Harmonic analysis of compact and non-compact Riemannian symmetric spaces is well known: on semisimple symmetric spaces, which possess an invariant semi-Riemannian metric, it is investigated by various people. The author treats Grassmannian bundles; they are examples of affine symmetric spaces which possess no invariant semi-Riemannian metrics and are realized as limits of compact or semisimple symmetric spaces (§ 6).
The Grassmann bundle \(P_{n,k}\) of k-dimensional affine subspaces of \({\mathbb{R}}^ n\) is a homogeneous space: \(G\times V/H\times W\), where \(G=O(m)\), \(V={\mathbb{R}}^ n\), \(H=O(k)\times O(m)\) \((k+m=n)\) and W the subspace spanned by the first k basis elements. Let \(\Lambda\) be an index set of H-orbits \(\lambda\) in \(W^{\perp}\); dz, \(d\mu\) (\(\lambda)\) Lebesgue measure on \(W^{\perp}\), the invariant measure on \(\lambda\), respectively. For a point \(z_{\lambda}\in \lambda\) let \(G_{\lambda}\) and \(H_{\lambda}\) be the isotropy subgroups of \(z_{\lambda}\) in G and H, respectively; \(\Delta_{\lambda}\) the set of irreducible representations of \(G_{\lambda}\) that occur in \(L^ 2(G_{\lambda}/H_{\lambda})\). Then in abstract sense (§ 2) each right \(H\times W\) invariant function f(g,x) has the Fourier decomposition: \[ f(g,x)=\int_{\Lambda}\sum_{\delta \in \Delta_{\lambda}}F_{\lambda,\delta}(g,x) d\nu (\lambda), \] where \(dz=d\mu (\lambda)d\nu (\lambda)\). In § 3 the explicit form of \(F_{\lambda,\delta}\) is given as \[ F_{\lambda,\delta}(g,x)=\int_{O(m)}M_{\delta}(g\left( \begin{matrix} 0\\ s\end{matrix} \right))e^{i\lambda <x,g(se_ n)>} ds\quad, \] where \(M_{\delta}\) is an explicit function defined in terms of determinants and \((se_ n)\) means we let \(s\in O(m)\) act on the last m components.
The abstract method is generalized for a semidirect product \(G\times V\); G, V are locally compact groups, V is abelian (§ 4), and the explicit form is used to obtain the diagonalization of various generalizations of the Radon transform between bundles (§ 5).