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Ferdaous Kellil; Guy Rousseau
Opérateurs invariants sur certains immeubles affines de rang 2
Annales de la faculté des sciences de Toulouse Sér. 6, 16 no. 3 (2007), p. 591-610, doi: 10.5802/afst.1160
Article PDF | Reviews MR 2379053
[C 01] Cartwright (D.I.).— Spherical harmonic analysis on buildings of type $\widetilde{A_n}$, Monatsh. Math. 133, 93-109 (2001). MR 1860293 | Zbl 1008.51019
[CM 94] Cartwright (D.I.), Mlotkowski (W.).— Harmonic analysis for groups acting on triangle buildings, J. Austral. Math. Soc. (Series A). 56, 345-383 (1994). MR 1271526 | Zbl 0808.51014
[CW 04] Cartwright (D.I.), Woess (W.).— Isotropic random walks in a building of type $\widetilde{A_d}$, Math. Z. 247, 101-135 (2004). MR 2054522 | Zbl 1060.60070
[GL 99] Gerardin (P.), Lai (K.F.).— Opérateurs invariants sur les immeubles affines de type A, C. R. Acad. Sci. Paris ser. I, 329, 1-4 (1999). MR 1703307 | Zbl 0928.22011
[McD 71] Mac Donald (I.G.).— Spherical functions on a group of p-adic type, Ramanujan Inst. Publ. 2 Université de Madras (1971). MR 435301 | Zbl 0302.43018
[MZ 00] Mantero (A. M.), Zappa (A.).— Eigenfunctions of the Laplace operators for a building of type $\widetilde{A_2}$, J. Geom. Analysis. 10, 339-363 (2000). MR 1766487 | Zbl 0986.22009
[MZ 02] Mantero (A. M.), Zappa (A.).— Eigenfunctions of the Laplace operators for buildings of type $\widetilde{B_2}$, Boll. Unione Mat. Ital. Ser B Artic. Ric. Mat. (8)5, 163-195 (2002). MR 1881930 | Zbl 05019631
[R 89] Ronan (M.).— Lectures on buildings, Perspect. in Math. 7. Acad. Press. New York (1989). MR 1005533 | Zbl 0694.51001
[S 63] Satake (I.).— Theory of spherical functions on reductive algebraic groups over p-adic fields. Pub. Math. I. H. E. S, 18, 5-69 (1963).
Numdam | MR 195863 | Zbl 0122.28501
Ferdaous Kellil; Guy Rousseau
Opérateurs invariants sur certains immeubles affines de rang 2
Annales de la faculté des sciences de Toulouse Sér. 6, 16 no. 3 (2007), p. 591-610, doi: 10.5802/afst.1160
Article PDF | Reviews MR 2379053
Résumé - Abstract
We consider a building $\Delta $ of type $\widetilde{A_2}$ or $\widetilde{B_2}$ , different subsets $\mathcal{S}^{\prime}$ of the set $\mathcal{S}$ of vertices in $\Delta $ and different automorphism groups $G$, very strongly transitive on $\Delta $. We prove that the algebra of $G$-invariant operators acting on the space of functions on $\mathcal{S}^{\prime}$ is often not commutative (contrarily to the classical results). In some cases we describe its structure, determine its radial eigunfunctions and deduce that the Helgason conjecture is not verified in this context
Bibliography
[CM 94] Cartwright (D.I.), Mlotkowski (W.).— Harmonic analysis for groups acting on triangle buildings, J. Austral. Math. Soc. (Series A). 56, 345-383 (1994). MR 1271526 | Zbl 0808.51014
[CW 04] Cartwright (D.I.), Woess (W.).— Isotropic random walks in a building of type $\widetilde{A_d}$, Math. Z. 247, 101-135 (2004). MR 2054522 | Zbl 1060.60070
[GL 99] Gerardin (P.), Lai (K.F.).— Opérateurs invariants sur les immeubles affines de type A, C. R. Acad. Sci. Paris ser. I, 329, 1-4 (1999). MR 1703307 | Zbl 0928.22011
[McD 71] Mac Donald (I.G.).— Spherical functions on a group of p-adic type, Ramanujan Inst. Publ. 2 Université de Madras (1971). MR 435301 | Zbl 0302.43018
[MZ 00] Mantero (A. M.), Zappa (A.).— Eigenfunctions of the Laplace operators for a building of type $\widetilde{A_2}$, J. Geom. Analysis. 10, 339-363 (2000). MR 1766487 | Zbl 0986.22009
[MZ 02] Mantero (A. M.), Zappa (A.).— Eigenfunctions of the Laplace operators for buildings of type $\widetilde{B_2}$, Boll. Unione Mat. Ital. Ser B Artic. Ric. Mat. (8)5, 163-195 (2002). MR 1881930 | Zbl 05019631
[R 89] Ronan (M.).— Lectures on buildings, Perspect. in Math. 7. Acad. Press. New York (1989). MR 1005533 | Zbl 0694.51001
[S 63] Satake (I.).— Theory of spherical functions on reductive algebraic groups over p-adic fields. Pub. Math. I. H. E. S, 18, 5-69 (1963).
Numdam | MR 195863 | Zbl 0122.28501
