부산대학교 도서관

Let $F$ be a non-archimedean local field with the ring of integers $\Cal{O}$ and a uniformizer $\pi$ such that its residue field $\Cal{O}/\pi \Cal{O}$ has cardinality $q$. Let $G=\roman{PGL}_3(F)$ and denote by $X$ the building attached to $G$, which is a contractible $2\roman{D}$ simplicial complex whose vertices are homothety classes of $\Cal{O}$ lattices of rank 3 in the $3\roman{D}$ vector space $F^3$. For a discrete cocompact torsion-free subgroup $\Gamma$ of $G$, denote by $X_{\Gamma}\coloneq \Gamma \backslash X $ a finite $2\roman{D}$ complex locally isomorphic to $X$. \par The authors study two Artin $L$-functions of $X_{\Gamma}$ attached to a $d$-dimensional representation $\rho$ of $\Gamma$ acting on the space $V_{\rho}$ over $\Bbb{C}$. Namely, for $i=1,2$ they study the functions $$ L_i(X_{\Gamma}, \rho,u)\coloneq \prod_{[\germ{p}]} \roman{\det}(I-\rho(\roman{Frob}_{[\germ{p}]})u^{l_{A}(\germ{p})})^{-1}, $$ where $I$ is the $d\times d$ identity matrix, $[\germ{p}]$ (which plays the role of an $i$-dimensional prime) runs through all equivalence classes of primitive uni-type closed $i$-dimensional geodesics $\germ{p}$ in $X_{\Gamma}$, $l_{A}(\germ{p})$ is the algebraic length of $\germ{p}$ and $\roman{Frob}_{[\germ{p}]}$ is a conjugacy class in $\Gamma$ associated to $[\germ{p}]$. \par In the case when $\rho$ is the trivial representation, the above $L$-functions coincide with the edge and the chamber zeta functions studied in [M.-H. Kang and W.-C.~W. Li, Adv. Math. {\bf 256} (2014), 46--103; MR3177290] and [M.-H. Kang, W.-C.~W. Li and C.-J. Wang, Israel J. Math. {\bf 177} (2010), 335--348; MR2684424]. \par In the paper under review, some of the key properties of functions $L_i(X_{\Gamma}, \rho,u)$, ${i=1,2}$, such as rationality (and thus meromorphic continuation to the whole $u$-plane), functional equation, and invariance under induction are derived. Moreover, the authors also establish the following relation between $L_i(X_{\Gamma}, \rho,u)$, $i=1,2$, and the unramified $L$-function of the induced representation $\roman{Ind}_{\Gamma}^{G}\rho$: $$ (1-u^3)^{\chi(X_{\Gamma})d}L(\roman{Ind}_{\Gamma}^{G}\rho, qu)= \frac{L_1(X_{\Gamma}, \rho,u)}{L_2(X_{\Gamma}, \rho,-u)}, $$ where $\chi(X_{\Gamma})$ is the Euler characteristic of the complex $X_{\Gamma}$.