Geodesic
Eigenfunctions of the Laplace operators for buildings of type $\tilde{B}_2$
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 1, pp. 163-195.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We consider for an affine building of type $\tilde{B}_2$ Helgason's conjecture with respect to Laplace operators defined over different types of vertices. We prove that there are cases in which the conjecture fails, since there exist eigenfunctions which are not the Poisson transform of finitely additive measures at the maximal boundary of the building.
Si considera per un palazzo affine di tipo $\tilde{B}_2$ la congettura di Helgason relativa a operatori di Laplace definiti su diversi tipi di vertici. Si prova che ci sono casi in cui la congettura non è verificata, in quanto esistono autofunzioni che non sono la trasformata di Poisson di misure finitamente additive sulla frontiera massimale del palazzo 
@article{BUMI_2002_8_5B_1_a7,
     author = {Mantero, A. M. and Zappa, A.},
     title = {Eigenfunctions of the {Laplace} operators for buildings of type $\tilde{B}_2$},
     journal = {Bollettino della Unione matematica italiana},
     pages = {163--195},
     publisher = {mathdoc},
     volume = {Ser. 8, 5B},
     number = {1},
     year = {2002},
     zbl = {1177.51010},
     mrnumber = {1463728},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_1_a7/}
}
TY  - JOUR
AU  - Mantero, A. M.
AU  - Zappa, A.
TI  - Eigenfunctions of the Laplace operators for buildings of type $\tilde{B}_2$
JO  - Bollettino della Unione matematica italiana
PY  - 2002
SP  - 163
EP  - 195
VL  - 5B
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_1_a7/
LA  - en
ID  - BUMI_2002_8_5B_1_a7
ER  - 
%0 Journal Article
%A Mantero, A. M.
%A Zappa, A.
%T Eigenfunctions of the Laplace operators for buildings of type $\tilde{B}_2$
%J Bollettino della Unione matematica italiana
%D 2002
%P 163-195
%V 5B
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_1_a7/
%G en
%F BUMI_2002_8_5B_1_a7
Mantero, A. M.; Zappa, A. Eigenfunctions of the Laplace operators for buildings of type $\tilde{B}_2$. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 1, pp. 163-195. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_1_a7/

[1] D. I. Cartwright, A Brief Introduction to Buildings, Contemporary Mathematics, Vol. 206 (1997), 45-77. | MR | Zbl

[2] W. Feit-G. Higman, The nonexistence of certain generalized polygons, J. Algebra, 1 (1964), 114-131. | MR | Zbl

[3] D. G. Higman, Invariant relations, coherent configurations and generalized polygons, Combinatorics part 3 (ed. M. Hall and J. Van Lint), Reidel, Dordrecht 1975, 247-263. | MR | Zbl

[4] S. Kato, On eigenspaces of the Hecke algebra with respect to a good maximal compact subgroup of a $p$-adic reductive group, Math. Ann., 257 (1981), 1-7. | MR | Zbl

[5] A. M. Mantero-A. Zappa, Eigenfunctions of the Laplace Operators for a Building of type $\tilde{A}_2$, J. of Geom. Anal., 10 (2000), 339-363. | MR | Zbl

[6] A. M. Mantero-A. Zappa, Eigenfunctions of the Laplace Operators for a Building of type $\tilde{G}_2$, preprint. | fulltext bdim | MR

[7] M. A. Ronan, Lectures on Buildings, Perspectives in Math. 7, Academic Press, London 1989. | MR | Zbl