Spherical functions and local densities on the space of p -adic quaternion Hermitian forms

We introduce the space X of quaternion Hermitian forms of size n on a -adic field with odd residual characteristic, and define typical spherical functions ω(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 Sn, and define a spherical Fourier transform on the Schwartz space (∖) which is Hecke algebra ℋ(,)-injective map into the symmetric Laurent polynomial ring of size n. Then, we determine the explicit formulas of ω(x; s) by a method of the author's former result. In the last section, we give precise generators of (∖) and determine all the spherical functions for n ≤ 4, and give the Plancherel formula for n = 2.