부산대학교 도서관

This paper is concerned with zeta functions of finite simplicial complexes, obtained as quotients of the Bruhat-Tits building of $G = \roman{PGL}_3(F)$, where $F$ is a non-Archimedean local field. Let $q$ denote the order of the residue field. Under mild assumptions, the vertices of such a complex $X$ are naturally partitioned into two types. \par Similar to graph zeta functions, the complex zeta function counts tailless closed geodesics up to homotopy: they are defined as $Z(X,u) = \prod_{[C]}(1-u^{\ell(C)})^{-1}$, where the product is over equivalence classes of homotopic primitive tailless closed geodesics $C$, and $\ell(C)$ is the length. \par Let $A_i$, $L_E$ and $L_B$ be the adjacency matrices for type $i$ vertices, for edges, and for directed chambers, and let $\chi(X)$ denote the Euler characteristic. \par The main result of this paper is a representation theoretic proof of the identity $$\multline Z(X,u) = \\(1-u^3)^{\chi(X)} \det(I-A_1 u + q A_2 u^2 - q^3 u^3 I)^{-1} \det(I+L_Bu)^{-1},\endmultline$$ which was proved by the first two authors by combinatorial methods [``Zeta functions of complexes arising from ${\rm PGL}(3)$'', preprint, \arx{0804.2305}], this time by computing the spectrum of each operator. \par The identity $Z(X,u) = \det(I-L_Eu)^{-1}\det(I-L_Eu^2)^{-1}$ follows from the definition. As a corollary to the results on eigenvalues of the various operators, it is shown that the property of $X$ to be Ramanujan (in the sense that the eigenvalues of each of $A_1$ and $A_2$ are contained in the respective spectrum on the building) can be characterized in terms of the eigenvalues of the operator $L_B$, and also in terms of the eigenvalues of the operator $L_E$.