Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
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 S-.F 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 7.h p is a closed
