arXiv subjects
Publications and source records attributed to Chian-Jen Wang.
In this article we prove that the gamma factors attached to generic representations of ${\rm U}_{2n+1}\times{\rm Res}_{E/F}{\rm GL}_r$ over a local field $F$ defined by various approaches coincide when $F$ is non-archimedean and under additional assumptions on $n,r$ and $E$ when $F$ is archimedean.
In this paper, we study relations between Langlands L-functions and zeta functions of geodesic walks and galleries for finite quotients of the apartments of G=PGL3 and PGSp4 over a nonarchimedean local field with q elements in its residue field. They give rise to an identity (Theorem 5.3) which can be regarded as a generalization of Ihara's theorem for finite quotients of the Bruhat-Tits trees. This identity is shown to agree with the q=1 version of the analogous identities for finite quotients of the building of G established in (KL1, KLW, FLW), verifying the philosophy of the field with one element by Tits. A new identity for finite quotients of the building of PGSp4 involving the standard $L$-function (Theorem 6.3), complementing the one in (FLW) which involves the spin L-function, is also obtained.
The zeta function attached to a finite complex $X_Γ$ arising from the Bruhat-Tits building for $\PGL_3(F)$ was studied in \cite{KL}, where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of $X_Γ$. In this paper we reprove the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed chambers, respectively.
Let $F$ be a non-archimedean local field with a finite residue field. To a 2-dimensional finite complex $X_Γ$ arising as the quotient of the Bruhat-Tits building $X$ associated to $\Sp_4(F)$ by a discrete torsion-free cocompact subgroup $Γ$ of $\PGSp_4(F)$, associate the zeta function $Z(X_Γ, u)$ which counts geodesic tailless cycles contained in the 1-skeleton of $X_Γ$. Using a representation-theoretic approach, we obtain two closed form expressions for $Z(X_Γ, u)$ as a rational function in $u$. Equivalent statements for $X_Γ$ being a Ramanujan complex are given in terms of vertex, edge, and chamber adjacency operators, respectively. The zeta functions of such Ramanujan complexes are distinguished by satisfying the Riemann Hypothesis.