Spherical functions on symmetric spaces of Friedberg-Jacquet type
Murilo Corato-Zanarella
TL;DR
The paper develops explicit models for spherical functions on p-adic symmetric spaces X=H(F0)\backslash G(F0) for Friedberg–Jacquet type pairs, providing a complete description of the associated Hecke module structure in several unramified settings. The authors reduce the problem to lattice-counting via a relative Cartan decomposition, and then construct explicit combinatorial models using lattice Typ data, incidence-algebra tools, and Lusztig–Kato formulas to relate Hecke actions to dual-group representations. Central innovations include the introduction of straightening relations Rel^flat/Rel^sharp, extension arguments that propagate local identifications to global results, and explicit bases for S(X/K) with ranks 2^r or 4^r depending on the case. These results yield concrete, implementable descriptions of spherical functions and their Hecke-action in the toy models, illuminating connections to dual Hamiltonian varieties and Satake transforms in a controlled setting. The approach yields practical formulas for spherical transforms and bases, with potential applications to Euler systems and local-to-global compatibility in related automorphic contexts.
Abstract
We give explicit models for spherical functions on $p$-adic symmetric spaces $X=H\backslash G$ for pairs of $p$-adic groups $(G,H)$ of the form $(\mathrm{U}(2r),\mathrm{U}(r)\times \mathrm{U}(r)),$ $(\mathrm{O}(2r),\mathrm{O}(r)\times \mathrm{O}(r)),$ $(\mathrm{Sp}(4r),\mathrm{Sp}(2r)\times\mathrm{Sp}(2r))),$ $(\mathrm{U}(2r+1),\mathrm{U}(r+1)\times \mathrm{U}(r)),$ and $ (\mathrm{O}(2r+1),\mathrm{O}(r+1)\times \mathrm{O}(r)).$ As an application, we completely describe their Hecke module structure.