# The $L_{p}$-dual space of a semisimple Lie group

Bachir Bekka

###### Abstract.

Let $G$ be a semisimple Lie group. We describe the irreducible representations
of $G$ by linear isometries on $L_{p}$-spaces for $p\in(1,+\infty)$ with
$p\neq 2.$ More precisely, we show that, for every such representation $\pi,$
there exists a parabolic subgroup $Q$ of $G$ such that $\pi$ is equivalent to
the natural representation of $G$ on $L_{p}(G/Q)$ twisted by a unitary
character of $Q.$ When $G$ is of real rank one, we give a complete
classification of the possible irreducible representations of $G$ on an
$L_{p}$-space for $p\neq 2,$ up to equivalence.

###### 2000 Mathematics Subject Classification:

22E46; 47L10; 37A40

The author acknowledges the support by the ANR (French Agence Nationale de la
Recherche) through the project Labex Lebesgue (ANR-11-LABX-0020-01) .

##  1\. Introduction

Let $G$ be a locally compact topological group and $E$ a Banach space. Denote
by $\mathcal{B}(E)$ the algebra of bounded operators on $E$, equipped with the
strong operator topology, that is, the weakest topology for which the map
$\mathcal{B}(E)\to E,T\mapsto Tv$ is continuous for every $v\in V.$ An
_isometric representation_ (for short, a _representation_) of $G$ on $E$ is a
continuous group homomorphism $\pi:G\rightarrow\textnormal{Iso}(E)$, where
$\textnormal{Iso}(E)$ is the subgroup of $GL(E)$ consisting of the linear
surjective isometries on $E$.

In the context of semisimple Lie groups, there is strong evidence (see [HC1],
[Godement1], [Warner1], [Fell1]) that the right notions of irreducibility and
equivalence of Banach representations are defined as follows. Recall that to a
representation $\pi:G\rightarrow\textnormal{Iso}(E)$ of $G$ there is
associated an algebra homomorphism $C_{c}(G)\to\mathcal{B}(E)$, denoted by
$\pi$ again, defined by

$\pi(f)\,=\,\int_{G}f(g)\pi(g)d\mu_{G}(g)\qquad\text{for all}\quad f\in
C_{c}(G),$

where $C_{c}(G)$ is the convolution algebra of continuous functions on $G$
with compact support and $\mu_{G}$ is a left Haar measure on $G.$

######  Definition 1.

(i) A representation $\pi:G\rightarrow\textnormal{Iso}(E)$ is _completely
irreducible_ if the algebra $\pi(C_{c}(G))$ is dense in $\mathcal{B}(E)$ for
the strong operator topology.

(ii) Two representations $\pi_{1}:G\rightarrow\textnormal{Iso}(E_{1})$ and
$\pi_{2}:G\rightarrow\textnormal{Iso}(E_{2})$ on Banach spaces $E_{1}$ and
$E_{2}$ are _Naimark equivalent_ (for short, _equivalent_) if, for
$i\in\\{1,2\\},$ there there exists a dense subspace $V_{i}$ of $E_{i}$ which
are $\pi_{i}(C_{c}(G)$-invariant and a closed injective linear map $T:V_{1}\to
V_{2}$ such that

$\pi_{2}(f)Tv=T\pi_{1}(f)v\qquad\text{for all}\quad v\in E_{1}\hskip
5.69046ptf\in C_{c}(G).$

(iii) Let $\mathcal{E}$ be a class of Banach spaces. The $\mathcal{E}$-_dual_
of $G$, denoted by $\widehat{G}_{\mathcal{E}},$ is the set of equivalence
classes of completely irreducible representations of $G$ on some Banach space
$E\in\mathcal{E};$

When $E$ is a Hilbert space, a representation
$G\rightarrow\textnormal{Iso}(E)$ is traditionally called a unitary
representation; a unitary representation $\pi$ is completely irreducible if
and only if $\pi$ is irreducible (that is, $\\{0\\}$ and $E$ are the only
$G$-invariant closed subspaces of $E$). Moreover, unitary representations
$\pi_{1}$ and $\pi_{2}$ are Naimark equivalent if and only if $\pi_{1}$ and
$\pi_{2}$ are unitarily equivalent (see [Warner1, 4.3]); in particular, if
$\mathcal{E}$ denotes the class of Hilbert spaces, then
$\widehat{G}_{\mathcal{E}}$ coincides with the usual unitary dual
$\widehat{G}$ of $G.$

Let $G$ a noncompact connected semisimple linear Lie group. For such a group,
we will be concerned with $\widehat{G}_{L_{p}}$ where $L_{p}$ is the class of
$L_{p}$-spaces for $p\in[1,+\infty[.$ By an $L_{p}$-space, we mean the usual
space $L_{p}(X,\mathcal{B},\mu)$ of equivalence classes (modulo null sets) of
measurable $p$-integrable functions $f:X\to\mathbf{C},$ , where $\mu$ is a
positive $\sigma$-finite measure defined on a standard Borel space
$(X,\mathcal{B}).$ The unitary dual space $\widehat{G}=\widehat{G}_{L_{2}}$ is
of course a classical much studied object (see for instance [Knapp1],
[Warner1]). We will deal here with $\widehat{G}_{L_{p}}$ for $p\neq 2.$

Let $\theta$ be a Cartan involution on the Lie algebra $\mathfrak{g}$ of $G$
and $\mathfrak{k}$ (respectively $\mathfrak{p}$) the eigenspace for the
eigenvalue $1$ (respectively $-1$) of $\theta.$ Let $\mathfrak{a}$ be a
maximal commutative subspace of $\mathfrak{p}$ and let
$\Sigma\subset\mathfrak{a}^{*}$ be the corresponding root system, where
$\mathfrak{a}^{*}=~{\mathrm{Hom}}_{\mathbf{R}}(\mathfrak{a},\mathbf{R})$ is
the real dual space of $\mathfrak{a}.$ Let $\Sigma^{+}$ be the set of positive
roots for an ordering of $\Sigma$ and $\mathfrak{n}$ the sum of the root
spaces $\mathfrak{g}_{\alpha}$ for $\alpha\in\Sigma^{+}.$ Let $K$, $A,$ and
$N$ be the subgroups of $G$ with Lie algebras $\mathfrak{k},\mathfrak{a}$ and
$\mathfrak{n},$ respectively. Then $K$ is a maximal compact, $A$ an abelian
and $N$ a nilpotent subgroup; moreover, we have an _Iwasawa decomposition_
$G=KAN.$

Let $M$ be the centralizer of $A$ in $K$. The group $MAN$ is a minimal
parabolic subgroup of $G.$ Recall (see [Knapp1, Section V.5] or [Warner1,
Chap. I, 1.2]) that a _parabolic subgroup_ of $G$ is a closed subgroup
containing a conjugate of $MAN.$ Every such group is conjugate to a _standard
parabolic subgroup_ $Q=Q_{F}$, which is parametrized by a subset $F$ of the
set $\Delta$ of simple positive roots; more precisely, let $\mathfrak{a}_{F}$
be the intersection of all $\ker\alpha$ for $\alpha\in F$ and
$\mathfrak{n}_{F}$ the sum of the root spaces $\mathfrak{g}_{\alpha}$ for
$\alpha\in\Sigma^{+}\setminus\rm{span}(F)$. We have a _Langlands
decomposition_

$Q=M_{Q}A_{Q}N_{Q},$

where $M_{Q}$ is reductive and contains $M$, $A_{Q}=\exp(\mathfrak{a}_{F})$
and $N_{Q}=\exp(\mathfrak{n}_{F})$. Moreover, $M_{Q}$ and $A_{Q}$ commute, and
both normalize $N_{Q}$. In particular, the standard minimal parabolic
parabolic subgroup is given by $Q_{\emptyset}=MAN.$

Fix a parabolic subgroup $Q=Q_{F}$ of $G$ and a real number $p\in[1,+\infty[.$
We are going to define the $L_{p}$ analog of the (unitary) principal series
representations associated to $Q$.

Let $K_{Q}:=K\cap M_{Q}$ and choose a Borel fundamental domain
$\Omega_{Q}\subset K$ for the coset space $K/K_{Q}$. Then $\Omega_{Q}$ is also
a fundamental domain for $G/Q,$ since $G=KQ$ and $K\cap Q=K_{Q}.$ Every $g\in
G$ has a unique decomposition

$g=\kappa(g)\mu(g)\exp(H(g))n(g)\qquad\text{for}\quad~\kappa(g)\in\Omega_{Q},\mu(g)\in
M_{Q},H(g)\in\mathfrak{a}_{F},n(g)\in N_{Q}.$

Let $\mu_{Q}$ be the unique quasi-invariant Borel probability measure on $G/Q$
which is $K$-invariant. We identify $G/Q$ with $\Omega_{Q}$ and transfer
$\mu_{Q}$ and the natural $G$-action on $G/Q$ to $\Omega_{Q}.$ Let
$\rho_{Q,p}\in\mathfrak{a}^{*}$ be defined by

$\rho_{Q,p}:=\dfrac{1}{p}\sum_{\alpha\in\Sigma^{+}\setminus\rm{span}(F)}\dim(\mathfrak{g}_{\alpha})\alpha.$

Observe that $\rho_{MAN,2}$ is the familiar half-sum of positive roots.

Fix a real linear form $\lambda\in\mathfrak{a}^{*}$ and a unitary character

$\chi:M_{Q}\to\mathbf{S}^{1}=\mathbf{U}(1)$

of the reductive group $M_{Q}.$ We define a representation
$\pi(Q,\chi,\lambda,p)$ of $G$ on

$L_{p}(G/Q,\mu_{Q})=L_{p}(\Omega_{Q},\mu_{Q})$

by

$\pi(Q,\chi,\lambda,p)(g)f(x)=\overline{\chi(\mu(g^{-1}x))}e^{-(i\lambda+\rho_{Q,p})H(g^{-1}x)}f(\kappa(g^{-1}x))$

for $f\in L_{p}(\Omega_{Q},\mu_{F}),x\in\Omega_{Q},g\in G.$ It is easily
verified (see Section 2) that $\pi(Q,\chi,\lambda,p)$ is indeed an isometric
representation of $G$ on $L_{p}(G/Q,\mu_{Q})$.

We denote by $\widehat{M_{Q}^{\rm Ab}}$ the group of unitary characters of
$M_{Q}.$ Here is our main result.

######  Theorem A.

Let $G$ be a noncompact connected semisimple linear Lie group, and let
$p\in(1,+\infty)$ with $p\neq 2.$ Let $\pi$ be a completely irreducible
representation of $G$ on an $L_{p}$-space. Then there exists a parabolic
subgroup $Q$ of $G$, a real linear form $\lambda\in\mathfrak{a}^{*},$ and a
unitary character $\chi\in\widehat{M_{Q}^{\rm Ab}}$ such that $\pi$ is
equivalent to the representation $\pi(Q,\chi,\lambda,p)$ on
$L_{p}(G/Q,\mu_{Q})$ described above.

In order to study the irreducibility of a representation
$\pi(Q,\chi,\lambda,p)$ as in the previous theorem, we consider its associated
Harish-Chandra $(\mathfrak{g},K)$-module. For a unitary representation
$\sigma$ of $M_{Q}$ and for
$\nu\in\mathfrak{a}^{*}_{\mathbf{C}}=~{\mathrm{Hom}}_{\mathbf{R}}(\mathfrak{a},\mathbf{C}),$
following [Wallach-book, 5.2], we denote by $I_{Q,\sigma,\nu}$ the associated
infinitesimal nonunitary principal series of $G,$ that is, the
$(\mathfrak{g},K)$-module underlying the induced representation

${\rm Ind}_{Q}^{G}(\sigma\otimes\nu\otimes
1_{N_{Q}})=\operatorname{Ind}_{M_{Q}A_{Q}N_{Q}}^{G}(\sigma\otimes\nu\otimes
1_{N_{Q}}).$

######  Proposition 2.

  * (i)

Let $\pi(Q,\chi,\lambda,p)$ be a representation of $G$ on $L_{p}(G/Q,\mu_{Q})$
as in Theorem A. Then $\pi(Q,\chi,\lambda,p)$ is completely irreducible if and
only if the principal series representation $I_{Q,\chi,\nu}$ is
(algebraically) irreducible, where

$\nu=i\lambda+\delta_{p}\rho_{Q,2}$

for $\delta_{p}=\dfrac{2}{p}-1\in(-1,1).$

  * (ii)

For $i=1,2,$ let $\pi(Q_{i},\chi_{i},\lambda_{i},p_{i})$ be representations of
$G$ on $L_{p_{i}}(G/Q_{i},\mu_{Q_{i}})$ as in Theorem A. Then
$\pi(Q_{1},\chi_{1},\lambda_{1},p_{1})$ and
$\pi(Q_{2},\chi_{2},\lambda_{2},p_{2})$ are equivalent if and only if the
corresponding principal series representations $I_{Q_{1},\chi_{1},\nu_{1}}$
and $I_{Q_{2},\chi_{2},\nu_{2}}$ as in (i) are (algebraically) equivalent.

Principal series representations $I_{Q,\sigma,\nu}$ are of course central
objects in the representation theory of semisimple Lie groups; their structure
has been much studied, especially in the case where $Q$ is minimal parabolic,
and it has been shown that “most” of them are irreducible (see [Gelfand-
Naimark], [Bruhat], [PRV], [Kostant1], [Wallach1], [Lepowski-Wallach]).

Using known irreducibility and equivalence results from the literature, we can
settle the case where $G$ is a simple Lie group with real rank one. So, $G$
belongs to one of the series $\mathrm{SO}_{0}(n,1),\mathrm{SU}(n,1),$
$\mathrm{Sp}(n,1)$ or $\mathrm{Sp}(n,1)$ for $n\geq 2$ or $G$ is the
exceptional Lie group $F_{4(-20)}$. We set $P:=MAN.$

######  Theorem B.

Let $p\in(1,+\infty)$ with $p\neq 2.$

  * (i)

Let $G=\mathrm{SO}_{0}(n,1)$ for $n\geq 2$. Then $\widehat{M^{\rm
Ab}}={1_{M}}$ for $n\neq 3$ and $\widehat{M^{\rm Ab}}\cong\mathbf{Z}$ for
$n=3.$ The representation $\pi(P,\chi,\lambda,p)$ is irreducible for every
$\lambda\in\mathfrak{a}^{*}$ and every $\chi\in\widehat{M^{\rm Ab}}.$

  * (ii)

Let $G=\mathrm{SU}(n,1)$ for $n\geq 2$. Then $\widehat{M^{\rm
Ab}}\cong\mathbf{Z}$.

    * (ii1)

The representation $\pi(P,\chi,\lambda,p)$ is irreducible for every
$\lambda\in\mathfrak{a}^{*}\setminus\\{0\\}$ and every $\chi\in\widehat{M^{\rm
Ab}}.$

    * (ii2)

After an appropriate identification of $\widehat{M^{\rm Ab}}$ with
$\mathbf{Z},$ the representation $\pi(P,m,0,p)=\pi(P,\chi,0,p)$ for
$m\in\mathbf{Z}$ is _not_ irreducible if and only if $p$ belongs to the finite
set

$\left\\{\dfrac{2n}{k}:k\in\\{1,\dots,2n-1\\},k\equiv m\ (\mathrm{mod}\
2),k\neq m,k\neq 2n+m\right\\}.$

  * (iii)

Let $G=\mathrm{Sp}(n,1)$ for $n\geq 2$. Then $\widehat{M^{\rm Ab}}={1_{M}}.$

    * (iii1)

The representation $\pi(P,1_{M},\lambda,p)$ is irreducible for every
$\lambda\in\mathfrak{a}^{*}\setminus\\{0\\}$

    * (iii2)

The representation $\pi(P,1_{M},0,p)$ is _not_ irreducible if and only if
$p=\dfrac{2n+1}{2n}$ or $p=2n+1.$

  * (iv)

Let $G=\mathrm{F}_{4(-20)}.$ Then $\widehat{M^{\rm Ab}}={1_{M}}.$

    * (iv1)

The representation $\pi(P,1_{M},\lambda,p)$ is irreducible for every
$\lambda\in\mathfrak{a}^{*}\setminus\\{0\\}$

    * (iv2)

The representation $\pi(P,1_{M},0,p)$ is _not_ irreducible if and only if $p$
belongs to the set
$\left\\{\dfrac{11}{10},\dfrac{11}{9},\dfrac{11}{8},\dfrac{11}{3},\dfrac{11}{2},11\right\\}.$

######  Corollary C.

Let $G$ be one of the groups as in Theorem B.

  * (i)

Let $p\in(1,+\infty)$ with $p\neq 2.$ Every completely irreducible
representation of $G$ on an $L_{p}$-space is equivalent either to the trivial
representation $1_{G}$ or to one of the irreducible representations of the
form $\pi(P,\chi,\lambda,p)$ from Theorem B.

  * (ii)

Two irreducible representations $\pi(P,\chi,\lambda,p)$ and
$\pi(P,\chi^{\prime},\lambda^{\prime},p^{\prime})$ from Theorem B with
$(\chi^{\prime},\lambda^{\prime},p^{\prime})\neq(\chi,\lambda,p)$ are
equivalent if and only if

$(\chi^{\prime},\lambda^{\prime},p^{\prime})=(\overline{\chi},\lambda,p)\quad\text{or}\quad(\chi^{\prime},\lambda^{\prime},p^{\prime})=(\overline{\chi},-\lambda,q),$

where $q$ is the conjugate exponent of $p.$.

The paper is organized as follows. The proofs of Theorem A, Proposition 3, and
Theorem B are is given in Sections 2, 3, and 4 respectively. As an example,
the case of the group $G=SL_{2}(\mathbf{R}),$ which is a twofold cover of
$SO_{0}(2,1),$ is treated in Section 5.

##  2\. Proof of Theorem A

Let $G$ be a noncompact connected semisimple linear Lie group, Let
$(X,\mathcal{B},\mu)$ be a standard Borel space equipped with a
$\sigma$-finite measure $\mu$ on $\mathcal{B}.$ Let

$\pi:G\to\textnormal{Iso}(L_{p}(X,\mu))$

be a completely irreducible representation of $G$ by linear isometries on
$L_{p}(X,\mu)$ for $p\in(1,+\infty)$ with $p\neq 2.$ Assume that $\pi$ is not
the trivial representation $1_{G}$ of $G.$

Observe that $L_{p}(X,\mu)$ is isometrically isomorphic to
$L_{p}(X,\mu^{\prime})$ for every $\sigma$-finite measure $\mu^{\prime}$ on
$X$ which is equivalent to $\mu;$ indeed, if
$\varphi=\frac{d\mu^{\prime}}{d\mu}$ denotes the Radon-Nikodym derivative of
$\mu^{\prime}$ with respect to $\mu,$ then

$f\mapsto f\varphi^{1/p}$

is a bijective linear isometry from $L_{p}(X,\mu^{\prime})$ to $L_{p}(X,\mu).$
So, upon choosing a probability measure in the measure class of $\mu,$ we can
and will assume in the sequel that $\mu$ is a probability measure.

$\bullet$ _First step._ We claim that there exists a measure-class preserving
measurable action

$G\times X\to X,\qquad(g,x)\mapsto gx$

of $G$ on $X$ and a measurable map $c:G\times X\to\mathbf{S}^{1}$ such that,
for $f\in L_{p}(X,\mu),g\in G,$ and $x\in X$ we have

(1)
$\pi(g)f(x)=c(g^{-1},x)\left(\frac{dg^{-1}_{*}(\mu)}{d\mu}(x)\right)^{1/p}f(g^{-1}x),$

where $\mathbf{S}^{1}$ is the set of complex numbers of modulus 1 and
$g_{*}(\mu)$ is the image of $\mu$ under the map $x\mapsto gx;$ moreover $c$
satisfies the cocycle relation

(2) $c(g_{1}g_{2},x)=c(g_{1},g_{2}x)c(g_{2},x)\qquad\text{for all}\quad
g_{1},g_{2}\in G,\ \text{for almost all}~\ x\in X.$

Indeed, let $g\in G.$ By the Banach-Lamperti theorem (see [Banach],
[Lamperti]), there exists a measure-class preserving measurable map
$\varphi(g):X\to X$ and a measurable map $c(g):X\to\mathbf{S}^{1}$ such that

$\pi(g)f(x)=c(g)(x)\left(\frac{d\varphi(g)_{*}(\mu)}{d\mu}(x)\right)^{1/p}f(\varphi(g)(x))$

for $f\in L_{p}(X,\mu),g\in G,$ and $x\in X.$ Since
$\pi(g_{1}g_{2})=\pi(g_{1})\pi(g_{2})$ for all $g_{1},g_{2}\in G$ and since
$g\mapsto\pi(g)f$ is measurable for every $f\in L_{p}(X,\mu),$ it is readily
checked that $(g,x)\mapsto\varphi(g^{-1})x$ is a measure-class preserving
$G$-action on $X$ and that

$c:(g,x)\to c(g^{-1})x$

is a measurable map satisfying Equation (2).

Recall that another measurable cocycle $c^{\prime}:G\times X\to\mathbf{S^{1}}$
is said to be cohomologous to $c$ if there exists a measurable map
$b:X\to\mathbf{S^{1}}$ such that

$c^{\prime}(g,x)=b(gx)c(g,x)b(x)^{-1}\qquad\text{for all}\quad g\in G,\
\text{for almost all}~\ x\in X.$

Let $\pi^{\prime}$ be the representation of $G$ on $L_{p}(X,\mu)$ defined by
the same formula (1), with $c^{\prime}$ instead of $c.$ Then $\pi^{\prime}$ is
equivalent to $\pi;$ indeed, the map $U:L_{p}(X,\mu)\to L_{p}(X,\mu),$ defined
by

$Uf(x)=b(x)f(x)\qquad\text{for all}\quad f\in L_{p}(X,\mu),x\in X,$

is a bijective isometry which intertwines $\pi^{\prime}$ and $\pi.$

$\bullet$ _Second step._ We claim that the measure $\mu$ is continuous (that
is, $\mu$ has no atoms).

Indeed, let $A$ be the (at most countable) set of atoms of $\mu.$ The
decomposition of $\mu$ into its atomic and continuous parts is
$\mu=\mu_{a}+\mu_{c}$, where $\mu_{a}=\mu|_{A}$ and $\mu_{c}=\mu-\mu_{a}.$ We
can write

$L_{p}(X,\mu)=L_{p}(X,\mu_{c})\oplus\ell_{p}(A)$

and it follows from the Banach-Lamperti theorem that $L_{p}(X,\mu_{c})$ are
$\ell_{p}(A)$ are $\pi(G)$-invariant and so define subrepresentations of
$\pi.$ Since $\pi$ is irreducible, it follows that either $\mu$ is purely
atomic or $\mu$ is continuous.

Assume by contradiction that $\mu$ is purely atomic, that is,
$L_{p}(X,\mu)=\ell_{p}(A)$. Let $G\curvearrowright A$ be the action of $G$ on
$A$ and $c:G\times A\to\mathbf{S}^{1}$ the cocycle as above. Since $\pi$ is
continuous, the map

$G\to\ell_{p}(A),\qquad g\mapsto c(g,x)\delta_{g\cdot x}$

is continuous for every $x\in A.$ As $G$ is connected, this implies that $G$
fixes pointwise $A$. By irreducibility of $\pi,$ it follows that $A$ is a
singleton $\\{x_{0}\\}.$ Equation (2) shows that then that $g\mapsto
c(g,x_{0})$ is homomorphism from $G$ to $\mathbf{S}^{1}.$ Since $G$ is
semisimple, $G$ has a trivial abelianization and hence $c(g,x_{0})=1$ for all
$g\in G.$ So, $\pi=1_{G};$ this is a contradiction and the claim is proved.

As is well-known (see [Zimmer1, Theorem 2.1.19]), the measurable $G$-system
$(X,\mu,G)$ admits a continuous compact model, that is, there exists a compact
metric space $Y$ on which $G$ acts continuously, a probability measure $\nu$
on the Borel subsets of $Y$ and a $G$-equivariant Borel isomorphism $\Phi:Y\to
X$ with $\Phi_{*}(\nu)=\mu.$

Let $c^{\prime}:G\times Y\to\mathbf{S}^{1}$ be defined by
$c^{\prime}(g,y)=c(g,\Phi(y))$. The representation $\pi^{\prime},$ defined on
$L_{p}(Y,\nu)$ by the formula

$\pi^{\prime}(g)f(y)=c^{\prime}(g^{-1},y)\left(\frac{dg^{-1}_{*}(\nu)}{d\nu}(y)\right)^{1/p}f(g^{-1}y),$

is equivalent to $\pi.$ So, without loss of generality, we may assume that $X$
is a compact metric space on which $G$ acts continuously and that $\mu$ is a
quasi-invariant probability measure on the Borel subsets of $X.$

Next, let $K$ be a maximal compact subgroup of $G$ and denote by $dk$ the
normalized Haar measure on $K.$ Then $\mu^{\prime}=\int_{K}k_{*}(\mu)dk,$
defined by

$\mu^{\prime}(A)=\int_{K}\mu(kA)dk$

for all Borel subsets $A$ of $X,$ is a probability measure on $X$ which is
$K$-invariant; moreover, $\mu^{\prime}$ is equivalent to $\mu.$ So, we can and
will assume in the sequel that $\mu$ is $K$-invariant.

$\bullet$ _Third step._ We claim that there are countably many $K$-orbits
$\mathcal{O}_{1},\mathcal{O}_{2},\cdots,$ in $X$ such that

$\mu\left(\bigcup_{i\geq 1}\mathcal{O}_{i}\right)=1.$

Indeed, let $Y:=X/K$ be the space of $K$-orbits, equipped with the quotient
topology structure given by the quotient map $r:X\to Y$ and with the
probability measure $\nu:=r_{*}(\mu).$

Since $K$ is compact, $Y$ is a Hausdorff space. So, $Y$ is a second countable
compact space and is therefore a standard Borel space. Let
$\nu=\nu_{a}+\nu_{c}$ be the decomposition of $\nu$ into its atomic and
continuous parts. The claim will be proved if we show that $\nu_{c}=0.$

By a well-known disintegration lemma (see e.g. Lemma 11.1 [MackeyInd] or
Theorem 5.14 in [EinsiedlerWard]), there exists a map

$\theta:Y\to{\rm Prob}(X),y\mapsto\theta_{y},$

where ${\rm Prob}(X)$ is the set of Borel probability measures on $X,$ with
the following properties:

  * (a)

for every $y\in Y,$ we have $\theta_{y}(r^{-1}(y))=1;$

  * (b)

for every $f\in L_{1}(X,\mu),$ the map $y\mapsto\int_{X}f(x)d\theta_{y}(x)$ is
measurable and we have

$\int_{X}fd\mu=\int_{Y}\left(\int_{X}f(x)d\theta_{y}(x)\right)d\nu(y).$

Moreover, if a second map $\theta^{\prime}:Y\to{\rm Prob}(X)$ satisfies (a)
and (b), then $\theta_{y}=\theta^{\prime}_{y}$ for $\nu$-almost every $y\in
Y.$

Since $\mu$ is $K$-invariant, it follows from the uniqueness of $\theta$ that
$\theta_{y}$ is the unique $K$-invariant probability measure on the $K$-orbit
$y\subset X,$ for $\nu$-almost every $y.$

For $f\in C(X)$ and $y\in Y,$ let $f|_{y}\in C(y)$ denote the restriction of
$f$ to $y$. The map $f\mapsto(y\mapsto f_{y})$ extends to an isometric
isomorphism

$S:L_{p}(X,\mu)\to\int^{\oplus}_{Y}L_{p}(y,\theta_{y})d\nu(y),$

where $\int^{\oplus}_{Y}L_{p}(y,\theta_{y})d\nu(y)$ is the $L_{p}$-direct
integral of the family $(L_{p}(y,\theta_{y}))_{y\in Y}$ in the sense of
[DeJeu-Roz]. Let $\pi_{y}$ denote the representation of $K$ on
$L_{p}(y,\theta_{y})$ given by

$\pi_{y}(k)f(x)=c(k^{-1},x)f(k^{-1}x))\qquad\text{for all}\quad f\in
L_{p}(y,\theta_{y}),k\in K,x\in y.$

The direct integral $\int^{\oplus}_{Y}~\pi_{y}d\nu(y)$ of the $\pi_{y}$’s is a
representation of $K$ defined on $\int^{\oplus}_{Y}L_{p}(y,\theta_{y})d\nu(y)$
and the map $S$ intertwines $\pi|_{K}$ with
$\int^{\oplus}_{Y}~\pi_{y}d\nu(y)$; for this, see [DeJeu-Roz, Theorem 4.9]
(the proof given there extends to representations twisted by a cocycle as in
our situation).

For $\delta\in\widehat{K}$ (recall that $\widehat{K}$ is the unitary dual of
$K$), set

$\chi_{\delta}=d(\delta){\rm ch}_{\delta},$

where ${\rm ch}_{\delta}=\mathrm{Tr}\circ\delta$ is the usual character of
$\delta$, and define

$P_{\pi}(\delta):=\pi(\overline{\chi_{\delta}})=\int_{K}\overline{\chi_{\delta}}(k)\pi(k)dk.$

Then $P_{\pi}(\delta)$ is a continuous projection of $L_{p}(X,\mu)$ onto the
isotypical $K$-submodule of $L_{p}(X,\mu)$ of type $\delta.$ Similarly, for
every $y\in Y,$

$P_{\pi_{y}}(\delta):=\pi_{y}(\overline{\chi_{\delta}})=\int_{K}\overline{\chi_{\delta}}(k)\pi_{y}(k)dk$

is a continuous projection of $L_{p}(y,\theta_{y})$ onto the isotypical
$K$-submodule of $L_{p}(y,\theta_{y})$ of type $\delta.$

The field $y\to P_{\pi_{y}}(\delta)$ defines a decomposable operator
$\int^{\oplus}_{Y}~P_{\pi_{y}}(\delta)d\nu(y)$ on
$\int^{\oplus}_{Y}L_{p}(y,\theta_{y})d\nu(y)$ and, since $S$ intertwines
$\pi|_{K}$ and $\int^{\oplus}_{Y}~\pi_{y}d\nu(y),$ we have

(3) $SP_{\pi}(\delta)S^{-1}=\int^{\oplus}_{Y}~P_{\pi_{y}}(\delta)d\nu(y)$

Let $\Omega$ be a Borel subset of $Y$. Denote by $V_{\Omega}$ the range of the
multiplication operator on $\int^{\oplus}_{Y}L_{p}(y,\theta_{y})d\nu(y)$ by
$\mathbf{1}_{\Omega}$, that is,

$\displaystyle V_{\Omega}$ $\displaystyle=\left\\{f=(f_{y})_{y\in
Y}\in\int^{\oplus}_{Y}L_{p}(y,\theta_{y})d\nu(y):\mathbf{1}_{\Omega}f=f\right\\}$
$\displaystyle=\left\\{f=(f_{y})_{y\in
Y}\in\int^{\oplus}_{Y}L_{p}(y,\theta_{y})d\nu(y):f_{y}=0\quad\text{for}\quad\nu\text{-almost
every}\quad y\notin\Omega\right\\}.$

Observe that

(4) $\int^{\oplus}_{Y}~P_{\pi_{y}}(\delta)d\nu(y)\quad\text{leaves
invariant}\quad V_{\Omega}.$

Assume, by contradiction, that the continuous part $\nu_{c}$ of $\nu$ is non-
zero; so, denoting by $Y_{c}$ the complement in $Y$ of set of atoms of $\nu,$
we have $\nu(Y_{c})>0$ and the measure $\nu|_{Y_{c}}$ is continuous.

For $\delta\in\widehat{K}$, the subset

$D_{\delta}:=\\{y\in Y:P_{\pi_{y}}(\delta)=0\\}$

of $Y$ is measurable. Indeed, let $(f_{n})_{n}$ be a dense sequence in $C(X)$
for the uniform convergence; then, by Urysohn lemma, $(f_{n}|_{y})_{n}$ is
dense in $C(y)$ and hence in $L_{p}(y,\theta_{y})$ for every $y\in Y,$ and so

$D_{\delta}=\bigcap_{n\in\mathbf{N}}\\{y\in
Y:P_{\pi_{y}}(\delta)(f_{n}|_{y})=0\\}.$

Since $y\mapsto P_{\pi_{y}}(\delta)(f_{n}|_{y})$ belongs to
$\int^{\oplus}_{Y}L_{p}(y,\theta_{y})d\nu(y)$ and is therefore a measurable
field, $D_{\delta}$ is measurable.

Observe that $\widehat{K}$ is countable, since $K$ is a separable compact
group. So, the set

$D:=\bigcap_{\delta\in\widehat{K}}D_{\delta}$

is measurable. We claim that

(5) $\nu(D)=0.$

Indeed, assume, by contradiction that $\nu(D)>0.$ Then $V_{D}\neq\\{0\\}.$
However, for every $(f_{y})_{y\in Y}\in V_{D},$ we have

$P_{\pi_{y}}(\delta)(f_{y})=0\qquad\text{for all}\quad\delta\in\widehat{K}$

and hence $f_{y}=0$ for every $y\in D.$ This contradiction shows that (5)
holds.

It follows from (5) that there exists $\delta\in\widehat{K}$ such that
$\nu\left(Y_{c}\cap N_{\delta}\right)<\nu(Y_{c}).$ So, for

$\Omega:=Y_{c}\setminus N_{\delta}$

we have $\nu(\Omega)>0$. Since $\nu|_{Y_{c}}$ is continuous, we can therefore
find a partition

$\Omega=\coprod_{n\in\mathbf{N}}\Omega_{n}$

of $\Omega$ in countably many Borel subsets $\Omega_{n}$ with
$\nu(\Omega_{n})>0$ for all $n\in\mathbf{N}.$ We have

$\left(\int^{\oplus}_{Y}~P_{\pi_{y}}(\delta)d\nu(y)\right)V_{\Omega_{n}}\neq\\{0\\}\qquad\text{for
all}\quad n\in\mathbf{N}.$

since $P_{\pi_{y}}(\delta)\neq 0$ for every $y\in\Omega.$ Hence, for every
$n\in\mathbf{N},$ we can find

$f_{n}\in\left(\int^{\oplus}_{Y}~P_{\pi_{y}}(\delta)d\nu(y)\right)V_{\Omega_{n}}\qquad\text{with}\qquad
f_{n}\neq 0.$

Since $\Omega_{n}\cap\Omega_{m}=\emptyset$ for $n\neq m,$ it follows from (4)
that the family $(f_{n})_{n\in\mathbf{N}}$ is linearly independent. Therefore,
on the one hand, the family $(S^{-1}f_{n})_{n\in\mathbf{N}}$ of elements in
$L_{p}(X,\mu)$ is linearly independent. On the other hand, it follows from (3)
that

$S^{-1}f_{n}\in P_{\pi}(\delta)(S^{-1}(V_{\Omega_{n}}))\subset
P_{\pi}(\delta)(S^{-1}(V_{\Omega}))\qquad\text{for all}\quad n\in\mathbf{N}.$

This implies that the subspace $P_{\pi}(\delta)(S^{-1}(V_{\Omega}))$ of
$L_{p}(X,\mu)$ is infinite dimensional. So, the projection $P_{\pi}(\delta)$
does not have finite-dimensional range. This is not possible since $\pi$ is
completely irreducible (see [Godement1, Theorem 2] or [HC1, Lemma 33]). This
is a contradiction shows that $\nu$ is an atomic measure.

$\bullet$ _Fourth step._ We claim that there exists a point $x_{0}\in X$ such
that $\mu(Gx_{0})=1$. Indeed, it follows from the third step that there exists
a point $x_{0}\in X$ such that $\mu(Kx_{0})>0$ and therefore $\mu(Gx_{0})>0.$
If we show that the $G$-action of $(X,\mu)$ is ergodic, then it will follow
that $\mu(Gx_{0})=1.$

Assume, by contradiction, that there exists a partition $X=X_{1}\coprod X_{2}$
into two $G$-invariant Borel subsets $X_{i}$ with $\mu(X_{i})>0$ for $i=1,2.$
Then

$L_{p}(X,\mu)=V_{X_{1}}\oplus V_{X_{2}},$

where

$V_{X_{i}}=\\{f\in L_{p}(X,\mu):\mathbf{1}_{X_{i}}f=f\\}\cong
L_{p}(X_{i},\mu|_{X_{i}});$

moreover, Formula (1) shows that $V_{X_{i}}$ is $\pi(G)$-invariant. Since
$V_{X_{i}}\neq\\{0\\}$ for $i=1,2$, this contradicts the fact that $\pi$ is
irreducible and the claim is proved.

The stabilizer of $x_{0}\in X$ as above is a closed subgroup $H$. Hence, upon
disregarding a set of measure zero, we may and will identify $X$, as _Borel
$G$-space,_ with the space $G/H$, equipped with its quotient Borel structure.

Next, we draw some consequences for the cocycle $c:G\times X\to\mathbf{S^{1}}$
over the transitive $G$-space $X$. First, upon passing to a cohomologous
cocycle, we can assume that $c$ is a strict cocycle, that is, Equation (2)
holds for all $g_{1},g_{2}\in G$ and $x\in X.$ Moreover, choose a Borel
section $s:G/H\to G$ for the projection $G\to G/H$ with $s(H)=e.$ Observe that
$gs(x)s(gx)^{-1}$ belongs to $H,$ for every $g\in G$ and $x\in X.$ Then there
exists a continuous homomorphism $\chi:H\to\mathbf{S^{1}}$ such that $c$ is
cohomologous to the cocycle $c_{\chi}:G\times X\to\mathbf{S^{1}}$ given by

$c_{\chi}(g,x)=\chi(gs(x)s(gx)^{-1})\qquad\text{for all}\quad g\in G,x\in X;$

for all this, see Theorem 5.27 in [Varadarajan1].

We may and will in the sequel assume that the cocycle $c$ coincides with
$c_{\chi}.$ Also, as is well-known (see e.g. [MackeyInd, Theorem 1.1]), all
$G$-quasi-invariant $\sigma$-finite Borel measures on $G/H$ are mutually
equivalent; so, we may assume that the $G$-quasi-invariant $\mu$ on $G/H$ is
one of the standard $G$-invariant Borel measures on $G/H$ constructed in
[MackeyInd, §.1].

$\bullet$ _Fifth step._ We claim that $K$ acts transitively on the space
$G/H.$

Indeed, on the one hand, since $G$ is a connected Lie group, $X=G/H$ is a
$C^{\infty}$\- smooth connected manifold; moreover, the measure $\mu$ on $X$
is locally Lebesgue measure, that is, $\mu$ is given by a smooth density times
Lebesgue measure in any local coordinates on $X;$ indeed, this follows from
the fact that the maps $x\mapsto gx$ are diffeomorphisms of $X$ for every
$g\in G.$

On the other hand, it follows from the third step that there exists a
$K$-orbit $\mathcal{O}$ in $X$ with $\mu(\mathcal{O})>0.$ As is well-known,
$\mathcal{O}$ is a submanifold of $X$, diffeomorphic to $K/L$ where $L$ is the
stabilizer in $K$ of a point in $K$ (see e.g. [Bourbaki1, III, §1, 7.]). In
view of what we said about $\mu,$ this is possible only if
$\dim\mathcal{O}=\dim X,$ that is, if $\mathcal{O}$ is open in $X.$ Since $X$
is connected, it follows that $\mathcal{O}=X$.

$\bullet$ _Sixth step._ We claim that $H$ is contained as a finite index
subgroup in a parabolic subgroup of $G$, as defined in the Introduction.
Indeed, it follows from the fifth step and Lemma 3 below, that there exists a
parabolic subgroup $Q$ which contains $H$ and which has the following
properties; let $Q=M_{Q}A_{Q}N_{Q}$ be the Langlands decomposition of $Q$.
Write $Q^{0}=CSA_{Q}N_{Q}$ for the connected identity component $Q^{0}$ of
$Q,$ where $C$ is the maximal compact factor of $M_{Q}^{0}$ and $S$ is the
product of all the non-compact simple factors of $M_{Q}^{0}$. Then

$H^{0}=C^{\prime}SA_{Q}N_{Q},$

where $C^{\prime}$ is a connected closed subgroup of $C.$

We claim $H$ has finite index in $Q.$ Assume, by contradiction, that $H$ has
infinite index in $Q.$ Observe that $Q$ is a direct product $Q=ZQ^{0}$ for a
finite subgroup $Z$ of its center (see [Knapp2, Theorem 7.53]). So, $H$ is of
the form $H=TSA_{Q}N_{Q}$ for a subgroup $T$ of infinite index in the compact
group $L:=ZC.$

Let $\chi$ be the unitary character of $H$ associated to $c.$ Observe that
$\chi$ is trivial on the semisimple Lie group $S$. Moreover, we have
$[\mathfrak{a}_{F},\mathfrak{n}_{F}]=\mathfrak{n}_{F}$ for the Lie algebras
$\mathfrak{a}_{F}$ and $\mathfrak{n}_{F}$ of $A_{Q}$ and $N_{Q}$, where $F$ is
the set of simple roots associated to $Q$ as in the Introduction; it follows
that the commutator subgroup of $A_{Q}N_{Q}$ coincides with $N_{Q}$ and hence
that $\chi$ is also trivial on $N_{Q}$.

Choose a Borel fundamental domain $\Omega_{0}\subset G$ for $G/Q$ and a
fundamental domain $\Omega_{1}\subset L$ for $L/T.$ Then
$\Omega_{0}\Omega_{1}$ is a fundamental domain for $G/H$ and every $g\in G$
has unique decompositions

$g=\omega_{0}(g)q(g)=\omega_{0}(g)\omega_{1}(g)h(g)\qquad\text{for}\quad\omega_{i}(g)\in\Omega_{i},q(g)\in
Q,h(g)\in H.$

Let $\nu_{1}$ be the $L$-invariant probability measure on $\Omega_{1}\cong
L/T.$ Identify $L_{p}(G/H,\mu)$ with
$L_{p}(\Omega_{0}\times\Omega_{1},\nu_{0}\otimes\nu_{1})$ for a $G$-quasi-
invariant probability measure $\nu_{0}$ on $\Omega_{0}\cong G/Q.$ For $g\in G$
and $F\in L_{p}(\Omega_{0}\times\Omega_{1},\nu_{0}\otimes\nu_{1}),$ we have

$\pi(g)F(x)=\chi(h(g^{-1}x))F(\omega_{0}(g^{-1}x)\omega_{1}(g^{-1}x))\qquad\text{for
all}\quad x\in\Omega_{0}\times\Omega_{1}.$

Define a representation $\pi_{Q}$ of $Q$ on $L_{p}(\Omega_{1},\nu_{1})$ by

$\pi_{Q}(q)f(x_{1})=\chi(h(q^{-1}x_{1}))f(\omega_{1}(q^{-1}x_{1}))$

for $f\in L_{p}(\Omega_{1},\nu_{1}),x_{1}\in\Omega_{1},q\in Q.$

Since $\Omega_{1}\cong Q/H$ is infinite, $L_{p}(\Omega_{1},\nu_{1})$ is
infinite dimensional; therefore, the restriction $\pi_{Q}|_{L}$ of $\pi_{Q}$
to the compact group $L$ is not irreducible (see [Warner1, 4.2.2.4]). Hence,
there exists a closed $\pi_{Q}(L)$-invariant subspace $V$ of
$L_{p}(\Omega_{1},\nu_{1})$ with $V\neq\\{0\\}$ and $V\neq
L_{p}(\Omega_{1},\nu_{1}).$ Observe that $\pi_{Q}(q)$ is a multiple of
$\chi(q)$ for $q\in A_{Q}$ and $\pi_{Q}(q)$ is the identity for $q\in SN_{Q}.$
Since $Q=LSA_{Q}N_{Q},$ it follows that $V$ is $Q$-invariant.

For $F\in L_{p}(\Omega_{0}\times\Omega_{1},\nu_{0}\otimes\nu_{1})$ and
$x_{0}\in\Omega_{0},$ let $F_{x_{0}}\in L_{p}(\Omega_{1},\nu_{1})$ be defined
by

$F_{x_{0}}(x_{1})=F(x_{0}x_{1}).$

Let $\widetilde{V}$ be subspace of
$L_{p}(\Omega_{0}\times\Omega_{1},\nu_{0}\otimes\nu_{1})$ of all $F\in
L_{p}(\Omega_{0}\times\Omega_{1},\nu_{0}\otimes\nu_{1})$ such that
$F_{x_{0}}\in V$ for $\nu_{0}$-almost every $x_{0}\in\Omega_{0}.$ Then
$\widetilde{V}$ is a proper non trivial closed subspace of
$L_{p}(\Omega_{0}\times\Omega_{1},\nu_{0}\otimes\nu_{1}).$

We claim that $\widetilde{V}$ is $\pi(G)$-invariant. Indeed, let
$F\in\widetilde{V},g\in G$ and $F^{\prime}=\pi(g)F;$ for
$x_{0}\in\Omega_{0},x_{1}\in\Omega_{1},$ set $x=x_{0}x_{1}$ and let
$q=q(x_{0}^{-1}g)$ be the $Q$-component of $x_{0}^{-1}g.$ We have
$\omega_{0}(g^{-1}x)=\omega_{0}(g^{-1}x_{0})$ and

$\displaystyle F^{\prime}_{x_{0}}(x_{1})$
$\displaystyle=F^{\prime}(x_{0}x_{1})=\chi(h(g^{-1}x))F(\omega_{0}(g^{-1}x_{0})\omega_{1}(g^{-1}x))$
$\displaystyle=\chi(h(((x_{0}^{-1}g)^{-1}x_{1}))F_{\omega_{0}(g^{-1}x_{0})}(\omega_{1}((x_{0}^{-1}g)^{-1}x_{1}))$
$\displaystyle=\chi(h(q^{-1}x_{1}))F_{\omega_{0}(g^{-1}x_{0})}(\omega_{1}(q^{-1}x_{1})))$
$\displaystyle=(\pi_{Q}(q)F_{\omega_{0}(g^{-1}x_{0})})(x_{1})$

Since $F_{\omega_{0}(g^{-1}x_{0})}\in V$ for almost every $x_{0}$ and since
$V$ is $Q$-invariant, it follows that $F^{\prime}\in\widetilde{V}$ and the
claim is proved.

Now, $\pi$ is irreducible. So, we have obtained a contradiction which shows
that $H$ has finite index in $Q.$

$\bullet$ _Seventh step._ We claim that $H=Q.$ Indeed, $H$ contains $Q^{0},$
since $H$ has finite index in $Q.$ As mentioned in the sixth step, $Q$ is a
direct product $Q=ZQ^{0}$ for a finite abelian subgroup $Z$. Hence, we have
$Q=Z^{\prime}H$ for a subgroup $Z^{\prime}$ of $Z.$ We have to show that
$Z^{\prime}$ is trivial.

Assume, by contradiction, that $Z^{\prime}\neq\\{e\\}.$ Then there exists a
non trivial character $\delta$ of $Z^{\prime}.$ We proceed as in the sixth
step. Let $\Omega_{0}\subset G$ a fundamental domain for $G/Q$. We identify
$L_{p}(G/H,\mu)$ with $L_{p}(\Omega_{0}\times
Z^{\prime},\nu_{0}\otimes\nu_{1})$ for a convenient $G$-quasi-invariant
probability measure $\nu_{0}$ on $\Omega_{0}\cong G/Q$ and a probability
measure $\nu_{1}$ on $Z^{\prime}.$

For $F\in L_{p}(\Omega_{0}\times Z^{\prime},\nu_{0}\otimes\nu_{1})$ and
$x_{0}\in\Omega_{0},$ let $F_{x_{0}}\in L_{p}(\Omega_{1},\nu_{1})$ be defined
by $F_{x_{0}}(x_{1})=F(x_{0}x_{1}).$ Let $\widetilde{V}_{\delta}$ be subspace
of $L_{p}(\Omega_{0}\times\Omega_{1},\nu_{0}\otimes\nu_{1})$ of all $F$ such
that $F_{x_{0}}(zx_{1})=\delta(z)F_{x_{0}}(x_{1})$ for all $z\in Z^{\prime}$
and $\nu_{0}$-almost every $x_{0}\in\Omega_{0}.$ Then $\widetilde{V}_{\delta}$
is a proper non trivial closed subspace of
$L_{p}(\Omega_{0}\times\Omega_{1},\nu_{0}\otimes\nu_{1}).$ Moreover, one
checks that $\widetilde{V}_{\delta}$ is $\pi(G)$-invariant; this contradicts
the irreducibility of $\pi.$

$\bullet$ _Eighth step._ There exists a a real linear form
$\lambda\in\mathfrak{a}^{*}$ and a unitary character $\chi$ of $M_{Q}$ such
that $\pi$ is equivalent to the representation $\pi(Q,\chi,\lambda,p)$ on
$L_{p}(G/Q,\mu_{Q})$ described in Theorem A.

Indeed, by the seventh step, we have $H=Q=M_{Q}A_{Q}N_{Q}.$ The unitary
character of $Q$ associated to the cocycle $c$ is trivial on $N_{Q}$ and
$M_{Q}$ centralizes $A_{Q};$ hence, this character is of the form
$\chi\otimes\chi^{\prime}$ for unitary characters $\chi$ of $M_{Q}$ and
$\chi^{\prime}$ of $A_{Q}.$ Let $F\subset\Sigma$ be the set of simple positive
roots associated to $Q$ There exists $\lambda\in\mathfrak{a}^{*}$ such that

$\chi^{\prime}(\exp X)=e^{-i\lambda(X)}\qquad\text{for all}\quad
X\in\mathfrak{a}_{F}=\log(A_{Q}).$

Let $\mu$ be the unique quasi-invariant Borel probability measure on $G/Q$
which is $K$-invariant. For $g\in G,$ decompose $g$ according to
$G=KM_{Q}A_{Q}N_{Q}$ as

$g=\kappa(g)\mu(g)\exp(H(g))n(g)\quad\text{for}\quad~\kappa(g)\in\Omega_{Q},\mu(g)\in
M_{Q},H(g)\in\mathfrak{a}_{F},n(g)\in N_{Q}.$

The Radon-Nikodym derivative of $g_{*}\mu$ with respect to $\mu$ is given by
(see e.g. [Knapp2, Proposition 8.44])

$\frac{dg_{*}(\mu)}{d\mu}(x)=e^{-\rho_{F}(H(g))}$

for

$\rho_{F}=\sum_{\alpha\in\Sigma^{+}\setminus\rm{span}(F)}\dim(\mathfrak{g}_{\alpha})\alpha.$

This finishes the proof of Theorem A.

The following lemma, which was used in the proof above, is a consequence of
the description of cocompact subgroups of semisimple Lie groups from [Witte]
(see also [Goto-Wang]).

######  Lemma 3.

Let $G$ be a noncompact connected semisimple linear Lie group with maximal
compact subgroup $K.$ Let $H$ be a closed subgroup of $G$ with the property
that the natural action of $K$ on $G/H$ is transitive. Upon replacing $H$ by
one of its conjugate, there exists a standard parabolic subgroup $Q$ which
contains $H$ and which has the following properties; denote by
$Q=M_{Q}A_{Q}N_{Q}$ the Langlands decomposition of $Q;$ write
$Q^{0}=CSA_{Q}N_{Q}$ for the identity component $Q^{0}$ of $Q,$ where $C$ is
the maximal compact factor of $M_{Q}^{0}$ and $S$ is the product of all the
non-compact simple factors of $M_{Q}^{0}$. Then there exists a connected
closed subgroup $C^{\prime}$ of $C$ such that

$H^{0}=C^{\prime}SA_{Q}N_{Q}.$

###### Proof.

Since $K$ acts transitively on $G/H$, we have $G=KH.$ In particular, $H$ is a
cocompact subgroup of $G.$ By [Witte, Theorem 1.2] and upon conjugating $H$,
there exists a standard parabolic subgroup $Q$ of $G$ with the following
properties:

  * •

$Q=M_{Q}A_{Q}N_{Q}$ contains $H;$

  * •

there exist a connected, normal subgroup $X$ of $S$ and a connected closed
subgroup $Y$ of $CA_{Q}$ such that $H^{0}=YXN_{Q}$, where $S$ and $C$ are as
in the statement of the Lemma.

Since $H/H^{0}$ is at most countable, we have

$G=\bigcup_{n\geq 1}KH_{0}g_{n}$

for some sequence $(g_{n})_{n\geq 1}$ of elements in $G.$ As $K$ is compact
and $H_{0}$ is closed, $KH_{0}g_{n}$ is a closed subset of $G$ for every $n.$
Hence, by Baire’s category theorem, $KH_{0}$ has a non empty interior. By
homogeneity, it follows that $KH_{0}$ is open in $G$. Since $G$ is connected,
we have therefore $G=KH_{0},$ that is,

(6) $G=KYXN_{Q}.$

Recall that the multiplication map $K\times A\times N\to G$ from the Iwasawa
decomposition $G=KAN$ is a diffeomorphism. The reductive Lie group $M_{Q}$ has
also an Iwasawa decomposition

$M_{Q}=K_{1}A_{1}N_{1},$

for $K_{1}:=K\cap M_{Q};$ we have $A=A_{1}A_{Q}$ as a direct product and
$N=N_{1}N_{Q}$ as semi-direct product (see [Knapp2, Proposition7.82]). In
particular, we have $C\subset K_{1}.$ Let $\Omega\subset K$ be a fundamental
domain for $K/K_{1}.$ So,

(7) $\text{the product map}\qquad\Omega\times K_{1}\times A_{1}\times
A_{Q}\times N_{1}\times N_{Q}\to G\qquad\text{is a bijection.}$

The Iwasawa decomposition of the semisimple Lie group $X\subset S$ is
$X=K_{2}A_{2}N_{2},$ where $K_{2}:=K_{1}\cap X$ and where $A_{2}$ and $N_{2}$
are subgroups of $A_{1}$ and $N_{2}$. By (6), we have therefore

$G=KYXN_{Q}=KK_{2}YA_{2}N_{2}N_{Q}.$

Since $Y\subset CA_{Q},$ it follows from (7) that

$\dim A_{2}=\dim A_{1},\quad\dim N_{2}=\dim N_{1},\quad\text{and}\quad
A_{Q}\subset Y.$

So, we have $S=X$ and $Y=C^{\prime}A_{Q}$ for a connected closed subgroup
$C^{\prime}$ of $C.$

∎

##  3\. Proof of Proposition 2

Let $Q=M_{Q}A_{Q}N_{Q}$ be a parabolic subgroup of $G$; for a real linear form
$\lambda\in\mathfrak{a}^{*}$ and a unitary character
$\chi:M_{Q}\to\mathbf{S}^{1}$, consider the representation
$\pi(Q,\lambda,\chi,p)$ of $G$ on $L_{p}(G/Q,\mu)$ as in the Introduction.

We may realize $\pi(Q,\lambda,\chi,p)$ in the so-called compact picture as
follows. Let $C(K,\chi)$ be the vector space of continuous functions
$F:K\to\mathbf{C}$ such that

$F(xk)=\overline{\chi(k)}F(x)\qquad\text{for all}\quad k\in K\cap M_{Q},\ x\in
K;$

equip $C(K,\chi)$ with the norm

$\|F\|_{p}=\left(\int_{K/(K\cap M_{Q})}|F(x)|^{p}d\mu(x))\right)^{1/p},$

where $\mu$ is the unique $K$-invariant probability measure on $K/(K\cap
M_{Q}).$ Let $L_{p}(K,\chi)$ be the completion of $(C(K,\chi),\|\cdot\|_{p}).$
Let $\rho_{Q,p}\in\mathfrak{a}^{*}$ be defined as in the Introduction. Then
$\pi(Q,\lambda,\chi,p)$ acts on $L_{p}(K,\chi)$ through

$\pi(Q,\chi,\lambda,p)(g)F(x)=\overline{\chi(\mu(g^{-1}x))}e^{-(i\lambda+\rho_{Q,p})H(g^{-1}x)}F(\kappa(g^{-1}x))$

for $F\in L_{p}(K,\chi),x\in K$ and $g\in G$ with decomposition

$g=\kappa(g)\mu(g)\exp(H(g))n(g),\qquad~\kappa(g)\in K,\mu(g)\in
M_{Q},H(g)\in\log(A_{Q}),n(g)\in N_{Q}$

Observe that, for $g\in K,$ the operator $\pi(Q,\chi,\lambda,p)(g)$ is simply
left translation on $L_{p}(K,\mu)$ by $g^{-1}$.

Set $E:=L_{p}(K,\chi)$. Recall that the subspace $E_{K}$ of $K$-finite vectors
in $E$ is the space of functions $F\in E$ such that

$\pi(Q,\chi,\lambda,p)(K)F=\left\\{\pi(Q,\chi,\lambda,p)(g)F:g\in K\right\\}$

spans a finite-dimensional subspace of $E.$ The space $E^{\infty}$ of
$C^{\infty}$-vectors in $E$ consists of smooth functions $F\in C(K,\chi).$ As
a $K$-module, $E^{\infty}$ is the representation of $K$ differentiably induced
by $\chi|_{K\cap M_{Q}}$ (in the sense of [Warner1, 5.3.1]). Therefore, by
Frobenius reciprocity (which remains valid in this context), the multiplicity
of $\delta\in\widehat{K}$ in $E^{\infty}$ is equal to the multiplicity of
$\chi$ in $\delta|_{K\cap M_{Q}}$ and is therefore finite, since $\delta$ is
finite dimensional. Hence, the multiplicity of $\delta$ in $E_{K}\cap
E^{\infty}$ is finite. Since $E_{K}\cap E^{\infty}$ is dense in $E$ (see e.g.
[Warner1, Theorem 4.4.5.16]), it follows that the multiplicity of $\delta$ in
$E$ is finite. In other words, $\pi(Q,\chi,\lambda,p)$ is an _admissible_
representation of $G$ (as defined [Knapp1, p.207] and [Wallach1, 3.3.5]) or is
a _$K$ -finite_ representation (as defined in [Warner1, 4.5.1]).

As is well-known, $E_{K}$ is a module over the universal enveloping algebra
$\mathcal{U}(\mathfrak{g}_{\mathbf{C}})$ of the complexification of the Lie
algebra $\mathfrak{g}$ of $G.$ Set

(8)
$\nu:=i\lambda+\rho_{Q,p}-\rho_{Q,2}=i\lambda+\left(\dfrac{2}{p}-1\right)\rho_{Q,2};$

then $E_{K}$ coincides with the $(\mathfrak{g},K)$-module associated to the
(nonunitary) induced representation ${\rm Ind}_{Q}^{G}(\chi\otimes\nu\otimes
1_{N_{Q}})$. This $(\mathfrak{g},K)$-module consists of the functions
$f:G\to\mathbf{C}$ which are $C^{\infty}$ and have the following properties:

  * •

$f(gman)=\overline{\chi(m)}e^{-(i\lambda+\nu+\rho_{Q,2})(\log a)}~f(g)$ for
all $g\in G$ and $man\in Q;$

  * •

the linear span of $K\cdot f$ is finite dimensional space, where $K\cdot f$ is
the space of translates $g\mapsto f(k^{-1}g)$ for $k\in K.$

(for this, see [Wallach1, 5.2.1]). So, the $(\mathfrak{g},K)$ module
associated to $\pi(Q,\chi,\lambda,p)$ is the principal series representation
denoted $I_{Q,\chi,\nu}$ in [Wallach1] for $\nu$ as in (8).

Now, since $\pi(Q,\chi,\lambda,p)$ is admissible, $\pi(Q,\chi,\lambda,p)$ is
completely irreducible if and only if its associated $(\mathfrak{g},K)$-module
$I_{Q,\chi,\nu}$ is algebraically irreducible. Moreover, two such
representations $\pi(Q_{1},\chi_{1},\lambda_{1},p_{1})$ and
$\pi(Q_{1},\chi_{1},\lambda_{1},p_{1})$ are Naimark equivalent if and only if
the $(\mathfrak{g},K)$-modules $I_{Q_{1},\chi_{1},\nu_{1}}$ and
$I_{Q_{2},\chi_{2},\nu_{2}}$ are algebraically equivalent (for all this, see
Theorems 4.5.5.4 and 4.5.5.2 in [Warner1]).

##  4\. Proof of Theorem B

We assume from now on that $G$ is a simple Lie group with real rank one. In
this case, we have $\dim\mathfrak{a}=1.$

Choose $\alpha\in\Sigma^{+}$ and $H\in\mathfrak{a}$ such that
$\frac{1}{2}\alpha\notin\Sigma$ and $\alpha(H)=1.$ We identify
$\mathfrak{a}^{*}_{\mathbf{C}}$ with $\mathbf{C}$ by means of the map
$\lambda\mapsto\lambda(H).$

There is, up to conjugation, only one parabolic subgroup, namely $P:=MAN.$ For
$p\in(1+\infty),$ we will write $\rho_{p}$ instead of $\rho_{P,p}$. In view of
Proposition 2, we have to study, given $\lambda\in\mathbf{R}$ and
$\chi\in\widehat{M^{\rm Ab}},$ the irreducibility of the (non unitary)
principal series representation $I_{P,\chi,\nu_{\lambda,p}}$ for

$\nu_{\lambda,p}=i\lambda+\left(\dfrac{2}{p}-1\right)\rho_{2}$

and to decide when two such representations are equivalent. We will apply
several times the following result of Kostant about the case where
$\nu=1_{M}.$ Set

$t_{\alpha}:=\begin{cases}~(\dim\mathfrak{g}_{\alpha})/2\quad\text{if}\quad
2\alpha\notin\Sigma\\\ (\dim\mathfrak{g}_{\alpha})/2+1\quad\text{if}\quad
2\alpha\in\Sigma,\end{cases}$

where $\mathfrak{g}_{\alpha}\subset\mathfrak{g}$ is the usual root space
corresponding to $\alpha;$ set also

$n_{\alpha}:=\begin{cases}~1\quad\text{if}\quad 2\alpha\notin\Sigma\\\
2\quad\text{if}\quad 2\alpha\in\Sigma.\end{cases}$

Recall that

$\rho_{2}=(\dim\mathfrak{g}_{\alpha})/2+\dim\mathfrak{g}_{2\alpha}.$

For $\nu\in\mathbf{C},$ we have (see Theorem 2 in [Kostant1] and Theorem 2.9.8
in [Kostant2]):

(9) $I_{P,1_{M},\nu}\ \text{is \emph{not}\
irreducible}~\Longleftrightarrow\nu\notin(-t_{\alpha},t_{\alpha})\ \text{and}\
\nu+\rho_{2}\in n_{\alpha}\mathbf{Z}.$

We will treat separately the different simple groups $G$ which may occur; for
the data we will use concerning these groups, we refer to [Helgason, Chap. X].

$\bullet$ Let $G=\mathrm{SO}_{0}(n,1)$ for $n\geq 2.$ Here, $K\cong SO(n)$ and
$M\cong SO(n-1)$. Moreover, we have $\dim\mathfrak{g}_{\alpha}=n-1$ and
$2\alpha\notin\Sigma.$ So,

$t_{\alpha}=\dfrac{n-1}{2},\qquad
n_{\alpha}=~1,\qquad\text{and}~\qquad\rho_{2}=\frac{n-1}{2};$

hence, for $\lambda\in\mathbf{R},$ we have

$\nu_{\lambda,p}=i\lambda+\left(\dfrac{2}{p}-1\right)\frac{n-1}{2}.$

Since $\left|\dfrac{2}{p}-1\right|<1,$ it follows from (9) that

$I_{P,1_{M},\nu_{\lambda,p}}\quad\text{is irreducible for all}\
\lambda\in\mathbf{R}\ \text{and}\ p\in(1,+\infty)\ \text{with}\ p\neq 2.$

The case where $n\neq 3$ is settled since the abelianization of $M$ is then
trivial and so $\widehat{M^{\rm Ab}}=1_{M}.$

Assume now that $n=3.$ Observe that in this case
$G\cong\mathrm{SL}_{2}(\mathbf{C})$ and $M\cong\mathrm{U}(1)$. We use here the
results from [Thieleker, §. 13, B]: identifying $\widehat{M^{\rm Ab}}$ with
$\frac{1}{2}\mathbf{Z}$, we have, for $m\in\widehat{M^{\rm Ab}}$ and
$\nu\in\mathbf{C},$

$I_{P,m,\nu}\ \text{is irreducible}~\Longleftrightarrow(\nu+l+1)(\nu-l-1)\neq
0$

for all $l\in\\{|m|+k:k=0,1,2,\cdots\\}.$ Since $(n-1)/2=1$, this condition is
clearly satisfied for $\nu_{\lambda,p}$ and so we obtain that
$I_{P,\chi,\nu_{\lambda,p}}$ is irreducible for all
$\lambda\in\mathbf{R},\chi\in\widehat{M^{\rm Ab}},$ and $p\in(1,+\infty)$ with
$p\neq 2.$

$\bullet$ Let $G=\mathrm{SU}(n,1)$ for $n\geq 2.$ Here, $K\cong\mathrm{SU}(n)$
and $M\cong\mathrm{U}(n-1)$. Moreover, we have
$\dim\mathfrak{g}_{\alpha}=2(n-1)$ and $\dim\mathfrak{g}_{2\alpha}=1$. So,

$t_{\alpha}=n,\qquad n_{\alpha}=~2,\qquad\text{and}~\qquad\rho_{2}=n;$

hence, for $\lambda\in\mathbf{R},$ we have

$\nu_{\lambda,p}=i\lambda+\left(\dfrac{2}{p}-1\right)n.$

Since $\left|\dfrac{2}{p}-1\right|<1,$ it follows from (9) that

$I_{P,1_{M},\nu_{\lambda,p}}\quad\text{is irreducible for all}\
\lambda\in\mathbf{R}\ \text{and}\ p\in(1,+\infty)\ \text{with}\ p\neq 2.$

In [Kraljevic, Proposition 1] precise necessary and sufficient conditions are
given on $\sigma\in\widehat{M}$ and $\nu\in\mathbf{C}$ for $I_{P,\sigma,\nu}$
to be irreducible. In the special case of $\sigma\in\widehat{M^{\rm Ab}}$,
this criterion reads as follows. Identify $\widehat{M^{\rm Ab}}$ with
$\frac{1}{n+1}\mathbf{Z}$ as in [Kraljevic] and for $\sigma\in\widehat{M^{\rm
Ab}}$ and $\nu\in\mathbf{C},$ define
$(s_{1},\dots,s_{n+1})\in\mathbf{C}^{n+1}$ by

$s_{i}=\begin{cases}\dfrac{1}{2}(\nu-(n-1)\sigma)\quad\text{if}\quad i=1\\\
\sigma+\dfrac{n}{2}-i+1\quad\text{if}\quad 2\leq i\leq n\\\
-\dfrac{1}{2}(\nu+(n-1)\sigma)\quad\text{if}\quad i=n+1.\end{cases}$

Then $I_{P,m,\nu}$ is not irreducible if and only if either
$s_{1}-s_{i}\in\mathbf{Z}\setminus\\{0\\}$ for all $2\leq i\leq n$ or
$s_{n+1}-s_{i}\in\mathbf{Z}\setminus\\{0\\}$ for all $2\leq i\leq n$.

It follows immediately that $I_{P,\sigma,\nu_{\lambda,p}}$ is irreducible if
$\lambda\neq 0.$ So, we may assume that $\lambda=0.$

Writing $\sigma=\dfrac{m}{n+1}$ for $m\in\mathbf{Z},$ the criterion above
implies that $I_{P,\sigma,\nu_{0,p}}$ is not irreducible if and only if either

$\dfrac{1}{2}\left(\nu_{p,0}-(n-1)\dfrac{m}{n+1}\right)-\dfrac{m}{n+1}+\dfrac{n}{2}\in\mathbf{Z}\setminus\\{0\\}$

or

$-\dfrac{1}{2}\left(\nu_{p,0}+(n-1)\dfrac{m}{n+1}\right)-\dfrac{m}{n+1}+\dfrac{n}{2}\in\mathbf{Z}\setminus\\{0\\}$

that is, if and only if

$\text{either}\quad\nu_{p,0}-(m+n)\in
2\mathbf{Z}\setminus\\{0\\}\quad\text{or}\quad\nu_{p,0}-(m+n)\in
2\mathbf{Z}\setminus\\{0\\}$

As $\nu_{\lambda,0}=\left(\dfrac{2}{p}-1\right)n,$ we see that
$I_{P,\sigma,\nu_{0,p}}$ is not irreducible if only if

(10) $\text{either}\quad 2\left(\dfrac{1}{p}-1\right)n-m\in
2\mathbf{Z}\setminus\\{0\\}\quad\text{or}\quad\dfrac{2n}{p}-m\in
2\mathbf{Z}\setminus\\{0\\}.$

It is clear that (10) is equivalent to

$p\in\left\\{\dfrac{2n}{k}:k\in\\{1,\dots,2n-1\\},k\equiv m\ (\mathrm{mod}\
2),k\neq m,k\neq 2n+m\right\\}.$

$\bullet$ Let $G=\mathrm{Sp}(n,1)$ for $n\geq 2.$ Here,
$K\cong\mathrm{Sp}(n)\times\mathrm{Sp}(1)$ and
$M\cong\mathrm{Sp}(n-1)\times\mathrm{Sp}(1)$. Moreover, we have
$\dim\mathfrak{g}_{\alpha}=4(n-1)$ and $\dim\mathfrak{g}_{2\alpha}=3$. So,

$t_{\alpha}=2n-1,\qquad n_{\alpha}=~2,\qquad\text{and}~\qquad\rho_{2}=2n+1;$

hence, for $\lambda\in\mathbf{R},$ we have

$\nu_{\lambda,p}=i\lambda+\left(\dfrac{2}{p}-1\right)(2n+1).$

Observe that the abelianization of $M$ is trivial and so $\widehat{M^{\rm
Ab}}=\\{1_{M}\\}.$ Moreover, we have

$\left|\dfrac{2}{p}-1\right|(2n+1)\geq 2n-1~\Longleftrightarrow
p\in(1,\frac{2n+1}{2n}]\cup[2n+1,+\infty).$

and

$\left(\dfrac{2}{p}-1\right)(2n+1)+2n-1~\in 2\mathbf{Z}\Longleftrightarrow
p\in\left\\{\dfrac{2n+1}{k}:k\in\mathbf{{\mathbf{N}}^{*}}\right\\}.$

It follows from (9) that

  * •

$I_{P,1_{M},\nu_{\lambda,p}}$ is irreducible for every
$\lambda\in\mathfrak{a}^{*}\setminus\\{0\\}$

  * •

$I_{P,1_{M},\nu_{0,p}}$ is not irreducible if and only if $p=\dfrac{2n+1}{2n}$
or $p=2n+1.$

$\bullet$ Let $G=\mathrm{F}_{4(-20)}.$ Here, $K\cong\mathrm{Spin}(9)$ and
$M\cong\mathrm{Spin}(7).$ Moreover, we have $\dim\mathfrak{g}_{\alpha}=8$ and
$\dim\mathfrak{g}_{2\alpha}=7$. So,

$t_{\alpha}=5,\qquad n_{\alpha}=~2,\qquad\text{and}~\qquad\rho_{2}=11;$

hence, for $\lambda\in\mathbf{R},$ we have

$\nu_{\lambda,p}=i\lambda+11\left(\dfrac{2}{p}-1\right).$

The abelianization of $M$ is trivial and so $\widehat{M^{\rm
Ab}}=\\{1_{M}\\}.$ Moreover, we have

$11\left|\dfrac{2}{p}-1\right|\geq 5~\Longleftrightarrow
p\in(1,\frac{11}{8}]\cup[\frac{11}{3},+\infty).$

and

$11\left(\dfrac{2}{p}-1\right)+5~\in 2\mathbf{Z}\Longleftrightarrow
p\in\left\\{\frac{11}{k}:k\in\mathbf{{\mathbf{N}}^{*}}\right\\}.$

It follows from (9) that

  * •

$I_{P,1_{M},\nu_{\lambda,p}}$ is irreducible for every
$\lambda\in\mathfrak{a}^{*}\setminus\\{0\\}$

  * •

$I_{P,1_{M},\nu_{0,p}}$ is not irreducible if and only if
$p\in\left\\{\dfrac{11}{10},\dfrac{11}{9},\dfrac{11}{8},\dfrac{11}{3},\dfrac{11}{2},11\right\\}.$

In all the cases above, if two representations $I_{P,\chi,\nu}$ and
$I_{P,\chi^{\prime},\nu^{\prime}}$ are irreducible, then they are equivalent
if and only if

$(\chi^{\prime},\nu)=(\overline{\chi},\nu)\quad\text{or}\quad(\chi^{\prime},\nu^{\prime})=(\overline{\chi},-\nu).$

Now,

$-\nu_{\lambda,p}=-(i\lambda+\left(\dfrac{2}{p}-1\right)\rho_{2}=(-i\lambda+\left(\dfrac{2}{q}-1\right)\rho_{2}=\nu_{\lambda,q}.$

Therefore, $I_{P,\chi^{\prime},\nu_{\lambda^{\prime},p^{\prime}}}$ and
$I_{P,\chi,\nu_{\lambda,p}}$ are equivalent if and only if

$(\chi^{\prime},\lambda^{\prime},p^{\prime})=(\overline{\chi},\lambda,p)\quad\text{or}\quad(\chi^{\prime},\lambda^{\prime},p^{\prime})=(\overline{\chi},-\lambda,q).$

##  5\. An example: $SL_{2}(\mathbf{R})$

Let $G=\mathrm{SL}_{2}(\mathbf{R})$, with maximal compact subgroup $K=SO(2)$.
The standard minimal parabolic subgroup is

$P\,=\left\\{\begin{pmatrix}a&b\\\ 0&a^{-1}\end{pmatrix}:a\in\mathbf{R},a\neq
0,b\in\mathbf{R}\right\\}.$

We have $P=MAN$ for

$A\,=\left\\{\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}:a\in\mathbf{R},a>0\right\\},$
$M\,=\left\\{\pm\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\right\\},$

and

$N\,=\left\\{\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}:b\in\mathbf{R}\right\\}.$

Identifying $\mathfrak{a}^{*}_{\mathbf{C}}$ with $\mathbf{C}$ as above, we
have

$\rho_{P,p}=\dfrac{1}{p}$

for every $p\in(1+\infty).$ We identify $G/P$ as $G$-space with the real
projective line $\mathbf{P}(\mathbf{R}),$ with $G$ acting by Möbius
transformations on this latter space. The Lebesgue measure $\mu$ is the unique
$K$-invariant probability Borel measure on $\mathbf{P}(\mathbf{R}).$ The
measurable space $G/P$ can further be identified with
$\mathbf{R}=\mathbf{P}(\mathbf{R})\setminus\\{\infty\\}$.

The Radon-Nikodym derivative for $g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$
acting on $\mathbf{R}$ is given by

$\frac{dg_{*}(\mu)}{d\mu}(x)=\dfrac{1}{(cx+d)^{2}}.$

Let $\varepsilon$ denote the non trivial character of $M$ and let
$\lambda\in\mathbf{R}.$ The representation $\pi(P,\chi,\lambda,p)$ of $G$ by
isometries on $L_{p}(\mathbf{R},\mu)$ is defined by

$\pi(P,1_{M},\lambda,p)(g)f(x)=|cx+d|^{-i\lambda-\frac{2}{p}}f\left(\frac{ax+b}{cx+d}\right)\quad\text{if}\quad
g^{-1}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$

and

$\pi(P,\varepsilon,\lambda,p)(g)f(x)={\rm
sgn}(cx+d)|cx+d|^{-i\lambda-\frac{2}{p}}f\left(\frac{ax+b}{cx+d}\right)\quad\text{if}\quad
g^{-1}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$

The Harish-Chandra $(\mathfrak{g},K)$-module underlying
$\pi(P,\varepsilon,\lambda,p)$ is $I_{P,\chi,\nu}$ for

$\nu=i\lambda+\delta_{p}\rho_{P,2}=i\lambda+\dfrac{2}{p}-1$

Assume that $p\neq 2.$ It is well-known that $I_{P,\chi,\nu}$ is irreducible
(see e.g. Proposition 1.3.3 in [Vogan]; observe that the irreducibility of
$I_{P,\varepsilon,\nu}$ depends on the fact that $\dfrac{2}{p}-1$ is not an
integer). Moreover, the modules $I_{P,\chi,\nu}$ are pairwise non equivalent
for fixed $p.$

In summary, the $L_{p}$-dual space of $G=\mathrm{SL}_{2}(\mathbf{R})$ for
$p\in(1,+\infty)$ with $p\neq 2$ is

$\widehat{G}_{L_{p}}=\left\\{\pi(P,1_{M},\lambda,p):\lambda\in\mathbf{R}\right\\}\cup\left\\{\pi(P,\varepsilon,\lambda,p):\lambda\in\mathbf{R}\right\\}\cup\\{1_{G}\\}.$

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author = {Varadarajan, V. S.}, title = {Geometry of quantum theory}, edition =
{2}, publisher = {Springer-Verlag, New York}, date = {1985}, pages =
{xviii+412}, isbn = {0-387-96124-0}, review = {\MR{0805158}}} VoganJr.David
A.Representations of real reductive lie groupsProgress in
Mathematics15Birkhäuser, Boston, MA1981xvii+754ISBN 3-7643-3037-6Review
MathReviews@book{Vogan, author = {Vogan, David A., Jr.}, title =
{Representations of real reductive Lie groups}, series = {Progress in
Mathematics}, volume = {15}, publisher = {Birkh\"{a}user, Boston, MA}, date =
{1981}, pages = {xvii+754}, isbn = {3-7643-3037-6}, review = {\MR{0632407}}}
WallachNolan R.Cyclic vectors and irreducibility for principal series
representationsTrans. Amer. Math. Soc.1581971107–113ISSN 0002-9947Review
MathReviewsDocument@article{Wallach1, author = {Wallach, Nolan R.}, title =
{Cyclic vectors and irreducibility for principal series representations},
journal = {Trans. Amer. Math. Soc.}, volume = {158}, date = {1971}, pages =
{107–113}, issn = {0002-9947}, review = {\MR{0281844}}, doi =
{10.2307/1995774}} WallachNolan R.Real reductive groups. iPure and Applied
Mathematics132Academic Press, Inc., Boston, MA1988xx+412ISBN
0-12-732960-9Review MathReviews@book{Wallach-book, author = {Wallach, Nolan
R.}, title = {Real reductive groups. I}, series = {Pure and Applied
Mathematics}, volume = {132}, publisher = {Academic Press, Inc., Boston, MA},
date = {1988}, pages = {xx+412}, isbn = {0-12-732960-9}, review =
{\MR{0929683}}} WarnerGarthHarmonic analysis on semi-simple lie groups. iDie
Grundlehren der mathematischen WissenschaftenBand 188Springer-Verlag, New
York-Heidelberg1972xvi+529Review MathReviews@book{Warner1, author = {Warner,
Garth}, title = {Harmonic analysis on semi-simple Lie groups. I}, series =
{Die Grundlehren der mathematischen Wissenschaften}, volume = {Band 188},
publisher = {Springer-Verlag, New York-Heidelberg}, date = {1972}, pages =
{xvi+529}, review = {\MR{0498999}}} WitteDaveCocompact subgroups of semisimple
lie groupstitle={Lie algebra and related topics}, address={Madison, WI},
date={1988}, series={Contemp. Math.}, volume={110}, publisher={Amer. Math.
Soc., Providence, RI}, ISBN 0-8218-5119-51990309–313Review
MathReviewsDocument@article{Witte, author = {Witte, Dave}, title = {Cocompact
subgroups of semisimple Lie groups}, conference = {title={Lie algebra and
related topics}, address={Madison, WI}, date={1988}, }, book =
{series={Contemp. Math.}, volume={110}, publisher={Amer. Math. Soc.,
Providence, RI}, }, isbn = {0-8218-5119-5}, date = {1990}, pages = {309–313},
review = {\MR{1079114}}, doi = {10.1090/conm/110/1079114}} ZimmerRobert
J.Ergodic theory and semisimple groupsMonographs in Mathematics81Birkhäuser
Verlag, Basel1984x+209ISBN 3-7643-3184-4Review
MathReviewsDocument@book{Zimmer1, author = {Zimmer, Robert J.}, title =
{Ergodic theory and semisimple groups}, series = {Monographs in Mathematics},
volume = {81}, publisher = {Birkh\"{a}user Verlag, Basel}, date = {1984},
pages = {x+209}, isbn = {3-7643-3184-4}, review = {\MR{0776417}}, doi =
{10.1007/978-1-4684-9488-4}}