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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), Tôhoku 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)

##### MSC:
 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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