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Spherical harmonic analysis on affine buildings
Authors:James Parkinson
Institution:(1) School of Mathematics and Statistics, University of Sydney, NSW, 2006, Australia
Abstract:Let ></img> </span> be a locally finite regular affine building with root system <em>R</em>. There is a commutative algebra <span class= ></img> </span> spanned by averaging operators <em>A</em> <sub> <em>λ</em> </sub>, λ ∈ <em>P</em> <sup>+</sup>, acting on the space of all functions <em>f</em>:<em>V</em> <sub> <em>P</em> </sub>→<span class= ></img> </span>, where <em>V</em> <sub> <em>P</em> </sub> is in most cases the set of all special vertices of <span class= ></img> </span>, and <em>P</em> <sup>+</sup> is a set of dominant coweights of <em>R</em>. This algebra is studied in 6] and 7] for Ã<sub> <em>n</em> </sub> buildings, and the general case is treated in 15]. In this paper we show that all algebra homomorphisms <em>h</em>:<span class= ></img> </span> may be expressed in terms of the <em>Macdonald spherical functions</em>. We also provide a second formula for these homomorphisms in terms of an integral over the <em>boundary</em> of <span class= ></img> </span>. We may regard <span class= ></img> </span> as a subalgebra of the <em>C</em> <sup>*</sup>-algebra of bounded linear operators on ?<sup>2</sup>(<em>V</em> <sub> <em>P</em> </sub>), and we write <span class= ></img> </span> for the closure of <span class= ></img> </span> in this algebra. We study the Gelfand map <span class= ></img> </span>, where <em>M</em> <sub>2</sub>=<span class= ></img> </span>, and we compute <em>M</em> <sub>2</sub> and the <em>Plancherel measure</em> of <span class= ></img> </span>. We also compute the ?<sup>2</sup>-operator norms of the operators <em>A</em> <sub> <em>λ</em> </sub>, λ ∈ <em>P</em> <sup>+</sup>, in terms of the Macdonald spherical functions.</td> </tr> <tr> <td align=
Keywords:Affine buildings  Macdonald spherical functions  harmonic functions
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