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\title[Charmenability of higher rank arithmetic groups]{Charmenability of higher rank arithmetic groups}
\alttitle{Carmoyennabilité des groupes arithmétiques de rang supérieur}
\subjclass{22D10, 22D25, 22E40, 37B05, 46L10, 46L30}
\keywords{Arithmetic groups, Characters, Lattices, Poisson boundaries, Simple algebraic groups, von Neumann algebras}
\author[\initial{U.} \lastname{Bader}]{\firstname{Uri} \lastname{Bader}}
\address{Faculty of Mathematics\\
and Computer Science,\\
The Weizmann Institute of Science,\\
234 Herzl Street,\\
Rehovot 7610001 (Israel)}
\email{bader@weizmann.ac.il}
\thanks{Uri Bader is supported by ISF Moked 713510 grant number 2919/19. Rémi Boutonnet is supported by ANR grant AODynG 19-CE40-0008. Cyril Houdayer is supported by Institut Universitaire de France.}
\author[\initial{R.} \lastname{Boutonnet}]{\firstname{Rémi} \lastname{Boutonnet}}
\address{Institut de Mathématiques\\
de Bordeaux, CNRS,\\
Université Bordeaux I,\\
33405 Talence (France)}
\email{remi.boutonnet@math.u-bordeaux.fr}
\author[\initial{C.} \lastname{Houdayer}]{\firstname{Cyril} \lastname{Houdayer}}
\address{Université Paris-Saclay,\\
Institut Universitaire de France,\\
CNRS, Laboratoire de mathématiques d'Orsay\\
91405, Orsay (France)}
\email{cyril.houdayer@universite-paris-saclay.fr}
\begin{abstract}
We complete the study of characters on higher rank semisimple lattices initiated in~\cite{BH19,BBHP20}, the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary characteristics. More precisely, we investigate dynamical properties of the conjugation action of such lattices on their space of positive definite functions. Our main results deal with the existence and the classification of characters from which we derive applications to topological dynamics, ergodic theory, unitary representations and operator algebras. Our key theorem is an extension of the noncommutative Nevo--Zimmer structure theorem obtained in~\cite{BH19} to the case of simple algebraic groups defined over arbitrary local fields. We also deduce a noncommutative analogue of Margulis'\! factor theorem for von Neumann subalgebras of the noncommutative Poisson boundary of higher rank arithmetic groups.
\end{abstract}
\begin{altabstract}
Nous complétons l'étude des caractères sur les réseaux semisimples de rang supérieur initiée dans~\cite{BH19,BBHP20}, le cas manquant étant celui des réseaux dans les groupes algébriques simples en caractéristique quelconque. Plus précisément, nous étudions les propriétés dynamiques de l'action par conjugaison de tels réseaux sur l'espace des fonctions de type positif. Nos résultats principaux concernent l'existence et la classification des caractères desquels nous déduisons des applications en dynamique topologique, théorie ergodique, représentations unitaires et algèbres d'opérateurs. Notre théorème clé est une extension du théorème de structure Nevo--Zimmer noncommutatif obtenu dans~\cite{BH19} au cas des groupes algébriques simples définis sur des corps locaux quelconques. Nous déduisons aussi un analogue noncommutatif du théorème du facteur de Margulis pour les sous-algèbres de von Neumann de la frontière de Poisson noncommutative des groupes arithmétiques de rang supérieur.
\end{altabstract}
\datereceived{2021-12-17}
\daterevised{2022-12-06}
\dateaccepted{2023-02-01}
\editors{X. Caruso and V. Guirardel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\section{Introduction and statements of the main results}
For any countable discrete group $\Gamma$, we denote by $\cP(\Gamma) \subset \ell^\infty(\Gamma)$ the weak-$*$ compact convex set of all positive definite functions $\varphi : \Gamma \to \C$ normalized so that $\varphi(e) = 1$. We endow the space $\cP(\Gamma)$ with the affine conjugation action $\Gamma \curvearrowright \cP(\Gamma)$. Recall that for any positive definite function $\varphi \in \cP(\Gamma)$, there is a unique (up to unitary conjugation) GNS triple $(\pi_\varphi, H_\varphi, \xi_\varphi)$, where $\pi_\varphi : \Gamma \to \cU(H_\varphi)$ is a unitary representation and $\xi_\varphi \in H_\varphi$ is a unit cyclic vector such that $\varphi(\Gamma) = \langle \pi_\varphi(\Gamma)\xi_\varphi, \xi_\varphi\rangle$ for every $\Gamma \in \Gamma$. Thus, one can always regard any positive definite function as a coefficient of a unitary representation. For standard facts on operator algebras ($\rC^*$-algebras and von Neumann algebras), we refer the reader to~\cite{Ta02}.
A conjugation invariant positive definite function $\varphi \in \cP(\Gamma)$ is called a {\em character} and we denote by $\Char(\Gamma) \subset \cP(\Gamma)$ the weak-$\ast$ closed convex subset of all characters on $\Gamma$. Any non-trivial group $\Gamma$ always possesses at least two characters: the trivial character $1_\Gamma$ and the regular character $\delta_e$ whose GNS representation coincides with the left regular representation $\lambda : \Gamma \to \cU(\ell^2(\Gamma))$. Any finite dimensional unitary representation $\pi : \Gamma \to \cU(n)$ gives rise to the character $\tr_n \circ \pi$. Also, any probability measure preserving (pmp) action $\Gamma \curvearrowright (X, \nu)$ gives rise to the character $\varphi_\nu : \Gamma \to \C : g \mapsto \nu(\Fix(g))$ where $\Fix(g) = \{x \in X \mid g x = x\}$ for every $g \in \Gamma$. Note that $\varphi_\nu = \delta_e$ if and only if the pmp action $\Gamma \curvearrowright (X, \nu)$ is essentially free, that is, for almost every $x \in X$, we have $\Stab_\Gamma(x) = \{e\}$.
We now review the notions of charmenable and charfinite groups that were recently introduced in~\cite{BBHP20}. We denote by $\Rad(\Gamma)$ the amenable radical of $\Gamma$, that is, the largest normal amenable subgroup of $\Gamma$.
\begin{defi}[{\cite{BBHP20}}]\label{def:charmenable}
We say that $\Gamma$ is {\em charmenable} if it satisfies the following two properties:
\begin{enumerate}\Penumi
\item Every nonempty $\Gamma$-invariant weak-$\ast$ compact convex subset $\cC \subset \cP(\Gamma)$ contains a fixed point, that is, a character.
\item Every extremal character $\varphi \in \Char(\Gamma)$ is either supported on $\Rad(\Gamma)$ or its GNS von Neumann algebra $\pi_\varphi(\Gamma)\dpr$ is amenable.
\end{enumerate}
Moreover, we say that $\Gamma$ is {\em charfinite} if it also satisfies the following three properties:
\begin{enumerate}\Penumi
\setcounter{enumi}{2}
\item $\Rad(\Gamma)$ is finite.
\item $\Gamma$ has a finite number of isomorphism classes of unitary representations in each given finite dimension.
\item Every extremal character $\varphi \in \Char(\Gamma)$ is either supported on $\Rad(\Gamma)$ or its GNS von Neumann algebra $\pi_\varphi(\Gamma)\dpr$ is finite dimensional.
\end{enumerate}
\end{defi}
Recall that a tracial von Neumann algebra $M \subset \rB(\rL^2(M))$ is {\em amenable} if there exists a state $\varphi \in \rB(\rL^2(M))^*$ such that $\varphi(xT) = \varphi(Tx)$ for every $T \in \rB(\rL^2(M))$ and every $x \in M$ and $\varphi|_M$ is a faithful normal tracial state on $M$.
Before reviewing previous works and providing examples, let us first recall some striking properties of charmenable and charfinite groups. In that respect, denote by $\Sub(\Gamma) \subset 2^\Gamma$ the closed subset of all subgroups of $\Gamma$ endowed with the conjugation action $\Gamma \curvearrowright \Sub(\Gamma)$. The topology on $\Sub(\Gamma)$ induced from the product topology on $2^\Gamma$ is called the Chabauty topology. Following~\cite{AGV12}, an {\em Invariant Random Subgroup}, or IRS for short, is a $\Gamma$-invariant Borel probability measure on $\Sub(\Gamma)$. Following~\cite{GW14}, a {\em Uniformly Recurrent Subgroup}, or URS for short, is a nonempty $\Gamma$-invariant minimal closed subset of $\Sub(\Gamma)$.
Any charmenable group $\Gamma$ enjoys the following properties: any normal subgroup $N \lhd \Gamma$ is either amenable or coamenable; for any URS $X \subset \Sub(\Gamma)$, either all the elements of $X$ are contained in $\Rad(\Gamma)$ or $X$ carries an IRS; any nonamenable unitary representation $\pi : \Gamma \to \cU(H_\pi)$ weakly contains the left regular representation $\lambda$. Moreover, in that case, the $\rC^*$-algebra $\rC^*_\pi(\Gamma) \subset \rB(H_\pi)$ generated by $\pi(\Gamma)$ has a unique trace and a unique maximal proper ideal.
Any charfinite group enjoys the following properties: any normal subgroup $N \lhd \Gamma$ is either finite or has finite index; any ergodic IRS and any URS of $\Gamma$ is finite. Any charmenable group $\Gamma$ with property (T) and such that $\Rad(\Gamma)$ is finite is charfinite. For all these facts, we refer the reader to~\cite[Section~3]{BBHP20}. In particular, charfinite groups satisfy the conclusion of Margulis'\! normal subgroup theorem~\cite{Ma91} and Stuck--xZimmer's stabilizer rigidity theorem~\cite{SZ92}. The motivation for studying charmenable and charfinite groups comes from these fundamental results as well as other results regarding the classification of characters (see~\cite{Be06, CP13, Pe14, PT13}).
The first class of charfinite groups were obtained in~\cite{BH19}. More precisely, the main results of~\cite{BH19} show that lattices in higher rank connected simple Lie groups with finite center are charfinite. New classes of charmenable and charfinite groups were subsequently obtained in~\cite{BBHP20}. Indeed, \cite[Theorem~A]{BBHP20} shows that irreducible lattices in products of (at least two) simple algebraic groups are charmenable (resp.\! charfinite if one of the factors has property (T)). Moreover, for every $d \geq 2$ and every nonempty (possibly infinite) set of primes $S \subset \cP$, the $S$-arithmetic group $\SL_d(\Z[S^{-1}])$ is charfinite. We refer the reader to the recent survey~\cite{Ho21} for further details and results.
The goal of the present paper is to complete the above picture by showing that lattices in higher rank simple algebraic groups defined over a local field $k$ of arbitrary characteristic are charfinite.
\begin{theo} \label{thm:AG-simple}
Let $k$ be a local field. Let $\bfG$ be an almost $k$-simple connected algebraic $k$-group such that $\rk_k(\bfG) \geq 2$. Then every lattice $\Gamma < \bfG(k)$ is charfinite.
\end{theo}
Theorem~\ref{thm:AG-simple} provides several new classes of charfinite groups.
\begin{exam}
Let $d \geq 3$ be an integer and $k$ a local field. Then any lattice $\Gamma < \SL_d(k)$ is charfinite. In particular, let $p \in \cP$ be a prime and $q = p^r$ for $r \geq 1$. Denote by $\bfQ_p$ the local field of $p$-adic numbers and by $\bfF_q((t))$ the local field of formal power series in one variable $t$ over the finite field $\bfF_q$. Then any lattice $\Gamma < \SL_d(\bfQ_p)$ and any lattice $\Gamma < \SL_d(\bfF_q((t)))$ is charfinite. In particular, the lattice $\SL_d(\bfF_q[t^{-1}]) < \SL_d(\bfF_q((t)))$ is charfinite. We also refer to~\cite{LL20} for the character classification of $\SL_d(\bfF_q[t^{-1}])$.
\end{exam}
Before stating our next result regarding charmenability of higher rank arithmetic groups, let us review some terminology. Let $K$ be a global field and ${\bfG}$ an almost $K$-simple connected algebraic $K$-group. Let $S$ be a (possibly empty, possibly infinite) set of non-Archimedean inequivalent absolute values on $K$. Let $\cO 0$. By continuity of $f$, we may find a function $f_0 \in C(G/Q)$ which is equal to $1$ at the point $x = gQ$, and such $\Vert f f_0\Vert < \eps$. Thus $|\phi(f)| = |\phi(f_0)\phi(f)| = |\phi(f_0f)| \leq \Vert f_0f\Vert < \eps$. Since this holds for every $\eps > 0$, we find that $\phi(f) = 0$, and thus, $\phi$ vanishes on $I$. So $\phi$ factors to a state on $A$ via the evaluation map $e_g$. This implies that $\phi \in \cS$, as desired.
\begin{enumerate}\alphenumi
\item Since $C(G/Q)$ is abelian, the set of its extremal states is weak-* closed, equal to the set of multiplicative characters. So $\cS = \cS'$ is weak-* closed as well.
\item By Lemma~\ref{C*int}, the restriction of an extremal state on $B$ to the central subalgebra $C(G/Q)$ is still extremal.
\item Let $g \in G$, $\psi \in \cS(A)$ and $\phi = \psi \circ e_g \in \cS(B)$. Denote by $R$ the kernel of the action $Q \actson A$. Take $f \in C_b(G,A)^Q$ and $h \in gRg^{-1}$. Denoting by $r := g^{-1}h^{-1}g \in R$, we have
\[
\phi(\sigma_h(f)) = \psi\left(f\left(h^{-1}g\right)\right) = \psi(f(gr)) = \psi(\sigma_r(f(g))) = \psi(f(g)) = \phi(f).\qedhere
\]
\end{enumerate}
\end{proof}
\section{Reduction to the commutative setting}\label{reduce to commutative}
Our proof of Theorem~\ref{thm:NCNZ} is split in two halves. The first half is the following theorem, which simultaneously achieves two goals:
\begin{itemize}
\item it reduces to the case where $M = L^\infty(X,\nu)$ is a commutative von Neumann algebra;
\item and similarly to Nevo--Zimmer's approach, it reduces to the case where the stabilizer of almost every point of $X$ has positive dimension in $G$.
\end{itemize}
The second half will then be to deduce the conclusion of Theorem~\ref{thm:NCNZ} from there. This will be achieved in Section~\ref{Gauss}, by adapting the Gauss map trick from~\cite{NZ00} to the general setting of algebraic groups over local fields. In fact, to be able to use this Gauss map in positive characteristics, one needs to know a bit more than just positive dimension of point stabilizers.
\begin{theo}\label{First half}
Let $k$ be a local field. Let $\bfG$ be an almost $k$-simple connected algebraic $k$-group such that $\rk_k(\bfG) \geq 2$ and set $G = \bfG(k)$. Let $\bfP < \bfG$ be a minimal parabolic $k$-subgroup and set $P = \bfP(k)$. Let $M$ be a $G$-von Neumann algebra and $E : M \to L^\infty(G/P)$ a faithful normal ucp $G$-map. The following dichotomy holds:
\begin{itemize}
\item Either $E$ is $G$-invariant.
\item Or there exists a commutative $G$-von Neumann subalgebra $M_0 \subset M$ such that the action $G^+ \actson M_0$ is non-trivial and moreover, when writing $M_0 = L^\infty(X,\nu)$, the corresponding nonsingular action $G \actson (X,\nu)$ has the following property: for almost every $x \in X$, the stabilizer $G_x < G$ contains the $k$-points of a non-trivial $k$-split torus of $\bfG$.
\end{itemize}
\end{theo}
We note that there is no ergodicity assumption for the $G$-action on $M$ in this theorem. The rest of this section is devoted to proving Theorem~\ref{First half}. To this end, let us consider our almost simple $k$-group $\bfG$ with $k$-rank at least $2$, and setting $G := \bfG(k)$, let us consider a von Neumann action $G \actson M$, with a $G$-equivariant faithful normal ucp map $E: M \to L^\infty(G/P)$. Here $P = \bfP(k)$, where $\bfP$ denotes a minimal parabolic $k$-subgroup of $\bfG$.
More generally, we will freely use the notation introduced in Section~\ref{AGnot}.
Let us assume that $E$ is not $G$-invariant. This means that $E(M)$ is not contained in the scalars. Since $G$ has rank at least $2$, the intersection of all $L^\infty(G/P_\theta)$, for $\theta \subsetneq \Delta$, is equal to the scalars inside $L^\infty(G/P)$. Thus there exists a proper subset $\theta \subsetneq \Delta$ such that the range of $E$ is not contained in $L^\infty(G/P_\theta)$.
For notational simplicity, we write $Q := P_\theta$, $U := V_\theta$, $H := H_\theta$, so that $Q = HU$ and $S := T_\theta$. We denote by $\oQ$ and $\oU$ the opposite parabolic group and its unipotent radical.
\begin{lemm}
$E$ is not $H$-invariant on $M^{\oU}$.
\end{lemm}
\begin{proof}
To prove this, denote by $B \subset M$ the maximal compact model for the $\oU$-action. By Lemma~\ref{LUdec} we note that, as a $\oU$-algebra, $L^\infty(G/P)$ is isomorphic with $L^\infty(\oU) \ovt N_0$, with $N_0 := L^\infty(Q/P)$. Moreover in this isomorphism, $L^\infty(G/Q)$ identifies with $L^\infty(\oU) \ot 1$. So, when thinking only in terms of $\oU$-algebras, we may view $E$ as a ucp map $E: M \to L^\infty(\oU) \ovt N_0$, the range of which is not contained in $L^\infty(\oU) \ovt 1$.
Restricting to compact models, and using Lemma~\ref{MCM1}, we obtain a ucp map $E: B \to C_b(\overline U,N_0)$. Since $E$ is normal, we may find $b_0 \in B$, $u \in \oU$, such that $E(b_0)(u)$ is not a scalar inside $N_0$. By equivariance of $E$, the element $b _1:= \sigma_{u^{-1}}(b)$ satisfies $E(b_1)(e) \notin \C$.
Fix now a torus element $s \in S$ such that $\lim_{n\,\to\,+\infty} s^{-n} \ou s^n = e$, for all $\ou \in \oU$, and define a ucp map $E_s : M \to M$ as a point-ultraweak limit of the maps $E_s^n: x \mapsto \frac{1}{n} \sum_{k = 1}^n \sigma_{s^k}(x)$. We note that $E_s$ is not normal in general.
\emph{\textbf{Claim.}} For every $b \in B$, we have $E_s(b) \in M^{\oU} = B^{\oU}$ and $(E \circ E_s)(b) \in C_b(\oU,N_0)$ is the constant function equal to $E(b)(e)$.
For $\ou \in \oU$, the sequence $(\sigma_{s^{-n}\ou s^n}(b) - b)_{n\,\in\,\N}$, converges in norm to $0$ and hence so does the sequence of its Cesaro average. This implies that
\[
\left\Vert\sigma_{\ou}\left(E_s^n(b)\right) - E_s^n(b)\right\Vert \leq \frac{1}{n}\sum_{k = 1}^n \left\Vert\sigma_{s^{-k}\ou s^k}(b) - b\right\Vert \to 0.
\]
So a fortiori the ultraweak limit $\sigma_{\ou}(E_s(b)) - E_s(b)$ is $0$, proving the first part of the claim.
As explained in Lemma~\ref{LUdec}, $E$ carries the $s$-action on $M$ to the diagonal action on $L^\infty(\oU) \ovt N_0$ deduced from the conjugation action $s \actson \oU$ on the one hand, and the translation action $s \actson Q/P$ on the other hand. But since $s$ centralizes $H$ and belongs to $P$, the Levi decomposition $Q = HU$ implies that $s$ acts trivially on $Q/P$. So in summary, $E \circ \sigma_s = (\sigma_s \ot \id) \circ E$.
Now, for $f \in C_b(\oU,N_0)$, the sequence $((\sigma_{s^n} \ot \id)(f))_{n\,\in\,\N}$ converges pointwise to the constant function equal to $f(e)$. By Lebesgue convergence theorem, it converges a fortiori in the ultraweak topology inherited from the embedding $C_b(\oU,N_0) \subset L^\infty(\oU) \ovt N_0$. Hence the Cesaro average of this sequence also ultraweakly converges to $f(e)$. In view of the previous paragraph, in the special case where $f = E(b)$, the Cesaro average is precisely $E \circ E_s^n(b)$. Since $E$ is normal, taking ultraweak limits, we find $E \circ E_s(b) = \lim_n E \circ E_s^n(b) = E(b)(e)$, as claimed.
We can now conclude the proof of the lemma as follows. Assume that $E$ is $H$-invariant on $M^{\oU}$. Then since $\oQ$ acts ergodically on $L^\infty(G/P)$, we conclude that in fact $E$ maps $M^{\oU}$ into $\C$. In particular, for the element $b_1 \in B$ defined above, we have $E(E_s(b_1)) \in \C$. By the claim, we have $E(E_s(b_1)) = E(b_1)(e)$. This contradicts the choice of $b_1$.
\end{proof}
Denote by $R := SU$ and $\oR := S\oU$ the solvable radical of $Q$ and $\oQ$ respectively.
\begin{lemm}\label{non-invariant}
In fact, $E$ is not $H$-invariant on $M^{\oR}$.
\end{lemm}
\begin{proof}
Consider the non-empty compact convex space
\[
\cC:= \left\{\Phi: M^{\oU} \to M^{\oU} \text{ ucp map } \,\middle|\, E \circ \Phi = E|_{M^{\oU}} \text{ and } \Phi \text{ is $H$-equivariant}\right\},
\]
with respect to the point-ultraweak topology. Since $S$ centralizes $H$ and acts trivially on $L^\infty(G/P)^{\oU}$ (by Lemma~\ref{LUdec}), we may define an action of $S$ on $\cC$ by the formula $s \cdot \Phi := \sigma_s \circ \Phi$, for all $s \in S$, $\Phi \in \cC$. This is a continuous affine action. So by amenability of $S$, it admits a fixed point $\Phi$. The range of $\Phi$ is contained in $M^{\oR}$ and $E \circ \Phi = E$ on $M^{\oU}$. Since $E$ is not $H$-invariant on $M^{\oU}$, it cannot be $H$-invariant on $\Phi(M^{\oU})$, and a fortiori on $M^{\oR}$.
\end{proof}
By Proposition~\ref{embedding}, we may find a $P$-von-Neumann algebra $N$ with a $P$-invariant state $\psi$ and a normal $G$-equivariant embedding into the induced algebra $M \subset \tN := (L^\infty(G) \ovt N)^P$, such that $E$ is the restriction to $M$ of $\tE = \id \ot \psi$.
We prove that $\tN^{\oR}$ is actually contained in a nice $G$-invariant subalgebra of $\tN$. This unfortunately only holds after cutting down by a suitable projection. In the commutative setting, this projection is not relevant. This consideration already appeared in~\cite{BH19}.
For this, we denote by $q \in N$ the support projection of $\psi$ restricted to $(N^R)' \cap N$. Since $R$ is normal inside $P$, $q$ is $P$-invariant. Set $p = 1 \ot q \in \tN$ and observe that $p \in \tN$ is $G$-invariant. In view of Lemma~\ref{LUdec}, $\tN = \Ind_P^G(N) = \Ind_Q^G(\Ind_P^Q(N))$ may be identified with
\[
L^\infty(\oU) \ovt N_Q\quad\text{where }N_Q = \Ind_P^Q(N) = \left(L^\infty(Q) \ovt N\right)^P.
\]
This isomorphism maps $\tN^{\oR}$ to $1 \ot N_Q^S$ and the projection $p$ is mapped to $1 \ot (1 \ot q)$, with $1 \ot q \in N_Q$.
\begin{lemm}\label{projection p}
The projection $p \in \cZ(\tN^{\oR})$ is $G$-invariant and satisfies $\tE(p) = 1$ and $p\tN^{\oR} \subset p(L^\infty(G) \ovt N^R)^P$.
\end{lemm}
\begin{proof}
Using the identifications $\tN^{\oR} = 1 \ot N_Q^S$ and $p = 1 \otimes (1 \otimes q)$, to prove the lemma, it suffices to prove the following claim.
\emph{\textbf{Claim.}} The projection $(1 \ot q) \in N_Q$ commutes with $N_Q^S$ and $(1 \ot q)N_Q^S = (1\ot q)N_Q^R = (1 \ot q)(L^\infty(Q) \ovt N^R)^P$.
Since $R = S U$ is normal in $Q$ and contained in $P$, the subalgebra $N^R \subset N$ is $P$-invariant and $R$ acts trivially on $Q/P$. This implies that $N_Q^R = \Ind_P^Q(N)^R = \Ind_P^Q(N^R) = (L^\infty(Q) \ovt N^R)^P$ and that the ucp map $E_Q = \id \ot \psi : N_Q \to L^\infty(Q/P)$ is $R$-invariant. Moreover, $1 \ot q$ is precisely the support projection of $E_Q$ on the von Neumann subalgebra $(L^\infty(Q) \ovt ((N^R)' \cap N))^P = (N_Q^R)' \cap N_Q$. So, composing $E_Q$ with any faithful normal state on $L^\infty(Q/P)$, we get an $R$-invariant state $\psi_Q$ on $N_Q$ such that $1 \ot q$ is the support of $\psi_Q$ restricted to $(N_Q^R)' \cap N_Q$. In the GNS representation of $(N_Q,\psi_Q)$, $1 \otimes q$ is the orthogonal projection onto the closed linear span of $(N_Q)' N_Q^R\xi_{\psi_Q}$.
Let now $x \in N_Q^S$ and $g \in U$. Pick $s_n \in S$ such that $\lim_n s_ngs_n^{-1} = e$. For all $a \in (N_Q)'$ and $y \in N_Q^R$, $n \in \N$, we have
\[
\left\Vert(\sigma_g(x) - x)ay\xi_{\psi_Q}\right\Vert \leq \Vert a \Vert \cdot \left\Vert(\sigma_g(x) - x)y\right\Vert_{\psi_Q} = \Vert a \Vert \cdot \left\Vert \left(\sigma_{s_ngs_n^{-1}}(x) - x\right)y\right\Vert_{\psi_Q}
\]
which tends to $0$ as $n \to \infty$. This proves that $\sigma_g(x (1 \otimes q)) = \sigma_g(x)(1 \otimes q) = x (1\otimes q)$. Hence $x(1 \otimes q) \in N_Q^U \cap N_Q^S = N_Q^R$. Since $1 \otimes q$ commutes with $N_Q^R$, we further get $x(1 \otimes q) = (1 \otimes q)x(1 \otimes q)$. The same equality for $x^*$ leads to $(1 \otimes q)x (1 \otimes q) = (1 \otimes q)x$. Hence $(1 \otimes q) \in (N_Q^S)'$ and $(1 \otimes q)N_Q^S = (1 \otimes q)N_Q^R$.
This finishes the proof of the claim and the proof of the Lemma~\ref{projection p}.
\end{proof}
Keep the notation $N_Q$ as in the proof of Lemma~\ref{projection p} and observe that $\tN$ is naturally identified with $\Ind_Q^G(N_Q) = (L^\infty(G) \ovt N_Q)^Q$.
We now denote by $M_1 \subset M$ the $G$-invariant von Neumann subalgebra generated by $M^{\oR}$. When viewed inside $\tN$, $M_1$ commutes with $p$, because $M^{\oR}$ commutes with $p$ and $p$ is $G$-invariant. By the previous lemma, we have $M_1p \subset (L^\infty(G) \ovt qN_Q^R)^Q$.
Since $\tE(p) = 1$ and $E$ is faithful on $M_1 \subset M$, the central support of $p \in M_1'$ inside $M_1$ is $1$. Hence the cut-down morphism $M_1 \to M_1p$ is injective. So abusing with notation, we view $M_1$ as a $G$-invariant von Neumann subalgebra of $(L^\infty(G) \ovt N_1)^Q$, with $N_1 = qN_Q^R$. We then define the commutative $G$-invariant von Neumann subalgebra $M_0 := \cZ(M_1) \subset M$.
\begin{lemm}\label{lem-non-trivial}
The action of $G^+ \curvearrowright M_0$ is non-trivial and hence $M_0$ is non-trivial.
\end{lemm}
\begin{proof}
By Lemma~\ref{LUdec}, $(L^\infty(G) \ovt N_1)^Q$ is identified with $L^\infty(\oU) \ovt N_1$, and the $\oR = S\oU$-action is explicit in this description. Denote by $A_1 \subset M_1$ the maximal compact model for the $G$-action on $M_1$. It is in particular contained in the maximal compact model for the $\oU$-action, and thus, it is contained inside $C_b(\oU,N_1)$.
Denote by $N_2 \subset N_1$ the von Neumann subalgebra generated by all the values of the functions inside $A_1 \subset C_b(\oU,N_1)$. Then we have $A_1 \subset C_b(\oU,N_2) \subset L^\infty(\oU) \ovt N_2$, so that $M_1 \subset L^\infty(\oU) \ovt N_2$.
%{\bf Claim~1.}
\setcounter{claimcount}{0}
\begin{claim}
$M_1$ contains $1 \ovt N_2$.
Take $f \in A_1$ and $\ou \in \oU$. It suffices to check that $1 \ot f(\ou) \in M_1$. Replacing $f$ by $\sigma_{\ou^{-1}}(f)$, we may assume that $\ou = e$. Then choose a sequence $(s_n)_{n\,\in\,\N}$ in $S$ such that $\lim_n s_n^{-1}\ov s_n = e$ for all $\overline v \in \oU$. Then the sequence $\sigma_{s_n}(f) \in A_1$ converges pointwise to $1 \ot f(e) \in C_b(\oU,N_2)$. So a fortiori, it converges to $1 \ot f(e)$ ultraweakly, which implies that $1 \ot f(e) \in M_1$, as claimed.
So thanks to the claim we have inclusions
\[
1 \ot N_2 \subset M_1 \subset L^\infty(\oU) \ovt N_2.
\]
In particular, $1 \ot \cZ(N_2) \subset M_0 \subset L^\infty(\oU) \ovt \cZ(N_2)$.
\end{claim}
%{\bf Claim~2.}
\begin{claim}\label{claim2}
$M_1 \neq 1 \ot N_2$.
Indeed, otherwise $\oU$ would act trivially on $M_1$. By Tits'\ simplicity theorem~\cite{Ti64}, this would imply that $G^+$ itself acts trivially on $M_1$. So in this case $E$ would be $G^+$-invariant on $M_1$. Since $G^+$ acts ergodically on $G/P$, we would conclude that $E$ is in fact $G$-invariant on $M_1$, contradicting Lemma~\ref{non-invariant}. This proves Claim~\ref{claim2}.
To conclude the lemma, it suffices to prove that $\oU$ acts non-trivially on $M_0$. Let us assume the contrary and derive a contradiction. We could do this by combining a result of Ge-Kadison from~\cite{GK95} and a direct integral argument as it was done in~\cite{BH19}; we will provide a different argument, not involving direct integrals.
If $\oU$ acts trivially on $M_0$, then $M_0 = 1 \ot \cZ(N_2)$. Denote by $\Phi : N_2 \to \cZ(N_2)$ a proper normal conditional expectation. Here proper means that $\Phi(x)$ belongs to the ultraweak closure of $\{uxu^* \mid u \in \cU(N_2)\}$ for all $x \in N_2$. Such a conditional expectation exists by~\cite[Theorem~C]{GK95}. Because $\Phi$ is proper, the map $\id \ot \Phi: L^\infty(\oU) \ovt N_2 \to L^\infty(\oU) \ovt \cZ(N_2)$ maps $M_1$ into $M_1 \cap (L^\infty(\oU) \ovt \cZ(N_2)) = M_0 = 1 \ot \cZ(N_2)$.
Take a normal faithful state $\psi$ on $\cZ(N_2)$, and denote by $\phi := \psi \circ \Phi$. Then $\phi$ is faithful on $\cZ(N_2)$. Moreover, $(\id \ot \phi)(M_1) \subset (\id \ot \psi)(1 \ot \cZ(N_2)) = \C$.
\end{claim}
%{\bf Claim~3.}
\begin{claim}
The set $\Lambda := \{a\phi b \mid a,b \in N_2\}$ is norm dense in $(N_2)_*$.
The annihilator of $\Lambda$ in $N_2$ is a weak-* closed two sided ideal of $N_2$, hence of the form $zN_2$ for some $z \in \cZ(N_2)$. Since $\phi$ is faithful on $\cZ(N_2)$ we must have $z = 0$. So the claim follows from Hahn--Banach theorem.
Since $M_1$ contains $1 \ot N_2$, we find that $(\id \ot \psi')(M_1) = \C$, for every $\psi' \in \Lambda$. Hence this also holds for every $\psi' \in (N_2)_*$. It follows from~\cite[Theorem~B]{GK95} that $M_1 = 1 \ot N_2$. This contradicts Claim~\ref{claim2}; the proof of the Lemma~\ref{lem-non-trivial} is complete.\qedhere
\end{claim}\let\qed\relax
\end{proof}
Denote by $A_0 \subset M_0$ the maximal compact model for the $G$-action. and denote by $X$ the Gelfand spectrum of $A_0$, so that $A_0 \simeq C(X)$. Then $M_0 = L^\infty(X,\nu)$ for some Borel measure $\nu$ on $X$\footnote{We can require that $\nu$ is a probability measure if we replace $M_0$ by a $G$-invariant separable subalgebra on which the $G^+$-action is still non-trivial.}.
\begin{lemm}
Every $x \in X$ is fixed by a conjugate of the torus $S$.
\end{lemm}
\begin{proof}
We want to show that every extremal state on $A_0$ is fixed by some conjugate of $S$. We note that $A_0$ is naturally embedded in the maximal compact model $B_0$ of the $G$-von Neumann algebra $(L^\infty(G) \ovt N_1)^Q$. By Lemma~\ref{C*1}, we know that every extremal state on $A_0$ is the restriction to $A_0$ of an approximately extremal state on $B_0$. Since the embedding $A_0 \subset B_0$ is $G$-equivariant, all we have to do is to check that every approximately extremal state on $B_0$ is fixed by some conjugate of~$S$.
By Lemma~\ref{MCM2}, we know that $B_0$ is equal to $C_b(G,C)^Q$, where $C \subset N_1$ is the maximal compact model for the $Q$-action. Since the torus $S$ is contained in the kernel of the action $Q \actson C$, the result follows from Lemma~\ref{C*2}.
\end{proof}
So $M_0 = L^\infty(X,\nu)$ satisfies all the desired properties, which ends the proof of Theorem~\ref{First half}.
\section{Algebraic factors}\label{Gauss}
Whenever $X$ is a standard Borel space, we denote by $\Prob(X)$ the space of all Borel probability measures on $X$. A standard Borel space $(X, \nu)$ endowed with a Borel probability measure is called a standard probability space. All probability spaces we consider are assumed to be standard. Recall that for any lcsc group $G$,\linebreak a Borel probability measure $\mu \in \Prob(G)$ is said to be \emph{admissible} if the following three conditions are satisfied:
\begin{enumerate}
\item $\mu$ is absolutely continuous with respect to the Haar measure;
\item $\supp(\mu)$ generates $G$ as a semigroup;
\item $\supp(\mu)$ contains a neighborhood of the identity element.
\end{enumerate}
In this section, we let $k$ be a local field and ${\bfG}$ an algebraic $k$-group which is assumed to be connected and absolutely almost simple (in particular, we make no rank assumption in this section). We will use the notation from Section~\ref{AGnot}.\linebreak We consider the central isogenies $\tilde{\pi}:\tilde{\bfG}\to {\bfG}$ and $\bar{\pi}:{\bfG}\to \bar{\bfG}$, where $\tilde{\bfG}$ and $\bar{\bfG}$ denote the simply connected cover and adjoint quotient of ${\bfG}$ correspondingly, and let $\pi=\bar{\pi}\circ\tilde{\pi}:\tilde{\bfG}\to \bar{\bfG}$.
Our goal is to prove the following theorem.
\begin{theo} \label{thm:gauss-stat}
Assume ${\bfG}$ is absolutely almost simple. Let $\mu$ be an admissible probability measure on $G$ and let $(X,\nu)$ be an ergodic $(G, \mu)$-stationary space. If for a.e.\! $x\in X$ the stabilizer $G_x$ contains the $k$-points of a non-trivial split torus in ${\bfG}$, then exactly one of the following two options holds.
\begin{itemize}
\item The measure $\nu$ is $G$-invariant and there exists an almost everywhere defined measurable $G$-map $G/G^+ \to X$.
\item The measure $\nu$ is not $G$-invariant and there exists a proper parabolic $k$-subgroup ${\bfQ}<{\bfG}$ and an almost everywhere defined measurable $G$-map $X\to ({\bfG}/{\bfQ})(k)$.
\end{itemize}
\end{theo}
Our proof of Theorem~\ref{thm:gauss-stat} relies on some results which might be of independent interest in the theory of algebraic groups over local fields. It is a general fact that non-discrete closed subgroups of $G$ ``tend to be algebraic''. In Proposition~\ref{prop:ZdenseG+} we make this vague statement more precise. In order to prove this over a local field of arbitrary characteristic we use the above assumption that the group contains the $k$-points of a split torus, but we comment here that this assumption could be dramatically relaxed in most situations. However, we choose not to elaborate further on this point here. Proposition~\ref{prop:ZdenseG+} implies Theorem~\ref{thm:gauss-general}, which is a non-stationary version of Theorem~\ref{thm:gauss-stat}. Proposition~\ref{prop:cocompact} describes cocompact algebraic subgroups of $G$ and Proposition~\ref{prop:XtoV} uses the latter to describe stationary measures on $G$-algebraic varieties. In turn, Proposition~\ref{prop:XtoV} is used to deduce Theorem~\ref{thm:gauss-stat} from Theorem~\ref{thm:gauss-general}. We note that the proofs below are sometimes intricate, due to phenomena of positive characteristic.
For every closed subfield of finite index, $k_0