# On the spectral theory of groups of automorphisms of $S$-adic nilmanifolds

Bachir Bekka  Bachir Bekka
IRMAR
UMR-CNRS 6625 Université de Rennes 1
Campus Beaulieu
F-35042 Rennes Cedex
France<EMAIL_ADDRESS>and  Yves Guivarc’h  Yves Guivarc’h
IRMAR
UMR-CNRS 6625
Université de Rennes 1
Campus Beaulieu
F-35042 Rennes Cedex
France<EMAIL_ADDRESS>
###### Abstract.

Let $S=\\{p_{1},\dots,p_{r},\infty\\}$ for prime integers $p_{1},\dots,p_{r}.$
Let $X$ be an $S$-adic compact nilmanifold, equipped with the unique
translation invariant probability measure $\mu.$ We characterize the countable
groups $\Gamma$ of automorphisms of $X$ for which the Koopman representation
$\kappa$ on $L^{2}(X,\mu)$ has a spectral gap. More specifically, we show that
$\kappa$ does not have a spectral gap if and only if there exists a non-
trivial $\Gamma$-invariant quotient solenoid (that is, a finite-dimensional,
connected, compact abelian group) on which $\Gamma$ acts as a virtually
abelian group.

###### 1991 Mathematics Subject Classification:

37A05, 22F30, 60B15, 60G50

##  1\. Introduction

Let $\Gamma$ be a countable group acting measurably on a probability space
$(X,\mu)$ by measure preserving transformations. Let $\kappa=\kappa_{X}$
denote the corresponding Koopman representation of $\Gamma$, that is, the
unitary representation of $\Gamma$ on $L^{2}(X,\mu)$ given by

$\kappa(\gamma)\xi(x)=\xi(\gamma^{-1}x)\qquad\text{for all}\quad\xi\in
L^{2}(X,\mu),x\in X,\gamma\in\Gamma.$

We say that the action $\Gamma\curvearrowright(X,\mu)$ of $\Gamma$ on
$(X,\mu)$ has a spectral gap if the restriction $\kappa_{0}$ of $\kappa$ to
the $\Gamma$-invariant subspace

$L^{2}_{0}(X,\mu)=\\{\xi\in L^{2}(X,\mu)\ :\ \int_{X}\xi(x)d\mu(x)=0\\}$

does not weakly contain the trivial representation $1_{\Gamma};$ equivalently,
if $\kappa_{0}$ does not have almost invariant vectors, that is, there is no
sequence $(\xi_{n})_{n}$ of unit vectors in $L^{2}_{0}(X,\mu)$ such that

$\lim_{n}\|\kappa_{0}(\gamma)\xi_{n}-\xi_{n}\|=0\qquad\text{for
all}\quad\gamma\in\Gamma.$

The existence of a spectral gap admits the following useful quantitative
version. Let $\nu$ be a probability measure on $\Gamma$ and $\kappa_{0}(\nu)$
the convolution operator defined on $L^{2}_{0}(X,\mu)$ by

$\kappa_{0}(\nu)\xi=\sum_{\gamma\in\Gamma}\nu(\gamma)\kappa_{0}(\gamma)\xi\qquad\text{for
all}\quad\xi\in L^{2}_{0}(X,\mu).$

Observe that we have $\|\kappa_{0}(\nu)\|\leq 1$ and hence
$r(\kappa_{0}(\nu))\leq 1$ for the spectral radius $r(\kappa_{0}(\nu))$ of
$\kappa_{0}(\mu)$. Assume that $\nu$ is aperiodic, that is, the support of
$\nu$ is not contained in the coset of a proper subgroup of $\Gamma$. Then the
action of $\Gamma$ on $X$ has a spectral gap if and only if
$r(\kappa_{0}(\nu))<1$ and this is equivalent to $\|\kappa_{0}(\nu)\|<1$; for
more details, see the survey [Bekk16].

In this paper, we will be concerned with the case where $X$ is an $S$-adic
nilmanifold, to be introduced below, and $\Gamma$ is a subgroup of
automorphisms of $X.$

Fix a finite set $\\{p_{1},\dots,p_{r}\\}$ of integer primes and set
$S=\\{p_{1},\dots,p_{r},\infty\\}.$ For an integer $d\geq 1,$ the product

$\mathbf{Q}_{S}:=\prod_{p\in
S}\mathbf{Q}_{p}=\mathbf{Q}_{\infty}\times\mathbf{Q}_{p_{1}}\times\cdots\times\mathbf{Q}_{p_{r}}$

is a locally compact ring, where $\mathbf{Q}_{\infty}={\mathbf{R}}$ and
$\mathbf{Q}_{p}$ is the field of $p$-adic numbers for a prime $p$. Let
${\mathbf{Z}}[1/S]={\mathbf{Z}}[1/p_{1},\cdots,1/p_{r}]$ denote the subring of
$\mathbf{Q}$ generated by $1$ and $\\{1/p_{1},\dots,1/p_{r}\\}.$ Through the
diagonal embedding

${\mathbf{Z}}[1/S]\to\mathbf{Q}_{S},\qquad b\mapsto(b,\cdots,b),$

we may identify ${\mathbf{Z}}[1/S]$ with a discrete and cocompact subring of
$\mathbf{Q}_{S}.$

If $\mathbf{G}$ is a linear algebraic group defined over $\mathbf{Q},$ we
denote by $\mathbf{G}(R)$ the group of elements of $\mathbf{G}$ with
coefficients in $R$ and determinant invertible in $R,$ for every subring $R$
of an overfield of $\mathbf{Q}.$

Let $\mathbf{U}$ be a linear algebraic unipotent group defined over
$\mathbf{Q},$ that is, $\mathbf{U}$ is an algebraic subgroup of the group of
$n\times n$ upper triangular unipotent matrices for some $n\geq 1,$ The group
$\mathbf{U}(\mathbf{Q}_{S})$ is a locally compact group and
$\Lambda:=\mathbf{U}({\mathbf{Z}}[1/S])$ is a cocompact lattice in
$\mathbf{U}(\mathbf{Q}_{S})$. The corresponding $S$-adic compact nilmanifold

$\mathbf{Nil}_{S}=\mathbf{U}(\mathbf{Q}_{S})/\mathbf{U}({\mathbf{Z}}[1/S])$

will be equipped with the unique translation-invariant probability measure
$\mu$ on the Borel subsets of $\mathbf{Nil}_{S}$.

For $p\in S,$ let ${\rm Aut}(\mathbf{U}(\mathbf{Q}_{p}))$ be the group of
continuous automorphisms of $\mathbf{U}(\mathbf{Q}_{s}).$ Set

${\rm Aut}(\mathbf{U}(\mathbf{Q}_{S})):=\prod_{p\in S}{\rm
Aut}(\mathbf{U}(\mathbf{Q}_{p}))$

and denote by ${\rm Aut}(\mathbf{Nil}_{S})$ the subgroup

$\\{g\in{\rm Aut}(\mathbf{U}(\mathbf{Q}_{S}))\mid g(\Lambda)=\Lambda\\}.$

Every $g\in{\rm Aut}(\mathbf{Nil}_{S})$ acts on $\mathbf{Nil}_{S}$ preserving
the probability measure $\mu.$

The abelian quotient group

$\overline{\mathbf{U}(\mathbf{Q}_{S})}:=\mathbf{U}(\mathbf{Q}_{S})/[\mathbf{U}(\mathbf{Q}_{S}),\mathbf{U}(\mathbf{Q}_{S})]$

can be identified with $\mathbf{Q}_{S}^{d}$ for some $d\geq 1$ and the image
$\Delta$ of $\mathbf{U}({\mathbf{Z}}[1/S])$ in
$\overline{\mathbf{U}(\mathbf{Q}_{S})}$ is a cocompact and discrete subgroup
of $\overline{\mathbf{U}(\mathbf{Q}_{S})};$ so,

$\mathbf{Sol}_{S}:=\overline{\mathbf{U}(\mathbf{Q}_{S})}/\Delta$

is a solenoid (that is, is a finite-dimensional, connected, compact abelian
group; see [HeRo63, §25]). We refer to $\mathbf{Sol}_{S}$ as the $S$-adic
solenoid attached to the $S$\- adic nilmanifold $\mathbf{Nil}_{S}.$ We equip
$\mathbf{Sol}_{S}$ with the probability measure $\nu$ which is the image of
$\mu$ under the canonical projection $\mathbf{Nil}_{S}\to\mathbf{Sol}_{S}.$

Observe that ${\rm Aut}(\mathbf{Q}_{S}^{d})$ is canonically isomorphic to
$\prod_{s\in S}GL_{d}(\mathbf{Q}_{s})$ and that ${\rm Aut}(\mathbf{Sol}_{S})$
can be identified with the subgroup $GL_{d}({\mathbf{Z}}[1/S]).$ The group
${\rm Aut}(\mathbf{Nil}_{S})$ acts naturally by automorphisms of
$\mathbf{Sol}_{S}$; we denote by

$p_{S}:{\rm Aut}(\mathbf{Nil}_{S})\to GL_{d}({\mathbf{Z}}[1/S])\subset
GL_{d}(\mathbf{Q})$

the corresponding homomorphism.

######  Theorem 1.

Let $\mathbf{U}$ be an algebraic unipotent group defined over $\mathbf{Q}$ and
$S=\\{p_{1},\dots,p_{r},\infty\\},$ where $p_{1},\dots,p_{r}$ are integer
primes. Let
$\mathbf{Nil}_{S}=\mathbf{U}(\mathbf{Q}_{S})/\mathbf{U}({\mathbf{Z}}[1/S]$ be
the associated $S$-adic nilmanifold and let $\mathbf{Sol}_{S}$ be the
corresponding $S$-adic solenoid, equipped with respectively the probability
measures $\mu$ and $\nu$ as above. Let $\Gamma$ be a countable subgroup ${\rm
Aut}(\mathbf{Nil}_{S})$. The following properties are equivalent:

  * (i)

The action $\Gamma\curvearrowright(\mathbf{Nil}_{S},\mu)$ has a spectral gap.

  * (ii)

The action $p_{S}(\Gamma)\curvearrowright(\mathbf{Sol}_{S},\nu)$ has a
spectral gap, where $p_{S}:{\rm Aut}(\mathbf{Nil}_{S})\to
GL_{d}({\mathbf{Z}}[1/S])$ is the canonical homomorphism.

Actions by groups of automorphisms (or more generally groups of by affine
transformations) on the $S$-adic solenoid $\mathbf{Sol}_{S}$ have been
completely characterized in [BeFr20, Theorem 5]. The following result is an
immediate consequence of this characterization and of Theorem 1. For a subset
$T$ of $GL_{d}(\mathbf{K})$ for a field $\mathbf{K},$ we denote by
$T^{t}=\\{g^{t}\mid g\in T\\}$ the set of transposed matrices from $T$.

######  Corollary 2.

With the notation as in Theorem 1, the following properties are equivalent:

  * (i)

The action of $\Gamma$ on the $S$-adic nilmanifold $\mathbf{Nil}_{S}$ does not
have a spectral gap.

  * (ii)

There exists a non-zero linear subspace $W$ of $\mathbf{Q}^{d}$ which is
invariant under $p_{S}(\Gamma)^{t}$ and such that the image of
$p_{S}(\Gamma)^{t}$ in $GL(W)$ is a virtually abelian group.

Here is an immediate consequence of Corollary 2.

######  Corollary 3.

With the notation as in Corollary 1, assume that the linear representation of
$p_{S}(\Gamma)^{t}$ in $\mathbf{Q}^{d}$ is irreducible. Then the action
$\Gamma\curvearrowright(\mathbf{Nil}_{S},\mu)$ has a spectral gap.

Recall that the action of a countable group $\Gamma$ by measure preserving
transformations on a probability space $(X,\mu)$ is strongly ergodic (see
[Schm81]) if every sequence $(B_{n})_{n}$ of measurable subsets of $X$ which
is asymptotically invariant (that is, which is such that $\lim_{n}\mu(\gamma
B_{n}\bigtriangleup B_{n})=0$ for all $\gamma\in\Gamma$) is trivial (that is,
$\lim_{n}\mu(B_{n})(1-\mu(B_{n}))=0$). It is straightforward to check that the
spectral gap property implies strong ergodicity and it is known that the
converse does not hold in general.

The following corollary is a direct consequence of Theorem 1 (compare with
Corollary 2 in [BeGu15]).

######  Corollary 4.

With the notation as in 1, the following properties are equivalent:

  * (i)

The action $\Gamma\curvearrowright(\mathbf{Nil}_{S},\mu)$ has the spectral gap
property.

  * (ii)

The action $\Gamma\curvearrowright(\mathbf{Nil}_{S},\mu)$ is strongly ergodic.

Theorem 1 generalizes our previous work [BeGu15], where we treated the real
case (that is, the case $S=\infty$). We had to extend our methods to the
$S$-adic setting. There are three main tools we use in the proof:

  * •

a canonical decomposition of the Koopman representation of $\Gamma$ in
$L^{2}(\mathbf{Nil}_{S})$ as a direct sum of certain representations of
$\Gamma$ induced from stabilizers of representations of
$\mathbf{U}(\mathbf{Q}_{S})$ (see Proposition 9);

  * •

a result of Howe and Moore [HoMo79] about the decay of matrix coefficients of
algebraic groups (see Proposition 11);

  * •

the fact that the irreducible representations of $\mathbf{U}(\mathbf{Q}_{S})$
appearing in the decomposition of $L^{2}(\mathbf{Nil}_{S})$ are rational, in
the sense that the Kirillov data associated to each one of them are defined
over $\mathbf{Q}$ (see Proposition 13).

Another tool we constantly use is a generalized version of Herz’s majoration
principle (see Lemma 7).

Given a probability measure $\nu$ on $\Gamma,$ our approach does not seem to
provide quantitative estimates for the operator norm of the convolution
operator $\kappa_{0}(\nu)$ acting on $L^{2}_{0}(\mathbf{Nil}_{S},\mu)$ for a
general unipotent group $\mathbf{U}.$ However, using known bounds for the so-
called metaplectic representation of $Sp_{2n}(\mathbf{Q}_{p})$, we give such
estimates in the case of $S$-adic Heisenberg nilmanifolds (see Section 11).

######  Corollary 5.

For an integer $n\geq 1,$ let $\mathbf{U}=\mathbf{H}_{2n+1}$ be the $(2n+1)$\-
dimensional Heisenberg group and
$\mathbf{Nil}_{S}=\mathbf{H}_{2n+1}(\mathbf{Q}_{S})/\mathbf{H}_{2n+1}({\mathbf{Z}}[1/S]).$
Let $\nu$ be a probability measure on

$Sp_{2n}({\mathbf{Z}}[1/S])\subset{\rm Aut}(\mathbf{Nil}_{S}).$

Then

$\|\kappa_{0}(\nu)\|\leq\max\\{\|\lambda_{\Gamma}(\nu)\|^{1/2n+2},\|\kappa_{1}(\nu)\|\\},$

where $\kappa_{1}$ is the restriction of $\kappa_{0}$ to
$L^{2}_{0}(\mathbf{Sol}_{S})$ and $\lambda_{\Gamma}$ is the regular
representation of the group $\Gamma$ generated by the support of $\nu.$ In
particular, in the case $n=1,$ the action of $\Gamma$ on $\mathbf{Nil}_{S}$
has a spectral gap if and only if $\Gamma$ is non-amenable.

##  2\. Extension of representations

Let $G$ be a locally compact group which we assume to be second countable. We
will need the notion of a projective representation. Recall that a mapping
$\pi:G\to U(\mathcal{H})$ from $G$ to the unitary group of the Hilbert space
$\mathcal{H}$ is a projective representation of $G$ if the following holds:

  * •

$\pi(e)=I$,

  * •

for all $g_{1},g_{2}\in G,$ there exists $c(g_{1},g_{2})\in\mathbf{C}$ such
that

$\pi(g_{1}g_{2})=c(g_{1},g_{2})\pi(g_{1})\pi(g_{2}),$

  * •

the function $g\mapsto\langle\pi(g)\xi,\eta\rangle$ is measurable for all
$\xi,\eta\in\mathcal{H}.$

The mapping $c:G\times G\to{\mathbf{S}}^{1}$ is a $2$-cocycle with values in
the unit cercle ${\mathbf{S}}^{1}.$ Every projective unitary representation of
$G$ can be lifted to an ordinary unitary representation of a central extension
of $G$ (for all this, see [Mack76] or [Mack58]).

Let $N$ be a closed normal subgroup of $G$. Let $\pi$ be an irreducible
unitary representation of $N$ on a Hilbert space $\mathcal{H}.$ Consider the
stabilizer

$G_{\pi}=\\{g\in G\mid\pi^{g}\text{ is equivalent to }\pi\\}$

of $\pi$ in $G$ for the natural action of $G$ on the unitary dual
$\widehat{N}$ given by $\pi^{g}(n)=\pi(g^{-1}ng).$ Then $G_{\pi}$ is a closed
subgroup of $G$ containing $N.$ The following lemma is a well-known part of
Mackey’s theory of unitary representations of group extensions.

######  Lemma 6.

Let $\pi$ be an irreducible unitary representation of $N$ on the Hilbert space
$\mathcal{H}.$ There exists a projective unitary representation
$\widetilde{\pi}$ of ${G}_{\pi}$ on $\mathcal{H}$ which extends $\pi$.
Moreover, $\widetilde{\pi}$ is unique, up to scalars: any other projective
unitary representation $\widetilde{\pi}^{\prime}$ of ${G}_{\pi}$ extending
$\pi$ is the form $\widetilde{\pi}^{\prime}=\lambda\widetilde{\pi}$ for a
measurable function $\lambda:G_{\pi}\to{\mathbf{S}}^{1}.$

Proof For every $g\in{G}_{\pi}$, there exists a unitary operator
$\widetilde{\pi}(g)$ on $\mathcal{H}$ such that

$\pi(g(n))=\widetilde{\pi}(g)\pi(n)\widetilde{\pi}(g)^{-1}\qquad\text{for
all}\quad n\in N.$

One can choose $\widetilde{\pi}(g)$ such that $g\mapsto\widetilde{\pi}(g)$ is
a projective unitary representation of ${G}_{\pi}$ which extends $\pi$ (see
Theorem 8.2 in [Mack58]). The uniqueness of $\pi$ follows from the
irreducibility of $\pi$ and Schur’s lemma. $\blacksquare\vskip 6.0pt plus
2.0pt minus 2.0pt$

##  3\. A weak containment result for induced representations

Let $G$ be a locally compact group with Haar measure $\mu_{G}.$ Recall that a
unitary representation $(\rho,\mathcal{K})$ of $G$ is weakly contained in
another unitary representation $(\pi,\mathcal{H})$ of $G,$ if every matrix
coefficient $g\mapsto\langle\rho(g)\eta\mid\eta\rangle$ of $\pi$ (for
$\eta\in\mathcal{K}$) is the limit, uniformly of compact subsets of $G,$ of a
finite sum of matrix coefficients of $\pi.$ Equivalently, if
$\|\rho(f)\|\leq\|\pi(f)\|$ for every $f\in C_{c}(G),$ where $C_{c}(G)$ is the
space of continuous functions with compact support on $G$ and where the
operator $\pi(f)\in\mathcal{B}(\mathcal{H})$ is defined by the integral

$\pi(f)\xi=\int_{G}f(g)\pi(g)d\mu_{G}(g)\qquad\text{for
all}\quad\xi\in\mathcal{H}.$

The trivial representation $1_{G}$ is weakly contained in $\pi$ if and only if
there exists, for every compact subset $Q$ of $G$ and every $\varepsilon>0,$
there exists a unit vector $\xi\in\mathcal{H}$ which is
$(Q,\varepsilon)$-invariant, that is such that

$\sup_{g\in Q}\|\pi(g)\xi-\xi\|\leq\varepsilon.$

Let $H$ be a closed subgroup of $G.$ We will always assume that the coset
space $H\backslash G$ admits a non-zero $G$-invariant (possibly infinite)
measure on its Borel subsets. Let $(\sigma,\mathcal{K})$ be a unitary
representation of $H.$ We will use the following model for the induced
representation $\pi:={\rm Ind}_{H}^{G}\sigma$. Choose a Borel fundamental
domain $X\subset G$ for the action of $G$ on $H\backslash G$. For $x\in X$ and
$g\in G,$ let $x\cdot g\in X$ and $c(x,g)\in H$ be defined by

$xg=c(x,g)(x\cdot g).$

There exists a non-zero $G$-invariant measure on $X$ for the action
$(x,g)\mapsto x\cdot g$ of $G$ on $X.$ The Hilbert space of $\pi$ is the space
$L^{2}(X,\mathcal{K},\mu)$ of all square-integrable measurable mappings
$\xi:X\to\mathcal{K}$ and the action of $G$ on $L^{2}(X,\mathcal{K},\mu)$ is
given by

$(\pi(g)\xi)(x)=\sigma(c(x,g))(\xi(x\cdot g)),\qquad g\in G,\ \xi\in
L^{2}(X,\mathcal{K},\mu),\ x\in X.$

Observe that, in the case where $\sigma$ is the trivial representation
$1_{H},$ the induced representation ${\rm Ind}_{H}^{G}1_{H}$ is equivalent to
quasi-regular representation $\lambda_{H\backslash G},$ that is the natural
representation of $G$ on $L^{2}(H\backslash G,\mu)$ given by right
translations.

We will use several times the following elementary but crucial lemma, which
can be viewed as a generalization of Herz’s majoration principle (see
Proposition 17 in [BeGu15]).

######  Lemma 7.

Let $(H_{i})_{i\in I}$ be a family of closed subgroups of $G$ such that
$H_{i}\backslash G$ admits a non-zero $G$-invariant measure. Let
$(\sigma_{i},\mathcal{K}_{i})$ be a unitary representation of $H_{i}.$ Assume
that $1_{G}$ is weakly contained in the direct sum $\oplus_{i\in I}{\rm
Ind}_{H_{i}}^{G}\sigma_{i}.$ Then $1_{G}$ is weakly contained in $\oplus_{i\in
I}\lambda_{H_{i}\backslash G}.$

Proof Let $Q$ be a compact subset of $G$ and $\varepsilon.$ For every $i\in
I,$ let $X_{i}\subset G$ be a Borel fundamental domain for the action of $G$
on $H_{i}\backslash G$ and $\mu_{i}$ a non-zero $G$-invariant measure on
$X_{i}.$ There exists a family of vectors $\xi_{i}\in
L^{2}(X_{i},\mathcal{K}_{i},\mu_{i})$ such that $\sum_{i}\|\xi_{i}\|^{2}=1$
and

$\sup_{g\in Q}\sum_{i}\|{\rm
Ind}_{H_{i}}^{G}\sigma_{i}(g)\xi_{i}-\xi_{i}\|^{2}\leq\varepsilon,$

Define $\varphi_{i}$ in $L^{2}(X_{i},\mu_{i})$ by
$\varphi_{i}(x)=\|\xi_{i}(x)\|$. Then $\sum_{i}\|\varphi_{i}\|^{2}=1$ and, for
every $g\in Q,$ we have

$\displaystyle\|\left({\rm
Ind}_{H_{i}}^{G}\sigma_{i}(g)\right)\xi_{i}-\xi_{i}\|^{2}$
$\displaystyle=\int_{X_{i}}\|\sigma(c_{i}(x,g))(\xi_{i}(x\cdot_{i}g))-\xi_{i}(x)\|^{2}d\mu_{i}(x)$
$\displaystyle\geq\int_{X_{i}}\left|\|\sigma_{i}(c_{i}(x,g))(\xi_{i}(x\cdot_{i}g))\|-\|\xi_{i}(x)\|\right|^{2}d\mu_{i}(x)$
$\displaystyle=\int_{X_{i}}\left|\|\xi_{i}(x\cdot_{i}g)\|-\|\xi_{i}(x)\|\right|^{2}d\mu_{i}(x)$
$\displaystyle=\int_{X_{i}}\left|\varphi_{i}(x\cdot_{i}g)-\varphi(x)\right|^{2}d\mu_{i}(x)$
$\displaystyle=\|\lambda_{{H_{i}}\backslash
G}(g)\varphi_{i}-\varphi_{i}\|^{2}$

and the claim follows. $\blacksquare\vskip 6.0pt plus 2.0pt minus 2.0pt$

##  4\. Decay of matrix coefficients of unitary representations

We recall a few general facts about decay of matrix coefficients of unitary
representations, Recall that the projective kernel of a (genuine or
projective) representation $\pi$ of the locally compact group $G$ is the
closed normal subgroup $P_{\pi}$ of $G$ consisting of the elements $g\in G$
such that $\pi(g)$ is a scalar multiple of the identity operator, that is such
that $\pi(g)=\lambda_{\pi}(g)I$ for some
$\lambda_{\pi}(g)\in{\mathbf{S}}^{1}.$

Observe also that, for $\xi,\eta\in\mathcal{H},$ the absolute value of the
matrix coefficient

$C^{\pi}_{\xi,\eta}:g\mapsto\langle\pi(g)\xi,\eta\rangle$

is constant on cosets modulo $P_{\pi}.$ For a real number $p$ with $1\leq
p<+\infty,$ the representation $\pi$ is said to be strongly $L^{p}$ modulo
$P_{\pi}$, if there is dense subspace $D\subset\mathcal{H}$ such that
$|C^{\pi}_{\xi,\eta}|\in L^{p}(G/P_{\pi})$ for all $\xi,\eta\in D.$

######  Proposition 8.

Assume that the unitary representation $\pi$ of the locally compact group $G$
is strongly $L^{p}$ modulo $P_{\pi}$ for $1\leq p<+\infty.$ Let $k$ be an
integer $k\geq p/2.$ Then the tensor power $\pi^{\otimes k}$ is contained in
an infinite multiple of ${\rm Ind}_{P_{\pi}}^{G}\lambda_{\pi}^{k}$, where
$\lambda_{\pi}$ is the unitary character of $P_{\pi}$ associated to $\pi.$

Proof Observe $\sigma:=\pi^{\otimes k}$ is square-integrable modulo $P_{\pi}$
for every integer $k\geq p/2.$ It follows (see Proposition 4.2 in [HoMo79] or
Proposition 1.2.3 in Chapter V of [HoTa92]) that $\sigma$ is contained in an
infinite multiple of ${\rm Ind}_{P_{\sigma}}^{G}\lambda_{\sigma}={\rm
Ind}_{P_{\pi}}^{G}\lambda_{\pi}^{k}$. $\blacksquare\vskip 6.0pt plus 2.0pt
minus 2.0pt$

##  5\. The Koopman representation of the automorphism group of a homogeneous
space

We establish a decomposition result for the Koopman representation of a group
of automorphisms of an $S$-adic compact nilmanifold. We will state the result
in the general context of a compact homogeneous space.

Let $G$ be a locally compact group and $\Lambda$ a lattice in $G.$ We assume
that $\Lambda$ is cocompact in $G.$ The homogeneous space $X:=G/\Lambda$
carries a probability measure $\mu$ on the Borel subsets $X$ which is
invariant by translations with elements from $G.$ Every element from

${\rm Aut}(X):=\\{\gamma\in{\rm Aut}(G)\mid\gamma(\Lambda)\subset\Lambda\\}$

induces a Borel isomorphism of $X$, which leaves $\nu$ invariant, as follows
from the uniqueness of $\nu$.

Given a subgroup $\Gamma$ of ${\rm Aut}(X),$ the following crucial proposition
gives a decomposition of the associated Koopman $\Gamma$ on $L^{2}(X,\mu)$ as
direct sum of certain induced representations of $\Gamma$.

######  Proposition 9.

Let $G$ be a locally compact group and $\Lambda$ a cocompact lattice in $G,$
and let $\Gamma$ be a countable subgroup of ${\rm Aut}(X)$ for $X:=G/\Lambda$
Let $\kappa$ be the Koopman representation of $\Gamma$ associated to the
action $\Gamma\curvearrowright X.$ There exists a family $(\pi_{i})_{i\in I}$
of irreducible unitary representations of $G$ such that $\kappa$ is equivalent
to a direct sum

$\bigoplus_{i\in I}{\rm
Ind}_{\Gamma_{i}}^{\Gamma}(\widetilde{\pi_{i}}|_{\Gamma_{i}}\otimes W_{i}),$

where $\widetilde{\pi}_{i}$ is an irreducible projective representation of the
stabilizer $G_{i}$ of $\pi_{i}$ in ${\rm Aut}(G)\ltimes G$ extending
$\pi_{i}$, and where $W_{i}$ is a finite dimensional projective unitary
representation of $\Gamma_{i}:=\Gamma\cap G_{i}$.

Proof We extend $\kappa$ to a unitary representation, again denoted by
$\kappa,$ of $\Gamma\ltimes G$ on $L^{2}(X,\mu)$ given by

$\kappa(\gamma,g)\xi(x)=\xi(\gamma^{-1}(gx))\qquad\text{for
all}\quad\gamma\in\Gamma,g\in G,\xi\in L^{2}(X,\mu),\ x\in X.$

Identifying $\Gamma$ and $G$ with subgroups of $\Gamma\ltimes G$, we have

$None$
$\kappa({\gamma^{-1}})\kappa(g)\kappa({\gamma})=\kappa({\gamma^{-1}(g)})\qquad\text{for
all}\quad\gamma\in\Gamma,\ n\in N.$

Since $\Lambda$ is cocompact in $G,$ we can consider the decomposition of
$L^{2}(X,\mu)$ into $G$–isotypical components: we have (see Theorem in Chap.
I, §3 of [GGPS69])

$L^{2}(X,\mu)=\bigoplus_{\pi\in\Sigma}\mathcal{H}_{\pi},$

where $\Sigma$ is a certain set of pairwise non-equivalent irreducible unitary
representations of $G;$ for every $\pi\in\Sigma$, the space
$\mathcal{H}_{\pi}$ is the union of the closed $\kappa(G)$-invariant subspaces
$\mathcal{K}$ of $\mathcal{H}$ for which the corresponding representation of
$G$ in ${\mathcal{K}}$ is equivalent to $\pi$; moreover, the multiplicity of
every $\pi$ is finite, that is, every $\mathcal{H}_{\pi}$ is a direct sum of
finitely many irreducible unitary representations of $G$.

Let $\gamma$ be a fixed automorphism in $\Gamma.$ Let $\kappa^{\gamma}$ be the
conjugate representation of $\kappa$ by $\gamma,$ that is,
$\kappa^{\gamma}(g)=\kappa(\gamma g\gamma^{-1})$ for all $g\in\Gamma\ltimes
N$. On the one hand, for every $\pi\in\Sigma,$ the isotypical component of
$\kappa^{\gamma}|_{G}$ corresponding to $\pi$ is
$\mathcal{H}_{\pi^{\gamma^{-1}}}$. On the other hand, relation $(*)$ shows
that $\kappa(\gamma)$ is a unitary equivalence between $\kappa|_{G}$ and
$\kappa^{\gamma}|_{G}.$ It follows that

$\kappa({\gamma})(\mathcal{H}_{\pi})=\mathcal{H}_{\pi^{\gamma}}\qquad\text{for
all}\quad\gamma\in\Gamma;$

so, $\Gamma$ permutes the $\mathcal{H}_{\pi}$’s among themselves according to
its action on $\widehat{G}.$

Write $\Sigma=\bigcup_{i\in I}\Sigma_{i},$ where the $\Sigma_{i}$’s are the
$\Gamma$-orbits in $\Sigma,$ and set

$\mathcal{H}_{\Sigma_{i}}=\bigoplus_{\pi\in\Sigma_{i}}\mathcal{H}_{\pi}.$

Every $\mathcal{H}_{\Sigma_{i}}$ is invariant under $\Gamma\ltimes G$ and we
have an orthogonal decomposition

$\mathcal{H}=\bigoplus_{i}\mathcal{H}_{\Sigma_{i}}.$

Fix $i\in I.$ Choose a representation ${\pi}_{i}$ in $\Sigma_{i}$ and set
$\mathcal{H}_{i}=\mathcal{H}_{\pi_{i}}.$ Let $\Gamma_{i}$ denote the
stabilizer of $\pi_{i}$ in $\Gamma.$ The space $\mathcal{H}_{i}$ is invariant
under $\Gamma_{i}.$ Let $V_{i}$ be the corresponding representation of
$\Gamma_{i}$ on $\mathcal{H}_{i}.$

Choose a set $S_{i}$ of representatives for the cosets in

$\Gamma/\Gamma_{i}=(\Gamma\ltimes G)/(\Gamma_{i}\ltimes G)$

with $e\in S_{i}.$ Then $\Sigma_{i}=\\{\pi_{i}^{s}\ :\ s\in S_{i}\\}$ and the
Hilbert space $\mathcal{H}_{\Sigma_{i}}$ is the sum of mutually orthogonal
spaces:

$\mathcal{H}_{\Sigma_{i}}=\bigoplus_{s\in S_{i}}\mathcal{H}_{i}^{s}.$

Moreover, $\mathcal{H}_{i}^{s}$ is the image under $\kappa(s)$ of
$\mathcal{H}_{i}$ for every $s\in S_{i}.$ This means that the restriction
$\kappa_{i}$ of $\kappa$ to $\mathcal{H}_{\Sigma_{i}}$ of the Koopman
representation $\kappa$ of $\Gamma$ is equivalent to the induced
representation ${\rm Ind}_{\Gamma_{i}}^{\Gamma}{V_{i}}.$

Since every $\mathcal{H}_{i}$ is a direct sum of finitely many irreducible
unitary representations of $G,$ we can assume that $\mathcal{H}_{i}$ is the
tensor product

$\mathcal{H}_{i}=\mathcal{K}_{i}\otimes\mathcal{L}_{i}$

of the Hilbert space $\mathcal{K}_{i}$ of $\pi_{i}$ with a finite dimensional
Hilbert space $\mathcal{L}_{i},$ in such a way that

$None$ $V_{i}(g)=\pi_{i}(g)\otimes I_{\mathcal{L}_{i}}\qquad\text{for
all}\quad g\in G.$

Let $\gamma\in\Gamma_{i}.$ By $(*)$ and $(**)$ above, we have

$None$ $V_{i}(\gamma)\left(\pi_{i}(g)\otimes
I_{\mathcal{L}_{i}}\right)V_{i}(\gamma)^{-1}=\pi_{i}(\gamma
g\gamma^{-1})\otimes I_{\mathcal{L}_{i}}\qquad\text{for all}\quad\ g\in G.$

On the other hand, let $G_{i}$ be the stabilizer of $\pi_{i}$ in ${\rm
Aut}(G)\ltimes G$; then $\pi_{i}$ extends to an irreducible projective
representation $\widetilde{\pi}_{i}$ of $G_{i}$ (see Section 2). Since

$\widetilde{\pi_{i}}(\gamma)\pi_{i}(g)\
\widetilde{\pi_{i}}(\gamma^{-1})=\pi_{i}(\gamma g\gamma^{-1})\qquad\text{for
all}\quad g\in G,$

it follows from $(***)$ that $\left(\widetilde{\pi_{i}}(\gamma^{-1})\otimes
I_{\mathcal{L}_{i}}\right)V_{i}(\gamma)$ commutes with $\pi_{i}(g)\otimes
I_{\mathcal{L}_{i}}$ for all $g\in\widetilde{G}_{i}.$ As $\pi_{i}$ is
irreducible, there exists a unitary operator $W_{i}(\gamma)$ on
$\mathcal{L}_{i}$ such that

$V_{i}(\gamma)=\widetilde{\pi_{i}}(\gamma)\otimes W_{i}(\gamma).$

It is clear that $W_{i}$ is a projective unitary representation of
$\Gamma_{i}\ltimes G$, since $V_{i}$ is a unitary representation of
$\Gamma_{i}\ltimes G$.$\blacksquare\vskip 6.0pt plus 2.0pt minus 2.0pt$

##  6\. Unitary dual of solenoids

Let $p$ be either a prime integer or $p=\infty.$ Define an element $e_{p}$ in
the unitary dual group $\widehat{\mathbf{Q}_{p}}$ of the additive group of
$\mathbf{Q}_{p}$ by $e_{p}(x)=e^{2\pi ix}$ if $p=\infty$ and
$e_{p}(x)=\exp(2\pi i\\{x\\}),$ where $\\{x\\}=\sum_{j=m}^{-1}a_{j}p^{j}$
denotes the “fractional part” of a $p$-adic number
$x=\sum_{j=m}^{\infty}a_{j}p^{j}$ for integers $m\in{\mathbf{Z}}$ and
$a_{j}\in\\{0,\dots,p-1\\}$. Observe that ${\rm Ker}(e_{p})={\mathbf{Z}}$ in
case $p=\infty$ and that ${\rm Ker}(e_{p})={\mathbf{Z}}_{p}$ in case $p$ is a
prime integer, where ${\mathbf{Z}}_{p}$ is the ring of $p$-adic integers. The
map

$\mathbf{Q}_{p}\to\widehat{\mathbf{Q}_{p}},\qquad y\mapsto(x\mapsto
e_{p}(xy))$

is an isomorphism of topological groups (see [BeHV08, Section D.4]).

Fix an integer $d\geq 1.$ Then $\widehat{\mathbf{Q}_{p}^{d}}$ will be
identified with $\mathbf{Q}_{p}^{d}$ by means of the map

$\mathbf{Q}^{d}\to\widehat{\mathbf{Q}_{p}^{d}},\qquad y\mapsto x\mapsto
e_{p}(x\cdot y),$

where $x\cdot y=\sum_{i=1}^{d}x_{i}y_{i}$ for
$x=(x_{1},\dots,x_{d}),y=(y_{1},\dots,y_{d})\in\mathbf{Q}_{p}^{d}.$

Let $S=\\{p_{1},\dots,p_{r},\infty\\}$, where $p_{1},\dots,p_{r}$ are integer
primes. For an integer $d\geq 1$, consider the $S$-adic solenoid

$\mathbf{Sol}_{S}=\mathbf{Q}_{S}^{d}/{\mathbf{Z}}[1/p_{1},\cdots,1/p_{r}]^{d},$

where ${\mathbf{Z}}[1/p_{1},\cdots,1/p_{r}]^{d}$ is embedded diagonally in
$\mathbf{Q}_{S}=\prod_{p\in S}\mathbf{Q}_{p}.$

Then $\widehat{\mathbf{Sol}_{S}}$ is identified with the annihilator of
${\mathbf{Z}}[1/p_{1},\cdots,1/p_{r}]^{d}$ in $\mathbf{Q}_{S}^{d},$ that is,
with ${\mathbf{Z}}[1/p_{1},\cdots,1/p_{r}]^{d}$ embedded in
$\mathbf{Q}_{S}^{d}$ via the map

${\mathbf{Z}}[1/p_{1},\cdots,1/p_{r}]^{d}\to\mathbf{Q}_{S}^{d},\qquad
b\mapsto(b,-b\cdots,-b).$

Under this identification, the dual action of the automorphism group

${\rm Aut}(\mathbf{Q}_{S}^{d})\cong GL_{d}({\mathbf{R}})\times
GL(\mathbf{Q}_{p_{1}})\times\cdots\times GL(\mathbf{Q}_{p_{r}})$

on $\widehat{\mathbf{Q}_{S}^{d}}$ corresponds to the right action on
${\mathbf{R}}^{d}\times\mathbf{Q}_{p_{1}}^{d}\times\cdots\times\mathbf{Q}_{p_{r}}^{d}$
given by

$((g_{\infty},g_{1},\cdots,g_{r}),(a_{\infty},a_{1},\cdots,a_{r}))\mapsto(g_{\infty}^{t}a_{\infty},g_{1}^{t}a_{1},\cdots,g_{r}^{t}a_{r}),$

where $(g,a)\mapsto ga$ is the usual (left) linear action of
$GL_{d}(\mathbf{k})$ on $\mathbf{k}^{d}$ for a field $\mathbf{k}.$

##  7\. Unitary representations of unipotent groups

Let $\mathbf{U}$ be a linear algebraic unipotent group defined over
$\mathbf{Q}$. The Lie algebra $\mathfrak{u}$ is defined over $\mathbf{Q}$ and
the exponential map $\exp:\mathfrak{u}\to U$ is a bijective morphism of
algebraic varieties.

Let $p$ be either a prime integer or $p=\infty.$ The irreducible unitary
representations of $U_{p}:=\mathbf{U}(\mathbf{Q}_{p})$ are parametrized by
Kirillov’s theory as follows.

The Lie algebra of $U_{p}$ is
$\mathfrak{u}_{p}=\mathfrak{u}(\mathbf{Q})\otimes_{\mathbf{Q}}\mathbf{\mathbf{Q}}_{p},$
where $\mathfrak{u}(\mathbf{Q})$ is the Lie algebra over $\mathbf{Q}$
consisting of the $\mathbf{Q}$-points in $\mathfrak{u}.$

Fix an element $f$ in the dual space
${\mathfrak{u}}_{p}^{*}={\mathcal{H}om}_{\mathbf{Q}_{p}}({\mathfrak{u}}_{p},\mathbf{Q}_{p})$
of $\mathfrak{u}_{p}.$ There exists a polarization $\mathfrak{m}$ for $f,$
that is, a Lie subalgebra $\mathfrak{m}$ of ${\mathfrak{u}}_{p}$ such that
$f([{\mathfrak{m}},{\mathfrak{m}}])=0$ and which is of maximal dimension. The
induced representation ${\rm Ind}_{M}^{U_{p}}\chi_{f}$ is irreducible, where
$M=\exp(\mathfrak{m})$ and $\chi_{f}$ is the unitary character of $M$ defined
by

$\chi_{f}(\exp X)=e_{p}(f(X))\qquad\text{for all}\quad X\in{\mathfrak{m}},$

where $e_{p}\in\widehat{\mathbf{Q}_{p}}$ is as in Section 6. The unitary
equivalence class of ${\rm Ind}_{M}^{U_{p}}\chi_{f}$ only depends on the co-
adjoint orbit ${\rm Ad}^{*}(U_{p})f$ of $f$. The map

${\mathfrak{n}}_{p}^{*}/{\rm
Ad}^{*}(U_{p})\to\widehat{U_{p}},\qquad{\mathcal{O}}\mapsto\pi_{\mathcal{O}}$

called the Kirillov map, from the orbit space ${\mathfrak{u}}_{p}^{*}/{\rm
Ad}^{*}(U_{p})$ of the co-adjoint representation to the unitary dual
$\widehat{U}_{p}$ of $U_{p}$, is a bijection. In particular, $U_{p}$ is a so-
called type I locally compact group. For all of this, see [Kiri62] or [CoGr89]
in the case $p=\infty$ and [Moor65] in the case of a prime integer $p$.

The group ${\rm Aut}(U_{p})$ of continuous automorphisms of $U$ can be
identified with the group of $\mathbf{Q}_{p}$-points of the algebraic group
${\rm Aut}(\mathfrak{u})$ of automorphisms of the Lie algebra $\mathfrak{u}$
of $\mathbf{U}.$ Notice also that the natural action of ${\rm Aut}(U_{p})$ on
${\mathfrak{u}}_{p}$ as well as its dual action on ${\mathfrak{u}}_{p}^{*}$
are algebraic.

Let $\pi\in\widehat{U_{p}}$ with corresponding Kirillov orbit
$\mathcal{O}_{\pi}$ and $g\in{\rm Aut}(U_{p}).$ Then $g(\mathcal{O}_{\pi})$ is
the Kirillov orbit associated to the conjugate representation $\pi^{g}.$

######  Lemma 10.

Let $\pi$ be an irreducible unitary representation of $U_{p}.$ The stabilizer
$G_{\pi}$ of $\pi$ in ${\rm Aut}(U_{p})$ is an algebraic subgroup of ${\rm
Aut}(U_{p}).$

Proof Let ${\mathcal{O}}_{\pi}\subset{\mathfrak{u}}_{p}^{*}$ be the Kirillov
orbit corresponding to $\pi.$ Then $G_{\pi}$ is the set of $g\in{\rm
Aut}(U_{p})$ such that $g(\mathcal{O}_{\pi})=\mathcal{O}_{\pi}.$ As
$\mathcal{O}_{\pi}$ is an algebraic subvariety of ${\mathfrak{u}}_{p}^{*}$,
the claim follows. $\blacksquare\vskip 6.0pt plus 2.0pt minus 2.0pt$

##  8\. Decay of matrix coefficients of unitary representations of $S$-adic
groups

Let $p$ be an integer prime or $p=\infty$ and let $\mathbf{U}$ be a linear
algebraic unipotent group defined over $\mathbf{Q}_{p}$. Set
$U_{p}:=\mathbf{U}(\mathbf{Q}_{p}).$

Let $\pi$ be an irreducible unitary representation of $U_{p}$. Recall (see
Lemma 10) that the stabilizer $G_{\pi}$ of $\pi$ in ${\rm Aut}(U_{p})$ is an
algebraic subgroup of ${\rm Aut}(U_{p}).$ Recall also (see Lemma 6) that $\pi$
extends to a projective representation of $G_{\pi}$. The following result was
proved in Proposition 22 of [BeGu15] in the case where $p=\infty,$ using
arguments from [HoMo79]. The proof in the case where $p$ is a prime integer is
along similar lines and will be omitted.

######  Proposition 11.

Let $\pi$ be an irreducible unitary representation of $U_{p}$ and let
$\widetilde{\pi}$ be a projective unitary representation of ${G}_{\pi}$ which
extends ${\pi}.$ There exists a real number $r\geq 1,$ only depending on the
dimension of $G_{\pi},$ such that $\widetilde{\pi}$ is strongly $L^{r}$ modulo
its projective kernel.

We will need later a precise description of the projective kernel of a
representation $\widetilde{\pi}$ as above.

######  Lemma 12.

Let $\pi$ be an irreducible unitary representation of $U_{p}$ and
$\widetilde{\pi}$ a projective unitary representation of ${G}_{\pi}$ which
extends ${\pi}.$ Let ${\mathcal{O}}_{\pi}\subset{\mathfrak{u}}_{p}^{*}$ be the
corresponding Kirillov orbit of $\pi.$ For $g\in{\rm Aut}(U_{p}),$ the
following properties are equivalent:

  * (i)

$g$ belongs to the projective kernel $P_{\widetilde{\pi}}$ of
$\widetilde{\pi};$

  * (ii)

for every $u\in U_{p},$ we have

$g(u)u^{-1}\in\bigcap_{f\in{\mathcal{O}}_{\pi}}\exp({\rm Ker}(f)).$

Proof We can assume that $\pi={\rm Ind}_{M}^{U_{p}}\chi_{f_{0}}$, for
$f_{0}\in{\mathcal{O}}_{\pi},$ and $M=\exp\mathfrak{m}$ for a polarization
$\mathfrak{m}$ of $f_{0}$. For $g\in{\rm Aut}(U_{p}),$ we have $g\in
P_{\widetilde{\pi}}$ if and only if

$\pi(g(u))=\pi(u)\qquad\text{for all}\quad u\in U_{p}$

that is,

$g(u)u^{-1}\in{\rm Ker}(\pi)\qquad\text{for all}\quad u\in U_{p}.$

Now, we have (see [BeGu15, Lemma 18])

${\rm Ker}(\pi)=\bigcap_{f\in{\mathcal{O}}_{\pi}}{\rm Ker}(\chi_{f})$

and so, $g\in P_{\widetilde{\pi}}$ if and only if

$g(u)u^{-1}\in\bigcap_{f\in{\mathcal{O}}_{\pi}}{\rm
Ker}(\chi_{f})\qquad\text{for all}\quad u\in U_{p}.$

Let $g\in P_{\widetilde{\pi}}.$ Denote by $X\mapsto g(X)$ the automorphism of
$\mathfrak{u}_{p}$ corresponding to $g.$ Let $u=\exp(X)$ for
$X\in\mathfrak{u}_{p}$ and $f\in{\mathcal{O}}_{\pi}$. Set $u_{t}=\exp(tX).$ By
the Campbell Hausdorff formula, there exists $Y_{1},\dots
Y_{r}\in\mathfrak{u}_{p}$ such that

$g(u_{t})(u_{t})^{-1}=\exp\left(tY_{1}+t^{2}Y_{2}+\dots+t^{r}Y_{r}\right),$

for every $t\in\mathbf{Q}_{p};$ Since

$None$
$1=\chi_{f}(g(u_{t})(u_{t})^{-1})=e_{p}(f\left(tY_{1}+t^{2}Y_{2}+\dots+t^{r}Y_{2}\right)),$

it follows that the polynomial

$t\mapsto Q(t)=tf(Y_{1})+t^{2}f(Y_{2})+\dots+t^{r}f(Y_{r})$

takes its values in ${\mathbf{Z}}$ in case $p=\infty$ and in
${\mathbf{Z}}_{p}$ (and so $Q$ has bounded image) otherwise. This clearly
implies that $Q(t)=0$ for all $t\in\mathbf{Q}_{p}$; in particular, we have

$\log(g(u)u^{-1})=Y_{1}+Y_{2}+\cdots+Y_{r}\in{\rm Ker}(f).$

This shows that (i) implies (ii).

Conversely, assume that (ii) holds. Then clearly

$g(u)u^{-1}\in\bigcap_{f\in{\mathcal{O}}_{\pi}}{\rm
Ker}(\chi_{f})\qquad\text{for all}\quad u\in U_{p}$

and so $g\in P_{\widetilde{\pi}}$. $\blacksquare\vskip 6.0pt plus 2.0pt minus
2.0pt$

##  9\. Decomposition of the Koopman representation for a nilmanifold

Let $\mathbf{U}$ be a linear algebraic unipotent group defined over
$\mathbf{Q}$. Let $S=\\{p_{1},\dots,p_{r},\infty\\}$, where
$p_{1},\dots,p_{r}$ are integer primes. Set

$U:=\mathbf{U}(\mathbf{Q}_{S})=\prod_{s\in S}U_{p}.$

Since $U$ is a type I group, the unitary dual $\widehat{U}$ of $U$ can be
identified with the cartesian product $\prod_{s\in S}\widehat{U_{p}}$ via the
map

$\prod_{s\in S}\widehat{U_{p}}\to\widehat{U},\qquad(\pi_{p})_{p\in
S}\mapsto\otimes_{s\in S}\pi_{p},$

where $\otimes_{s\in
S}\pi_{p}=\pi_{\infty}\otimes\pi_{p_{1}}\otimes\dots\otimes\pi_{r}$ is the
tensor product of the $\pi_{p}$’s.

Let $\Lambda:=\mathbf{U}({\mathbf{Z}}[1/S])$ and consider the corresponding
$S$-adic compact nilmanifold

$\mathbf{Nil}_{S}:=U/\Lambda,$

equipped with the unique $U$-invariant probability measure $\mu$ on its Borel
subsets.

The associated $S$-adic solenoid is

$\mathbf{Sol}_{S}=\overline{U}/\overline{\Lambda},$

where $\overline{U}:=U/[U,U]$ is the quotient of $U$ by its closed commutator
subgroup $[U,U]$ and where $\overline{\Lambda}$ is the image of
$\mathbf{U}({\mathbf{Z}}[1/S])$ in $\overline{U}.$

Set

${\rm Aut}(U):=\prod_{s\in S}{\rm Aut}(\mathbf{U}(\mathbf{Q}_{s}))$

and denote by ${\rm Aut}(\mathbf{Nil}_{S})$ the subgroup of all $g\in{\rm
Aut}(U)$ with $g(\Lambda)=\Lambda.$

Let $\Gamma$ be a subgroup of ${\rm Aut}(\mathbf{Nil}_{S})$. Let $\kappa$ be
the Koopman representation of $\Gamma\ltimes U$ on $L^{2}(\mathbf{Nil}_{S})$
associated to the action $\Gamma\ltimes U\curvearrowright\mathbf{Nil}_{S}.$ By
Proposition 9, there exists a family $(\pi_{i})_{i\in I}$ of irreducible
representations of $U,$ such that $\kappa$ is equivalent to

$\bigoplus_{i\in I}{\rm Ind}_{\Gamma_{i}\ltimes U}^{\Gamma\ltimes
U}(\widetilde{\pi_{i}}\otimes W_{i}),$

where $\widetilde{\pi}_{i}$ is an irreducible projective representation
$\widetilde{\pi}_{i}$ of the stabilizer $G_{i}$ of $\pi_{i}$ in ${\rm
Aut}(U)\ltimes U$ extending $\pi_{i}$, and where $W_{i}$ is a projective
unitary representation of $G_{i}\cap(\Gamma\ltimes U)$.

Fix $i\in I.$ We have $\pi_{i}=\otimes_{p\in S}\pi_{i,p}$ for irreducible
representations $\pi_{i,p}$ of $U_{p}.$

We will need the following more precise description of $\pi_{i}.$ Recall that
$\mathfrak{u}$ is the Lie algebra of $\mathbf{U}$ and that
$\mathfrak{u}(\mathbf{Q})$ denotes the Lie algebra over $\mathbf{Q}$
consisting of the $\mathbf{Q}$-points in $\mathfrak{u}.$ Let
$\mathfrak{u}^{*}(\mathbf{Q})$ be the set of $\mathbf{Q}$-rational points in
the dual space $\mathfrak{u}^{*};$ so, $\mathfrak{u}^{*}(\mathbf{Q})$ is the
subspace of $f\in\mathfrak{u}^{*}$ with $f(X)\in\mathbf{Q}$ for all
$X\in\mathfrak{u}(\mathbf{Q}).$ Observe that, for
$f\in\mathfrak{u}^{*}(\mathbf{Q}),$ we have $f(X)\in\mathbf{Q}_{p}$ for all
$X\in\mathfrak{u}_{p}=\mathfrak{u}(\mathbf{Q}_{p})$.

A polarization for $f\in\mathfrak{u}^{*}(\mathbf{Q})$ is a Lie subalgebra
$\mathfrak{m}$ of ${\mathfrak{u}}(\mathbf{Q})$ such that
$f([{\mathfrak{m}},{\mathfrak{m}}])=0$ and which is of maximal dimension with
this property.

######  Proposition 13.

Let $\pi_{i}=\otimes_{p\in S}\pi_{i,p}$ be one of the irreducible
representations of $U=\mathbf{U}(\mathbf{Q}_{S})$ appearing in the
decomposition $L^{2}(\mathbf{Nil}_{S})$ as above. There exist
$f_{i}\in\mathfrak{u}^{*}(\mathbf{Q})$ and a polarization
$\mathfrak{m}_{i}\subset\mathfrak{u}(\mathbf{Q})$ for $f_{i}$ with the
following property: for every $p\in S,$ the representation $\pi_{i,p}$ is
equivalent to ${\rm Ind}_{M_{i,p}}^{U}\chi_{f_{i}},$ where
$M_{i,p}=\exp(\mathfrak{m}_{i,p})$ and $\chi_{f_{i}}$ is the unitary character
of $M_{i,p}$ given by

$\chi_{f_{i}}(\exp X)=e_{p}(f_{i}(X)),\qquad\text{for all}\quad
X\in{\mathfrak{m}}_{i,p}=\mathfrak{\mathfrak{m}}_{i}\otimes_{\mathbf{Q}}\mathbf{\mathbf{Q}}_{p},$

where $e_{p}\in\widehat{\mathbf{Q}_{p}}$ is as in Section 6.

Proof The same result is proved in Theorem 11 in [Moor65] (see also Theorem
1.2 in [Fox89]) for the Koopman representation of $\mathbf{U}(\mathbf{A})$ in
$L^{2}(\mathbf{U}(\mathbf{A})/\mathbf{U}(\mathbf{Q})),$ where $\mathbf{A}$ is
the ring of adeles of $\mathbf{Q}.$ We could check that the proof, which
proceeds by induction of the dimension of $\mathbf{U}$, carries over to the
Koopman representation on
$L^{2}(\mathbf{U}(\mathbf{Q}_{S})/\mathbf{U}(\mathbf{{\mathbf{Z}}}[1/S])$,
with the appropriate changes. We prefer to deduce our claim from the result
for $\mathbf{U}(\mathbf{A})$, as follows.

It is well-known (see [Weil74]) that

$\mathbf{A}=\left(\mathbf{Q}_{S}\times\prod_{p\notin
S}{\mathbf{Z}}_{p}\right)+\mathbf{Q}$

and that

$\left(\mathbf{Q}_{S}\times\prod_{p\notin
S}{\mathbf{Z}}_{p}\right)\cap\mathbf{Q}={\mathbf{Z}}[1/S].$

This gives rise to a well defined projection
$\varphi:\mathbf{A}/\mathbf{Q}\to\mathbf{Q}_{S}/{\mathbf{Z}}[1/S]$ given by

$\varphi\left((a_{S},(a_{p})_{p\notin
S})+\mathbf{Q}\right)=a_{S}+{\mathbf{Z}}[1/S]\qquad\text{for all}\quad
a_{S}\in\mathbf{Q}_{S},(a_{p})_{p\notin
S}\in\prod_{p\notin\mathcal{P}}{\mathbf{Z}}_{p};$

so the fiber over a point
$a_{S}+{\mathbf{Z}}[1/S]\in\mathbf{Q}_{S}/{\mathbf{Z}}[1/S]$ is

$\varphi^{-1}(a_{S}+{\mathbf{Z}}[1/S])=\\{(a_{S},(a_{p})_{p\notin
S})+\mathbf{Q}\mid a_{p}\in{\mathbf{Z}}_{p}\text{ for all }p\\}.$

This induces an identification of
$\mathbf{U}(\mathbf{\mathbf{Q}_{S}})/\mathbf{U}(\mathbf{{\mathbf{Z}}}[1/S])=\mathbf{Nil}_{S}$
with the double coset space
$K_{S}\backslash\mathbf{U}(\mathbf{A})/\mathbf{U}(\mathbf{Q}),$ where $K_{S}$
is the compact subgroup

$K_{S}=\prod_{p\notin S}\mathbf{U}({\mathbf{Z}}_{p})$

of $\mathbf{U}(\mathbf{A}).$ Observe that this identification is equivariant
under translation by elements from $\mathbf{U}(\mathbf{Q}_{S}).$ In this way,
we can view $L^{2}(\mathbf{Nil}_{S})$ as the
$\mathbf{U}(\mathbf{Q}_{S})$-invariant subspace
$L^{2}(K_{S}\backslash\mathbf{U}(\mathbf{A})/\mathbf{U}(\mathbf{Q}))$ of
$L^{2}(\mathbf{U}(\mathbf{A})/\mathbf{U}(\mathbf{Q})).$

Choose a system $T$ of representatives for the ${\rm
Ad}^{*}(\mathbf{U}(\mathbf{Q}))$-orbits in $\mathfrak{u}^{*}(\mathbf{Q})$. By
[Moor65, Theorem 11], for every $f\in T,$ we can find a polarization
$\mathfrak{m}_{f}\subset\mathfrak{u}(\mathbf{Q})$ for $f$ with the following
property: setting

$\mathfrak{m}_{f}(\mathbf{A})=\mathfrak{\mathfrak{m}}_{f}\otimes_{\mathbf{Q}}\mathbf{A},$

we have a decomposition

$L^{2}(\mathbf{U}(\mathbf{A})/\mathbf{U}(\mathbf{Q}))=\bigoplus_{f\in
T}\mathcal{H}_{f}$

into irreducible $\mathbf{U}(\mathbf{A})$-invariant subspaces
$\mathcal{H}_{f}$ such that the representation $\pi_{f}$ of
$\mathbf{U}(\mathbf{A})$ in $\mathcal{H}_{f}$ is equivalent to ${\rm
Ind}_{M_{f}(\mathbf{A})}^{\mathbf{U}(\mathbf{A})}\chi_{f},$ where

$M_{f}(\mathbf{A})=\exp(\mathfrak{m}_{f}(\mathbf{A}))$

and $\chi_{f,\mathbf{A}}$ is the unitary character of $M_{f}(\mathbf{A})$
given by

$\chi_{f,\mathbf{A}}(\exp X)=e(f(X)),\qquad\text{for all}\quad
X\in{\mathfrak{m}}_{f}(\mathbf{A});$

here, $e$ is the unitary character of $\mathbf{A}$ defined by

$e((a_{p})_{p})=\prod_{p\in\mathcal{P}\cup\\{\infty\\}}e_{p}(a_{p})\qquad\text{for
all}\quad(a_{p})_{p}\in\mathbf{A},$

where $\mathcal{P}$ is the set of integer primes.

We have

$L^{2}(K_{S}\backslash\mathbf{U}(\mathbf{A})/\mathbf{U}(\mathbf{Q}))=\bigoplus_{f\in
T}\mathcal{H}_{f}^{K_{S}},$

where $\mathcal{H}_{f}^{K_{S}}$ is the space of $K_{S}$-fixed vectors in
$\mathcal{H}_{f}.$ It is clear that the representation of
$\mathbf{U}(\mathbf{Q}_{S})$ in $\mathcal{H}_{f}^{K_{S}}$ is equivalent to

${\rm
Ind}_{M_{f}(\mathbf{\mathbf{Q}_{S}})}^{\mathbf{U}(\mathbf{Q}_{S})}(\otimes_{p\in
S}\chi_{f,p})=\otimes_{p\in S}\left({\rm
Ind}_{M_{f}(\mathbf{\mathbf{Q}_{p}})}^{\mathbf{U}(\mathbf{\mathbf{Q}_{p}})}\chi_{f,p}\right),$

where $\chi_{f,p}$ is the unitary character of $M_{f}(\mathbf{Q}_{p})$ given
by

$\chi_{f,p}(\exp X)=e_{p}(f(X)),\qquad\text{for all}\quad
X\in{\mathfrak{m}}_{f}(\mathbf{Q}_{p}).$

Since $M_{f}(\mathbf{\mathbf{Q}_{p}})$ is a polarization for $f$, each of the
$\mathbf{U}(\mathbf{Q}_{p})$-representations ${\rm
Ind}_{M_{f}(\mathbf{\mathbf{Q}_{p}})}^{\mathbf{U}(\mathbf{\mathbf{Q}_{p}})}\chi_{f,p}$
and, hence, each of the $\mathbf{U}(\mathbf{Q}_{S})$-representations

${\rm
Ind}_{M_{f}(\mathbf{\mathbf{Q}_{S}})}^{\mathbf{U}(\mathbf{Q}_{S})}(\otimes_{p\in
S}\chi_{f,p})$

is irreducible. This proves the claim. $\blacksquare\vskip 6.0pt plus 2.0pt
minus 2.0pt$

We establish another crucial fact about the representations $\pi_{i}$’s in the
following proposition.

######  Proposition 14.

With the notation of Proposition 13, let ${\mathcal{O}}_{\mathbf{Q}}(f_{i})$
be the co-adjoint orbit of $f_{i}$ under $\mathbf{U}(\mathbf{Q})$ and set

$\mathfrak{k}_{i,p}=\bigcap_{f\in{\mathcal{O}}_{\mathbf{Q}}(f_{i})}\mathfrak{k}_{p}(f),$

where $\mathfrak{k}_{p}(f)$ is the kernel of $f$ in $\mathfrak{u}_{p}.$ Let
$K_{i,p}=\exp(\mathfrak{k}_{i,p})$ and $K_{i}=\prod_{p\in S}K_{i,p}.$

  * (i)

$K_{i}$ is a closed normal subgroup of $U$ and
$K_{i}\cap\Lambda=K_{i}\cap\mathbf{U}({\mathbf{Z}}[1/S])$ is a lattice in
$K_{i}.$

  * (ii)

Let $P_{\widetilde{\pi}_{i}}$ be the projective kernel of the extension
$\widetilde{\pi}_{i}$ of $\pi_{i}$ to the stabilizer $G_{i}$ of $\pi$ in ${\rm
Aut}(U)\ltimes U$. For $g\in G_{i}$, we have $g\in P_{\widetilde{\pi}_{i}}$ if
and only if $g(u)\in uK_{i}$ for every $u\in U.$

Proof (i) Let

$\mathfrak{k}_{i,\mathbf{Q}}=\bigcap_{f\in{\mathcal{O}}_{\mathbf{Q}}(f_{i})}\mathfrak{k}_{\mathbf{Q}}(f),$

where $\mathfrak{k}_{\mathbf{Q}}(f)$ is the kernel of $f$ in
$\mathfrak{u}(\mathbf{Q}).$ Observe that $\mathfrak{k}_{i,\mathbf{Q}}$ is an
ideal in $\mathfrak{u}(\mathbf{Q}),$ since it is ${\rm
Ad}(\mathbf{U}(\mathbf{Q}))$-invariant. So, we have

$\mathfrak{k}_{i,\mathbf{Q}}=\mathfrak{k}_{i}(\mathbf{Q})$

for an ideal $\mathfrak{k}_{i}$ in $\mathfrak{u}.$ Since
$f\in\mathfrak{u}^{*}(\mathbf{Q})$ for
$f\in{\mathcal{O}}_{\mathbf{Q}}(f_{i}),$ we have

$\mathfrak{k}_{i,p}(f)=\mathfrak{k}_{i,\mathbf{Q}}(f)\otimes_{\mathbf{Q}}\mathbf{Q}_{p}$

and hence

$\mathfrak{k}_{i,p}=\mathfrak{k}_{i}(\mathbf{Q}_{p}).$

Let $\mathbf{K}_{i}=\log(\mathfrak{k}_{i}).$ Then $\mathbf{K}_{i}$ is a normal
algebraic $\mathbf{Q}$-subgroup of $\mathbf{U}$ and we have
$K_{i,p}=\mathbf{K}_{i}(\mathbf{Q}_{p})$ for every $p$; so,

$K_{i}=\prod_{s\in
S}\mathbf{K}_{i}(\mathbf{Q}_{p})=\mathbf{K}_{i}(\mathbf{Q}_{S})$

and $K_{i}\cap\Lambda=\mathbf{K}_{i}({\mathbf{Z}}[1/S])$ is a lattice in
$K_{i}.$ This proves Item (i).

To prove Item (ii), observe that

$P_{\widetilde{\pi}_{i}}=\bigcap_{p\in S}P_{i,p},$

where $P_{i,p}$ is the projective kernel of $\widetilde{\pi_{i,p}}$.

Fix $p\in S$ and let $g\in G_{i}$. By Lemma 12, $g\in P_{i,p}$ if and only if
$g(u)\in uK_{i,p}$ for every $u\in U_{p}=\mathbf{U}(\mathbf{Q}_{p}).$ This
finishes the proof. $\blacksquare\vskip 6.0pt plus 2.0pt minus 2.0pt$

##  10\. Proof of Theorem 1

Let $\mathbf{U}$ be a linear algebraic unipotent group defined over
$\mathbf{Q}$ and $S=\\{p_{1},\dots,p_{r},\infty\\}$, where $p_{1},\dots,p_{r}$
are integer primes. Set $U:=\mathbf{U}(\mathbf{Q}_{S})$ and
$\Lambda:=\mathbf{U}({\mathbf{Z}}[1/S])$.

Let $\mathbf{Nil}_{S}=U/\Lambda$ and $\mathbf{Sol}_{S}$ be the $S$-adic
nilmanifold and the associated $S$-adic solenoid as in Section 9. Denote by
$\mu$ the translation invariant probability measure on $\mathbf{Nil}_{S}$ and
let $\nu$ be the image of $\mu$ under the canonical projection
$\varphi:\mathbf{Nil}_{S}\to\mathbf{Sol}_{S}.$ We identify
$L^{2}(\mathbf{Sol}_{S})=L^{2}(\mathbf{Sol}_{S},\nu)$ with the closed ${\rm
Aut}(\mathbf{Nil}_{S})$-invariant subspace

$\\{f\circ\varphi\mid f\in L^{2}(\mathbf{Sol}_{S})\\}$

of $L^{2}(\mathbf{Nil}_{S})=L^{2}(\mathbf{Nil}_{S},\mu).$ We have an
orthogonal decomposition into ${\rm Aut}(\mathbf{Nil}_{S})$-invariant
subspaces

$L^{2}(\mathbf{Nil}_{S})={\mathbf{C}}\mathbf{1}_{\mathbf{Nil}_{S}}\oplus
L_{0}^{2}(\mathbf{Sol}_{S})\oplus\mathcal{H},$

where

$L_{0}^{2}(\mathbf{Sol}_{S})=\\{f\in
L^{2}(\mathbf{Sol}_{S})\mid\int_{\mathbf{Nil}_{S}}fd\mu=0\\}$

and where $\mathcal{H}$ is the orthogonal complement of
$L^{2}(\mathbf{Sol}_{S})$ in $L^{2}(\mathbf{Nil}_{S}).$

Let $\Gamma$ be a subgroup of $G:={\rm Aut}(U).$ Let $\kappa$ be the Koopman
representation of $\Gamma$ on $L^{2}(\mathbf{Nil}_{S})$ and denote by
$\kappa_{1}$ and $\kappa_{2}$ the restrictions of $\kappa$ to respectively
$L_{0}^{2}(\mathbf{Sol}_{S})$ and $\mathcal{H}.$

Let $\Sigma_{1}$ be a set of representatives for the $\Gamma$-orbits in
$\widehat{\mathbf{Sol}_{S}}\setminus\\{\mathbf{1}_{\mathbf{Sol}_{S}}\\}$. We
have

$\kappa_{1}\cong\bigoplus_{\chi\in\Sigma_{1}}\lambda_{\Gamma/\Gamma_{\chi}},$

where $\Gamma_{\chi}$ is the stabilizer of $\chi$ in $\Gamma$ and
$\lambda_{\Gamma/\Gamma_{\chi}}$ is the quasi-regular representation of
$\Gamma$ on $\ell^{2}(\Gamma/\Gamma_{\chi}).$

By Proposition 9, there exists a family $(\pi_{i})_{i\in I}$ of irreducible
representations of $G,$ such that $\kappa_{2}$ is equivalent to a direct sum

$\bigoplus_{i\in I}{\rm
Ind}_{\Gamma_{i}}^{\Gamma}(\widetilde{\pi_{i}}|_{\Gamma_{i}}\otimes W_{i}),$

where $\widetilde{\pi}_{i}$ is an irreducible projective representation of the
stabilizer $G_{i}$ of $\pi_{i}$ in ${\rm Aut}(G)\ltimes G$ extending
$\pi_{i}$, and where $W_{i}$ is a projective unitary representation of
$\Gamma_{i}:=\Gamma\cap G_{i}.$

######  Proposition 15.

For $i\in I,$ let $\widetilde{\pi}_{i}$ be the (projective) representation of
$G_{i}$ and let $\Gamma_{i}$ be as above. There exists a real number $r\geq 1$
such that $\widetilde{\pi_{i}}|_{\Gamma_{i}}$ is strongly $L^{r}$ modulo
$P_{\widetilde{\pi}_{i}}\cap\Gamma_{i}$, where $P_{\widetilde{\pi}_{i}}$ is
the projective kernel of $\widetilde{\pi}_{i}.$

Proof By Proposition 11, there exists a real number $r\geq 1$ such that the
representation $\widetilde{\pi}_{i}$ of the algebraic group $G_{i}$ is
strongly $L^{r}$ modulo $P_{\widetilde{\pi}_{i}}$. In order to show that
$\widetilde{\pi_{i}}|_{\Gamma_{i}}$ is strongly $L^{r}$ modulo
$P_{\widetilde{\pi}_{i}}\cap\Gamma_{i}$, it suffices to show that
$\Gamma_{i}P_{\widetilde{\pi}_{i}}$ is closed in $G_{i}$ (compare with the
proof of Proposition 6.2 in [HoMo79]).

Let $K_{i}$ be the the closed $G_{i}$-invariant normal subgroup $K_{i}$ of $U$
as described in Proposition 14. Then $\overline{\Lambda}=K_{i}\Lambda/K_{i}$
is a lattice in the unipotent group $\overline{U}=U/K_{i}$ By Proposition
14.ii, $P_{\widetilde{\pi}_{i}}$ coincides with the kernel of the natural
homomorphism $\varphi:{\rm Aut}(U)\to{\rm Aut}(\overline{U})$. Hence, we have

$\Gamma_{i}P_{\widetilde{\pi}_{i}}=\varphi^{-1}(\varphi(\Gamma_{i})).$

Now, $\varphi(\Gamma_{i})$ is a discrete (and hence closed) subgroup of ${\rm
Aut}(\overline{U})$, since $\varphi(\Gamma_{i})$ preserves
$\overline{\Lambda}$ (and so $\varphi(\Gamma_{i})\subset{\rm
Aut}(\overline{U}/\overline{\Lambda})).$ It follows from the continuity of
$\varphi$ that $\varphi^{-1}(\varphi(\Gamma_{i}))$ is closed in ${\rm
Aut}(U).\blacksquare\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof of Theorem 1

We have to show that, if $1_{\Gamma}$ is weakly contained in $\kappa_{2},$
then $1_{\Gamma}$ is weakly contained in $\kappa_{1}.$ It suffices to show
that, if $1_{\Gamma}$ is weakly contained in $\kappa_{2},$ then there exists a
finite index subgroup $H$ of $\Gamma$ such that $1_{H}$ is weakly contained in
$\kappa_{1}|_{H}$ (see Theorem 2 in [BeFr20]).

We proceed by induction on the integer

$n(\Gamma):=\sum_{p\in S}\dim{\rm Zc}_{p}(\Gamma),$

where ${\rm Zc}_{p}(\Gamma)$ is the Zariski closure of the projection of
$\Gamma$ in $GL_{n}(\mathbf{Q}_{p})$.

If $n(\Gamma)=0,$ then $\Gamma$ is finite and there is nothing to prove.

Assume that $n(\Gamma)\geq 1$ and that the claim above is proved for every
countable subgroup $H$ of ${\rm Aut}(\mathbf{Nil}_{S})$ with $n(H)<n(\Gamma).$

Let $I_{\rm fin}\subset I$ be the set of all $i\in I$ such that
$\Gamma_{i}=G_{i}\cap\Gamma$ has finite index in $\Gamma$ and set
$I_{\infty}=I\setminus I_{\rm fin}.$ With
$V_{i}=(\widetilde{\pi_{i}}|_{\Gamma_{i}}\otimes W_{i})$, set

$\kappa_{2}^{\rm fin}=\bigoplus_{i\in I_{\rm fin}}{\rm
Ind}_{\Gamma_{i}}^{\Gamma}V_{i}\qquad\text{and}\qquad\kappa_{2}^{\infty}=\bigoplus_{i\in
I_{\infty}}{\rm Ind}_{\Gamma_{i}}^{\Gamma}V_{i}.$

Two cases can occur.

$\bullet$ _First case:_ $1_{\Gamma}$ is weakly contained in
$\kappa_{2}^{\infty}.$

Observe that $n(\Gamma_{i})<n(\Gamma)$ for $i\in I_{\infty}.$ Indeed,
otherwise ${\rm Zc}_{p}(\Gamma_{i})$ and ${\rm Zc}_{p}(\Gamma)$ would have the
same connected component $C^{0}_{p}$ for every $p\in S,$ since
$\Gamma_{i}\subset\Gamma.$ Then

$C^{0}:=\bigcap_{p\in S}C^{0}_{p}$

would stabilize $\pi_{i}$ and $\Gamma\cap C^{0}$ would therefore be contained
in $\Gamma_{i}.$ Since $\Gamma\cap C^{0}$ has finite index in $\Gamma,$ this
would be a contradiction to the fact that $\Gamma_{i}$ has infinite index in
$\Gamma.$

By restriction, $1_{\Gamma_{i}}$ is weakly contained in
$\kappa_{2}|_{\Gamma_{i}}$ for every $i\in I.$ Hence, by the induction
hypothesis, $1_{\Gamma_{i}}$ is weakly contained in $\kappa_{1}|_{\Gamma_{i}}$
for every $i\in I_{\infty}.$ Now, on the one hand, we have

$\kappa_{1}|_{\Gamma_{i}}\cong\bigoplus_{\chi\in
T_{i}}\lambda_{{\Gamma_{i}}/\Gamma_{\chi}\cap{\Gamma_{i}}},$

for a subset $T_{i}$ of
$\widehat{\mathbf{Sol}_{S}}\setminus\\{\mathbf{1}_{\mathbf{Sol}_{S}}\\}$. It
follows that ${\rm
Ind}_{\Gamma_{i}}^{\Gamma}1_{\Gamma_{i}}=\lambda_{\Gamma/\Gamma_{i}}$ is
weakly contained in

$\bigoplus_{\chi\in T_{i}}{\rm
Ind}_{\Gamma_{i}}^{\Gamma}(\lambda_{\Gamma_{i}/\Gamma_{\chi}\cap\Gamma_{i}})=\bigoplus_{\chi\in
T_{i}}\lambda_{\Gamma/\Gamma_{\chi}\cap\Gamma_{i}},$

for every $i\in I_{\infty}.$ On the other hand, since $1_{\Gamma}$ is weakly
contained in

$\kappa_{2}\cong\bigoplus_{i\in I_{\infty}}{\rm
Ind}_{\Gamma_{i}}^{\Gamma}(\widetilde{\pi_{i}}|_{\Gamma_{i}}\otimes W_{i}),$

Lemma 7 shows that $1_{\Gamma}$ is weakly contained in $\bigoplus_{i\in
I_{\infty}}\lambda_{\Gamma/\Gamma_{i}}.$ It follows that $1_{\Gamma}$ is
weakly contained in

$\bigoplus_{i\in I_{\infty}}\bigoplus_{\chi\in
T_{i}}\lambda_{\Gamma/\Gamma_{\chi}\cap\Gamma_{i}}.$

Hence, by Lemma 7 again, $1_{\Gamma}$ is weakly contained in

$\bigoplus_{i\in I_{\infty}}\bigoplus_{\chi\in
T_{i}}\lambda_{\Gamma/\Gamma_{\chi}}.$

This shows that $1_{\Gamma}$ is weakly contained in $\kappa_{1}.$

$\bullet$ _Second case:_ $1_{\Gamma}$ is weakly contained in $\kappa_{2}^{\rm
fin}.$

By the Noetherian property of the Zariski topology, we can find finitely many
indices $i_{1},\dots,i_{r}$ in $I_{\rm fin}$ such that, for every $p\in S,$ we
have

${\rm Zc}_{p}(\Gamma_{i_{1}})\cap\cdots\cap{\rm
Zc}_{p}(\Gamma_{i_{r}})=\bigcap_{i\in I_{\rm fin}}{\rm Zc}_{p}(\Gamma_{i}),$

Set $H:=\Gamma_{i_{1}}\cap\cdots\cap\Gamma_{i_{r}}.$ Observe that $H$ has
finite index in $\Gamma.$ Moreover, it follows from Lemma 10 that ${\rm
Zc}_{p}(\Gamma_{i_{1}})\cap\cdots\cap{\rm Zc}_{p}(\Gamma_{i_{r}})$ stabilizes
$\pi_{i,p}$ for every $i\in I_{\rm fin}$ and $p\in S.$ Hence, $H$ is contained
in $\Gamma_{i}$ for every $i\in I_{\rm fin}$.

By Proposition 9, we have a decomposition of $\kappa_{2}^{\rm fin}|_{H}$ into
the direct sum

$\bigoplus_{i\in I_{\rm fin}}(\widetilde{\pi_{i}}\otimes W_{i})|_{H}.$

By Proposition 8 and Proposition 15, there exists a real number $r\geq 1,$
which is independent of $i,$ such that $(\widetilde{\pi_{i}}\otimes
W_{i})|_{H}$ is a strongly $L^{r}$ representation of $H$ modulo its projective
kernel $P_{i}$. Observe that $P_{i}$ is contained in the projective kernel
$P_{\widetilde{\pi}_{i}}$ of $\widetilde{\pi}_{i}.$ Hence (see Proposition 8),
there exists an integer $k\geq 1$ such that $\kappa_{2}^{\rm
fin}|_{H}^{\otimes k}$ is contained in a multiple of the direct sum

$\oplus_{i\in I_{\rm fin}}{\rm Ind}_{P_{\widetilde{\pi}_{i}}}^{H}\rho_{i},$

for representations $\rho_{i}$ of $P_{\widetilde{\pi}_{i}}.$ Since $1_{H}$ is
weakly contained in $\kappa_{2}^{\rm fin}|H$ and hence in $\kappa_{2}^{\rm
fin}|_{H}^{\otimes k},$ using Lemma 7, it follows that $1_{H}$ is weakly
contained in

$\oplus_{i\in I_{\rm fin}}\lambda_{H/(H\cap P_{\widetilde{\pi}_{i}})}.$

Let $i\in I.$ We claim that $P_{i}$ is contained in $\Gamma_{\chi}$ for some
character $\chi$ from
$\widehat{\mathbf{Sol}_{S}}\setminus\\{\mathbf{1}_{\mathbf{Sol}_{S}}\\}$. Once
proved, this will imply, again by Lemma 7, that $\oplus_{i\in I_{\rm
fin}}\lambda_{H/(H\cap P_{\widetilde{\pi}_{i}})}$ and, hence $1_{H}$, is
weakly contained in $\kappa_{1}|_{H}.$ Since $H$ has finite index in $\Gamma,$
this will show that $1_{\Gamma}$ is weakly contained in $\kappa_{1}$ and
conclude the proof.

To prove the claim, recall from Proposition 14 that there exists a closed
normal subgroup $K_{i}$ of $U$ with the following properties:
$K_{i}\Lambda/K_{i}$ is a lattice in the unipotent algebraic group $U/K_{i}$,
$K_{i}$ is invariant under $P_{\widetilde{\pi}_{i}}$ and
$P_{\widetilde{\pi}_{i}}$ acts as the identity on $U/K_{i}.$ Observe that
$K_{i}\neq U$, since $\pi_{i}$ is not a unitary character of $U$. We can find
a non-trivial unitary character $\chi$ of $U/K_{i}$ which is trivial on
$K_{i}\Lambda/K_{i}$. Then $\chi$ lifts to a non-trivial unitary character
which is fixed by $P_{\widetilde{\pi}_{i}}$ and hence by $P_{i}.$
$\blacksquare\vskip 6.0pt plus 2.0pt minus 2.0pt$

##  11\.  An example: the $S$-adic Heisenberg nilmanifold

As an example, we study the spectral gap property for group of automorphisms
of the $S$-adic Heisenberg nilmanifold. We will give a quantitative estimate
for the norm of associated convolution operators, as we did in [BeHe11] in the
case of real Heisenberg nilmanifolds (that is, in the case $S=\\{\infty\\}$).

Let $\mathbf{K}$ be an algebraically closed field containing $\mathbf{Q}_{p}$
for $p=\infty$ and for all prime integers $p$. For an integer $n\geq 1$,
consider the symplectic form $\beta$ on $\mathbf{K}^{2n}$ given by

$\beta((x,y),(x^{\prime},y^{\prime}))=(x,y)^{t}J(x^{\prime},y^{\prime})\qquad\text{for
all}\quad(x,y),(x^{\prime},y^{\prime})\in\mathbf{K}^{2n},$

where $J$ is the $(2n\times 2n)$-matrix

$J=\left(\begin{array}[]{cc}0&I\\\ -I_{n}&0\end{array}\ \right).$

The symplectic group

${Sp}_{2n}=\\{g\in GL_{2n}(\mathbf{K})\mid{{}^{t}g}Jg=J\\}$

is an algebraic group defined over $\mathbf{Q}.$

The $(2n+1)$-dimensional Heisenberg group is the unipotent algebraic group
$\mathbf{H}$ defined over $\mathbf{Q},$ with underlying set
$\mathbf{K}^{2n}\times\mathbf{K}$ and product

$((x,y),s)((x^{\prime},y^{\prime}),t)=\left((x+x^{\prime},y+y^{\prime}),s+t+\beta((x,y),(x^{\prime},y^{\prime}))\right),$

for $(x,y),(x^{\prime},y^{\prime})\in\mathbf{K}^{2n},\,s,t\in\mathbf{K}.$

The group $Sp_{2n}$ acts by rational automorphisms of $H_{2n+1},$ given by

$g((x,y),t)=(g(x,y),t)\qquad\text{for all}\quad g\in Sp_{2n},\
(x,y)\in\mathbf{K}^{2n},\ t\in\mathbf{K}.$

Let $p$ be either an integer prime or $p=\infty.$ Set
$H_{p}=\mathbf{H}(\mathbf{Q}_{p}).$ The center $Z$ of $H_{p}$ is
$\\{(0,0,t)\mid t\in\mathbf{Q}_{p}\\}.$ The unitary dual $\widehat{H}$ of $H$
consists of the equivalence classes of the following representations:

  * $\bullet$

the unitary characters of the abelianized group $H/Z$;

  * $\bullet$

for every $t\in\mathbf{Q}_{p}\setminus\\{0\\},$ the infinite dimensional
representation $\pi_{t}$ defined on $L^{2}(\mathbf{Q}_{p}^{n})$ by the formula

$\pi_{t}((a,b),s)\xi(x)=e_{p}({ts})e_{p}(\langle a,x-b\rangle)\xi(x-b)$

for $((a,b),s)\in H,$ $\xi\in L^{2}(\mathbf{Q}_{p}^{n}),$ and
$x\in\mathbf{Q}_{p}^{n},$ where $e_{p}\in\widehat{\mathbf{Q}_{p}}$ is as in
Section 6.

For $t\neq 0,$ the representation $\pi_{t}$ is, up to unitary equivalence, the
unique irreducible unitary representation of $H$ whose restriction to the
centre $Z$ is a multiple of the unitary character $s\mapsto e_{p}{(ts)}.$

For $g\in Sp_{2n}(\mathbf{Q}_{p})$ and $t\in\mathbf{Q}_{p}\setminus\\{0\\},$
the representation $\pi_{t}^{g}$ is unitary equivalent to $\pi_{t},$ since
both representations have the same restriction to $Z.$ This shows that
$Sp_{2n}(\mathbf{Q}_{p})$ stabilizes $\pi_{t}$. We denote the corresponding
projective representation of $Sp_{2n}(\mathbf{Q}_{p})$ by $\omega_{t}^{(p)}$.
The representation $\omega_{t}^{(p)}$ has different names: it is called
metaplectic representation, Weil’s representation,  or oscillator
representation. The projective kernel of $\omega_{t}^{(p)}$ coincides with the
(finite) center of $Sp_{2n}(\mathbf{Q}_{p})$ and $\omega_{t}^{(p)}$ is
strongly $L^{4n+2+\varepsilon}$ on $Sp_{2n}(\mathbf{Q}_{p})$ for every
$\varepsilon>0$ (see Proposition 6.4 in [HoMo79] or Proposition 8.1 in
[Howe82]).

Let $S=\\{p_{1},\dots,p_{r},\infty\\}$, where $p_{1},\dots,p_{r}$ are integer
primes. Set $U:=\mathbf{H}(\mathbf{Q}_{S})$ and

$\Lambda:=\mathbf{H}({\mathbf{Z}}[1/S])=\\{((x,y),s)\ :\
x,y\in{\mathbf{Z}}^{n}[1/S],s\in{\mathbf{Z}}[1/S]\\}.$

Let $\mathbf{Nil}_{S}=U/\Lambda$; the associated $S$-adic solenoid is
$\mathbf{Sol}_{S}=\mathbf{Q}_{S}^{2n}/{\mathbf{Z}}[1/S]^{2n}.$ The group
$Sp_{2n}({\mathbf{Z}}[1/S])$ is a subgroup of ${\rm Aut}(\mathbf{Nil}_{S})$.
The action of $Sp_{2n}({\mathbf{Z}}[1/S])$ on $\mathbf{Sol}_{S}$ is induced by
its linear representation by linear bijections on $\mathbf{Q}_{S}$.

Let $\Gamma$ be a subgroup of $Sp_{2n}({\mathbf{Z}}[1/S])$. The Koopman
representation $\kappa$ of $\Gamma$ on $L^{2}(\mathbf{Nil}_{S})$ decomposes as

$\kappa=\mathbf{1}_{\mathbf{Nil}_{S}}\oplus\kappa_{1}\oplus\kappa_{2},$

where $\kappa_{1}$ is the restriction of $\kappa$ to
$L_{0}^{2}(\mathbf{Sol}_{S})$ and $\kappa_{1}$ the restriction of $\kappa$ to
the orthogonal complement of $L^{2}(\mathbf{Sol}_{S})$ in
$L^{2}(\mathbf{Nil}_{S}).$ Since $Sp_{2n}(\mathbf{Q}_{p})$ stabilizes every
infinite dimensional representation of $H_{p},$ it follows from Proposition 14
that there exists a subset $I\subset\mathbf{Q}$ such that $\kappa_{2}$ is
equivalent to a direct sum

$\bigoplus_{t\in I}\left(\otimes_{p\in S}(\omega_{t}^{(p)}|_{\Gamma}\otimes
W_{i})\right),$

where $W_{i}$ is an projective representation $\Gamma.$

Let $\nu$ be a probability measure on $\Gamma.$ We can give an estimate of the
norm of $\kappa_{2}(\nu)$ as in [BeHe11] in the case of $S=\\{\infty\\}.$
Indeed, by a general inequality (see Proposition 30 in [BeGu15]), we have

$\|\kappa_{2}(\nu)\|\leq\|\left(\kappa_{2}\otimes\overline{\kappa_{2}}\right)^{\otimes
k}(\nu)\|^{1/2k},$

for every integer $k\geq 1,$ where $\overline{\kappa_{2}}$ denotes the
representation conjugate to $\kappa_{2}$. Since $\omega_{t}^{(p)}$ is strongly
$L^{4n+2+\varepsilon}$ on $Sp_{2n}(\mathbf{Q}_{p})$ for any $t\in I$ and $p\in
S,$ Proposition 8 implies that
$(\kappa_{2}\otimes\overline{\kappa_{2}})^{\otimes(n+1)}$ is contained in an
infinite multiple of the regular representation $\lambda_{\Gamma}$ of
$\Gamma.$ Hence,

$\|\kappa_{2}(\nu)\|\leq\|\lambda_{\Gamma}(\nu)\|^{1/2n+2}$

and so,

$\|\kappa_{0}(\nu)\|\leq\max\\{\|\lambda_{\Gamma}(\nu)\|^{1/2n+2},\|\kappa_{1}(\nu)\|\\},$

where $\kappa_{0}$ is the restriction of $\kappa$ to
$L^{2}_{0}(\mathbf{Nil}_{S}).$

Assume that the subgroup generated by the support of $\nu$ coincides with
$\Gamma.$ If $\Gamma$ is not amenable then $\|\lambda_{\Gamma}(\nu)\|<1$ by
Kersten’s theorem (see [BeHV08, Appendix G]); so, in this case, the action of
$\Gamma$ on $\mathbf{Nil}_{S}$ has a spectral gap if and only if
$\|\kappa_{1}(\nu)\|<1,$ as stated in Theorem 1.

Observe that, if $\Gamma$ is amenable, then the action of $\Gamma$ on
$\mathbf{Nil}_{S}$ or $\mathbf{Sol}_{S}$ does not have a spectral gap; indeed,
by a general result (see [JuRo79, Theorem 2.4]), no action of a countable
amenable group by measure preserving transformations on a non-atomic
probability space has a spectral gap.

Let us look more closely to the case $n=1.$ We have
$Sp_{2}({\mathbf{Z}}[1/S])=SL_{2}({\mathbf{Z}}[1/S]$ and the stabilizer of
every element in
$\widehat{\mathbf{Sol}_{S}}\setminus\\{\mathbf{1}_{\mathbf{Sol}_{S}}\\}$ is
conjugate to the group of unipotent matrices in $SL_{2}({\mathbf{Z}}[1/S]$ and
hence amenable. This implies that $\kappa_{1}$ is weakly contained in
$\lambda_{\Gamma};$ so, we have

$\|\kappa_{1}(\nu)\|<1\Longleftrightarrow\Gamma\text{ is not amenable.}$

As a consequence, we see that the action of $\Gamma$ on $\mathbf{Nil}_{S}$ has
a spectral gap if and only if $\Gamma$ is not amenable.

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