Endomorphisms ofspaces ofvirtual vectors fixed by a discrete group

A study is made of unitary representations π of a discrete group G that are of type II when restricted to an almost-normal subgroup Γ ⊆ G. The associated unitary representation π p of G on the Hilbert space of ‘virtual’ Γ0-invariant vectors is investigated, where Γ0 runs over a suitable class of finite-index subgroups of Γ. The unitary representation π p of G is uniquely determined by the requirement that the Hecke operators for all Γ0 are the ‘block-matrix coefficients’ of π p. If π|Γ is an integer multiple of the regular representation, then there is a subspace L of the Hilbert space of π that acts as a fundamental domain for Γ. In this case the space of Γ-invariant vectors is identified with L. When π|Γ is not an integer multiple of the regular representation (for example, if G = PGL(2, Z[1/p]), Γ is the modular group, π belongs to the discrete series of representations of PSL(2,R), and the Γ-invariant vectors are cusp forms), π is assumed to be the restriction to a subspace H0 of a larger unitary representation having a subspace L as above. The operator angle between the projection PL onto L (typically, the characteristic function of the fundamental domain) and the projection P0 onto the subspace H0 (typically, a Bergman projection onto a space of analytic functions) is the analogue of the space of Γ-invariant vectors. It is proved that the character of the unitary representation π p is uniquely determined by the character of the representation π. Bibliography: 53 titles. Keywords: unitary representations, Hecke