Quaternionic Spherical Fourier Multipliers for Gelfand Pairs

Let ${\mathbb{H}}_q$ denote the quaternionic Heisenberg group with ${\mathbb{R}}^4\times {\mathbb{R}}^3$ stratification and $K$ a compact subgroup of Automorphism. We construct the spherical Fourier multiplier related to a Gelfand Pairs on the Quaternionic Heisenberg group ${\mathbb{H}}_q$ via the motion group $G=K⋉{\mathbb{H}}_q$ of ${\mathbb{H}}_q$, where $K⋉{\mathbb{H}}_q$ is the semidirect product of ${\mathbb{H}}_q$ and its compact subgroup of automorphism $K\in Aut({\mathbb{H}}_q)$.