Nonzero edge eigenvalues of the Tits building for finite groups of Lie type G(Fq) become powers of q after raising to a bounded exponent k depending on the type of G, proved uniformly via Hecke algebras.
Zeta functions and applications of group-based complexes
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Edge Zeta Functions and Eigenvalues for Buildings of Finite Groups of Lie Type
Nonzero edge eigenvalues of the Tits building for finite groups of Lie type G(Fq) become powers of q after raising to a bounded exponent k depending on the type of G, proved uniformly via Hecke algebras.