August 8-10, 2007, 10:00 - 12:00 h
Short Course on Commutative Spaces
Joseph A. Wolf, Berkeley
The course takes place at VR II, room 158, Engineering and Science building of University of Iceland.
Commutative spaces (or Gelfand pairs) form a common generalization of a number of areas in differential geometry and harmonic analysis. The point of these lectures is to indicate (without proofs) some of the principal facts. This "Short Course on Commutative Spaces" will consist of three lectures:
- Basic Facts on Commutative Spaces
Definition of Gelfand pairs
Examples: Riemannian symmetric spaces, weakly symmetric spaces, locally compact abelian groups, compact groups and homogeneous trees
Algebra of invariant differential operators
Spherical measures and spherical functions
Positive definite functions, associated representations
Examples: Euclidean spaces, spheres, projective spaces, hyperbolic spaces, Heisenberg groups
Induced spherical functions, induced representations - Harmonic Analysis on Commutative Spaces
Spherical transform
Godement-Bochner theorem
Inverse spherical transform
Plancherel formulas
Direct integral decompositions
Symmetric space examples - Structure and Classification of Commutative Spaces
Reductive commutative spaces: work of Kraemer, Brion, Akhiezer, Vinberg, Mikityuk, Yakimova
Nilpotent commutative spaces: 2-step nilpotent theorem, Carcano's theorem, Kac' classification, Benson-Jenkins-Ratcliff classification, Vinberg classification, Yakimova classification, square integrable representations and spherical functions
Some remarks on the general classification