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AN EIGENSPACE OF LARGE DIMENSION FOR A HECKE ALGEBRA ON AN BUILDING

Part of: Abstract harmonic analysis Finite geometry and special incidence structures

Published online by Cambridge University Press:  28 September 2011

A. M. MANTERO  and
A. ZAPPA
A. M. MANTERO
Affiliation:
D. S. A., Facoltà di Architettura, Università di Genova, Salita Sant’Agostino 37,16123 Genova, Italy (email: [email protected])
A. ZAPPA*
Affiliation:
D. I. M. A., Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let Δ be an affine building of type and let 𝔸 be its fundamental apartment. We consider the set 𝕌0 of vertices of type 0 of 𝔸 and prove that the Hecke algebra of all W0-invariant difference operators with constant coefficients acting on 𝕌0 has three generators. This property leads us to define three Laplace operators on vertices of type 0 of Δ. We prove that there exists a joint eigenspace of these operators having dimension greater than ∣W0 ∣. This implies that there exist joint eigenfunctions of the Laplacians that cannot be expressed, via the Poisson transform, in terms of a finitely additive measure on the maximal boundary Ω of Δ.

Keywords

buildingsHecke algebrasLaplace operatorsPoisson kernel

MSC classification

Secondary: 51E24: Buildings and the geometry of diagrams 43A80: Analysis on other specific Lie groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 1 , August 2011 , pp. 29 - 54
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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