×

The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces. (English) Zbl 0871.43005

Let \(G\) be a connected noncompact semisimple Lie group with finite center, \(K\) a maximal compact subgroup, and \(G/K\) the corresponding Riemannian symmetric space of noncompact type. Let \(\tau\) be an irreducible unitary representation of \(K\) on a vector space \(V_\tau\) and let \(E^\tau\) be the homogeneous vector bundle over \(G/K\) defined by \(\tau\). In this paper harmonic analysis for homogeneous vector bundles \(E^\tau\) over \(G/K\) is investigated. By a radial system of sections we mean a map \(F:G\to\text{End} (V_\tau)\) such that \[ F(k_1gk_2) =\tau(k_2^{-1}) F(g)\tau (k_1^{-1}), \quad g \in G, \quad k_1,k_2\in K. \] The radial systems of sections generalize the notion of \(K\)-biinvariant functions on \(G\) in the scalar case. We denote by \(C_0^\infty (G,\tau,\tau)\) the space of radial systems of \(C^\infty\) sections with compact support. A function \(\Phi: G\to \text{End} (V_\tau)\) is called a spherical function of type \(\tau\) on \(G\) if it satisfies \[ \Phi (k_1gk_2) =\tau(k_1) \Phi(g) \tau(k_2), \quad g\in G, \quad k_1,k_2\in K \] and if the map \[ F\to \widehat F(\Phi)= {1\over d_\tau} \int_GTr \bigl[\Phi(x) F(x)\bigr]dx \] is a homomorphism of \(C_0^\infty (G,\tau,\tau)\) into \(\mathbb{C}\), where \(d_\tau\) is the dimension of \(\tau\). In this paper functional and differential properties of spherical functions of type \(\tau\) on \(G\) are investigated in the commutative case of the algebra \(C_0^\infty (G,\tau,\tau)\). The theory of spherical functions of type \(\tau\) on \(G\), as developed by Godement, Harish Chandra and Warner, is used to discuss the analysis of radial systems of sections of a homogeneous vector bundle \(E^\tau\) over \(G/K\). In this case the spherical transform and its inversion formula on \(C_0^\infty (G,\tau,\tau)\) are derived from the Plancherel formula on \(G\). An important example in the commutative case of the convolution algebra \(C_0^\infty(G,\tau,\tau)\) is provided by the pairs \((G,K)\) such that \(G/K\) is either a real or a complex hyperbolic space, and the spherical transform and the inversion formula for radial systems of sections of \(E^\tau\) are discussed in detail.
Reviewer: K.Saka (Akita)

MSC:

43A90 Harmonic analysis and spherical functions
53C35 Differential geometry of symmetric spaces
PDFBibTeX XMLCite
Full Text: EuDML