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 $G=K\ltimes \mathbb{H}_q$ of $\mathbb{H}_q$, where $K\ltimes \mathbb{H}_q$ is the semidirect product of $\mathbb{H}_q$ and its compact subgroup of automorphism $K\in Aut(\mathbb{H}_q)$.