tailieunhanh - Báo cáo toán học: "The problem of integral geometry and intertwining operators for a pair of real Grassmanian manifolds "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Vấn đề của hình học không tách rời và các nhà khai thác gắn liền với một cặp thực sự đa tạp Grassmanian. | J. OPERATOR THEORY 12 1984 359 - 383 Copyright by INCREST 1984 THE PROBLEM OF INTEGRAL GEOMETRY AND INTERTWINING OPERATORS FOR A PAIR OF REAL GRASSMANNIAN MANIFOLDS I. M. GELFAND M. I. GRAEV R. ROSU INTRODUCTION Let Gp Gq be the Grassmannian manifolds of the p respective q dimensional subspaces of the real linear space V p q . This paper is concerned with the remarkable integral transform associated with Gp and Gq. It can be defined as the transform which associates with each function on Gp a function on Gq namely if is a function on Gp and aeGq then v a is the value of the integral of the function f on Gp a the set of the subspaces b eGp which are contained in a. When so defining it we already assume that on each Gp á there is a given measure and by this a supplementary structure is being introduced on Gp. In order to avoid this we shall define the operator J slightly different we introduce the function spaces Fp on Gp the manifold of the pairs b P where b eGp and ỊÍ is a non-oriented volume element in b which satisfy the homogeneity condition f b tP P for any t 0. We shall define see 1 p. 3 J as an operator p- Fp which for the natural representation of SL K on Fpq and Fị is an intertwining operator that means by definition that it commutes with the operators of the representation . If an Euclidian structure is introduced on V and by this we have a measure on each Gp a then the definition of the operator coincides with the one at the beginning. The main purpose of this paper is to explicitly construct an operator Fp F which on Im y the image of coincides with the inverse of y - that is obtaining an inversion formula for y. We shall consider the casep q n when J is injective. Several approaches on the subject are already known in 6 it was constructed the operator which associates with every function p e Fp and every b eGp the p 7 p differential form xb p on the manifold Gq_p. It was proved that if p 360 I. M. GELFAND M. I. GRAEV R. ROSL then p is a closed