicic), AMS Proc. Symp. Pure Math. 68 (2000), 151-190.
• Verifying a conjecture of Kottwitz
  Journal of Representation Theory 4 (2000)
• Computations in real tori
  In Representation theory of real groups, Contemporary Mathematics, volume 472, AMS, 2007.
  A remark on an algorithm used, more or less, in Fokko du Cloux's ATLAS program.
• The computation of structure constants according to Jacques Tits
  A practical explanation of how to use Tits' work on structure constants to compute them.
• Notes on representations of real groups
  • Continuous representations
  • Weyl's character formula and n-homology
  • The theorem of Dixmier-Malliavin

• Automorphic forms and a Hodge theory for congruence subgroups of SL₂(Z), pp. 108--140 in
  Lie group representations II (proceedings of a conference at the University of Maryland),
  Lecture Notes in Mathematics 1041, 1983.
• Introduction to the Schwartz space of Γ \G
  Canadian Journal of Mathematics XLI (1989)
• Extended automorphic forms on the upper half-plane
  Mathematischen Annalen 296 (1993).

  There are some silly errors in this. The exponent of π in Riemann's function is -s/2 not s/2, and the residue of E_s is 1/2ξ(2), making the volume of the quotient SL(2,Z)\H equal to π/3.

• Geometric rationality of Satake compactifications
  Australian Mathematical Lecture Series 9 (1997) (the Roger Richardson memorial issue)
• On the Plancherel measure for the continuous spectrum of the modular group
  This is a minor revision of a paper originally published in an A.M.S. volume dedicated to Goro Shimura. The revisions correct a few minor typographical errors and (I hope) improve clarity.
• The L-group
  (from a conference in Tokyo on class field theory). There are many typographical errors in the publishd version, and a few misstatements in this as well. Alas.
• Stability of lattices and the partition of arithmetic quotients
  Asian Journal of Mathematics 8 (2004) (issue dedicated to Armand Borel)
• Truncation exercises
  Functional Analysis VIII (2003)
• Harmonic analysis of the Schwartz space of Γ \ SL₂(R)
  Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Press, 2004
• A conjecture on the analytical behaviour of Eisenstein series
  Pure and Applied Mathematics Quarterly 1 (2005)
• Notes on automorphic forms of SL(2)
  • Geometry of the upper half plane
  • Arithmetic subgroups
  • Classical and adelic automorphic forms
  • Stereographic projection
  • Admissible representations of SL(2,R)