# Limits of complementary series, Special representations and cohomology of
CAT(-1) groups.

###### Abstract.

In this paper we extend the construction of special representations [27] [18]
to convex-cocompact isometry groups of CAT(-1) spaces which admits
complementary series in the sense of [4]. We prove that these limits of
complementary series have a natural non-vanishing reduced cohomology class
$[c]$. As a by product we generalize Kuhn Vershik formula [18] and
characterize geometrically CAT(-1) groups that admit complementary series.
Investigating dynamical properties of the cohomology class $[c]$ we prove an
cocycle equidistribution theorem à la Roblin-Margulis [25] [19] (see also [5])
and deduce the irreducibility of the associated affine actions [20] [3] [1].
The constructions and results apply beyond the class of rank 1 linear groups
where the irreducibility of the affine actions associated to the canonical
class $[c]$ , even in the case of uniform lattices in $SO(n,1)$, $SU(n,1)$ or
$SL_{2}(\mathbb{Q}_{p})$ with $n\geq 1$ and $p$ prime, had not been
established.

KEVIN BOUCHER

Weizmann institute of science, 234 Herzl street, Rehovot 7610001, Israel

(e-mail<EMAIL_ADDRESS>
Key words: Affine actions, conditionally negative functions, complementary
series, harmonic cocycles, Knapp-Stein operators.
2020 Mathematics Subject Classification: 37A46 20F67 22D10

##  1\. Introduction

Even though most discrete groups [29] do not admit interesting isometric
affine actions it is known that those acting sufficiently well on negatively
curved spaces admit interesting ones [13] [11] [22]. Indeed, given a rank 1
Lie group $G$ and $X=G/P$ its symmetric space where $P$ stands for a minimal
parabolic subgroup. A remarkable example can be constructed by taking the
natural action of $G$ on its top degree forms with zero average,
$\Omega_{0}^{\text{top}}(X)$. This space carries a certain invariant quadratic
form $Q^{\prime}$ and a canonical $Q^{\prime}$-proper cocycle $c$ associated
to the extension:

(1)
$0\rightarrow\Omega_{0}^{\text{top}}(X)\rightarrow\Omega^{\text{top}}(X)\rightarrow{\mathbb{C}}\rightarrow
0$

When $G=SO_{o}(n,1)$ or $SU(n,1)$ it was proved in [17] that this quadratic
form is actually positive and the representation obtained might be seen as a
limit of complementary series.

As investigated in [13], the cohomology of this representation appears to have
a particular role in cohomology theory of rank 1 Lie groups. It also worth to
be mentioned that a certain uniformly bounded deformation of this
representation has non-vanishing cohomology even for the property (T) groups
$Sp(n,1)$ with $n\geq 2$ [23] .

The main purpose of this work is the extension of this construction over the
class of convex-cocompact CAT(-1) groups and investigate its relation with
complementary series representations as introduced in [4].

In order to formulate our results we set up some notations that are discussed
further in Section 2 (see also [4]).

Let $(X,d)$ be a proper CAT(-1) space and ${\partial}X$ its geometric
boundary. Let $G$ be a discrete non-elementary group of isometries that is
convex-compact and $\mu_{o}$ its Patterson-Sullivan measure over ${\partial}X$
at a basepoint $o\in X$. We denote $\Lambda(G)\subset{\partial}X$ the limit of
$G$ in $(X,d)$ and $\mathcal{Q}(\Lambda(G))$ its convex hull inside of
$\overline{X}=X\cup{\partial}X$.

Given $s\in{\mathbb{C}}$ the space of $s$-homogeneous functions, denoted
$\mathcal{F}_{s}$, is the Banach representation over
${\mathcal{C}}({\partial}X)$ defined as:

$\pi_{s}(g)\varphi(\xi)=e^{-s\delta b_{\xi}(g.o,o)}\varphi(g^{-1}\xi)$

where $b$ stands for the Busemann cocycle, $g\in G$,
$\varphi\in{\mathcal{C}}({\partial}X)$ and $\xi\in{\partial}X$. As observed in
[4] these representations extend over $L^{2}({\partial}X,\mu_{o})$ as linear
actions through bounded operators with $\pi_{s}^{*}=\pi_{1-s}$. Moreover,
under our assumptions, for all $\Re{s}>\frac{1}{2}$ the Knapp-Stein operator
$\mathcal{I}_{s}$ over $L^{2}({\partial}X,\mu_{o})$ given as:

$\mathcal{I}_{s}:L^{2}({\partial}X,\mu_{o})\rightarrow
L^{2}({\partial}X,\mu_{o}),\quad
v\mapsto\int_{{\partial}X}\frac{v(\xi)}{d^{2\delta(1-s)}(\xi,\bullet)}d\mu_{o}(\xi)$

for $v\in L^{2}({\partial}X,\mu_{o})$ is well defined, intertwines the dual
representations $\pi_{s}$ with $\pi_{1-s}$ and thus defines a unitary
structure over $\mathcal{F}_{s}$ whenever $\mathcal{I}_{s}$, as an operator
$L^{2}({\partial}X,\mu_{o})$, is positive.

Analogously to the rank 1 case we construct a Knapp-Stein type of operator
$\mathcal{I}^{\prime}$, that might be seen as the infinitesimal generator of
$(\mathcal{I}_{s})_{s\in{\mathbb{C}}}$, between the
$\pi_{1}$-subrepresentation $L^{2}({\partial}X,\mu_{o})\ominus{\mathbb{C}}$
and its dual $\pi_{0}$-representation
$L^{2}({\partial}X,\mu_{o})/{\mathbb{C}}$. Similarly this self-adjoint
operator defines a unitary structure over the $\pi_{1}$-sub-representation of
1-homogeneous functions with zero average whenever $\mathcal{I}^{\prime}$, as
an operator $L^{2}({\partial}X,\mu_{o})\ominus{\mathbb{C}}$, is positive.
As discussed in Section 4, this representation has a canonical $G$-equivariant
cocycle with coefficients in $\pi_{1}$:

$c:X\times X\rightarrow L^{2}({\partial}X,\mu_{o})\ominus{\mathbb{C}}$

associated to a similar exact sequence as 1 and satisfies:

######  Theorem 1.

Assume the operator $\mathcal{I}^{\prime}$ over
$L^{2}({\partial}X,\mu_{o})\ominus{\mathbb{C}}$ is positive. Then the
associated unitary representation is mixing and $c$ has non-trivial reduced
cohomology class.

The distance of a negatively curved $G$-space $(Z,d_{Z})$ is called roughly
conditionally negative if it decomposes as $d_{Z}=N_{0}+r$ over a $G$-subspace
$Z^{\prime}\subset Z$ with $\mathcal{Q}(\Lambda(G))\subset Z^{\prime}$ where:

  * •

$N_{0}$ is a $G$-invariant conditionally negative kernel over $Z^{\prime}$;

  * •

$r$ a continuous kernel over $Z^{\prime}$, for all $o^{\prime}\in
Z^{\prime}/G$, there exists $\kappa(G.o^{\prime})\in{\mathbb{R}}$ such that
$r|_{G.o^{\prime}\times G.o^{\prime}}$ satisfies

$r|_{G.o^{\prime}\times
G.o^{\prime}}(z,z^{\prime})\xrightarrow{d(z,z^{\prime})\rightarrow+\infty}\kappa(G.o^{\prime})\in{\mathbb{R}}$

######  Theorem ([4] Theorem 1).

If the distance $d$ of $(X,d)$ is roughly conditionally negative, then the
operators $\mathcal{I}_{s}$ with $s>\frac{1}{2}$ are positive over
$L^{2}({\partial}X,\mu_{o})$.

It appears that this result is tight. Indeed a explicit computation of the
cocycle $c$ norm leads to a generalization of Kuhn Vershik [18] theorem 5 and
as a byproduct, using [4] Theorem 1, allows us to characterizes geometrically
the existence of complementary series representations:

######  Theorem 2.

The following properties are equivalent:

  1. (1)

the distance $d$ over $X$ is roughly conditionally negative;

  2. (2)

the family of Knapp-Stein operators $(\mathcal{I}_{s})_{s}$ over
$L^{2}({\partial}X,\mu_{o})$ are positive for all $s>\frac{1}{2}$ and thus $G$
admits complementary series over the all interval $[0,1]$ in the sense of [4];

  3. (3)

the operator $\mathcal{I}^{\prime}$ is positive over
$L^{2}({\partial}X,\mu_{o})\ominus{\mathbb{C}}$.

Even if every a-T-menable group acts properly on a infinite dimensional real
hyperbolic space [21], which admits conditionally negative distance, it is a
open question whether all a-T-menable CAT(-1) groups act geometrically on a
proper Gromov hyperbolic space with a roughly conditionally negative distance.

From another perspective the cocycle $c$ might be seen as derived from the
complementary series in the sense:

$(\mathcal{I}^{\prime}[c(g)],\varphi)=\lim_{s\rightarrow
1}\frac{(\mathcal{I}_{s}[\pi_{s}(g){\bf 1}],\varphi)}{(1-s)\delta}$

, where $\varphi\in{\mathcal{C}}({\partial}X)$ has zero average, and thus
keeps track of the equidistribution phenomenons already observed for these
representations in [15] [4] [5]. This led us to the following cocycle analog
of the Roblin-Margulis equidistribution theorem:

######  Theorem 3.

Let $(\nu_{o,t})_{t}$ be an approximation over $\overline{X}=X\cup{\partial}X$
of the Patterson-Sullivan probability $\mu_{o}$ as in Corollary 2.6 . Then for
every $u,v\in\mathcal{H}$ and $f_{1},f_{2}\in{\mathcal{C}}(\overline{X})$ with
$\mu_{o}(f_{1})=\mu_{o}(f_{2})=0$ the sequence of averages:

$\int_{G}[f_{1}(g.o)Q^{\prime}_{1}(c(g),u)][f_{2}(g^{-1}.o)Q^{\prime}_{1}(c(g^{-1}),v)]d\nu_{o,t}(g)$

converges to

$Q^{\prime}(u,[f_{1}|_{{\partial}X}])Q^{\prime}(v,[f_{2}|_{{\partial}X}])$

when $t\rightarrow+\infty$.

As introduced by Neretin in [20] and investigated in [3] and [1], an isometric
$G$-affine action over a Hilbert space $\mathcal{H}$ is affine irreducible
(AI) if $\mathcal{H}$ is the only closed affine subspace $G$-stable.

On the way of proving the cocycle equidistribution theorem it appears that a
weaker version of it (cf. Proposition 5.2) implies:

######  Theorem 4.

The affine commutant of the $G$-affine action $(\pi_{1},c)$ is trivial. In
particular the action is irreducible.

###  1.1. Outlines

After introducing standard materials about Paterson-Sullivan theory in Section
2, we construct the special representations in Section 3. The Section 4 is
dedicated to cohomological aspects of these representations where Theorem 1
and 2 are proved. Section 5 is dedicated to the proof of the equidistribution
Theorem 3 and 4 concerning the irreducible of the affine actions
$(\pi_{1},[c])$.

###  1.2. Notations and conventions

Even though most of the arguments run the same way for orthogonal
representations, throughout this paper we only consider complex Hilbert
spaces.

Given a topological space $Z$, the space of continuous functions over $Z$ is
endowed with topology of uniform convergence over compact sets and is denoted
${\mathcal{C}}(Z)$. The space of Borel probabilities over $Z$ is considered
with its $*$-weak topology.

The notations $r$ and $R$ stand for specific constants introduced in
subsection 2.4 and $o\in X$ for a fixed basepoint.

In order to avoid the escalation of constants coming from estimates up to
controlled additive or multiplication error terms we will use the following
conventions. Given two real valued functions, $a,b$, over a set $Z$, we write
$a\preceq b$ if there exists $C>0$ such that $a(z)\leq Cb(z)$ for all $z\in Z$
and $a\asymp b$ if $a\preceq b$ and $b\preceq a$. Analogously we write
$a\lesssim b$ if there exists $c$ such that $a(z)\leq b(z)+c$ and $a\simeq b$
if $a\lesssim b$ and $b\lesssim a$.

##  2\. preliminaries

In this section we introduce basic materials needed throughout this paper as
well as some results and notations from [4].

A proper geodesic space $(X,d)$ is called CAT(-1) if for any geodesic triangle
$\Delta$ its comparison $\overline{\Delta}$ in $\mathbb{H}^{2}$ [9] satisfies:

$d(x,y)\leq d_{\mathbb{H}^{2}}(\overline{x},\overline{y})$

for all $x,y\in\Delta$.

###  2.1. Hyperbolic spaces and compactifications.

Given a basepoint $o\in X$, the Gromov product over $X$ at $o\in X$ is defined
as:

$\langle x,y\rangle_{o}=\frac{1}{2}(d(x,o)+d(y,o)-d(x,y))$

for $x,y\in X$. A space $(X,d)$ is called Gromov hyperbolic if one can find a
positive constant $c_{X}\geq 0$ such that:

$\langle x,y\rangle_{w}\geq\min\\{\langle x,z\rangle_{w},\langle
y,z\rangle_{w}\\}-c_{X}$

for all $x,y,z,w\in X$. Every proper geodesic CAT(-1) space satisfies this
condition [9].

####  2.1.1. Gromov boundaries of CAT(-1) spaces.

Given a CAT(-1) space $(X,d)$ its Gromov boundary ${\partial}X$ is constructed
as follows:
A sequence of points $(x_{n})_{n}$ in $X$ converges to infinity if
$\lim_{n,m}\langle x_{n},x_{m}\rangle_{o}\rightarrow+\infty$. Since $(X,d)$ is
hyperbolic the relation defined over the set of sequences which converge to
infinity, $X^{\infty}\subset X^{\mathbb{N}}$ as:

$(a_{n})\mathcal{R}(b_{n})\quad\text{if and only if}\quad\lim_{n,m}\langle
a_{n},b_{m}\rangle_{o}\rightarrow\infty$

with $(a_{n})_{n},(b_{n})_{n}\in X^{\infty}$, is a equivalence relation
independent of the basepoint $o$. The boundary ${\partial}X$ of $X$ is defined
as the set of equivalence classes for this relation and the class of a
sequence $(x_{n})_{n}\in X^{\infty}$ is denoted $\lim_{n}x_{n}=\xi$.
Note that the boundary ${\partial}X$ is visual in the sense that for every
distinct pair of points $\xi_{+},\xi_{-}\in{\partial}X$ there exists a
isometry $(\xi_{t})_{t}$ from ${\mathbb{R}}$ to $(X,d)$ such that
$\lim_{t\rightarrow+\infty}\xi_{t}=\xi_{+}$ and
$\lim_{t\rightarrow-\infty}\xi_{t}=\xi_{-}$ [9].

In the case of CAT(-1) spaces the Gromov product extends continuously [7] to
$\overline{X}=X\cup{\partial}X$ by the formula:

$\langle x,y\rangle_{o}=\lim_{n,m}\langle
x_{n},y_{m}\rangle_{o}\in[0,+\infty]$

where $(x_{n})_{n}$ and $(y_{m})_{m}$ are sequences that converge respectively
to $x$ and $y$ in $\overline{X}=X\cup{\partial}X$. Moreover, given $x\in X$,
the kernel:

$d_{x}:(\xi,\eta)\in{\partial}X\times{\partial}X\mapsto\begin{cases}e^{-\langle\xi,\eta\rangle_{x}}\,\text{when
$\xi\neq\eta$}\\\ 0\,\text{otherwise}\end{cases}$

defines a distance over ${\partial}X$ called visual distance at $x$ [6] which
induces the natural topology considered over ${\partial}X$ independent of
$x\in X$. The space $\overline{X}=X\cup{\partial}X$ admits a compact
metrizable topology compatible with the topology of $X$ and ${\partial}X$ such
that a sequence $(x_{n})_{n}$ in $\overline{X}$ converges to
$\xi\in{\partial}X$ if and only if $\langle
x_{n},\xi\rangle_{o}\rightarrow\infty$.

####  2.1.2. CAT(-1) spaces and Busemann cocycles

A essential object in our framework is the Busemann function $b$. Given
$\xi\in{\partial}X$, any geodesic $(\xi_{t})_{t}$ that converges to $\xi$ and
$x,y\in X$, $b_{\xi}(x,y)$ is defined as $\lim_{t}d(x,\xi_{t})-d(y,\xi_{t})$.
The map $b_{\bullet}(\bullet,\bullet):\overline{X}\times X\times
X\rightarrow{\mathbb{R}}$ is continuous and satisfies:

$\langle
x^{\prime},y^{\prime}\rangle_{o}=\frac{1}{2}\sup_{z}[b_{x^{\prime}}(o,z)+b_{y^{\prime}}(o,z)]=\frac{1}{2}[b_{x^{\prime}}(o,p)+b_{y^{\prime}}(o,p)]\in{\mathbb{R}}_{+}$

where $p$ belongs to the geodesic between
$x^{\prime},y^{\prime}\in\overline{X}$. Observe that for any
$x^{\prime}\in\overline{X}$ the map $b_{x^{\prime}}:X\times
X\rightarrow{\mathbb{R}}$ is a additive cocycle over $X$. As a consequence the
family of visual distances $\\{d_{x}\\}_{x\in X}$ satisfy the conformal
relation:

$d_{x}(\xi,\eta)=e^{\frac{1}{2}[b_{\xi}(x,x^{\prime})+b_{\eta}(x,x^{\prime})]}d_{x^{\prime}}(\xi,\eta)$

for any $\xi,\eta\in{\partial}X$ and $x,x^{\prime}\in X$.

The group of isometries, $\text{Is}(X)$, of $(X,d)$ is endowed with the
compact-open topology that makes it locally compact and second countable.
Under these assumptions the action of $\text{Is}(X)$ on $(X,d)$ is continuous,
has closed orbits and compact stabilizers. Its action extends continuously to
$\overline{X}$ by homeomorphisms and satisfies the invariance relations:

$b_{g.x}(g.y,g.z)=b_{x}(y,z)$

and

$d_{g.x}(g.\xi,g.\eta)=d_{x}(\xi,\eta)$

for any $g\in\text{Is}(X)$, $x\in\overline{X}$, $y,z\in X$ and
$\xi,\eta\in{\partial}X$.

###  2.2. Boundary retractions

The limit set of a discrete subgroup $G\subset\text{Is}(X)$ is defined as
$\Lambda(G)=\overline{G.o}\cap{\partial}X$. Given $\Lambda(G)$ the geodesic
hull $\mathcal{Q}(\Lambda(G))\subset X$ is defined as the union of geodesics
with endpoints in $\Lambda(G)$. The group $G$ is called convex-cocompact if it
acts cocompactly over $\mathcal{Q}(\Lambda(G))$ or equivalently if there
exists a uniform constant $R_{X,G}$ such that any geodesic from a given
basepoint $o\in X$ to $\xi\in\Lambda(G)$ stays within a $R_{X,G}$-neighborhood
of $G.o$. In particular cocompact subgroups are convex-cocompact [12] [26].

A measurable map $f:\overline{\mathcal{Q}(\Lambda(G))}\to\Lambda(G)$ is called
boundary retraction if one can find $c_{X,f}$ such that $\langle
f(x),x\rangle_{o}\geq d(o,x)-c_{X,f}$ for all $x\in\mathcal{Q}(\Lambda(G))$
and $f|_{\Lambda(G)}={\text{id}}_{\Lambda(G)}$. In particular $f$ is
continuous at any point of $\Lambda(G)$.

The shadow at $x\in X$ of radius $\rho$ from the basepoint $o\in X$ is defined
as:

$\mathcal{O}_{o}(x,\rho)=\\{\xi\in{\partial}X|\,\text{$\langle\xi,x\rangle_{o}\geq
d(x,o)-\rho$}\\}$

There exists $r_{X,o}>0$ such that $\mathcal{O}_{o}(x,r_{X,o})\neq\emptyset$
for all $x\in G.o$. Using a compact fundamental domain for the action of $G$
over $\mathcal{Q}(\Lambda(G))$ one can show that boundary retractions exist
(cf [4] [8]).

For the rest of this paper a boundary retraction, $f$, is fixed. Given
$x\in\mathcal{Q}(\Lambda(G))$ and $g\in\text{Isom}(X,d)$ we denote
$\widehat{x}=f(x)\in{\partial}X$, $\widehat{g}=f(g.o)\in{\partial}X$ and
$\widecheck{g}=f(g^{-1}.o)\in{\partial}X$.

###  2.3. The Patterson-Sullivan theory

Let $G\subset\text{Is}(X)$ be a discrete convex-cocompact subgroup of
isometries of a proper CAT(-1) space $(X,d)$ that is non-elementary, i.e. $G$
is not virtually abelian.

######  Definition 2.1.

The critical exponent of $G$ denoted $\delta\in[0,+\infty]$, is defined as the
infimum over all $s\geq 0$ such that the Poincaré integral at $s$:

$\int_{G}e^{-sd(g.o,o)}dg$

is finite.

It follows from [12] that $G$ has finite critical exponent and is of divergent
type, i.e. its Poincaré integral diverges at $s=\delta$.

######  Definition 2.2.

A $\alpha$-density, for some positive $\alpha$, is a continuous
$G$-equivariant map:

$\mu_{\bullet}:X\mapsto\text{Prob}({\partial}X);\quad x\mapsto\mu_{x}$

such that for all $x,y\in X$, $\mu_{x}\sim\mu_{y}$ and

$\frac{d\mu_{x}}{d\mu_{y}}(\xi)=e^{-\alpha b_{\xi}(x,y)}$

for $[\mu]$-almost every $\xi\in{\partial}X$.

Let us denote $m$ the push-forward of the Haar measure of $G$ over the orbit
$G.o$ where $o\in X$ stands for some basepoint. A $\delta$-density over
${\partial}X$ can be constructed from $m$ as follows.
Let

$\mu_{\bullet,t}:X\rightarrow\text{Prob}(\overline{X});\quad
x\mapsto\mu_{x,t}$

be the map defined as:

$d\mu_{x,t}(y)=\frac{1}{\mathcal{W}(t)}e^{-td(x,y)}dm(y)$

where

$\mathcal{W}(t)=\int_{X}e^{-td(x,y)}dm(y)$

and $t>\delta$.
The family $(\mu_{\bullet,t})_{t>\delta}$ is $G$-equivariant, equicontinuous
and bounded on compact sets for any $t\in(\delta,\delta+1]$ [10]. It is
therefore pre-compact as elements of
${\mathcal{C}}_{G}(X,\text{Prob}(\overline{X}))$ and any of its accumulation
point $\mu_{\bullet}$ defines a $\delta$-density supported on the limit set of
$G$, $\overline{G.o}\cap{\partial}X=\Lambda(G)$.

######  Theorem ([12]).

Under the assumptions over $G$ the map
$\mu_{\bullet}\in{\mathcal{C}}_{G}(X,\text{Prob}({\partial}{X}))$ is unique,
non-atomic and ergodic.

The following alternative form of the Shadow lemma for the family
$(\mu_{\bullet,t})_{t>\delta}$ is proved in [4]:

######  Lemma 2.3 (Shadow lemma).

There exist $T>\delta$ and $r_{0}$ such that:

$\mu_{o,t}(\mathcal{U}(o,x,\rho))\asymp e^{\delta\rho}e^{-\delta d(o,x)}$

where

$\mathcal{U}(o,x,\rho)=\\{z\in\overline{X}\,|\,\langle z,x\rangle_{o}\geq
d(o,x)-\rho\\}\subset\overline{X}$

$o\in X$, $x\in\mathcal{Q}(\Lambda(G))$, $\rho\geq r_{0}$ and $\delta<t\leq
T$.
In particular:

$\mu_{o,t}(\\{z\in\overline{X}\,|\,\langle z,\xi\rangle_{o}\geq\rho\\})\asymp
e^{-\delta\rho}$

for all $\xi\in\Lambda(G)$, $\rho\geq r_{0}$ and $\delta<t\leq T$.

As a consequence of Lemma 2.3 the probabilities $(\mu_{x})_{x\in X}$ are
Ahlfors regular and $\mu_{o}(\mathcal{O}_{o}(x,\rho))\neq 0$ for all
$x\in\mathcal{Q}(\Lambda(G))$ and $\rho>r_{0}$.

###  2.4. Boundary covering and equidistribution

The visual ball centered at $\xi\in{\partial}X$ of radius $e^{-\rho}$ is
defined as:

$B_{o}(\xi,e^{-\rho})=\\{\eta\in{\partial}X\,|\,\langle\xi,\eta\rangle_{o}\geq\rho\\}=\\{\eta\in{\partial}X\,|\,d_{o}(\xi,\eta)\leq
e^{-\rho}\\}$

Given $\varsigma>r_{0}$ (cf. Lemma 2.3), the straight and inverted cones from
$o\in X$ to $\xi\in{\partial}X$ of radius $\rho$ respectively as:

${\bf C}^{+}_{o}(\xi;\rho,\varsigma)=\\{g\in
G\,|\,\mathcal{O}_{o}(g.o,\varsigma)\cap B_{o}(\xi,e^{-\rho})\neq\emptyset\\}$
${\bf C}^{-}_{o}(\xi;\rho,\varsigma)=\\{g\in
G\,|\,\mathcal{O}_{o}(g^{-1}.o,\varsigma)\cap
B_{o}(\xi,e^{-\rho})\neq\emptyset\\}$

Since $G$ is convex-cocompact there exists $R_{X,G}>0$ such that every
geodesic from a basepoint $o\in X$ to $\xi\in\Lambda(G)$ is at distance at
most $R_{X,G}$ from $G.o$. Let us fix once for all a universal constant $R\geq
R_{X,G}$ such that for all $g,h\in G$, there exists $g^{\prime}\in G$ with
$d(g.o,g^{\prime}.o)\leq R$ and $d(gh.o,o)\geq d(g.o,o)+d(h.o,o)-2R$. The
existence of such a constant follow from Milnor lemma together with [15] Lemma
4.4. In the rest $R$ will refer to this particular constant. We denote:

$S_{o,G}(t)=\\{g\in G\,|\,tR\leq d(g.o,o)<(t+1)R\\}\subset G.$

The following counting lemma corresponds to Lemma 4.3 of [15]:

######  Lemma 2.4.

Let $r_{0}$ be as in Lemma 2.3. There exists $r\geq r_{0}$ such that for all
$\xi\in\Lambda(G)$ and $t\geq\frac{\rho}{R}$:

$|{\bf C}^{+}_{o}(\xi;\rho,r)\cap S_{o,G}(t)|\asymp_{o,r}e^{\delta
tR}e^{-\delta\rho}$

In the rest $r$ will refer to this particular constant.

The following results can be find in [4] Lemma 2.6 and [8] Theorem 3.2.

######  Lemma 2.5 (Vitali cover lemma).

There exists $r^{\prime}>r$ such that for all $t>0$ there exist
$S^{*}_{o,G}(t)\subset S_{o,G}(t)$ and a family of measurable subsets
$(O^{(2)}_{o}(g))_{g\in S^{*}_{o,G}(t)}$ of ${\partial}X\times{\partial}X$
such that:

  1. (1)

$|S^{*}_{o,G}(t)|\asymp e^{\delta tR}$;

  2. (2)

$\bigcup_{g\in
S^{*}_{o,G}(t)}O^{(2)}_{o}(g)\supseteq\Lambda(G)\times\Lambda(G)$;

  3. (3)

$O^{(2)}_{o}(g)\cap O^{(2)}_{o}(h)=\emptyset$ for all $g\neq h$ in
$S^{*}_{o,G}(t)$;

  4. (4)

$\mathcal{O}_{o}(g.o,R(t))\times\mathcal{O}_{o}(g^{-1}.o,R(t))\subset
O^{(2)}_{o}(g)$

and

$O^{(2)}_{o}(g)\subset\mathcal{O}_{o}(g.o,R^{\prime}(t))\times\mathcal{O}_{o}(g^{-1}.o,R(t)^{\prime})$

where $R(t)=\frac{1}{2}tR+r$ and $R^{\prime}(t)=\frac{1}{2}tR+r^{\prime}$.

As a consequence of the Lebesgue differentiation theorem for Ahlfors regular
measures:

######  Corollary 2.6.

The sequence of probabilities $(\nu_{o,t})_{t>0}$ over $G$ defined as:

$\nu_{o,t}=\int_{S^{*}_{o,G}(t)}\mu_{o}(O^{(2)}_{o}(g)).\mathcal{D}_{g}dg$

,where $\mathcal{D}_{g}$ stands for the Dirac mass at $g\in G$, are supported
over $S^{*}_{o,G}(t)$ for any $t>0$ and satisfy:

$\int_{G}\Psi(g.o,g^{-1}.o)d\nu_{o,t}(g)\rightarrow\int_{{\partial}X\times{\partial}X}\Psi(\xi,\eta)d\mu_{o}(\xi)d\mu_{o}(\eta)$

for all $\Phi\in{\mathcal{C}}({\partial}X\times{\partial}X)$.

##  3\. Special representations and infinitesimal generator of complementary
series.

Throughout the rest of this paper $G$ is assumed to be a non-elementary
convex-cocompact group acting over a CAT(-1) space $(X,d)$.
Under our assumptions the probability $\mu_{o}$ over ${\partial}X$ has
Hausdorff dimension $\delta$ which means that:

$\sup_{\xi}\int_{{\partial}X}\frac{1}{d_{o}^{\alpha}(\xi,\eta)}d\mu_{o}(\eta)<+\infty$

for all $0\leq\alpha<\delta$ [26].

We denote $Q$ the natural bilinear form over ${\mathcal{C}}({\partial}X)$
defined as

$Q(\varphi)=\int_{{\partial}X}|\varphi|^{2}(\xi)d\mu_{o}(\xi)$

with $\varphi\in{\mathcal{C}}({\partial}X)$. As proved in [4] $Q$ defines a
non-degenerated coupling between $\mathcal{F}_{s}$ and $\mathcal{F}_{1-s}$:

######  Lemma 3.1 ([4]).

The bilinear map:

$Q:\mathcal{F}_{s}\times\mathcal{F}_{1-s}\rightarrow{\mathbb{C}};\quad(f_{1},f_{2})\mapsto\int_{{\partial}X}f_{1}(\xi)\overline{f_{2}}(\xi)d\mu_{o}(\xi)$

is continuous and $G$-invariant. Moreover the operator
$Q:\mathcal{F}_{s}\rightarrow\mathcal{F}_{1-s}^{*}$ defines an $G$-intertwiner
with

$\text{Ker}[Q]=\\{f\in\mathcal{F}_{s}\,|\,f|_{\Lambda(G)}=0\\}.$

As a consequence, given any sub-representation
$\mathcal{T}_{1-s}\subset\mathcal{F}_{1-s}$ and
$\mathcal{T}_{s}\subset\mathcal{F}_{s}$ its $Q$-orthogonal in
$\mathcal{F}_{s}$, the operator:

$T\in\text{End}_{G}(\mathcal{T}_{s},\mathcal{F}_{1-s}/{\mathcal{T}_{1-s}})\mapsto
Q(T[\bullet],\bullet)\in\text{Bil}_{G}(\mathcal{T}_{s})$

that maps intertwiners between $\mathcal{T}_{s}$ and
$\mathcal{F}_{1-s}/{\mathcal{T}_{1-s}}$ to $G$-invariant bilinear forms over
$\mathcal{T}_{s}$ is well defined.

In our case we consider $\mathcal{F}_{1}^{0}\subset\mathcal{F}_{1}$ be the
$Q$-orthogonal space of $\pi_{0}$-fixed vectors in $\mathcal{F}_{0}$. Because
$G$ acts ergodically over ${\partial}X$ this sub-representation corresponds to
the space of $1$-homogeneous functions with zero integral. Observe that the
coupling $Q$ induces a correspondence between the Banach $G$-representations
$\mathcal{F}_{1}^{0}$ and
$\mathcal{F}_{0}^{0}=\mathcal{F}_{0}/{\mathbb{C}}{\bf 1}$.

The reader familiar with the subject should think of $\mathcal{F}_{1}^{0}$ as
the analogue of $\Omega_{0}^{\text{top}}(G/P)$, the space differential forms
of maximal degree with zero integral over the manifold $G/P$ where $G$ stands
for a rank 1 Lie group and $P$ a minimal parabolic subgroup of $G$.

In this section we construct dual unitary structures over the spaces
$\mathcal{F}_{1}^{0}$ and
$\mathcal{F}_{0}^{0}=\mathcal{F}_{0}/{\mathbb{C}}{\bf 1}$ and prove a
preliminary result about their relation with complementary series (cf.
Proposition 3.5).

Analogously to [4] Proposition 3.4 we have:

######  Lemma 3.2.

Given any $o\in X$ the operator $\mathcal{I}^{\prime}_{(o)}$ defined as:

$\mathcal{I}^{\prime}_{(o)}:{\mathcal{C}}({\partial}X)\rightarrow{\mathcal{C}}(\overline{X});\quad\varphi\mapsto[z\mapsto\int_{{\partial}X}\varphi(\xi)\langle\xi,z\rangle_{o}d\mu_{o}(\xi)]$

is well defined and bounded.

###### Proof.

Let $k_{1}\in{\mathcal{C}}({\mathbb{R}}_{+})$ be a positive function such that
$k_{1}|_{[0,1]}=1$ and $k_{1}|_{[2,+\infty)}=0$. Let us denote
$k_{\tau}\in{\mathcal{C}}({\mathbb{R}}_{+})$ with $\tau>0$ be the function
defined as $k_{\tau}(t)=k_{1}(\frac{t}{\tau})$ for $t\in{\mathbb{R}}_{+}$.
Then the kernel , $K_{\tau}$, on $\overline{X}\times{\partial}X$ defined as
$(x,\eta)\mapsto k_{\tau}(\langle x,\eta\rangle_{o}).\langle
x,\eta\rangle_{o}$ is continuous and bounded. In particular:

$\mathcal{I}^{\prime}_{(o),\tau}[\varphi]:\overline{X}\rightarrow{\mathbb{C}};\quad
x\mapsto\int_{{\partial}X}\varphi(\eta)K_{\tau}(x,\eta)d\mu_{o}(\eta)$

is continuous for any fixed $\tau\geq 0$ and
$\varphi\in{\mathcal{C}}({\partial}X)$.

On the other hand the Ahlfors regularity of $\mu_{o}$ (cf Lemma 2.3) implies:

$\displaystyle|\mathcal{I}^{\prime}_{(o),\tau}(\varphi)(x)-\mathcal{I}^{\prime}_{(o),\tau^{\prime}}(\varphi)(x)|\leq\|\varphi\|_{\infty}\int_{\min\\{\tau,\tau^{\prime}\\}\leq\langle
x,\bullet\rangle_{o}}\langle x,\eta\rangle_{o}d\mu_{o}(\eta)$
$\displaystyle\leq\|\varphi\|_{\infty}\int_{\min\\{\tau,\tau^{\prime}\\}}^{+\infty}t\mu_{o}(\\{\langle
x,\bullet\rangle_{o}>u\\})du\preceq\|\varphi\|_{\infty}\int_{\min\\{\tau,\tau^{\prime}\\}}^{+\infty}te^{-\delta
t}du$

which goes to zero when $\tau$ and $\tau^{\prime}$ go to infinity. In other
words $(\mathcal{I}^{\prime}_{(o),\tau}(\varphi))_{\tau}$ is Cauchy for the
uniform norm over $\overline{X}$ and its limit,
$\mathcal{J}^{\prime}_{(o)}[\varphi]$, is therefore continuous.

Since $K_{\tau}(x,\bullet)$ is dominated by the integrable function
$\eta\mapsto\langle x,\eta\rangle_{o}$ for any fixed $x\in\overline{X}$ the
dominated convergence theorem implies that:

$\mathcal{J}^{\prime}_{(o)}[\varphi](x)=\mathcal{I}^{\prime}_{(o)}[\varphi](x)$

for all $x\in\overline{X}$ and thus $\mathcal{I}^{\prime}_{(o)}$ is well
defined. Moreover:

$\displaystyle|\mathcal{I}^{\prime}_{(o)}[\varphi](x)|$
$\displaystyle\leq\|\varphi\|_{\infty}\int_{{\partial}X}\langle\xi,x\rangle_{o}d\mu_{o}(\xi)$
$\displaystyle\preceq\|\varphi\|_{\infty}\int_{0}^{d(o,x)}te^{-\delta t}dt$
$\displaystyle=\frac{1}{\delta^{2}}[e^{-\delta d(o,x)}(1-\delta
d(o,x))-1]\|\varphi\|_{\infty}\preceq\|\varphi\|_{\infty}$

and thus $\mathcal{I}^{\prime}_{(o)}$ is bounded. ∎

Similarly to the Knapp-Stein operators
$(\mathcal{I}_{s})_{\Re{s}>\frac{1}{2}}$, the operator
$\mathcal{I}^{\prime}_{(o)}$ can be seen as an intertwiner:

######  Proposition 3.3.

As an operator between $\mathcal{F}^{0}_{1}$ and $\mathcal{F}_{0}^{0}$,
$\mathcal{I}^{\prime}_{(x)}$ is independent of the choice of $x\in X$ and will
be denoted $\mathcal{I}^{\prime}$. Moreover $\mathcal{I}^{\prime}$ induces an
intertwiner between these Banach representations and the quadratic form over
$\mathcal{F}^{0}_{1}$ given as:

$Q^{\prime}_{1}(\varphi,\psi)=Q(\mathcal{I}^{\prime}[\varphi],\psi)=\int_{{\partial}X\times{\partial}X}\varphi(\xi)\overline{\varphi}(\eta)\langle\xi,\eta\rangle_{x}d\mu_{o}(\xi)d\mu_{o}(\eta)$

is independent of $x\in X$ and $G$-invariant.

###### Proof.

Given any $\varphi\in\mathcal{F}^{0}_{1}$ and $x,y\in X$ one has:

$\displaystyle\mathcal{I}^{\prime}_{(x)}[\varphi](\eta)=\int_{{\partial}X}\varphi(\xi)\langle\xi,\eta\rangle_{x}d\mu_{o}(\xi)$
$\displaystyle=\int_{{\partial}X}\varphi(\xi)[\langle\xi,\eta\rangle_{y}+\frac{1}{2}b_{\xi}(x,y)+\frac{1}{2}b_{\eta}(x,y)]d\mu_{o}(\xi)$
$\displaystyle=\int_{{\partial}X}\varphi(\xi)\langle\xi,\eta\rangle_{y}d\mu_{o}(\xi)+\frac{1}{2}\int_{{\partial}X}\varphi(\xi)b_{\xi}(x,y)d\mu_{o}(\xi)$
$\displaystyle=\mathcal{I}^{\prime}_{(y)}[\varphi](\eta)+\frac{1}{2}\int_{{\partial}X}\varphi(\xi)b_{\xi}(x,y)d\mu_{o}(\xi)$

for almost every $\eta\in{\partial}X$, where the third equality uses the zero
average assumption on $\varphi$. In other words
$\mathcal{I}^{\prime}_{(x)}[\varphi]-\mathcal{I}^{\prime}_{(y)}[\varphi]\in{\mathbb{C}}$
and thus
$\mathcal{I}^{\prime}_{(x)}[\varphi]=\mathcal{I}^{\prime}_{(y)}[\varphi]$ on
$\mathcal{F}_{0}^{0}$.
It follows:

$\displaystyle\mathcal{I}^{\prime}_{(o)}[\pi_{1}(g)\varphi](\eta)=\int_{{\partial}X}e^{-\delta
b_{\xi}(g.o,o)}\varphi(g^{-1}\xi)\langle\xi,\eta\rangle_{o}d\mu_{o}(\xi)$
$\displaystyle=\int_{{{\partial}X}}\varphi(\xi)\langle\xi,g^{-1}\eta\rangle_{g^{-1}o}d\mu_{o}(\xi)=\pi_{0}(g)\mathcal{I}^{\prime}_{(g^{-1}.o)}[\varphi](\eta)$
$\displaystyle=\pi_{0}(g)\mathcal{I}^{\prime}_{(o)}[\varphi](\eta)+\frac{1}{2}\int_{{{\partial}X}}\varphi(\xi)b_{\xi}(g^{-1}.o,o)d\mu_{o}(\xi)$

for any $\varphi\in\mathcal{F}^{0}_{1}$, $g\in G$ and almost every
$\eta\in{\partial}X$. In other words
$\mathcal{I}^{\prime}[\pi_{1}(g)\varphi]=\pi_{0}(g)\mathcal{I}^{\prime}[\varphi]$
on $\mathcal{F}_{0}^{0}$. The rest follows from the properties of the
quadratic form $Q$ described above.

∎

######  Remark 3.4.

Using the Cauchy-Schwarz inequality together with the Ahlfors regularity of
$\mu_{o}$, we have:

$\displaystyle\int|\mathcal{I}^{\prime}_{(x)}[\varphi](\eta)|^{2}d\mu_{o}(\eta)$
$\displaystyle\leq\int_{{\partial}X}\int_{{\partial}X}|\langle\xi^{\prime},\eta\rangle_{x}|^{2}d\mu_{o}(\xi^{\prime})\int_{{\partial}X}|\varphi(\xi)|^{2}|\langle\xi,\eta\rangle_{x}|^{2}d\mu_{o}(\xi)d\mu_{o}(\eta)$
$\displaystyle\preceq\int_{{\partial}X}2[\int_{0}^{+\infty}t\mu_{o}({\langle\xi,\bullet\rangle_{x}>t})dt]|\varphi(\xi)|^{2}d\mu_{o}(\xi)$
$\displaystyle\preceq 2[\int_{0}^{+\infty}te^{-\delta
t}dt]\int_{{\partial}X}|\varphi(\xi)|^{2}d\mu_{o}(\xi)\preceq\|\varphi\|^{2}_{2}$

and thus $\mathcal{I}^{\prime}_{(x)}$ extends as a self-adjoint bounded
operator over $L^{2}({\partial}X,\mu_{o})$ which satisfies
$\mathcal{I}^{\prime}_{(x)}\circ\pi_{1}(g)-\pi_{0}(g)\circ\mathcal{I}^{\prime}_{(x)}\in{\mathbb{C}}$
over $L^{2}({\partial}X,\mu_{o})\ominus{\mathbb{C}}$.

The following result partially establishes the relation between the spectrum
of the operators $\mathcal{I}^{\prime}_{(x)}$ and the family of Knapp-Stein
operators $(\mathcal{I}_{s})_{s}\in[\frac{1}{2},1]$ over
$L^{2}({\partial}X,\mu_{o})$:

######  Proposition 3.5.

Assume the family Knapp-Stein operators
$(\mathcal{I}_{s})_{s\in(\frac{1}{2},1]}$ are positive over
$L^{2}({\partial}X,\mu_{o})$ for $s$ in a neighborhood of $1$, then the
operator $\mathcal{I}^{\prime}_{(o)}$ is positive over
$L^{2}({\partial}X,\mu_{o})\ominus{\mathbb{C}}$ or equivalently the quadratic
form $Q^{\prime}_{1}$ over $\mathcal{F}^{0}_{1}$ is positive.

###### Proof.

Given $s>\frac{1}{2}$ and $\varphi\in L^{\infty}({\partial}X,\mu_{o})\subset
L^{2}({\partial}X,\mu_{o})$, the dominated convergence theorem implies:

$\displaystyle\frac{(\mathcal{I}_{s}[\varphi],\varphi)}{2(1-s)\delta}=\frac{1}{2(1-s)\delta}\int_{{\partial}X\times{\partial}X}\frac{\varphi(\xi)\overline{\varphi}(\eta)}{d_{o}^{2(1-s)\delta}(\xi,\eta)}d\mu_{o}(\xi)d\mu_{o}(\eta)$
$\displaystyle=\frac{1}{2(1-s)\delta}\int_{{\partial}X\times{\partial}X}\varphi(\xi)\overline{\varphi}(\eta)e^{2(1-s)\delta\langle\xi,\eta\rangle_{o}}d\mu_{o}(\xi)d\mu_{o}(\eta)$
$\displaystyle=\frac{1}{2(1-s)\delta}|\int_{{\partial}X}\varphi(\xi)d\mu_{o}(\xi)|^{2}+\int_{{\partial}X\times{\partial}X}\varphi(\xi)\overline{\varphi}(\eta)\langle\xi,\eta\rangle_{o}d\mu_{o}(\xi)d\mu_{o}(\eta)+o_{s\sim
1}(1)$
$\displaystyle=\frac{1}{2(1-s)\delta}|\int_{{\partial}X}\varphi(\xi)d\mu_{o}(\xi)|^{2}+(\mathcal{I}^{\prime}_{(o)}[\varphi],\varphi)+o_{s\sim
1}(1)$

for $s$ closed to $1$. If the operators $\mathcal{I}_{s}$ are positive on
$L^{2}({\partial}X,\mu_{o})$ for $s$ closes enough to $1$ and $\varphi$ has
zero integral it follows:

$\frac{(\mathcal{I}_{s}[\varphi],\varphi)}{2(1-s)\delta}=(\mathcal{I}^{\prime}_{(o)}[\varphi],\varphi)+o_{s\sim
1}(1)\geq 0$

which, using the density of $L^{\infty}({\partial}X,\mu_{o})\subset
L^{2}({\partial}X,\mu_{o})$, implies the positivity of the bounded operator
$\mathcal{I}^{\prime}_{(o)}$. ∎

In the rest $\mathcal{H}_{1}^{\prime}$ stands for the unitary representation
obtained from the Hilbert completion of the triple
$(\mathcal{F}_{1}^{0},Q^{\prime}_{1},\pi_{1})$.

######  Corollary 3.6.

There exists a unitary structure over the sub-representation
$\mathcal{I}^{\prime}[\mathcal{F}^{0}_{1}]$ of $\mathcal{F}^{0}_{0}$, denoted
$\mathcal{H}_{0}^{\prime}$, such that the intertwiner
$\mathcal{I}^{\prime}:\mathcal{F}_{1}^{0}\rightarrow\mathcal{I}^{\prime}[\mathcal{F}^{0}_{1}]$
extends as an isometric intertwiner between $\mathcal{H}_{1}^{\prime}$ and
$\mathcal{H}_{0}^{\prime}$. In addition $Q$ extends uniquely as a coupling
between $\mathcal{H}_{0}^{\prime}$ and $\mathcal{H}_{1}^{\prime}$ such that:

$|Q(v,w)|\leq\|v\|_{Q_{1}^{\prime}}\|w\|_{Q_{0}^{\prime}}$

for all $v\in\mathcal{H}_{1}^{\prime}$ and $w\in\mathcal{H}^{\prime}_{0}$.

###### Proof.

Using the self-adjointness of $\mathcal{I}^{\prime}_{(o)}$ over
$L^{2}({\partial}X,\mu_{o})$ it appears that the bilinear form over
$\mathcal{F}^{0}_{0}\subset L^{2}({\partial}X,\mu_{o})$, $Q^{\prime}_{0}$,
given as:

$Q_{0}^{\prime}(\varphi^{\prime},\phi^{\prime})=(\mathcal{I}^{\prime}_{(o)}[\varphi],\phi)$

for
$\varphi^{\prime}=\mathcal{I}^{\prime}_{(o)}[\varphi],\phi^{\prime}=\mathcal{I}^{\prime}_{(o)}[\phi]\in\mathcal{I}^{\prime}[\mathcal{F}^{0}_{1}]$
with $\varphi,\phi\in\mathcal{F}^{0}_{1}$ is well defined.

It follows

$Q_{0}^{\prime}(\varphi^{\prime})=(\mathcal{I}^{\prime}_{(o)}[\varphi],\varphi)\geq
0$

and

$Q_{0}^{\prime}(\pi_{0}(g)\varphi^{\prime})=(\mathcal{I}^{\prime}_{(o)}\pi_{1}(g)\varphi,\pi_{1}(g)\varphi)=(\mathcal{I}^{\prime}_{(o)}[\varphi],\varphi)=Q^{\prime}_{0}(\varphi^{\prime})$

for all
$\varphi^{\prime}=\mathcal{I}^{\prime}_{(o)}[\varphi]\in\mathcal{F}^{0}_{0}$
with $\varphi\in\mathcal{F}^{0}_{1}$ and $g\in G$.

By the very definition of $Q^{\prime}_{0}$ the operator
$\mathcal{I}^{\prime}_{(o)}:\mathcal{F}_{1}^{0}\rightarrow\mathcal{I}^{\prime}[\mathcal{F}^{0}_{1}]$
induces an unitary intertwiner between $\mathcal{H}_{1}^{\prime}$ and
$\mathcal{H}_{0}^{\prime}$ and using the Cauchy-Schwarz inequality we have:

$|Q(\varphi,\mathcal{I}^{\prime}_{(o)}[\phi])|\leq\|\varphi\|_{Q_{1}^{\prime}}\|\phi\|_{Q_{1}^{\prime}}=\|\varphi\|_{Q_{1}^{\prime}}\|\phi^{\prime}\|_{Q_{0}^{\prime}}$

for all $\varphi,\phi\in\mathcal{F}_{1}^{0}$ and
$\phi^{\prime}=\mathcal{I}^{\prime}_{(o)}[\phi]\in\mathcal{I}^{\prime}[\mathcal{F}^{0}_{1}]\subset\mathcal{F}^{0}_{0}$
and thus $Q$ extends as a coupling.∎

We conclude this section with the following result:

######  Lemma 3.7.

The representations $\mathcal{H}_{1}^{\prime}\simeq\mathcal{H}_{0}^{\prime}$
are mixing and thus do not admit finite dimensional sub-representations.

###### Proof.

Let $(g_{n})_{n}$ be a sequence in $G$ with $d(g_{n}.o,o)\rightarrow+\infty$,
$\varphi\in\mathcal{F}_{0}^{0}$ and $\phi\in\mathcal{F}_{1}^{0}$. Up to a
subsequence, one can assume that $(g_{n}.o)_{n}$ converges to
$\xi\in{\partial}X$. Then the dominated convergence implies:

$\displaystyle\lim_{n}Q(\pi_{0}(g_{n})\varphi,\phi)$
$\displaystyle=\lim_{n}(\pi_{0}(g_{n})\varphi,\phi)$
$\displaystyle=\varphi(\xi).({\bf 1},\phi)=0$

It follows, using Corollary 3.6 and the density of $\mathcal{F}_{1}^{0}$
(respectively, $\mathcal{F}_{0}^{0}$) in $\mathcal{H}_{1}^{\prime}$
(respectively $\mathcal{H}_{0}^{\prime}$), that $\lim_{g}Q(\pi_{0}(g)v,w)=0$
when $g.o$ goes to infinity with $v\in\mathcal{H}_{0}^{\prime}$ and
$w\in\mathcal{H}_{1}^{\prime}$. In addition, since $\mathcal{I}^{\prime}$ is
an unitary intertwiner between $\pi_{1}$ and $\pi_{0}$, the mixing of
$\mathcal{H}_{1}^{\prime}\simeq\mathcal{H}_{0}^{\prime}$ follows. ∎

##  4\. Cohomology and special representations

In this section we investigate the cohomology of the unitary representation
$\mathcal{H}_{1}^{\prime}\simeq\mathcal{H}_{0}^{\prime}$ and prove Theorem 1
and 2 that will respectively follows from Corollary 4.5 and 4.7.

Let us start with some notations and basic results about cohomology:

Given a unitary representation, $\pi$, the space of $\pi$-cocycles denoted
$Z^{1}(G,\pi)$ is defined as the subspace of maps
$b_{\pi}:G\rightarrow\mathcal{H}_{\pi}$ which satisfy:

$b(gh)=\pi(g)b(h)+b(g)$

for all $g,h\in G$ endowed with the compact-open topology. The subspace of co-
boundary $B^{1}(G,\pi)$ is defined as the range of operator:

${\partial}:\mathcal{H}_{\pi}\rightarrow Z^{1}(G,\pi);\quad
w\mapsto{\partial}_{w}(g)=w-\pi(g)w$

######  Definition 4.1.

Respectively the first cohomology and the reduced cohomology of $G$ with
coefficient in $\pi$ are defined as the quotient spaces:

$H^{1}(G,\pi)=Z^{1}(G,\pi)/{B^{1}(G,\pi)}$

respectively,

$\overline{H}^{1}(G,\pi)=Z^{1}(G,\pi)/\overline{B^{1}(G,\pi)}$

where $\overline{B^{1}(G,\pi)}$ stands for the closer of $B^{1}(G,\pi)$ inside
of $Z^{1}(G,\pi)$.

Given a cocycle $b_{\pi}\in Z^{1}(G,\pi)$, the formula:

$A_{b_{\pi}}(g)v=\pi(g)v+b_{\pi}(g)$

with $g\in G$ and $v\in\mathcal{H}_{\pi}$ defines an $G$-affine action with
linear part $\pi$ over $\mathcal{H}_{\pi}$. Conversely, any affine action with
linear part $\pi$ is induced by an element of $Z^{1}(G,\pi)$. The following
result is a standard fact:

######  Theorem ([2]).

Given a unitary representation $\pi$ and a cocycle $b\in Z^{1}(G,\pi)$, $b\in
B^{1}(G,\pi)$ if and only if its associated affine action has a fixed point or
equivalently has bounded orbits.

The first cohomology of $G$ with coefficient in $\pi$ classifies proper affine
actions with linear part $\pi$ up to conjugation by a affine translation.

As Proposition 3.5 proof shows, the quadratic forms associated to operators
$\mathcal{I}_{s}$ for $\frac{1}{2}<s\leq 1$ degenerate when $s$ goes to $1$ to
the $G$-spaces exact sequence:

$0\rightarrow\mathcal{F}^{0}_{1}\rightarrow\mathcal{F}_{1}\rightarrow{\mathbb{C}}\rightarrow
0$

Identifying the space $\mathcal{F}_{1}$ with continuous densities over
${\partial}X$ this exact sequence induces the cocycle:

$c:X\times X\rightarrow\mathcal{F}_{1}^{0};\quad(x,y)\mapsto\mu_{x}-\mu_{y}$

that is $G$-equivariant in the sense $c(g.x,g.y)=\pi_{1}(g)c(x,y)$ for all
$g\in G$ and $x,y\in X$. Because

$\displaystyle|c(x,y)|=|1-\frac{d\mu_{x}}{d\mu_{y}}|d\mu_{y}=\delta|b_{\xi}(x,y)+o(d(x,y))|d\mu_{y}\preceq
d(x,y)$

the cocycle $c$ defines a continuous map from $X\times X$ to
$\mathcal{F}_{1}^{0}$.

Even though the cocycle $c$ is our primer concern the following co-boundary
appears to be related to it. Let ${\partial}F$ be the
${\mathcal{C}}(\overline{\mathcal{Q}(\Lambda(G))})$-valued cocycle over
$\mathcal{Q}(\Lambda(G))$ defined as:

${\partial}F:\mathcal{Q}(\Lambda(G))\times\mathcal{Q}(\Lambda(G))\rightarrow{\mathcal{C}}(\overline{\mathcal{Q}(\Lambda(G))});\quad(x,y)\mapsto[z\in\overline{\mathcal{Q}(\Lambda(G))}\mapsto
F_{x}(z)-F_{y}(z)]$

where $F_{x}\in{\mathcal{C}}(\overline{\mathcal{Q}(\Lambda(G))})$ is given as:

$F_{x}(z)=\int_{{\partial}X}\langle\xi,z\rangle_{x}d\mu_{x}(\xi)$

Observe that:

$\displaystyle
F_{g.x}=\int_{{\partial}X}\langle\xi,z\rangle_{g.x}d\mu_{g.x}(\xi)=\int_{{\partial}X}\langle
g^{-1}\xi,g^{-1}z\rangle_{x}dg_{*}\mu_{x}(\xi)=\int_{{\partial}X}\langle\xi,g^{-1}z\rangle_{x}d\mu_{x}(\xi)=\pi_{o}(g)F_{x}$

and thus ${\partial}F(g.x,g.y)=\pi_{0}(g){\partial}F(x,y)$ for all $x,y\in X$
and $g\in G$.

######  Remark 4.2.

Given $x,y\in X$ we have:

$\displaystyle|{\partial}F(x,y)[z]|\leq|\int_{{\partial}X}\langle
z,\xi\rangle_{x}-\langle
z,\xi\rangle_{y}d\mu_{x}(\xi)|+|\int_{{\partial}X}\langle
z,\xi\rangle_{y}d[\mu_{y}-\mu_{x}](\xi)|$
$\displaystyle\leq\frac{1}{2}\int_{{\partial}X}|b_{\xi}(x,y)|+|b_{z}(x,y)|\mu_{x}(\xi)+\int_{{\partial}X}\langle
z,\xi\rangle_{y}|1-\frac{d\mu_{x}}{d\mu_{y}}(\xi)|d\mu_{y}(\xi)$
$\displaystyle\leq d(x,y)+\delta\int_{{\partial}X}\langle
z,\xi\rangle_{y}|b_{\xi}(x,y)+o(d(x,y))|d\mu_{y}(\xi)$ $\displaystyle\preceq
d(x,y)+\delta d(x,y)\int_{{\partial}X}\langle
z,\xi\rangle_{y}d\mu_{y}(\xi)\preceq d(x,y)$

and thus $(x,y)\mapsto{\partial}F(x,y)$ is a continuous map.
Moreover the shadow lemma implies:

$\sup_{x,y\in\overline{\mathcal{Q}(\Lambda(G))}}\|{\partial}F(x,y)\|_{\infty}\leq\|F_{x}\|_{\infty}+\|F_{y}\|_{\infty}\preceq
2\int_{0}^{+\infty}ue^{-\delta u}du$

is finite.

In the rest, given a fixed basepoint $o\in\mathcal{Q}(\Lambda(G))$, we denote
$c(g)=c(o,g.o)$ and $b(g)=b(o,g.o)$ for $g\in G$.

The following relation is extensively used in the rest:

######  Lemma 4.3.

$\mathcal{I}_{(o)}[c(x,y)]=b_{\bullet}(y,x)+{\partial}F|_{{\partial}X}(x,y)$

for all $x,y\in\mathcal{Q}(\Lambda(G))$. In particular whenever
$Q^{\prime}_{1}$ is positive $\mathcal{I}^{\prime}_{(o)}[c(x,y)]$ is
cohomologuous to the Busemann cocycle in $(\mathcal{H}_{0}^{\prime},\pi_{0})$.

###### Proof.

Given $m\in\mathcal{F}_{1}^{0}$ one has:

$\displaystyle
Q^{\prime}_{1}(c(x,y),m)=Q(\mathcal{I}^{\prime}_{(y)}[c(x,y)],m)=\int_{{\partial}X\times{\partial}X}\langle\xi,\eta\rangle_{y}d[\mu_{x}-\mu_{y}](\xi)dm(\eta)$
$\displaystyle=\int_{{\partial}X\times{\partial}X}[\langle\xi,\eta\rangle_{y}-\langle\xi,\eta\rangle_{x}]d\mu_{x}(\xi)dm(\eta)$
$\displaystyle+\int_{{\partial}X\times{\partial}X}\langle\xi,\eta\rangle_{x}d\mu_{x}(\xi)dm(\eta)-\int_{{\partial}X\times{\partial}X}\langle\xi,\eta\rangle_{y}d\mu_{y}(\xi)dm(\eta)$
$\displaystyle=\int_{{\partial}X\times{\partial}X}\frac{1}{2}[b_{\xi}(y,x)+b_{\eta}(y,x)]d\mu_{x}(\xi)dm(\eta)$
$\displaystyle+\int_{{\partial}X\times{\partial}X}\langle\xi,\eta\rangle_{x}d\mu_{x}(\xi)dm(\eta)-\int_{{\partial}X\times{\partial}X}\langle\xi,\eta\rangle_{y}d\mu_{y}(\xi)dm(\eta)$
$\displaystyle=\frac{1}{2}\int_{{\partial}X}b_{\eta}(y,x)dm(\eta)+\int_{{\partial}X\times{\partial}X}\langle\xi,\eta\rangle_{x}d\mu_{x}(\xi)dm(\eta)-\int_{{\partial}X\times{\partial}X}\langle\xi,\eta\rangle_{y}d\mu_{y}(\xi)dm(\eta)$
$\displaystyle=(b(y,x)+{\partial}F|_{{\partial}X}(x,y),m)$

which proves the statement. ∎

We deduce the following generalization of Kuhn and Vershik [18] Theorem 5
which as a byproduct establishes the properness of the cocycle $c$ relatively
to the quadratic form $Q^{\prime}_{1}$.

######  Proposition 4.4.

There exists a continuous, symmetric and bounded $G$-invariant kernel $r$ over
$\mathcal{Q}(\Lambda(G))$ and $\kappa\in
L^{\infty}(\mathcal{Q}(\Lambda(G))/G,{\mathbb{R}})$ with

$r(z,t)\xrightarrow{}\kappa(G.x)$

when $d(z,t)\rightarrow\infty$ with $z,t\in G.x$ and
$x\in\mathcal{Q}(\Lambda(G))$ such that

$Q_{1}^{\prime}(c(x,y))=d(x,y)+r(x,y)$

for all $x,y\in\mathcal{Q}(\Lambda(G))$.

###### Proof.

Lemma 4.3 implies:

$\displaystyle
Q^{\prime}(c(x,y))=(\mathcal{I}^{\prime}_{(o)}[c(x,y)],c(x,y))=\frac{1}{2}\int_{{\partial}X}b_{\eta}(y,x)d[\mu_{x}-\mu_{y}](\eta)+({\partial}F(x,y),c(x,y))$
$\displaystyle=\frac{1}{2}[\int_{{\partial}X}b_{\eta}(y,x)d\mu_{x}(\eta)-\int_{{\partial}X}b_{\eta}(y,x)d\mu_{y}(\eta)]+({\partial}F(x,y),c(x,y))$
$\displaystyle=d(x,y)+({\partial}F,c(x,y))-\int_{{\partial}X}\langle
y,\eta\rangle_{x}d\mu_{x}(\eta)-\int_{{\partial}X}\langle
x,\eta\rangle_{y}d\mu_{y}(\eta)$ $\displaystyle=d(x,y)+r(x,y)$

for all $x,y\in\mathcal{Q}(\Lambda(G))$, where

$r(x,y)=({\partial}F(x,y),c(x,y))-({\partial}F(x,y),\mathcal{D}_{y}-\mathcal{D}_{x})$

is real, symmetric and $G$-invariant. Moreover the continuity and boundedness
follow from remark 4.2.

On the other hand given $(g_{n})_{n}$ and $(g_{n}^{\prime})_{n}$ in $G$ such
that $d(g_{n}.o,g_{n}^{\prime}.o)\rightarrow+\infty$, up to a subsequence, one
can assume ${g^{\prime}_{n}}^{-1}g_{n}.o\rightarrow\xi_{+}\in{\partial}X$ and
${g_{n}}^{-1}g^{\prime}_{n}.o\rightarrow\xi_{-}\in{\partial}X$. Reorganizing
the terms we have:

$\displaystyle r(g_{n}.o,g_{n}^{\prime}.o)$
$\displaystyle=(F_{g_{n}.o},c(g_{n}.o,g_{n}^{\prime}.o))-(F_{g^{\prime}_{n}.o},c(g_{n}.o,g_{n}^{\prime}.o))-(F_{g_{n}.o},\mathcal{D}_{g_{n}.o}-\mathcal{D}_{g_{n}^{\prime}.o})$
$\displaystyle+(F_{g^{\prime}_{n}.o},\mathcal{D}_{g_{n}.o}-\mathcal{D}_{g_{n}^{\prime}.o})$
$\displaystyle=(F_{o},c(o,g_{n}^{-1}.g_{n}^{\prime}.o))-(F_{o},c({g_{n}^{\prime}}^{-1}g_{n}.o,o))-(F_{o},-\mathcal{D}_{g_{n}^{-1}.g_{n}^{\prime}.o})$
$\displaystyle+(F_{o},\mathcal{D}_{{g_{n}^{\prime}}^{-1}.g_{n}.o})$
$\displaystyle=F_{o}({g_{n}}^{-1}g^{\prime}_{n}.o)-\int_{{\partial}X}F_{o}(\eta)d\mu_{{g_{n}}^{-1}g^{\prime}_{n}.o}(\eta)$
$\displaystyle+F_{o}({g^{\prime}_{n}}^{-1}g_{n}.o)-\int_{{\partial}X}F_{o}(\eta)d\mu_{{g^{\prime}_{n}}^{-1}g_{n}.o}(\eta)+2\int_{{\partial}X}F_{o}(\eta)d\mu_{o}(\eta)$

Observe that

$\frac{1}{2}\kappa:x\in\mathcal{Q}(\Lambda(G))\mapsto\int_{{\partial}X}F_{x}(\eta)d\mu_{x}(\eta)$

is bounded and constant over every $G$-orbits.

Together with the dominated convergence theorem and Lemma 3.2 it follows:

$\displaystyle\lim_{n}r(g_{n}.o,g_{n}^{\prime}.o)$
$\displaystyle=F_{o}(\xi_{-})-F_{o}(\xi_{-})+F_{o}(\xi_{+})-F_{o}(\xi_{+})+2\int_{{\partial}X}F_{o}(\eta)d\mu_{o}(\eta)$
$\displaystyle=\kappa(G.o)$

∎

A remarkable consequence of Proposition 4.4 together with [4] Theorem 1 is the
following converse of Proposition 3.5:

######  Corollary 4.5.

The family of Knapp-Stein operators $(\mathcal{I}_{s})_{s>\frac{1}{2}}$ over
$L^{2}({\partial}X,\mu_{o})$ are positive if and only if
$\mathcal{I}^{\prime}_{(o)}$ is positive over
$L^{2}({\partial}X,\mu_{o})\ominus{\mathbb{C}}$. In this case the unitary
representation $(\mathcal{H}_{1}^{\prime},\pi_{1})$ is well defined and
satisfies $H^{1}(\mathcal{H}_{1}^{\prime},\pi_{1})\neq 0$.

In other words the operators $(\mathcal{I}_{s})_{s>\frac{1}{2}}$ over
$L^{2}({\partial}X,\mu_{o})$ are positive if and only if $d$ is roughly
conditionally negative. In particular [4] Theorem 1 is tight.

###  4.1. Special representations and reduced cohomology

Before investigating the reduced cohomology of
$(\mathcal{H}^{\prime}_{1},\pi_{1})$ we recall some materials concerning
harmonic cocycles [24] [14] [1].

A probability $\beta$ over $G$ will be called admissible if it is finitely
supported, symmetric such that the $\beta$-random walk over $(X,d)$ has non-
zero drift. If $(X_{n})_{n}$ denotes a $\beta$-random walk over $(X,d)$ the
last assumption is equivalent to say that the random variable
$Z_{n}=\frac{d(X_{n}.o,o)}{n}$ converges in $L^{1}(\Omega,{\mathbb{P}})$ to a
non-zero constant $\ell$ [28].

Given a unitary representation $\pi$ and a admissible probability $\beta$; a
cocycle with coefficient in $\pi$, $b_{\pi}$, is called $\beta$-harmonic if it
satisfies:

$\int_{G}b_{\pi}(g)d\beta(g)=0$

or equivalently

$\int_{G}b_{\pi}(hg)d\beta(g)=b_{\pi}(h)$

for all $h\in G$.

The norm over the space of cocycles
$Z^{1}(G,\pi)\subset\text{Map}(G,\mathcal{H}_{\pi})$ defined as:

$\|b_{\pi}\|^{2}_{\beta}=\int_{G}\|b_{\pi}(g)\|_{\pi}^{2}d\beta(g)$

induces a equivalent topology as the compact-open topology [14] and the
following $\|\,.\,\|_{\pi}$-orthogonal decomposition holds:

$Z^{1}(G,\pi)=B^{1}(G,\pi)^{\perp}\oplus\overline{B^{1}(G,\pi)}$

Moreover, since

$\int_{G}(b(g),\xi-\pi(g)\xi)d\beta(g)=2(\int_{G}b(g)d\beta(g),\xi)$

for all $b\in Z^{1}(G,\pi)$, we have the following isometric identification:

$B^{1}(G,\pi)^{\perp}=Z_{\text{Harm}}^{1}(G,\pi)$

where $Z_{\text{Harm}}^{1}(G,\pi)$ stands for the closed space of
$\beta$-harmonic cocycles and thus:

$Z_{\text{Harm}}^{1}(G,\pi)\simeq\overline{H}^{1}(G,\pi)=Z^{1}(G,\pi)/{\overline{B^{1}(G,\pi)}}$

The following criterion was established by Erschler and Ozawa in [14]:

######  Lemma 4.6.

Given a unitary representation $\pi$ and $b_{\pi}\in Z^{1}(G,\pi)$:

$\lim_{n}\frac{{\mathbb{E}}[\|b_{\pi}(X_{n})\|_{\pi}^{2}]}{n}=\|\mathcal{P}_{\text{Harm}}[b_{\pi}]\|_{\beta}^{2}$

where $\mathcal{P}_{\text{Harm}}:Z^{1}(G,\pi)\rightarrow
B^{1}(G,\pi)^{\perp}=Z_{\text{Harm}}^{1}(G,\pi)$ stands for the orthogonal
projection over the $\beta$-harmonic part of $b_{\pi}$ and $(X_{n})_{n}$ for a
$\beta$-random walk over $G$.

######  Corollary 4.7.

Let $\beta$ be a admissible probability over $G$ and $\Lambda\subset G$ the
subgroup generated by its support. Then the reduced cohomology class
$c|_{\Lambda}$ is non-trivial in
$\overline{H}^{1}(\Lambda,\pi_{1}|_{\Lambda})$.

###### Proof.

As a consequence of Kingman subadditive theorem [28] together with Proposition
4.4 one has:

$\displaystyle\lim_{n}\frac{1}{n}{\mathbb{E}}[\|c(X_{n})\|_{Q_{1}^{\prime}}^{2}]$
$\displaystyle=\lim_{n}\frac{1}{n}{\mathbb{E}}[d(o,X_{n}.o)-r(o,X_{n}.o)]$
$\displaystyle=\lim_{n}\frac{{\mathbb{E}}[d(o,X_{n}.o)]}{n}=\ell(\beta)$

where $\ell(\beta)$ stands for the drift of $\beta$ over $(X,d)$. It follows
that $\|\mathcal{P}_{\text{Harm}}[c]\|_{\beta}^{2}=\ell(\beta)>0$ and because
$\overline{H}^{1}(\Lambda,{\mathbb{R}})\simeq Z_{\text{Harm}}^{1}(G,\pi)$ the
cocycle $c$ has non trivial reduced cohomology class. ∎

##  5\. Special representations and irreducible affine actions

In this section we investigate the irreducibility of the affine action
$(\mathcal{H}_{1}^{\prime},\pi_{1},c)\simeq(\mathcal{H}_{0}^{\prime},\pi_{0},b)$
and prove Theorem 4.

The following criterion was proved by Bekka, Pillon and Valette in [3]:

######  Proposition 5.1 (B-P-V [3]).

An isometric affine action $(\mathcal{H},\pi,b_{\pi})$ is irreducible if and
only if for every $w\in\mathcal{H}$, $\overline{{\bf
Span}\\{b_{\pi}(g)+{\partial}_{w}(g)\,|\,g\in G\\}}=\mathcal{H}$ or
equivalently if and only if the affine commutant of the $G$-affine action is
reduced to translations by $\pi$-fixed vectors.

The main result of this subsection is the following von Neumann type theorem
for the $\pi_{1}$-cocycle $c$:

######  Proposition 5.2 (Cocycle von Neumann theorem).

Given $f\in{\mathcal{C}}(\overline{X})$ with $\mu_{o}(f)=0$ and
$w,u\in\mathcal{H}^{\prime}_{1}$:

$\lim_{t}\frac{1}{2}\int_{G}f(g.o)Q_{1}^{\prime}(c(g)+{\partial}_{w}(g),u)d\nu_{o,t}(g)=Q_{1}^{\prime}([f|_{{\partial}X}],u)$

Since $\mathcal{F}_{1}^{0}$ is dense in $\mathcal{H}^{\prime}_{1}$,
Proposition 5.2 together with the B-P-V criterion imply the irreducibility of
the affine action $(\mathcal{H}_{1}^{\prime},\pi_{1},c)$. Moreover, since
$\mathcal{H}^{\prime}_{1}$ does not have any fixed vector due to Lemma 3.7,
Theorem 4 follows.

The main step consists to reduce Proposition 5.2 proof to the case where
$u\in\mathcal{F}_{1}^{0}$ and $w=0$. This will be a consequence of the
following technical lemmas:

######  Lemma 5.3.

Given any $\alpha\geq 1$ the map:

$h\in G\mapsto\int_{G}|Q(b(g),c(h))|^{\alpha}d\nu_{o,t}(g)$

is uniformly bounded over $G$.

###### Proof.

As a consequence of the Jensen inequality:

$\displaystyle\int_{G}|Q(b(g),c(h))|^{\alpha}d\nu_{o,t}(g)\leq\int_{G}|\int_{{\partial}X}2\langle\xi,g.o\rangle_{o}d[\mu_{h.o}+\mu_{o}](\xi)|^{\alpha}d\nu_{o,t}(g)$
$\displaystyle\leq
4^{\alpha}\int_{G}\int_{{\partial}X}|\langle\xi,g.o\rangle_{o}|^{\alpha}d\mu_{h.o}(\xi)d\nu_{o,t}(g)+4^{\alpha}\int_{G}\int_{{\partial}X}|\langle\xi,g.o\rangle_{o}|^{\alpha}d\mu_{o}(\xi)d\nu_{o,t}(g)$
$\displaystyle\preceq
4^{\alpha}\int_{{\partial}X}[\int_{G}|\langle\xi,g.o\rangle_{o}|^{\alpha}d\nu_{o,t}(g)]d\mu_{h.o}(\xi)+\alpha
4^{\alpha}\int_{0}^{+\infty}u^{\alpha-1}e^{-\delta u}du$

On the other hand, using Lemma 2.4 and 2.5, we have:

$\displaystyle\int_{G}|\langle\xi,g.o\rangle_{o}|^{\alpha}d\nu_{o,t}(g)=\alpha\int_{0}^{+\infty}u^{\alpha-1}\nu_{o,t}(\\{g\in
G\,|\,\langle\xi,g.o\rangle_{o}\geq u\\})du$
$\displaystyle\preceq\alpha\int_{0}^{t.R}u^{\alpha-1}e^{-t\delta R}|{\bf
C}^{+}_{o}(\xi;u,r)\cap
S_{o,G}(t)|du\preceq\alpha\int_{0}^{+\infty}u^{\alpha-1}e^{-\delta u}du$

and thus

$\displaystyle\int_{{\partial}X}[\int_{G}|\langle\xi,g.o\rangle_{o}|^{\alpha}d\nu_{o,t}(g)]d\mu_{h.o}(\xi)\preceq\alpha\int_{0}^{+\infty}u^{\alpha-1}e^{-\delta
u}du$

which is finite and independent of $h\in G$. ∎

######  Corollary 5.4.

Given $w\in\mathcal{H}_{1}^{\prime}$ the family of operators:

$\mathcal{A}^{\prime}_{t}[w]:{\mathcal{C}}(\overline{X})\rightarrow\mathcal{H}_{1}^{\prime};\quad
f\mapsto\int_{G}f(g.o)[c(g)+{\partial}_{w}(g)]d\nu_{o,t}(g)$

is uniformly bounded over $t\geq 0$.

###### Proof.

Since the co-boundary ${\partial}_{w}$ is bounded we can assume $w=0$.

Observe that:

$\displaystyle\|$
$\displaystyle\int_{G}f(g.o)c(g)d\nu_{o,t}(g)\|_{Q_{1}^{\prime}}^{2}=\int_{G\times
G}f(g.o)f(h.o)Q^{\prime}_{1}(c(g),c(h))d\nu_{o,t}(g)d\nu_{o,t}(h)$
$\displaystyle\leq\|f\|_{\infty}^{2}\int_{G\times
G}|Q_{1}^{\prime}(c(g),c(h))|d\nu_{o,t}(g)d\nu_{o,t}(h)$
$\displaystyle=\|f\|_{\infty}^{2}\int_{G\times
G}|(b(g)+{\partial}F(g),c(h))|d\nu_{o,t}(g)d\nu_{o,t}(h)$
$\displaystyle=\|f\|_{\infty}^{2}\int_{G\times
G}|(b(g),c(h))|d\nu_{o,t}(g)d\nu_{o,t}(h)+\|f\|_{\infty}^{2}\int_{G\times
G}|({\partial}F(g),c(h))|d\nu_{o,t}(g)d\nu_{o,t}(h)$

which is uniformly bounded over $t$ due to Lemma 5.3 and because ${\partial}F$
is bounded.∎

Another observation due to the fact that $(\pi_{1},\mathcal{H}_{1}^{\prime})$
is mixing:

######  Lemma 5.5.

Given any $u,v\in\mathcal{H}_{1}^{\prime}$ and
$f_{1},f_{2}\in{\mathcal{C}}(\overline{X})$ with
$\mu_{o}(f_{1})\mu_{o}(f_{2})=0$:

$\lim_{t}\int_{G}f_{1}(g.o)f_{2}(g^{-1}.o)Q_{1}^{\prime}({\partial}_{u}(g),v)d\nu_{o,t}(g)=0$

###### Proof.

Observe that:

$\displaystyle\int_{G}f_{1}(g.o)f_{2}(g^{-1}.o)$ $\displaystyle
Q_{1}^{\prime}({\partial}_{u}(g),v)d\nu_{o,t}(g)=Q_{1}^{\prime}(u,v)\int_{G}f_{1}(g.o)f_{2}(g^{-1}.o)d\nu_{o,t}(g)$
$\displaystyle-\int_{G}[f_{1}(g.o)Q_{1}^{\prime}(\pi_{1}(g)u,v)]f_{2}(g^{-1}.o)d\nu_{o,t}(g)$

for all $u,v\in\mathcal{H}_{1}^{\prime}$ and
$f_{1},f_{2}\in{\mathcal{C}}(\overline{X})$ with
$\mu_{o}(f_{1})\mu_{o}(f_{2})=0$. It follows from Lemma 3.7 and Corollary 2.6:

$\displaystyle\lim_{t}Q_{1}^{\prime}(u,v)\int_{G}$ $\displaystyle
f_{1}(g.o)f_{2}(g^{-1}.o)d\nu_{o,t}(g)-\int_{G}[f_{1}(g.o)Q_{1}^{\prime}(\pi(g)u,v)]f_{2}(g^{-1}.o)d\nu_{o,t}(g)$
$\displaystyle=Q_{1}^{\prime}(u,v)\mu_{o}(f_{1})\mu_{o}(f_{2})-\mu_{o}(0)\mu_{o}(f_{2})=0$

∎

Eventually Proposition 5.2 proof is reduced to the following lemma:

######  Lemma 5.6 ($C^{0}$-equidistribution Lemma 1).

Given any $f\in{\mathcal{C}}(\overline{X})$ with $\mu_{o}(f)=0$ and
$\varphi\in\mathcal{F}_{1}^{0}$:

$\lim_{t}\int_{G}f(g.o)Q_{1}^{\prime}(c(g),\varphi)d\nu_{o,t}(g)=2Q^{\prime}_{1}([f|_{{\partial}X}],\varphi)$

Indeed, using Lemma 5.5, we can assume $w=0$. On the other hand Corollary 5.4
allows to extend the weak convergence of

$v_{t}=\int_{G}f(g.o)c(g)d\nu_{o,t}(g)$

with $t\geq 0$, over the dense subspace $\mathcal{F}_{1}^{0}$ to
$\mathcal{H}_{1}^{\prime}$.

###### Proof of Lemma 5.6.

Observe that, as consequence of Lemma 4.3, we have:

$Q^{\prime}_{1}(c(g),\varphi)=2\mathcal{I}_{(o)}^{\prime}[\varphi](g.o)+({\partial}F(g),\varphi)$

for $g\in G$ and $\varphi\in\mathcal{F}_{1}^{0}$. It follows:

$\displaystyle I_{t}:=\int_{G}f(g.o)Q^{\prime}_{1}(c(g),\varphi)d\nu_{o,t}(g)$
$\displaystyle=2\int_{G}f(g.o)\mathcal{I}_{(o)}^{\prime}[\varphi](g.o)d\nu_{o,t}(g)+\int_{G}f(g.o)({\partial}F(g),\varphi)d\nu_{o,t}(g)$
$\displaystyle=2\int_{G}f(g.o)\mathcal{I}_{(o)}^{\prime}[\varphi](g.o)d\nu_{o,t}(g)+(\mathcal{I}_{(o)}^{\prime}[{\bf
1}],\varphi)\int_{G}f(g.o)d\nu_{o,t}(g)$
$\displaystyle-\int_{G}f(g.o)(\pi_{0}(g)\mathcal{I}_{(o)}^{\prime}[{\bf
1}],\varphi)d\nu_{o,t}(g)$

and, using Lemma 5.5 and Corollary 2.6, we have:

$\displaystyle\lim_{t}I_{t}=2Q^{\prime}_{1}([f|_{{\partial}X}],\varphi)$

for all $\varphi\in\mathcal{F}_{1}^{0}$. ∎

###  5.1. Cocycle Margulis-Roblin equidistribution

This subsection is dedicated to prove the following affine analog of the
Margulis-Roblin equidistribution theorem 3 (see [25] [5]):

######  Proposition 5.7 (Double cocycle von Neumann theorem).

Given $f_{1},f_{2}\in{\mathcal{C}}(\overline{X})$ with
$\mu_{o}(f_{1})=\mu_{o}(f_{2})=0$ and
$w,w^{\prime},u,u^{\prime}\in\mathcal{H}^{\prime}_{1}$, the sequence of
averages

$\frac{1}{4}\int_{G}[f_{1}(g.o)Q_{1}^{\prime}(c(g)+{\partial}_{w}(g),u)][f_{2}(g^{-1}.o)Q^{\prime}_{1}(c(g^{-1})+{\partial}_{w^{\prime}}(g^{-1}),u^{\prime})]d\nu_{o,t}(g)$

converges to

$Q^{\prime}_{1}([f_{1}|_{{\partial}X}],u)Q^{\prime}_{1}([f_{2}|_{{\partial}X}],u^{\prime})$

The proof is similar as Proposition 5.2 proof and will follow directly from
Lemma 5.8 and 5.10 below.

######  Lemma 5.8 ($C^{0}$-equidistribution Lemma 2).

Given $f_{1},f_{2}\in{\mathcal{C}}(\overline{X})$ with
$\mu_{o}(f_{1})=\mu_{o}(f_{2})=0$, $\varphi,\phi\in\mathcal{F}_{1}^{0}$ and
$w,w^{\prime}\in\mathcal{H}^{\prime}_{1}$, the sequence of averages

$\int_{G}[f_{1}(g.o)Q_{1}^{\prime}(c(g)+{\partial}_{w}(g),\varphi)][f_{2}(g^{-1}.o)Q^{\prime}_{1}(c(g^{-1})+{\partial}_{w^{\prime}}(g^{-1}),\phi)]d\nu_{o,t}(g)$

converges to

$Q^{\prime}_{1}([f_{1}|_{{\partial}X}],\varphi)Q^{\prime}_{1}([f_{2}|_{{\partial}X}],\phi)$

when $t\rightarrow+\infty$.

###### Proof.

Let $f_{1},f_{2}\in{\mathcal{C}}(\overline{X})$ with
$\mu_{o}(f_{1})=\mu_{o}(f_{2})=0$, $\varphi,\phi\in\mathcal{F}_{1}^{0}$,
$w,w^{\prime}\in\mathcal{H}^{\prime}_{1}$ and denote

$I_{t}=\int_{G}f_{1}(g.o)Q^{\prime}_{1}(c(g)+{\partial}_{w}(g),\varphi)f_{2}(g^{-1}.o)Q^{\prime}(c(g^{-1})+{\partial}_{w^{\prime}}(g^{-1}),\phi)d\nu_{o,t}(g)$

for $t\geq 0$.

Using Lemma 4.3 $I_{t}$ decomposes as
$I_{t}=4I_{1,t}+2I_{2,t}+2I_{3,t}+I_{4,t}$ with

$I_{1,t}=\int_{G}f_{1}(g.o)\mathcal{I}_{(o)}^{\prime}[\varphi](g.o)f_{2}(g^{-1}.o)\mathcal{I}_{(o)}^{\prime}[\phi](g^{-1}.o)d\nu_{o,t}(g)$
$I_{2,t}=\int_{G}f_{1}(g.o)({\partial}_{\mathcal{I}_{(o)}^{\prime}[{\bf
1}]+\mathcal{I}_{(o)}^{\prime}[w]}(g),\varphi)f_{2}(g^{-1}.o)\mathcal{I}_{(o)}^{\prime}[\phi](g^{-1}.o)d\nu_{o,t}(g)$
$I_{3,t}=\int_{G}f_{1}(g.o)\mathcal{I}_{(o)}^{\prime}[\varphi](g.o)f_{2}(g^{-1}.o)({\partial}_{\mathcal{I}_{(o)}^{\prime}[{\bf
1}]+\mathcal{I}_{(o)}^{\prime}[w^{\prime}]}(g^{-1}),\phi)d\nu_{o,t}(g)$

and

$I_{4,t}=\int_{G}f_{1}(g.o)({\partial}_{\mathcal{I}_{(o)}^{\prime}[{\bf
1}]+\mathcal{I}_{(o)}^{\prime}[w]}(g),\varphi)f_{2}(g^{-1}.o)({\partial}_{\mathcal{I}_{(o)}^{\prime}[{\bf
1}]+\mathcal{I}_{(o)}^{\prime}[w^{\prime}]}(g^{-1}),\phi)d\nu_{o,t}(g)$

It follows from Lemma 5.5 that:

$\lim_{t}I_{2,t}=\lim_{t}I_{3,t}=\lim_{t}I_{4,t}=0$

and eventually, using Corollary 2.6, we have:

$\lim_{t}I_{t}=\lim_{t}4I_{1,t}=4Q_{1}^{\prime}(f_{1},\varphi)Q_{1}^{\prime}(f_{2},\phi)$

∎

######  Remark 5.9.

Given $\alpha\geq 1$, using Lemma 4.3 and the Hölder inequality, we have:

$\displaystyle[\int_{G}$
$\displaystyle|Q_{1}^{\prime}(c(g),c(h))|^{\alpha}d\nu_{o,t}(g)]^{1/\alpha}=[\int_{G}|(b(g)+{\partial}_{\mathcal{I}^{\prime}[{\bf
1}]}(g),c(h))|^{\alpha}d\nu_{o,t}(g)]^{1/\alpha}$
$\displaystyle\leq[\int_{G}|(b(g),c(h))|^{\alpha}d\nu_{o,t}(g)]^{1/\alpha}+[\int_{G}|({\partial}_{\mathcal{I}^{\prime}[{\bf
1}]}(g),c(h))|^{\alpha}d\nu_{o,t}(g)]^{1/\alpha}$

where the second term is bounded due to the boundedness of co-boundary
${\partial}_{\mathcal{I}^{\prime}[{\bf 1}]}(g)$ and that $c(h)$, as a signed
measure, has bounded total variation by $2$ for every $h\in G$. Eventually,
using Lemma 5.3, it follows that the map:

$h\in G\mapsto\int_{G}|Q(c(g),c(h))|^{\alpha}d\nu_{o,t}(g)$

is uniformly bounded over $G$.

######  Lemma 5.10 (Uniform boundness Lemma).

Given $w,w^{\prime}\in\mathcal{H}^{\prime}_{1}$ the family of bilinear
operators:

$\mathcal{A}^{\prime(2)}_{t}[w,w^{\prime}]:{\mathcal{C}}(\overline{X})\times{\mathcal{C}}(\overline{X})\rightarrow\mathcal{H}^{\prime}_{1}\otimes\mathcal{H}^{\prime}_{1}$

defined as

$(f_{1},f_{2})\mapsto\int_{G}f_{1}(g.o)[c(g)+{\partial}_{w}(g)]\otimes
f_{2}(g^{-1}.o)[c(g^{-1})+{\partial}_{w^{\prime}}(g^{-1})]d\nu_{o,t}(g)$

is uniformly bounded over $t\geq 0$.

###### Proof.

Observe that:

$\displaystyle\int_{G}f_{1}(g.o)[c(g)]\otimes
f_{2}(g^{-1}.o)[{\partial}_{w^{\prime}}(g^{-1})]d\nu_{o,t}(g)=[\int_{G}f_{1}(g.o)f_{2}(g^{-1}.o)c(g)d\nu_{o,t}(g)]\otimes
w^{\prime}$
$\displaystyle-\int_{G}f_{1}(g.o)f_{2}(g^{-1}.o)[c(g)]\otimes[\pi_{1}(g^{-1})w^{\prime}]d\nu_{o,t}(g)$

where the first term is uniformly bounded due to Corollary 5.4. Regarding the
second term we have:

$\displaystyle\|\int_{G}f_{1}(g.o)f_{2}(g^{-1}.o)[c(g)]\otimes[\pi_{1}(g^{-1})w^{\prime}]d\nu_{o,t}(g)\|_{Q_{1}^{\prime}}^{2}$
$\displaystyle=\int_{G\times
G}f_{1}(g.o)f_{2}(g^{-1}.o)\overline{f_{1}}(h.o)\overline{f_{2}}(h^{-1}.o)Q_{1}^{\prime}(c(g),c(h))Q_{1}^{\prime}(\pi_{1}(g^{-1})w^{\prime},\pi_{1}(h^{-1})w^{\prime})d\nu_{o,t}\otimes\nu_{o,t}(g,h)$
$\displaystyle\leq\|f_{1}\|_{\infty}^{2}\|f_{2}\|_{\infty}^{2}\|w^{\prime}\|_{Q_{1}^{\prime}}^{2}\int_{G\times
G}|Q_{1}^{\prime}(c(g),c(h))|d\nu_{o,t}\otimes\nu_{o,t}(g,h)$

that is uniformly bounded due to Lemma 5.3.

It is therefore enough to prove that:

$\int_{G}f_{1}(g.o)f_{2}(g^{-1}.o)c(g)\otimes c(g^{-1})d\nu_{o,t}(g)$

is uniformly bounded over $t$. Observe that:

$\displaystyle\|\int_{G}f_{1}(g.o)f_{2}(g^{-1}.o)c(g)\otimes
c(g^{-1})d\nu_{o,t}(g)\|_{Q_{1}^{\prime}}^{2}$ $\displaystyle=\int_{G\times
G}f_{1}(g.o)f_{2}(g^{-1}.o)\overline{f_{1}}(h.o)\overline{f_{2}}(h^{-1}.o)Q^{\prime}_{1}(c(g),c(h))Q^{\prime}_{1}(c(g^{-1}),c(h^{-1}))d\nu_{o,t}(g)d\nu_{o,t}(h)$
$\displaystyle\leq\|f_{1}\|_{\infty}^{2}\|f_{2}\|_{\infty}^{2}\int_{G\times
G}|Q^{\prime}_{1}(c(g),c(h))Q^{\prime}_{1}(c(g^{-1}),c(h^{-1}))|d\nu_{o,t}\otimes\nu_{o,t}(g,h)$
$\displaystyle\preceq\sqrt{\int_{G\times
G}|Q^{\prime}(c(g),c(h))|^{2}d\nu_{o,t}\otimes\nu_{o,t}(g,h)}\sqrt{\int_{G\times
G}|Q^{\prime}(c(g^{-1}),c(h^{-1}))|^{2}d\nu_{o,t}\otimes\nu_{o,t}(g,h)}$

which is again bounded due to remark 5.9. ∎

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