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A note on automorphic forms. (English) Zbl 0702.11027

Notations; G a locally compact unimodular group, K a compact subgroup of G, \(\Gamma\) a closed unimodular subgroup of G, A a closed subgroup of \(\Gamma\cap Z(G)\) (Z(G) is the center of G), \(\chi\) a continuous unitary character of A, (\(\rho\),V) a finite dimensional unitary representation of \(\Gamma\) on V such that \(\rho |_ A=\chi\), (\(\pi\),H) an irreducible unitary representation of G on H such that \(\pi |_ A=\chi\), \(\delta\) an irreducible unitary representation of K. We will suppose that the multiplicity of \(\delta\) in \(\pi |_ K\) is equal to one. We denote by \({\mathcal A}_{\delta}(\Gamma \setminus G,\rho,\pi)\) the complex vector space consisting of the measurable functions f: \(G\to Hom_ C(H(\delta),V)\) such that \[ f(\gamma x)=\rho (\gamma)\circ f(x)\text{ for all } \gamma \in \Gamma,\quad \int_{\Gamma \setminus G}| f(x)|^ 2 d_{\Gamma \setminus G}(\dot x)<\infty, \]
\[ f(xk)=f(x)\circ \pi (k)\text{ for all } k\in K, \]
\[ \int_{\Gamma /A}f(xy)\phi (y)d_{G/A}(\dot y)={\hat \psi}_{\pi,\delta}(\phi)f(x)\text{ for all } \phi \in C_ c(G/A,\chi,\delta)^ 0. \] Here H(\(\delta\)) is the \(\delta\)-isotypic component of \(\pi |_ K\), \(C_ c(G/A,\chi,\delta)^ 0\) is the Hecke algebra with K-type \(\delta\) and central character \(\chi\), and \({\hat \psi}{}_{\pi,\delta}(\phi)\) is the Fourier transform of \(\phi\) with respect to the spherical function \(\psi_{\pi,\delta}\) of \(\pi\) with K-type \(\delta\) in the sense of R. Godement [Trans. Am. Math. Soc. 73, 496-556 (1952; Zbl 0049.201)]. We will prove that the dimension of \({\mathcal A}_{\delta}(\Gamma \setminus G,\rho,\pi)\) is equal to the multiplicity of \(\pi\) in \(Ind^ G_{\Gamma} \rho\). If \(\pi\) is integrable, the space \({\mathcal A}_{\delta}(\Gamma \setminus G,\rho,\pi)\) has a simpler description which is convenient for the explicit calculation. In this case, we will give a dimension formula for \({\mathcal A}_{\delta}(\Gamma \setminus G,\rho,\pi).\)
For the group Sp(n,\({\mathbb{R}})\) (resp. the Jacobi group, i.e. the semi- direct product of Sp(n,\({\mathbb{R}})\) and the Heisenberg group) and the integrable holomorphic discrete series of the group, the space \({\mathcal A}_{\delta}(\Gamma \setminus G,\rho,\pi)\) coincides with the space of the Siegel cusp forms (resp. the cuspidal Jacobi forms).
Reviewer: K.Takase

MSC:

11F12 Automorphic forms, one variable
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

Citations:

Zbl 0049.201
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