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Spherical functions and local densities on the space of \(p\)-adic quaternion Hermitian forms. (English) Zbl 1483.11110

Summary: 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 define a spherical Fourier transform on the Schwartz space \(\mathcal{S}(\mathcal{K}\setminus\mathcal{X})\) which is Hecke algebra \(\mathcal{H}(\mathcal{G},\mathcal{K})\)-injective map into the symmetric Laurent polynomial ring of size \(n\). Then, we determine the explicit formulas of \(\omega(x;s)\) by a method of the author’s former result. In the last section, we give precise generators of \(\mathcal{S}(\mathcal{K}\setminus\mathcal{X})\) and determine all the spherical functions for \(n\leq 4\), and give the Plancherel formula for \(n=2\).

MSC:

11F85 \(p\)-adic theory, local fields
11E95 \(p\)-adic theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)

Software:

Macaulay2
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References:

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