Endomorphisms of spaces of virtual vectors fixed by a discrete group

Abstract

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