Representing some II$_1$ factors in $L^2(\Lambda \backslash G)$

TL;DR

This paper demonstrates how the pointwise limit multiplicity property can be used with a generalized theorem to embed II$_1$ factors associated with lattices into $L^2$ spaces of quotient groups, extending previous results.

Contribution

It introduces a method to represent II$_1$ factors of certain lattices in $L^2$ spaces of quotient groups, generalizing earlier work and expanding the understanding of these representations.

Findings

Representation of II$_1$ factors in $L^2$ spaces of quotient groups.

Extension of previous results to broader classes of lattices.

Connection between limit multiplicity property and operator algebra representations.

Abstract

Let $G$ be $PGL(n,F)$, $n \geq 3$, $F$ a certain non-archimedean local field; or let $G$ be $PSL(2,\mathbb{R}) \times \cdots \times PSL(2,\mathbb{R})$. Let $\Gamma$ be a lattice in $G$, and let $( \Lambda_n )$ be a sequence of lattices in $G$ satisfying the pointwise limit multiplicity property. In this note, we explain how the pointwise limit multiplicity property can be combined with a generalization of a theorem in \cite{ghj} to give representations of the II$_1$ factor $R \Gamma$ on a subspace of $L^2(\Lambda_i \backslash G)$ for some $\Lambda_i$ in $( \Lambda_n )$. This extends a result in the author's dissertation \cite{ruthphd}.

Equations

$${\text{dim}}_{R\Gamma }(\mathcal{H})=\text{covol}(\Gamma )\cdot {\text{d}}_{\pi }$$

$$l^2(\Gamma )$$

$${\int }_G⟨\pi (g)u,v⟩\overline{⟨\pi (g)u^{\prime },v^{\prime }⟩}dg={\text{d}}_{\pi }^{-1}⟨u,u^{\prime }⟩\overline{⟨v,v^{\prime }⟩}$$

$$\underset{n\to \infty }{\lim }\frac{\text{mult}(\pi ,{\Lambda }_n)}{\text{vol}({\Lambda }_n\backslash G)}=\{\begin{array}{ll}{\text{d}}_{\pi }& \pi \in \Pi (G{)}_{\text{d}}\\ 0& \pi \in \Pi (G)\backslash \Pi (G{)}_{\text{d}}\end{array}$$

Full text

1. Introduction

The main purpose of this note is to explain:

Observation 1.1**.**

Let $G$ and $(\Lambda_{n})$ be as in one of the following two cases.

• Case 1.

$G$* is $PGL(n,F)$, $n\geq 3$, $F$ a non-archimedean local field of characteristic [math] and residue field of order relatively prime to $n$, and $(\Lambda_{n})$ is any sequence of pairwise non-conjugate lattices in $G$; or,*

• Case 2.

$G$* is $PSL(2,{\mathbb{R}})^{r}$, $r\geq 2$, and $(\Lambda_{n})$ is a sequence of lattices in $G$ with each $\Lambda_{i}$ of the form $PSL(2,{\mathcal{O}}_{F})$, where ${\mathcal{O}}_{F}$ is the ring of integers of a real number field $F$ with $r$ real embeddings.*

Let $\Gamma$ be a lattice in $G$. Then there exists a representation of the II1 factor $R\Gamma$ on a subspace of $L^{2}(\Lambda_{i}\backslash G)$ for some $\Lambda_{i}$ in $(\Lambda_{n})$.

The content of the proof of this observation lies in work on the limit multiplicity property carried out in [ABB*+*17], [GL18], and [Mat19], as well as work on representations of II1 factors in [GdlHJ89] rooted in Atiyah’s $L^{2}$-index theory in [Ati76], Atiyah and Schmid’s work on discrete series representations in [AS77], and Murray and von Neumann’s foundational work [MVN36]. We give the proof in Section 3.

Remark 1.2**.**

We could have featured groups besides $PGL(n,F)$, $n\geq 3$, and $PSL(2,{\mathbb{R}})\times\ldots\times PSL(2,{\mathbb{R}})$, since there are more general results on the pointwise limit multiplicity property than those we mention here. But these two groups, along with the group $PSL(2,{\mathbb{R}})$ in Theorem 2.1 below, already demonstrate that representations of the II1 factor $R\Gamma$ on subspaces of $L^{2}(\Lambda\backslash G)$ can be obtained regardless of whether the ambient group $G$ is connected or totally disconnected; higher-rank or rank-1; and Property (T) or a-T-menable.

2. Background

First we give background on representations of $R\Gamma$, then we give background on the pointwise limit multiplicity property.

In [Rut18], we showed:

Theorem 2.1**.**

(Theorem 35 in [Rut18].) Let $G=PSL(2,{\mathbb{R}})$, and let $\Gamma$ and $\Lambda$ be lattices in $G$. Then there exists a representation of the II1 factor $R\Gamma$ on a subspace of $L^{2}(\Lambda\backslash G)$.

To prove this result, we started with the representation of $R\Gamma$ on a discrete series representation in $L^{2}(G)$ given by:

Theorem 2.2**.**

(Theorem 3.3.2 in [GdlHJ89].) Let $G$ be a connected semi-simple real Lie group with Haar measure $dg$, let $\Gamma$ be a discrete subgroup in $G$, and let $\pi:G\rightarrow{\mathcal{U}}({\mathcal{H}})$ be an irreducible representation in the discrete series. Assume that every non-trivial conjugacy class of $\Gamma$ has infinitely many elements, so that the group von Neumann algebra $R\Gamma$ is a II1 factor. Then the representation $\pi$ of $G$ restricted to $\Gamma$ extends to a representation of the II1 factor $R\Gamma$, with von Neumann dimension

[TABLE]

where $\emph{d}_{\pi}$ is the formal dimension of $\pi$.

For background on Theorem 2.2 and each piece of formula (1), including exposition on discrete series representations and their formal dimensions, on lattices in algebraic groups and their covolumes, and on II1 factors, see [Rut19], where we computed von Neumann dimensions in the setting of $PGL(2,F)$ for $F$ a non-archimedean local field, extending Theorem 61 in [Rut18].

Remark 2.3**.**

The proof of Theorem 2.2 applies to a wider class of groups than the theorem states: The only necessary ingredients are a locally compact unimodular group $G$, a lattice $\Gamma<G$ whose non-trivial conjugacy classes have infinitely many elements, and an irreducible unitary representation of $G$ that is a subrepresentation of the right regular representation of $G$ on $L^{2}(G)$.

For example, $GL(n,F)$, $n\geq 2$, $F$ a non-archimedean local field of characteristic [math] and residue field characteristic relatively prime to $n$, has irreducible unitary representations with matrix coefficients that are compactly-supported modulo the center of $GL(n,F)$, hence square-integrable modulo the center of $GL(n,F)$ (see the introduction of [CMS90] for more about these representations); and if we consider only those with trivial central character, we obtain irreducible unitary representations of $PGL(n,F)$ with square-integrable matrix coefficients, which are (by Theorem 16.2 in [Rob83]) equivalent to subrepresentations of the right regular representation of $PGL(n,F)$ on $L^{2}(PGL(n,F))$.

For another example, the group $PSL(2,{\mathbb{R}})\times\cdots\times PSL(2,{\mathbb{R}})$ has representations equivalent to subprepresentations of the right regular representation of

$PSL(2,{\mathbb{R}})\times\cdots\times PSL(2,{\mathbb{R}})$ on $L^{2}(PSL(2,{\mathbb{R}})\times\cdots\times PSL(2,{\mathbb{R}}))$: tensor products of discrete series representations of $PSL(2,{\mathbb{R}})$ (by which we mean discrete series representation of $SL(2,{\mathbb{R}})$ with trivial central character).

The left side of the diagram below illustrates the situation in Theorem 2.2, while the right side of the diagram illustrates the situation in Theorem 2.1, and we use the notation ${\mathcal{H}}$ for a discrete series representation, $\alpha$ for a basis vector in $L^{2}(G/\Gamma)$, $\beta$ for a basis vector in $l^{2}(\Lambda)$, $\Phi$ for the intwertwiner between $G$-representations, and $\bar{\Phi}$ for the extension to $R\Gamma$ of the restriction of $\Phi$ to $\Gamma$.

[TABLE]

To obtain the representation of $R\Gamma$ on a discrete series representation of $G$ in $L^{2}(\Lambda\backslash G)$, instead of in $L^{2}(G)$, we had to take care of the right-hand side of the diagram above — that is, we needed to show that some discrete series representation of $G$ occurs in $L^{2}(\Lambda\backslash G)$. Since we were working in the setting $PSL(2,{\mathbb{R}})$, we used the fact that the multiplicity of a discrete series representation of weight $k\geq 2$ is equal to the dimension of the space of cusp forms of weight $k$ (see [Gel75] Theorem 2.10; the proof given there works for all lattices in $SL(2,{\mathbb{R}})$, not just $SL(2,{\mathbb{Z}})$). Then formulas for dimensions of spaces of cusp forms (see [Miy06] Theorem 2.5.2) show that for $k$ high enough, the dimension of the space of cusp forms of weight $k$ for $\Lambda$ is non-zero. (We only considered $k=2,4,6,\ldots$ because the even $k$ correspond to the discrete series representations of $SL(2,{\mathbb{R}})$ that factor through $PSL(2,{\mathbb{R}})$.)

In the proof of Observation 1.1, the picture is the same, and a non-zero intertwiner $\Phi$ is guaranteed by the pointwise limit multiplicity property, which we now describe.

For background on the general limit multiplicity property, see [M1̈6], Section 4. In this note, we are concerned only with the “pointwise” limit multiplicity property, since it is all we need to guarantee occurrence of discrete series representations. If a sequence of lattices has the limit multiplicity property, then it has the pointwise limit multiplicity property.

Let $\Pi(G)$ denote the unitary dual of $G$ (set of irreducible unitary representations of $G$), and let $\Pi(G)_{\text{d}}$ denote the discrete part of the unitary dual (the representations equivalent to subrepresentations of the right regular representation on $L^{2}(G)$, i.e. discrete series representations).

Each $\pi\in\Pi(G)_{\text{d}}$ has a formal dimension $\text{d}_{\pi}$ defined by

[TABLE]

Definition 2.4**.**

We say that a sequence of lattices $(\Lambda_{n})$ in a locally compact group $G$ has the pointwise limit multiplicity property if

[TABLE]

where $\text{mult}(\pi,\Lambda_{n})$ denotes the multiplicity of $\pi$ in $L^{2}(\Lambda_{n}\backslash G)$.

(Both $\text{d}_{\pi}$ and $\text{vol}(\Lambda_{n}\backslash G)$ depend on the choice of Haar measure on $G$.)

So, if $\pi$ is a discrete series representation of $G$, and if $(\Lambda_{n})$ is a sequence of lattices in $G$ having the pointwise limit multiplicity property, then $\pi$ will “eventually” occur in $L^{2}(\Lambda_{n}\backslash G)$.

3. Proof of Observation 1.1

Proof.

The generalization of Theorem 2.2 discussed in Remark 2.3 guarantees that a discrete series representation of $G$ in $L^{2}(G)$ restricted to $\Gamma$ extends to a representation of $R\Gamma$. (Now $R\Gamma$ is represented on a subspace $L^{2}(G)$.)

The sequence $(\Lambda_{n})$ in Case 1 has the pointwise limit multiplicity property, by Corollary 1.5 and Remark 1.6 of [GL18].

The sequence $(\Lambda_{n})$ in Case 2 has the pointwise limit multiplicity property, by Theorem 1 of [Mat19].

The pointwise limit multiplicity property guarantees that the discrete series representation of $G$ occurs in $L^{2}(\Lambda_{i}\backslash G)$ for some $\Lambda_{i}$ in $(\Lambda_{n})$. Let $\Phi$ denote the intertwiner between the representation of $G$ on a subspace of $L^{2}(G)$ and on a subspace of $L^{2}(\Lambda_{i}\backslash G)$. Then $\Phi$ restricts to an intertwiner between representations of $\Gamma$, and this extends to an intertwiner $\bar{\Phi}$ between representations of $R\Gamma$; see Lemmma 30 of [Rut18] for the complete argument. (Now $R\Gamma$ is represented on a subspace $L^{2}(\Lambda_{i}\backslash G)$.) ∎

Remark 3.1**.**

In [Mat19], the limit multiplicity property was proven to hold using Sauvageot’s density principle and the trace formula. In [GL18], the limit multiplicity property was proven to hold using Sauvageot’s density principle and convergence of invariant random subgroups, similarly to a theorem in [ABB*+*17] that connects Benjamini–Schramm convergence (a geometric formulation of convergence of invariant random subgroups) to convergence of relative Plancherel measures. We note that [Lev17] proved Benjamini–Schramm convergence for sequences of lattices including the sequences considered in [Mat19], but the theorems on convergence of relative Plancherel measures in [ABB*+*17] and [GL18] cannot be applied to give the limit multiplicity property, as the sequences in [Mat19] consist of non-uniform lattices, while the sequences in [ABB*+*17] and [GL18] consist of uniform lattices.

Question 1**.**

All the representations of II1 factors $R\Gamma$ in this note start from restricting discrete series representations of $G$ to $\Gamma$. Is it possible to represent $R\Gamma$ on a principal series representation or a complementary series representation of $G$, rather than on a discrete series representation of $G$? Such a representation would not come from restricting a representation of $G$ to $\Gamma$.

Acknowledgments

We thank Alain Valette for many enlightening correspondences.