Abstract

We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a $\mathrm{PGL}<!--FUNCTION\; APPLICATION-->(3,{\mathbb{𝔽}}_q[t])$ quotient of an ${Ã}_2$-type building. We prove that the set of nontrivial approximate eigenvalues $({\lambda }^{+},{\lambda }^{-})$ of the weighted adjacency operators $A_w^{\pm }$ on the quotient induced from the colored adjacency operators $A^{\pm }$ on the building for ${\mathrm{PGL}<!--FUNCTION\; APPLICATION-->}_3$ contains the simultaneous spectrum of $A^{\pm }$ and another hypocycloid with three cusps. As a byproduct, we reestablish a proof of the fact that $\mathrm{PGL}<!--FUNCTION\; APPLICATION-->(3,{\mathbb{𝔽}}_q[t])\setminus \mathrm{PGL}<!--FUNCTION\; APPLICATION-->(3,{\mathbb{𝔽}}_q((t^{-1})))∕\mathrm{PGL}<!--FUNCTION\; APPLICATION-->(3,{\mathbb{𝔽}}_q[[t^{-1}]])$ is not a Ramanujan complex, from a combinatorial aspect.

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