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A Duality for Symmetric Spaces with Applications to Group Representations

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A Duality for Symmetric Spaces with Applications to Group Representations ADVANCES IN MATHEMATICS 5, 1-154 (1970) A Duality for Symmetric Spaces with Applications to Group Representations SIGURDUR HELGASON~ Massachusetts Institute of Technology Table of Contents Notation and Preliminaries ........................ 3 Chapter I. Summary .......................... 8 01. The Space of Horocycles in a Symmetric Space ........... 8 $2. Function Theory on X and S ................... 12 93. Two Dual Models for the Spherical Representations ......... 21 Chapter II. The Radon Transform and its Dual .............. 26 $1. The Injectivity ......................... 26 92. Digression on the Spherical Fourier Transform ........... 28 53. The Inversion Formula and the Plancherel Formula for the Radon Transform. .......................... 39 Chapter III. Conical Functions, Conical Distributions and RepresentationTheory 48 $1. Totally Geodesic Submanifolds of Xand of S. Reduction to the Hermitian Hyperbolic Case ........................ 48 $2. Conical Distributions. Elementary Properties ............ 65 $3. Conical Functions and Finite-Dimensional Representations ...... 73 $4. The Determination of the Conical Distributions ........... 79 $5. The Spherical Representations and the Fourier Transform on X .... 110 96. Construction of the Intertwining Operators ............. 121 Chapter IV. Eigenspaces of Invariant Differential Operators as Models for Irreducible Representations ................. 129 $1. The Eigenspaces of the Laplacian d on a Symmetric Space of Rank One 129 92. The Irreducibility of the Eigenspaces of d ............. 142 $3. The Eigenspaces of the Invariant Differential Operators on B ..... 144 $4. The Eigenspaces of the Invariant Differential Operators on G/N (G Complex) ........................... 145 Bibliography .............................. 152 i Present address: Institut Mittag-Leffler, Djursholm, Sweden. 1 0 1970 by Academic Press, Inc. 2 HELGASON Preface En aldrei sd n&n pann sem augad gaf. E.B. Let X be a symmetric space of the noncompact type, B the (dual) space of horocycles in X. The largest connected group G of isometries of X acts transitively both on X and on 8. The present paper deals with various topics in analysis on X and on B and relates these to the repre- sentation theory of G. We give now a very rough indication of the con- tents of the paper; Chapter I contains a detailed summary of the main results. The proofs are given in the subsequent chapterswheretherelation- ship to past and current work in representation theory is also explained. The analogies between the spaces X and 8 are first studied in some detail and function theory on B developed from the point of view of these analogies. The principal concepts here are the conical functions and conical distributions on E; they are the counterparts to the spherical functions on X. The joint eigenspaces of the G-invariant differential operators on E give models for irreducible representation of G; similarly for X. This provides two distinct models for the Gelfand-Naimark principal series of class one representations of G, the connection between the two models being provided by the Radon transform (mapping functions on X into functions on E) an d its dual (mapping distributions on B mto’ distributions on X). On the other hand when the class one principal series is realized by means of the models over 8, certain equivalences take place and the corresponding intertwining operators are constructed by means of the conical distributions. These distributions also furnish extreme weight vectors for the representations in the class one principal series. The two dual models of the class one principal series occur when the natural representations of G on L2(X) and on L”(E) are decomposed into a direct integral of irreducibIe representations. The decomposition is most explicitly exhibited by means of a suitable Fourier transform on X and on E. The Plancherel measure on X is given by Harish-Chandra’s function c(X) which is given by a ratio of certain Gamma factors. Whereas the spherical functions on X involve the c-function alone we find that the numerator itself enters significantly in the construction of the conical distributions. The denumerator on the other hand turns out to be intimately connected with the principal series of representations men- tioned above. GROUP REPRESENTATIONS 3 This paper is fairly self-contained but uses many results from Harish- Chandra’s papers [21(i), (j)] on spherical functions and a few results from [22(c)] w h ere the Radon transform on X is applied to differential equations. Many of the results have been given in research announce- ments [22(d), (g), (h), (k)]. Th e material in Chapter II was presented at the Scandinavian Mathematical Congress, Oslo, 1968 and the material in Chapter III in lectures at Eidgenossische Technische Hochschule, Zurich whose hospitality I enjoyed during the summer 1969. I wish also to acknowledge support by the National Science Foundation during the preparation of this paper. Finally, I wish to express my appreciation to G. Schiffmann for showing me his proof of his formula reducing intertwining operators to the rank one case [40(a)] and to B. Kostant for enlightenment about his work [28(b)]. Notation and Preliminaries $1. General Notation. We shall use the standard notation 2, R, C for the ring of integers, the field of real numbers and the field of complex numbers, respectively; Z+ is the set of nonnegative integers, Rf the set of nonnegative real numbers. If S is a set, T a subset and f a function on S, the restriction off to T is denoted f 1 T. If S is a topological space C1( T) denotes the closure of T in S. The space of continuous functions on S is denoted by C(S), C,(S) th e set of those of compact support. Composition of functions and operators will often be denoted by o. $2. Manifolds. If M is a manifold (satisfying the second countability axiom) and m E M the tangent space to Mat m is denoted Mm . Following Schwartz [41] we write 9(M) for the space of C” functions on M of compact support, topologized by means of uniform convergence of functions along with their derivatives; 9’(M) denotes the dual space of all distributions on M. The space b(M) denotes the space of all C” functions on M topologized in a similar way as 9(M) and b’(M) denotes the dual space of distributions on M of compact support. If I’ is a vector space over R, 9(V) d enotes the space of rapidly decreasing functions on V (Schwartz [41]). Let T be a diffeomorphism of M onto itself, and let f E b(M), T E L@‘(M) and D a differential operator on M. We put fTW = f(+w)Y TilEi Vf) = qf+)> f c WW D(f) = (Of 7-y) f E B(M) 4 HELGASON Thenp E b(M), TT E 9’(M) and D is another differential operator on M. The value of Df at a point m will usually be denoted by (of)(m) but sometimes it is convenient to write D,,(f(m)). If @ is a differentiable mapping from a manifold M into another manifold d@,,& (and sometimes d@) denotes the differential of @ at m. 93. Lie Groups. If A is a group and a E A, L(a) denotes the left translation x 4 ax and R(a) denotes the right translation x 4 xa on A. If B C A is a subset we write B” = nBn-l; if B is a subgroup A/B denotes the set of left cosets aB, a E A. The transformation xB ---f axB of A/B will always be denoted by ~(a). Lie groups will be denoted by Latin capital letters and their Lie algebras by corresponding lower case German letters. If G is a Lie group and g its Lie algebra the adjoint representation of G is denoted by Ad (or Ad,) and the adjoint representation of g by ad (or ad,). We shall now list some standard notation concerning semisimple Lie groups which will be utilized throughout the paper. Let G be a connected semisimple Lie group with finite center, g the Lie algebra of G, and ( , ) (sometimes B) the Killing form of g. Let 8 be a Cartan involution of g, that is an involutive automorphism such that the form (X, Y) ---f -(X, 0Y) is strictly positive definite on g x g. Let g = f + p be the decomposition of g into eigenspaces of 19(a Curtan decomposition) and K the analytic subgroup of G with Lie algebra I. Let a C p be a maximal abelian subspace, a* its dual, a,* the complexification of a*, i.e. the space of R-linear maps of a into C. Let A = exp a and log the inverse of the map exp: a ---f A. For A E a* put g,a= ix E 9 I If& Xl = ww, for all HEa}. IfX # Oandg, # (01th en X is called a (restricted) root and m, = dim(g,) is called its multiplicity. Let gc denote the complexification of g and if s is any subspace of g let 5, denote the complex subspace of gc spanned by 5. If A, p E a,* let HA E a, be determined by X(H) = (HA , H) for HE a and put (A, p) = (H, , H,). S ince ( , ) is positive definite on p we put 1 X 1 = (A, h)1/2 for h E a* and 1 X 1 = (X, X)li2 for X E p. Let a’ be the open subset of a where all restricted roots are # 0. The com- ponents of a’ are called Weyl chambers. Fix a Weyl chamber a+ and call a (restricted) root a: positive if it is positive on a+. Let a+* denote the corresponding Weyl chamber in a*, that is the preimage of a+ under the mapping A --+ HA .
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