SO(p, q). Another property of the singular representations is that they persist in the limit as q --+ oo. Bibliography: 70 titles.Since the works of V. Bargmasm [32] and I. M. Gelfand and Naimark [54], which initiated the study of infinite-dimensional representations, Hilbert spaces with scalar products of the formwhere M is a manifold and K(x, y) is a distribution on M x M, are usual in the theory of unitary representations of semisimple groups. Hermitian forms of type (0.1) a priori need not be positive definite. Thus, in each specific case, the verification of the positive definiteness is a more or less difficult problem (sometimes very difficult).The above function spaces did not fix the attention of analysts (although, some objects of this type were studied, e.g., the scalar product in Sobolev spaces is written in the form (0.1) with M = R n and g(x, y) = I[x -y[lX). On the other hand, in representation theory, such Hilbert spaces are regarded as a "receptacle" of algebraic structure, but not as an object worth special study.The aim of the present paper is to show that there are phenomena in noncommutative harmonic analysis and representation theory (some of them are usually regarded as algebraic phenomena of complicated nature) which are based on the assertions of the "trace theorem" type. We mean that in a function space 7"l with a scalar product of the form (0.1) there is a well-defined restriction operator of a function f to a submanifold N C M of sufficiently large dimension (arid suitably located). We note that the value of f at an individual point x 6 N can be undefined. Such phenomena are well known in analysis, e.g., the restriction to the boundary in Hardy spaces; the restriction to a submanifold in Sobolev spaces (see, e.g., [8], w and [26], w the Nagel-Rudin theorem about the restriction of a function from H ~ in the ball to a suitably located curve on the sphere, see [25], w some questions in the theory of exceptional sets (the Salem-Zygmund theorem), see [65,10]; the results of Stein and Fefferman on Fourier transform of functions f 6 L p ([70]). The space E(N) of all functions on N which are the restrictions of functions f E 7"l to N is a quotient space of 7"(. Therefore, E(N) is a Hilbert space itself.Dually, it can happen that 7-/contains distributions with support lying in N. Then the space EO(N) of all such distributions is also a Hilbert space as a subspace of 7"/.