Zeta functions of ${\mathrm{PGL}}_n$ over non-Archimedean local fields

Abstract

Let $\mathcal{B}$ be the Bruhat--Tits building of ${\mathrm{PGL}}_n(F)$, where $F$ is a non-Archimedean local field. We introduce geometric $k$-geodesics in $\mathcal{B}$ by means of CAT(0) convexity and combinatorial $k$-geodesics by a local successor relation on pointed $k$-facets. We prove that the two notions coincide. This allows us to use the local combinatorial definition on quotients $\Gamma \backslash \mathcal{B}$, without referring to the universal covering. When $\Gamma$ is discrete, torsion-free, cocompact, and type-preserving, the primitive closed $k$-geodesics define zeta functions $Z_k$ and their $\epsilon$-twisted variants $Z_k^{\epsilon }$. Our main result identifies an alternating product of these zeta functions with the unramified $L$-function of $L^2(\Gamma \backslash {\mathrm{PGL}}_n(F))$: $(1-u^n{)}^{\chi (\Gamma \backslash \mathcal{B})}L(\Gamma ,q^{(n-1)/2}u)={\prod }_{k=1}^{n-1}{Z}_k^{\epsilon }(\Gamma \backslash \mathcal{B},u{)}^{(-1{)}^{k+1}}$. This gives a uniform Ihara-type identity for all ${\mathrm{PGL}}_n$. We also extend the construction and the identity to ${\mathrm{PGL}}_n(D)$, where $D$ is a central division algebra over $F$; in that setting the residue parameter is $Q=|{\mathcal{O}}_D/{\mathfrak{p}}_D|$.

47 pages