Abstract
The zeta function attached to a finite complex X Γ arising from the Bruhat-Tits building for PGL3(F) was studied in [KL], where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of X Γ. In this paper we re-establish the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed chambers, respectively.
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References
A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Inventiones Mathematicae 35 (1976), 233–259.
W. Casselman, The unramified principal series of p-adic groups. I. The spherical function, Compositio Mathematica 40 (1980), 387–406.
H. Garland, p-adic curvature and the cohomology of discrete subgroups of p-adic groups, Annals of Mathematics 97 (1973), 375–423.
K. Hashimoto, Zeta functions of finite graphs and representations of p-adic groups, in Automorphic Forms and Geometry of Arithmetic Varieties, in Advanced Studies in Pure Mathematics 15, Academic Press, Boston, MA, 1989, pp. 211–280.
H. Jacquet, I. I. Piatetski-Shapiro and J. Shalika, Automorphic forms on GL(3), I & II, Annals of Mathematics 109 (1979), 169–258.
M.-H. Kang and W.-C. W. Li, Zeta functions of complexes arising from PGL(3), (2008), submitted.
W.-C. W. Li, Ramanujan hypergraphs, Geometric and Functional Analysis 14 (2004), 380–399.
A. Lubotzky, B. Samuels and U. Vishne, Ramanujan complexes of type Ad, Israel Journal of Mathematics 149 (2005), 267–299.
M. Tadić, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Annales Scientifiques de l’École Normale Supérieure. Quatriéme Série 19 (1986), 335–382.
A. V. Zelevinsky, Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n), Annales Scientifiques de l’École Normale Supérieure. Quatriéme Série 13 (1980), 165–210.
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The research of the first and second authors are supported in part by the DARPA grant HR0011-06-1-0012. The second author is also supported in part by the NSF grants DMS-0457574 and DMS-0801096. She would like to thank Jiu-Kang Yu for inspiring conversations.
Part of the work was done when the third author visited the Pennsylvania State University. She would like to thank the Penn State University for its hospitality, and National Center for Theoretical Sciences, Hsinchu, Taiwan, for its support.
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Kang, MH., Li, WC.W. & Wang, CJ. The zeta functions of complexes from PGL(3): A representation-theoretic approach. Isr. J. Math. 177, 335–348 (2010). https://doi.org/10.1007/s11856-010-0049-2
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DOI: https://doi.org/10.1007/s11856-010-0049-2