부산대학교 도서관

In the paper under review, two topics (representation theory andgeometry) play an important role in the study of a discrete subgroup$\Gamma$ of an algebraic group $G$ over a nonarchimedean local field$F$ with $q$ elements in its residue field. The Langlands $L$-functionsattached to unramified irreducible subrepresentations of the regularrepresentations $G$ on $L^2(\Gamma \backslash G)$ are the main objectsin the representation side, while the quotient of the Bruhat-Titsbuilding $\Cal{B}$ of $G$ by $\Gamma$ and zeta functions of geodesicsare the main objects in the geometric side. It was Ihara who firststudied relations (called zeta identities) between Langlands$L$-functions and zeta functions (called Ihara or graph zeta functions)for the case of ${\rm PGL}_2$ [Y. Ihara, J. Math. Soc. Japan {\bf 18} (1966), 219--235; MR0223463].\par The authors generalize the Ihara zeta functions to higher rankalgebraic groups and study the zeta identities involving Langlands$L$-functions and zeta functions of geodesic walks and galleries forfinite quotients of the apartment $\Cal{A}$ of ${\rm PGL}_3$ and ${\rmPGSp}_4$. The authors also study the case of a simple split algebraicgroup $G$ of adjoint type. For example, Section 3 studies $L$-functionsassociated to finite index subgroups $\Gamma$ of the extended affineWeyl group $W_{ext}$ of a simple split algebraic group $G$ of adjointtype and studies the zeta identities involving those $L$-functions andzeta functions of geodesic walks.\par The authors further study the building case for ${\rm PGL}_3$ and${\rm PGSp}_4$ and compare those with the previously known results in [M.-H. Kang and W.-C.~W. Li, Adv. Math. {\bf 256} (2014), 46--103;MR3177290; M.-H. Kang, W.-C.~W. Li and C.-J. Wang,Israel J. Math. {\bf 177} (2010), 335--348; MR2684424;Y. Fang, W.-C.~W. Li and C.-J. Wang, Int. Math. Res. Not. IMRN {\bf 2013}, no.~4, 886--923; MR3024268]. The proof of zetaidentities in the three papers just cited relies on calculations withlattice models of the Bruhat-Tits building and some results fromrepresentation theory. In the paper under review, the authors recastthose expressions using the method used for the apartment case.Furthermore, the authors establish a new zeta identity involving adegree 5 standard $L$-function in the case of ${\rm PGSp}_4$,complementing the one in [Y. Fang, W.-C.~W. Li and C.-J. Wang, op.cit.] involving the spin $L$-function of ${\rm Spin}_5(\Bbb{C})$.