부산대학교 도서관

In the paper under review, two topics (representation theory and geometry) 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 $L$-functions attached to unramified irreducible subrepresentations of the regular representations $G$ on $L^2(\Gamma \backslash G)$ are the main objects in the representation side, while the quotient of the Bruhat-Tits building $\Cal{B}$ of $G$ by $\Gamma$ and zeta functions of geodesics are the main objects in the geometric side. It was Ihara who first studied 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 rank algebraic groups and study the zeta identities involving Langlands $L$-functions and zeta functions of geodesic walks and galleries for finite quotients of the apartment $\Cal{A}$ of ${\rm PGL}_3$ and ${\rm PGSp}_4$. The authors also study the case of a simple split algebraic group $G$ of adjoint type. For example, Section 3 studies $L$-functions associated to finite index subgroups $\Gamma$ of the extended affine Weyl group $W_{ext}$ of a simple split algebraic group $G$ of adjoint type and studies the zeta identities involving those $L$-functions and zeta 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 zeta identities in the three papers just cited relies on calculations with lattice models of the Bruhat-Tits building and some results from representation theory. In the paper under review, the authors recast those expressions using the method used for the apartment case. Furthermore, the authors establish a new zeta identity involving a degree 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 ${\rm Spin}_5(\Bbb{C})$.