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Spherical functions and local densities on Hermitian forms. (English) Zbl 0936.11024
The aim of this paper is to complete the theory of the spherical functions on the space \(X\) of unramified hermitian forms over a \({\mathfrak p}\)-adic field \(k\) [Jap. J. Math., New Ser. 14, 203-223 (1988; Zbl 0674.43006), ibid. 15, 15-51 (1989; Zbl 0714.43012), TĂ´hoku Math. J., II. Ser. 40, 651-671 (1988; Zbl 0674.43007), Comment. Math. Univ. St. Pauli 39, 157-193 (1990; Zbl 0718.11016)]. A nonzero \(K\)-invariant function on \(X\) is called a spherical function on \(X\) if it is a common eigenfunction under the action of the Hecke algebra \({\mathcal H} (G,K)\) by the convolution product, where \(G= GL_n(k)\) and \(K= GL_n ({\mathcal O}_k)\).
The author gives an explicit expression for the typical spherical functions and defines the spherical Fourier transform on the space \({\mathcal S}(K\setminus X)\) of \(K\)-invariant Schwartz-Bruhat functions on \(X\). The structure of \({\mathcal S}(K\setminus X)\) as \({\mathcal H}(G,K)\)-modules, the Plancherel formula and the inversion formula of the spherical Fourier transform, and a parametrization of all spherical functions on \(X\) are given explicitly. Finally, as an application of the theory, the author gives explicit expressions of local densities of integral representations of hermitian forms.
Reviewer: Y.Hironaka (Tokyo)

11E95 \(p\)-adic theory
43A90 Harmonic analysis and spherical functions
11E39 Bilinear and Hermitian forms
11E08 Quadratic forms over local rings and fields
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