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Zeta function for kernels of parabolic actions on building links

Determine, for a locally finite building Δ of type (W,S) with uniform thickness q+1 and a Weyl-transitive closed subgroup G ≤ Aut₀(Δ), whether for each spherical parabolic subgroup P_J (stabilizer of a face D of a fixed chamber) the series ζ_{G,U_J}(s) associated to the compact open subgroup U_J = Ker(P_J ↷ Link(D)) admits a meromorphic continuation that is a rational function of q^{-s}, and whether the identity χ̃_G = ζ_{G,U_J}(-1)^{-1} · μ_{U_J} holds.

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Background

Beyond chamber stabilizers and parahoric subgroups, the authors propose replacing O by U_J, the kernel of the action of a spherical parabolic P_J on the link of a face. They ask if the same analytic and Euler characteristic identities established for B (Iwahori) or P_J (and its pro-p radical) extend to these kernels.

This would generalize their χ–ζ relationship in building settings and potentially connect to finer subgroup structures beyond standard parahorics.

References

Let $\Delta$ be a locally finite building of type $(W,S)$ and let $G$ be a Weyl-transitive closed subgroup of $Aut_0(\Delta)$. Assume that $\Delta$ has uniform thickness $q+1$, i.e., all the entries of the thickness vector of $\Delta$ equal $q$, and choose a chamber $C$. Let $P_J$ be a spherical parabolic subgroup, i.e., $P_J$ is the stabiliser of a face of the chamber $C$. Denote by $U_J$ the kernel of the action of $P_J$ on the link of $D$. Does the series $\zeta_{G,U_J}(s)$ define a meromorphic function that is a rational function in $q{-s}$? Additionally, does one have ${G}={G,U_J}(-1){-1}\cdot\mu_{U_J}$?

— The Hattori-Stallings rank, the Euler-Poincaré characteristic and zeta functions of totally disconnected locally compact groups (2405.08105 - Castellano et al., 13 May 2024) in End of Section 6 (Double coset zeta functions)