Endomorphisms of spaces of virtual vectors fixed by a discrete group

Abstract

Consider a unitary representation $\pi$ of a discrete group $G$, which, when restricted to an almost normal subgroup $\Gamma\subseteq G$, is of type II. We analyze the associated unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ on the Hilbert space of "virtual" $\Gamma_0$-invariant vectors, 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, there exists a subspace $L$ of the Hilbert space of the representation $\pi$, acting 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, (e.g. if $G=PGL(2,\mathbb Z[\frac{1}{p}])$, $\Gamma$ is the modular group, $\pi$ belongs to the discrete series of representations of $PSL(2,\mathbb R)$, and the $\Gamma$-invariant vectors are the cusp forms) we assume that $\pi$ 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 $\Gamma$- invariant vectors. We prove that the character of the unitary representation $\overline{\pi}^{\rm{p}}$ is uniquely determined by the character of the representation $\pi$.