Spherical functions and local densities on the space of $p$-adic quaternion hermitian matrices

We introduce the space $X$ of quaternion hermitian forms of size $n$ on a ${\mathfrak p}$-adic field with odd residual characteristic, and define typical spherical functions $\omega(x;s)$ on $X$ and give their induction formula on sizes by using local densities of quaternion hermitian forms. Then we give functional equation of spherical functions with respect to $S_n$, and determine the explicit formulas of $\omega(x;s)$. On the other hand, we define the spherical transform on the Schwartz space ${\mathcal S}(K\backslash X)$ based on $\omega(x; s)$ and study the Hecke module structure of ${\mathcal S}(K \backslash X)$.