×

The problem of integral geometry and intertwining operators for a pair of real Grassmannian manifolds. (English) Zbl 0551.53034

Let \(G_ p\), \(G_ q\) be the Grassmannian manifolds of the p, q dimensional subspaces of the linear space V \((p<q)\). The authors introduce the function spaces \(F_ p^{\lambda}\) on the manifold \(G^*_ p\) of the pairs (b,\(\beta)\), where \(b\in G_ p\) and \(\beta\) is a non-oriented volume element in b which satisfy the homogeneity condition \(f(b,t^{\beta})=t^{\lambda}f(b,\beta)\) for any \(t>0\). The main subject of the paper is the intertwining operator \(J: F^ q_ p\to F^ p_ q\) \((1\leq p<q<n)\) that is, an operator which commutes with the operators of the natural representation of SL(V) on \(F^ p_ q\) and \(F^ q_ p\). Roughly speaking, (Jf)(a), \(a\in G_ q(V)\) is equal to the value of the integral of the function f on the set of the p subspaces that are contained in a. The authors construct explicitly an operator \(F^ p_ q\to F^ q_ p\) which on the image of J coincides with the inverse of J. The case \(p+q\leq n\) is considered when J is injective. Several particular cases have been considered previously by the first of the authors and collaborators. The present paper uses some usual techniques of the authors and several other new ones, interesting by themselves, but difficult to abridge in a limited space.
Reviewer: L.A.Santaló

MSC:

53C65 Integral geometry
44A05 General integral transforms
58C30 Fixed-point theorems on manifolds
PDFBibTeX XMLCite