Endomorphisms of spaces of virtual vectors fixed by a discrete group

Abstract

Consider a unitary representation $π$ of a discrete group $G$, which, when restricted to an almost normal subgroup $Γ\subseteq G$, is of type II. We analyze the associated unitary representation $\overlineπ^{\rm{p}}$ of $G$ on the Hilbert space of "virtual" $Γ_0$-invariant vectors, where $Γ_0$ runs over a suitable class of finite index subgroups of $Γ$. The unitary representation $\overlineπ^{\rm{p}}$ of $G$ is uniquely determined by the requirement that the Hecke operators, for all $Γ_0$, are the "block matrix coefficients" of $\overlineπ^{\rm{p}}$. If $π|_Γ$ is an integer multiple of the regular representation, there exists a subspace $L$ of the Hilbert space of the representation $π$, acting 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, (e.g. if $G=PGL(2,\mathbb Z[\frac{1}{p}])$, $Γ$ is the modular group, $π$ belongs to the discrete series of representations of $PSL(2,\mathbb R)$, and the $Γ$-invariant vectors are the cusp forms) we assume that $π$ is 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 $Γ$- invariant vectors. We prove that the character of the unitary representation $\overlineπ^{\rm{p}}$ is uniquely determined by the character of the representation $π$.