TY - JOUR

T1 - Spherical functions and local densities on the space of p -adic quaternion Hermitian forms

AU - Hironaka, Yumiko

N1 - Publisher Copyright:
© 2022 World Scientific Publishing Company.

PY - 2022/4/1

Y1 - 2022/4/1

N2 - 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.

AB - 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.

KW - Plancherel formula

KW - Spherical function

KW - orthogonal polynomials

KW - quaternion Hermitian forms

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U2 - 10.1142/S1793042122500324

DO - 10.1142/S1793042122500324

M3 - Article

AN - SCOPUS:85116872780

VL - 18

SP - 575

EP - 612

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

IS - 3

ER -