We establish a strong connection between buildings and Hecke algebras through the study of two algebras of averaging operators on buildings. To each locally finite regular building we associate a natural algebra B of chamber set averaging operators, and when the building is affine we also define an algebra A of vertex set averaging operators. In the affine case, it is shown how the building gives rise to a combinatorial and geometric description of the Macdonald spherical functions, and of the centers of affine Hecke algebras. The algebra homomorphisms from A into the complex numbers are studied, and some associated spherical harmonic analysis is conducted. This generalises known results concerning spherical functions on groups of p-adic type. As an application of this spherical harmonic analysis we prove a local limit theorem for radial random walks on affine buildings.
| Identifer | oai:union.ndltd.org:ADTP/220691 |
| Date | January 2005 |
| Creators | Parkinson, James William |
| Publisher | University of Sydney. Mathematics and Statistics |
| Source Sets | Australiasian Digital Theses Program |
| Language | English, en_AU |
| Detected Language | English |
| Rights | Copyright Parkinson, James William;http://www.library.usyd.edu.au/copyright.html |
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