Meromorphicity of ζ_{G,O}(s) for p-adic analytic groups

Ascertain whether, for every p-adic analytic group G and every compact open subgroup O ≤ G, the double-coset Dirichlet series ζ_{G,O}(s) = ∑_{r∈R} μ_O(OrO)^{-s} admits a meromorphic continuation to the complex plane.

Background

The authors isolate a narrower setting of interest for Question G: p-adic analytic groups. In many arithmetic and representation-theoretic contexts, such groups exhibit good analytic properties, and the authors ask whether the meromorphicity of ζ_{G,O}(s) holds uniformly in this class.

Positive evidence is provided in the paper for algebraic groups over local fields and certain parahoric/pro-p radical choices of O, yet a general result for all p-adic analytic groups is not known.