Endomorphisms of spaces of virtual vectors fixed by a discrete group

TL;DR

This paper studies the structure and properties of unitary representations of discrete groups, focusing on virtual vectors fixed by subgroups and their relation to Hecke operators, with applications to automorphic forms and representation theory.

Contribution

It introduces a novel framework for analyzing unitary representations on virtual vector spaces fixed by subgroups, connecting Hecke operators to block matrix coefficients.

Findings

Character of the induced representation is uniquely determined by the original representation.

Identifies the space of invariant vectors with a fundamental domain in certain cases.

Provides a new perspective on the relationship between subgroup restrictions and larger group representations.

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 $\pi$.