Explicit formulas for spherical functions on symmetric spaces of type AII

1992 ◽  
Vol 26 (1) ◽  
pp. 59-61 ◽  
Author(s):  
A. P. Veselov ◽  
O. A. Chalykh
Keyword(s):  
Symmetric Spaces ◽  
Spherical Functions ◽  
Explicit Formulas

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

2021 ◽  
pp. 1-38
Author(s):  
Yumiko Hironaka
Keyword(s):  
Laurent Polynomial ◽  
Spherical Functions ◽  
Plancherel Formula ◽  
Explicit Formulas ◽  
Hermitian Forms ◽  
Spherical Fourier Transform ◽  
Local Densities ◽  
Laurent Polynomial Ring ◽  
Injective Map ◽  
Residual Characteristic

We introduce the space [Formula: see text] of quaternion Hermitian forms of size [Formula: see text] on a [Formula: see text]-adic field with odd residual characteristic, and define typical spherical functions [Formula: see text] on [Formula: see text] 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 [Formula: see text], and define a spherical Fourier transform on the Schwartz space [Formula: see text] which is Hecke algebra [Formula: see text]-injective map into the symmetric Laurent polynomial ring of size [Formula: see text]. Then, we determine the explicit formulas of [Formula: see text] by a method of the author’s former result. In the last section, we give precise generators of [Formula: see text] and determine all the spherical functions for [Formula: see text], and give the Plancherel formula for [Formula: see text].

Spherical functions on mixed symmetric spaces

2001 ◽  
Vol 5 (3) ◽  
pp. 43-92 ◽  
Author(s):  
Bernhard Krötz ◽  
Karl-Hermann Neeb ◽  
Gestur Ólafsson
Keyword(s):  
Symmetric Spaces ◽  
Spherical Functions
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