The zeta functions of complexes from PGL(3): A representation-theoretic approach |
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Authors: | Ming-Hsuan Kang Wen-Ching Winnie Li Chian-Jen Wang |
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Affiliation: | (1) Lycée Anna de Noailles, 2, Avenue Anna de Noailles, 74500 Evian les Bains, France;(2) Institut Fourier, Université Grenoble I, BP 74, 38402 Saint-Martin dHères, France |
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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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