Meromorphicity of the double-coset zeta function ζ_{G,O}(s)

Determine, for totally disconnected locally compact groups G that satisfy the double coset property with respect to a compact open subgroup O, whether the Dirichlet series ζ_{G,O}(s) = ∑_{r∈R} μ_O(OrO)^{-s}, where R is a set of O-double coset representatives and μ_O is the Haar measure with μ_O(O)=1, admits a meromorphic continuation to the complex plane.

Background

The paper defines, for a t.d.l.c. group G and compact open subgroup O with the double coset property, the Dirichlet series ζ{G,O}(s)=∑{r∈R} μ_O(OrO)^{-s}. In many examples (e.g., Weyl-transitive actions on buildings, algebraic groups over non-archimedean fields), this series is shown to coincide with a known rational function in q^{-s} and hence is meromorphic.

The authors ask in general for which G and O this analytic continuation holds. The question is motivated by several positive cases (e.g., Theorems E and F) but no general criterion is known.