On Representations and Compactifications of Symmetric Riemannian Spaces

Abstract

In the recent study of automorphic functions [2], [4], [7], it has become necessary to take boundaries of symmetric bounded domains (or more generally, of symmetric Riemannian spaces) into account and to consider explicitly the behavior of the functions on these boundaries. Now the notion of boundary presupposes naturally a compactification of the space. The purpose of this paper is to give a general method of compactifying symmetric Riemannian spaces by means of certain isometries of them into the space of all positive definite hermitian matrices. Let S be a symmetric Riemannian space of non-compact type, i.e., S = GIK, G being a (connected) semi-simple Lie group with finite center such that all its simple factors are non-compact and Ka maximal compact subgroup of G. Let g, f be the Lie algebras of G and K, respectively. Then for any faithful irreducible representation p of g in a (complex) vector space V of dimension n, we can take a suitable base in V such that p(zX) =p(X) for X e g, z denoting the automorphism of g defined by the symmetry of S around x0 = K; we have then the corresponding irreducible projective representation, denoted also by p, of G into PSL(n, C) such that the matrices p(k) corresponding to k e K are unitary. Then making x = gK e S correspond to the hermitian matrix p(g)p(g), we obtain an injection, denoted also by p, of S into the space ?P1 of all positive definite hermitian matrices of degree n and of determinant 1. This mapping p from S into 91 is an isometry (with respect to some invariant metrics in S and in ?E1) and satisfies the following conditions: (i ) p(S) is totally geodesic in ?Pl, (ii) p(S) is an irreducible set of matrices, (iii) p(x0) i_ n (the identity matrix of degree n). Conversely, it can be proved easily that any isometry p from S into ?n satisfying these conditions is obtained by the above method; we call such an isometry an irreducible representation of S into 9nF. Now denote by P(iCn) the (real) projective space associated with the (real) vector space Mn of all hermitian matrices of degree n. On1 being imbedded canonically in P(3Cn), p(S) can be considered as a subset of P(?n);