Endomorphisms of spaces of virtual vectors fixed by a discrete group

A study is made of unitary representations $\pi$ of a discrete group $G$ that are of type II when restricted to an almost-normal subgroup $\Gamma\subseteq G$. The associated unitary representation $\overline{\pi}^{\,\rm{p}}$ of $G$ on the Hilbert space of ‘virtual’ $\Gamma_0$-invariant vectors is investigated, where $\Gamma_0$ runs over a suitable class of finite-index subgroups of $\Gamma$. The unitary representation $\overline{\pi}^{\,\rm{p}}$ of $G$ is uniquely determined by the requirement that the Hecke operators for all $\Gamma_0$ are the ‘block-matrix coefficients’ of $\overline{\pi}^{\,\rm{p}}$. If $\pi|^{}_\Gamma$ is an integer multiple of the regular representation, then there is a subspace $L$ of the Hilbert space of $\pi$ that acts as a fundamental domain for $\Gamma$. In this case the space of $\Gamma$-invariant vectors is identified with $L$. When $\pi|^{}_\Gamma$ is not an integer multiple of the regular representation (for example, if $G=\operatorname{PGL}(2,\mathbb Z[1/p])$, $\Gamma$ is the modular group, $\pi$ belongs to the discrete series of representations of $\operatorname{PSL}(2,\mathbb R)$, and the $\Gamma$-invariant vectors are cusp forms), $\pi$ is assumed to be the restriction to a subspace $H_0$ of a larger unitary representation having a subspace $L$ as above. The operator angle between the projection $P_L$ onto $L$ (typically, the characteristic function of the fundamental domain) and the projection $P_0$ onto the subspace $H_0$ (typically, a Bergman projection onto a space of analytic functions) is the analogue of the space of $\Gamma$-invariant vectors. It is proved that the character of the unitary representation $\overline{\pi}^{\,\rm{p}}$ is uniquely determined by the character of the representation $\pi$.
Bibliography: 53 titles.