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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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