Essays on representations and automorphic forms by Bill Casselman

• Geometry of the tree
• Analysis on the tree
• Conjugacy classes in local GL(2) and SL(2)

• Analysis on profinite groups
• Smooth representations
• Jacquet modules

Written to accompany a reprinting of Ian Macdonald's book on spherical functions on a p-adic group, originally published in Madras and difficult to locate. The book will indeed be reprinted in Springer's series of Lecture Notes in Mathematics, but with no contribution from me.

Real groups

• Structure
  • Maximal compact subgroups
  • On Chevalley's formula for structure constants
  • A simple way to compute structure constants
• Representations
  • Continuous representations
  • Unitary representations
  • Weyl's character formula and n-homology
  • Weight multiplicities
  • The theorem of Dixmier-Malliavin
  • Admissible representations of SL(2,R)
  • Langlands' classification of characters of real tori
  • Compact groups as algebraic groups

Automorphic forms

• Automorphic forms for SL(2)
  • Geometry of the upper half plane
  • Arithmetic subgroups
  • Classical automorphic forms and representations of SL(2)
  • Classical and adelic automorphic forms
  • Stereographic projection
  • Growth conditions and the constant term
  • The Schwartz space of Ehrenpreis and Mautner
  • Hecke's precursor to endoscopy
  • Maass forms
  • Langlands' combinatorial lemma

Root systems

• Introduction to finite root systems
• Finite root systems of rank two
• A simple way to compute structure constants

Coxeter groups

• Reflections
• Bruhat closures
• Hecke algebras
• Symmetry in the algorithm of Robinson-Schensted-Knuth
• Remarks on Knuth equivalence
• Coxeter elements
• Kazhdan-Lusztig polynomials
• Automata
• Non-Euclidean graphics
• Associates

Real analysis

• The Fourier transform
• The Gamma function
• Fourier series and elliptic regularity
• The Stone-Weierstrass Theorem
• A theorem of Émile Borel
• Integration on real manifolds
• Ordinary differential equations
• Complex analytic ordinary differential equations (with emphasis on regular singularities)
• The discrete Hardy-Littlewood inequality

Topological vector spaces and linear operators

• Introduction to topological vector spaces
• Quasi-complete vector spaces
• Hilbert spaces and self-adjoint operators
• Compact operators
• Hölder's inequality

Algebra

• Representations of finite groups
• Partitions and nilpotent matrices
• Introduction to Lie algebras
• Jordan decompositions
• Galois descent
• Free Lie algebras
• Non-commutative determinants
• Newton polygons
• Classical reductive groups
• Computing in reductive groups
• Symmetric power decompositions for GL(2)

Quadratic forms

• Introduction to quadratic forms
• Quadratic forms over finite fields
• Clifford algebras and spinors
• Hensel's Lemma

Mathematical astronomy

• The physics of orbits
• Planetary motion and Kepler's equation

Convexity

• Excerpt from a paper by Minkowski
• Digest of a paper by Herman Weyl
• Notes on the simplex method

Algebraic number theory

• Bernoulli numbers
• The geometry of continued fractions
• Integral square roots
• Continued fractions of quadratic irrationals
• Quadratic forms and quadratic extensions
• Indefinite quadratic forms and quadratic fields