text:
\begin{document}

\title{Horn problem for quasi-hermitian Lie groups}

\author{Paul-Emile Paradan\footnote{IMAG, Univ Montpellier, CNRS, email : paul-emile.paradan@umontpellier.fr}}

\maketitle

\date{}

\begin{abstract}
In this paper, we prove some convexity results associated to orbit projection of non-compact real reductive Lie groups.
\end{abstract}

\tableofcontents

\section{Introduction}

This paper is concerned with convexity properties associated to orbit projection.

Let us consider two Lie groups $G\subset \wG$ with Lie algebras $\ggot\subset\wg$ and corresponding projection $\pi_{\ggot,\wg}: \wg^*\to\ggot^*$. A
longstanding problem has been to understand how a coadjoint orbit $\tilde{\Ocal}\subset \wg^*$ decomposes under the projection
$\pi_{\ggot,\wg}$. For this purpose, we may define
$$
\Delta_G(\tilde{\Ocal})=\{\Ocal\in \ggot^*/G\, ; \ \Ocal\subset \pi_{\ggot,\wg}(\tilde{\Ocal})\}.
$$

When the Lie group $G$ is compact and connected, the set $\ggot^*/G$ admits a natural identification with a Weyl chamber $\tgot^*_{\geq 0}$. In this context,
we have the well-known convexity theorem \cite{Heckman82,Atiyah82,Guillemin-Sternberg82.bis,Kirwan.84.bis,Hilgert-Neeb-Plank-94,sjamaar98,L-M-T-W}.

\begin{theo}\label{theo:convexity-1}
Suppose that $G$ is compact connected and that the projection $\pi_{\ggot,\wg}$ is proper when restricted to $\tilde{\Ocal}$. Then
$\Delta_G(\tilde{\Ocal})=\{\xi\in \tgot^*_{\geq 0}\, ; \ G\xi\subset \pi_{\ggot,\wg}(\tilde{\Ocal})\}$ is a closed convex locally polyhedral subset of $\tgot^*$.
\end{theo}

When the Lie group $\tilde{G}$ is also compact and connected, we may consider
\begin{equation}\label{eq:pi-G-tilde-G}
\Pi(\tilde{G},G):=\left\{(\tilde{\xi},\xi)\in \tilde{\tgot}^*_{\geq 0}\times\tgot^*_{\geq 0} ; \ G\xi\,\subset \,\pi_{\ggot,\tilde{\ggot}}\big(\tilde{G}\tilde{\xi}\,\big)\right\}.
\end{equation}

Here is another convexity theorem \cite{Horn,Klyachko,Berenstein-Sjamaar,Belkale06,BK1,Montagard-Ressayre-07,Knutson-Tao-Woodward-04,Ressayre-10}.
\begin{theo}\label{theo:convexity-2}
Suppose that $G\subset \wG$ are compact connected Lie groups. Then $\Pi(\tilde{G},G)$ is a closed convex polyhedral cone and we can parametrize its facets by cohomological means (i.e. Schubert calculus).
\end{theo}

In this article we obtain an extension of Theorems \ref{theo:convexity-1} and \ref{theo:convexity-2} in the case where
the two groups $G$ and $\wG$ are not compact.

Let us explain in which framework we work. We suppose that $\wG/\wK$ is a Hermitian symmetric space of a non-compact type. Among the elliptic coadjoint orbits of $\wG$, some of them are naturally K\"{a}hler $\wK$-manifolds. These orbits are called the holomorphic coadjoint orbits of $\wG$. They are the strongly elliptic coadjoint orbits closely related to the holomorphic discrete series of Harish-Chandra. These orbits intersect the Weyl chamber $\tilde{\tgot}^*_{\geq 0}$ of $\wK$ into a sub-chamber $\wchol$ called the holomorphic chamber. The basic fact here is that the union
$$
\Ccal_{\wG/\wK}^0:=\bigcup_{\tilde{a}\in\wchol} \wG\tilde{a}
$$
is an open invariant convex cone of $\wg^*$. See \S \ref{sec:cone-elliptic} for more details.

Let $G/K\subset \wG/\wK$ be a sub-Hermitian symmetric space of a non-compact type. As the projection $\pi_{\ggot,\wg}: \wg^*\to\ggot^*$ sends the convex cone
$\Ccal_{\wG/\wK}^0$ inside the convex cone $\Ccal_{G/K}^0$, it is natural to study the following object reminiscent of (\ref{eq:pi-G-tilde-G}) :
\begin{equation}
\Pi_{\rm hol}(\tilde{G},G):=\left\{(\tilde{\xi},\xi)\in \wchol\times\chol ; \ G\xi\,\subset \,\pi_{\ggot,\tilde{\ggot}}\big(\tilde{G}\tilde{\xi}\,\big)\right\}.
\end{equation}

Let $\tilde{\mu}\in \wchol$. We will also give a particular attention to the intersection of $\Pi_{\rm hol}(\tilde{G},G)$ with the hyperplane $\tilde{\xi}=\tilde{\mu}$,
that is to say
\begin{equation}
\Delta_G(\wG\tilde{\mu}):=\left\{\xi\in \chol ; \ G\xi\,\subset \,\pi_{\ggot,\tilde{\ggot}}\big(\tilde{G}\tilde{\mu}\,\big)\right\}.
\end{equation}

Consider the case where $G$ is embedded diagonally in $\wG:=G^s$ for $s\geq 2$. The corresponding set $\Pi_{\rm hol}(G^s,G)$
is called the holomorphic Horn cone, and it is defined as follows
$$
\Horn^s(G):=\Big\{(\xi_1,\cdots,\xi_{s+1})\in \chol^{s+1} ; \ G\xi_{s+1}\,\subset \sum_{j=1}^s G\xi_j\Big\}.
$$

The first result of this article is the following Theorem.

\medskip

\noindent {\bf Theorem A.}
\begin{itemize}
\item $\Pi_{\rm hol}(\tilde{G},G)$ is a closed convex cone of $\wchol\times\chol$.
\item $\Horn^s(G)$ is a closed convex cone of $\chol^{s+1}$ for any $s\geq 2$.
\end{itemize}

\medskip

We obtain the following corollary which corresponds to a result of Weinstein \cite{Weinstein01}.

\medskip

\noindent {\bf Corollary.}{\em \
For any $\tilde{\mu}\in \wchol$, $\Delta_G(\wG\tilde{\mu})$ is a closed and convex subset of $\chol$.
}

\medskip

A first description of the closed convex cone $\Pi_{\rm hol}(\tilde{G},G)$ goes as follows. The quotient $\qgot$ of the tangent spaces
$\T_{e}G/K$ and $\T_{e}\wG/\wK$ has a natural structure of a Hermitian $K$-vector space. The symmetric algebra ${\rm Sym}(\qgot)$ of
$\qgot$ defines an admissible $K$-module. The irreducible representations of $K$ (resp. $\wK$) are parametrized by a semi-group
$\wedge^*_+:$ (resp. $\tilde{\wedge}^*_+$). For any $\lambda\in\wedge^*_+$ (resp.  $\tilde{\lambda}\in\tilde{\wedge}^*_+$), we denote by $V_\lambda^K$ (resp. $V_{\tilde{\lambda}}^{\wK}$) the irreducible representation of $K$ (resp. $\wK$) with highest weight $\lambda$ (resp. $\tilde{\lambda}$). If $E$ is a representation of
$K$, we denote by $\left[V^K_\lambda : E\right]$ the multiplicity of $V^K_\lambda$ in $E$.

\begin{defi}\label{defi:horn-cone-general}
\begin{enumerate}
\item $\Pi^\Zbb(\wG,G)$ is the semigroup of $\tilde{\wedge}^*_+\times\wedge^*_+$ defined by the conditions:
$$
(\tilde{\lambda},\lambda)\in \Pi^\Zbb(\wG,G)\Longleftrightarrow \Big[V^K_\lambda\,:\,V_{\tilde{\lambda}}^{\wK}\otimes {\rm Sym}(\qgot)\Big]\neq 0.
$$
\item $\Pi(\wG,G)$ is the convex cone defined as the closure of
$\Qbb^{>0}\cdot \Pi^\Zbb(\wG,G)$.
\end{enumerate}
\end{defi}

The second result of this article is the following Theorem.

\medskip

\medskip

\noindent {\bf Theorem B.} {\em We have the equality
\begin{equation}\label{eq:theo-B}
\Pi_{\rm hol}(\wG,G)=\Pi(\wG,G)\, \bigcap\,  \wchol\!\times\!\chol.
\end{equation}
}

\medskip

A natural question is the description of the facets of the convex cone $\Pi_{\rm hol}(\tilde{G},G)$. In order to do that,
we consider the group $\wK$ endowed with the following $\wK\times K$-action : $(\tilde{k},k)\cdot \tilde{a}=\tilde{k}\tilde{a}k^{-1}$.
The cotangent space $\T^*\wK$ is then a symplectic manifold equipped with a Hamiltonian action of $\wK\times K$.
We consider now the Hamiltonian $\wK\times K$-manifold $\T^*\wK\times \qgot$, and we denote by $\Delta(\T^*\wK\times \qgot)$
 the corresponding  Kirwan polyhedron.

Let $W=N(T)/T$ be the Weyl group of $(K,T)$ and let $w_0$ be the longest Weyl group element. Define an involution $*:\tgot^*\to\tgot^*$ by
$\xi^*=- w_0\xi$. A standard result permits to affirm that $(\tilde{\xi},\xi)\in \Pi(\wG,G)$ is an only if
$(\tilde{\xi},\xi^*)\in\Delta(\T^*\wK\times \qgot)$  (see \S \ref{sec:affine}).

We obtain then another version of Theorem B.

\medskip

\medskip

\noindent {\bf Theorem B, second version.} {\em An element $(\tilde{\xi},\xi)$ belongs to  $\Pi_{\rm hol}(\wG,G)$ if and only if
$$
(\tilde{\xi},\xi)\in\wchol\times\chol\quad {\rm and}\quad (\tilde{\xi},\xi^*)\in\Delta(\T^*\wK\times \qgot).
$$
}

\medskip

Thanks to the second version of Theorem B, a natural way to describe the facets of the cone $\Pi_{\rm hol}(\wG,G)$ is to exhibit those of
the Kirwan polyhedron $\Delta(\T^*\wK\times \qgot)$. In this later case, it can be done using Ressayre's data (see \S \ref{sec:cone:faces}).

The second version of Theorem B  permits also the following description of the convex subsets $\Delta_G(\wG\tilde{\mu})$, $\tilde{\mu}\in \wchol$.
Let $\Delta_K(\wK\tilde{\mu}\times\overline{\qgot})$  be the Kirwan polyhedron associated to the Hamiltonian action of
$K$ on $\wK\tilde{\mu}\times \overline{\qgot}$. Here $\overline{\qgot}$ denotes the
$K$-module $\qgot$ with opposite complex structure.

\medskip

\medskip

\noindent {\bf Theorem C.} {\em For any $\tilde{\mu}\in \wchol$, we have $\Delta_G(\wG\tilde{\mu})=\Delta_K(\wK\tilde{\mu}\times\overline{\qgot})$.
}

\medskip

\medskip

Let us detail Theorem C in the case where $G$ is embedded in $\wG=G\times G$ diagonally. We denote by
$\pgot$  the $K$-Hermitian space $\T_{e}G/K$. Let $\kappa$ be the Killing form of the Lie algebra $\ggot$. The vector space $\overline{\pgot}$ is equipped with the symplectic $2$-form $\Omega_{\bar{\pgot}}(X,Y)=-\kappa(z,[X,Y])$ and the compatible complex structure $-{\rm ad}(z)$.

Let us denote by $\Delta_K(K\mu_1\times K\mu_2\times\overline{\pgot})$ and by $\Delta_K(\overline{\pgot})$ the Kirwan polyhedrons relative to the Hamiltonian actions of
$K$ on $K\mu_1\times K\mu_2\times\overline{\pgot}$ and on $\overline{\pgot}$. Theorem C says that for any
$\mu_1,\mu_2\in\chol$, the convex set $\Delta_G(G\mu_1\times G\mu_2)$ is equal to the Kirwan polyhedron
$\Delta_K(K\mu_1\times K\mu_2\times\overline{\pgot})$.

To any non-empty subset $C$ of a real vector space $E$, we may associate its asymptotic cone ${\rm As}(C) \subset E$ which is  the set formed by the limits
$y=\lim_{k\to\infty} t_k y_k$, where $(t_k)$ is a sequence of non-negative reals converging to $0$ and $y_k\in C$.

We finally get the following description of the asymptotic cone of $\Delta_G(G\mu_1\times G\mu_2)$.

\medskip

\medskip

\noindent {\bf Corollary D.} {\em
For any $\mu_1,\mu_2\in\chol$, the asymptotic cone of $\Delta_G(G\mu_1\times G\mu_2)$ is equal to $\Delta_K(\overline{\pgot})$.
}

\medskip

\medskip

In \cite{pep-discrete-2008} \S 5,  we explained how to describe the cone $\Delta_K(\overline{\pgot})$ in terms of strongly orthogonal roots.

\medskip

Let us finish this introduction with few remarks on related works:

\begin{enumerate}
\item[$-$] When $G$ is compact, equal to the maximal compact subgroup $\wK$ of $\wG$, the results of Theorems B and C were already obtained by G. Deltour in his thesis
\cite{Deltour-these,Deltour-transf-group}.

\item[$-$] In \cite{Eshmatov-Foth14}, A. Eshmatov and P. Foth proposed a description of the set $\Delta_G(G\mu_1\times G\mu_2)$. {\bf But
their computations give not the same result as ours}. From their main result (Theorem 3.2) it follows that the asymptotic cone of $\Delta_G(G\mu_1\times G\mu_2)$
is equal to the intersection of the Kirwan polyhedron $\Delta_T(\overline{\pgot})$ with the Weyl chamber $\tgot^*_{\geq 0}$. But
since $\Delta_K(\overline{\pgot})\neq \Delta_T(\overline{\pgot})\cap \tgot^*_{\geq 0}$ in general, it is in contradiction with Corollary D.
\end{enumerate}

\subsection*{Notations}

 In this paper, $G$ denotes a connected real reductive Lie group : we take here
the convention of Knapp \cite{Knapp-book}. We have a Cartan involution $\Theta$ on $G$, such
that the fixed point set $K:=G^\Theta$ is a connected maximal compact subgroup. We have Cartan
decompositions at the level of Lie algebras $\ggot=\kgot\oplus \pgot$ and at the level of the group
$G\simeq K\times \exp(\pgot)$. We denote by $b$ a $G$-invariant non-degenerate bilinear form
on $\ggot$ that is equal to the Killing form on $[\ggot,\ggot]$, and that defines a $K$-invariant
scalar product $(X,Y):=-b(X,\Theta(Y))$. We will use the $K$-equivariant identification
$\xi\mapsto \tilde{\xi},\ \ggot^*\simeq \ggot$ defined by
$(\tilde{\xi},X):=\langle\xi,X\rangle$ for $\xi\in\ggot^*$ and $X\in\ggot$.

When a Lie group $H$ acts on a manifold $N$, the stabilizer subgroup of $n\in N$ is denoted by $H_n=\{g\in G,gn=n\}$, and its Lie algebra by $\hgot_n$.
Let us define
\begin{equation}\label{eq:dim-K}
\dim_H(\Xcal)=\min_{n\in \Xcal} \dim(\hgot_n)
\end{equation}
for any subset $\Xcal\subset N$.

\section{The cone $\Pi_{\rm hol}(\tilde{G},G)$: first properties}\label{sec:cone}

We assume here that $G/K$ is a Hermitian symmetric space, that is to say, there exists a $G$-invariant complex structure on the manifold
$G/K$, or equivalently there exists a $K$-invariant element $z\in\kgot$ such that $\ad(z)\vert_{\pgot}$ defines a complex structure on $\pgot$ :
$(\ad(z)\vert_{\pgot})^2= -{\rm Id}_{\pgot}$. This condition imposes that the ranks of $G$ and $K$ are equal.

We are interested in the following closed invariant convex cone of $\ggot^*$:
$$
\Ccal_{G/K}=\left\{\xi\in\ggot^*,\langle \xi,g z\rangle \geq 0,\ \forall g\in G\right\}.
$$

\subsection{The holomorphic chamber}\label{sec:cone-elliptic}

Let $T$ be a maximal torus of $K$, with Lie algebra $\tgot$. Its dual $\tgot^*$ can be seen as the subspace of $\ggot^*$ fixed by $T$.
Let us denote by $\ggot^*_e$ the set formed by the elliptic elements: in other words $\ggot^*_e:={\rm Ad}^*(G)\cdot \tgot^*$.

Following \cite{Weinstein01}, we consider the invariant open subset
$\ggot^*_{se}=\{\xi\in\ggot^*\,\vert\, G_\xi \ \mathrm{is\  compact} \}$
of {\em strongly elliptic} elements.  It is non-empty since the groups $G$ and $K$ have the same rank.

We start with the following basic facts.

\begin{lem}\label{lem:C-se}
\begin{itemize}
\item $\ggot^*_{se}$ is contained in $\ggot^*_{e}$.
\item  The interior $\Ccal_{G/K}^0$ of the cone $\Ccal_{G/K}$ is contained in $\ggot^*_{se}$.

\end{itemize}
\end{lem}
{\em Proof :}  The first point is due to the fact that every compact subgroup of $G$ is conjugate to a subgroup of $K$.

Let $\xi\in \Ccal_{G/K}^0$. There exists $\epsilon>0$ so that
$$
\langle \xi +\eta,g z\rangle \geq 0,\quad \forall g\in G,\quad \forall \|\eta\|\leq \epsilon.
$$
It implies that  $|\langle \eta,g z\rangle | \leq\langle \xi, z\rangle$, $\forall g\in G_\xi$ and $\forall \|\eta\|\leq \epsilon$. In other words, the adjoint orbit
$G_\xi\cdot z\subset \ggot$ is bounded. For any $g=e^X k$, with $(X,k)\in \pgot\times K$, a direct computation shows that $\|gz\|=\|e^X z\|\geq \|[z,X]\|=\|X\|$. Then,
there exists $\rho>0$ such that $\|X\|\leq\rho$ if $e^X k\in G_\xi$ for some $k\in K$. It follows that the stabilizer subgroup $G_\xi$ is compact.
$\Box$

\medskip

Let $\wedge^*\subset\tgot^*$ be the weight lattice : $\alpha\in\wedge^*$ if $i\alpha$ is the
differential of a character of $T$. Let $\Rgot\subset \wedge^*$ be the set of roots
for the action of $T$ on $\ggot\otimes\Cbb$. We have $\Rgot=\Rgot_c\cup \Rgot_n$
where $\Rgot_c$ and $\Rgot_n$ are respectively the set of roots for the action of $T$ on
$\kgot\otimes\Cbb$ and $\pgot\otimes\Cbb$. We fix a system of positive roots $\Rgot^+_c$ in
$\Rgot_c$, and we denote by $\tgot^*_{\geq 0}$ the corresponding Weyl chamber.

We have $\pgot\otimes\Cbb=\pgot^+\oplus\pgot^-$ where the $K$-module $\pgot^\pm$
is equal to $\ker({\rm ad}(z)\mp i)$.
Let $\Rgot^{\pm,z}_n$ be the set of roots for the action of $T$
on $\pgot^\pm$. The union
\begin{equation}\label{eq:R-plus}
\Rgot^+=\Rgot^+_c\cup\Rgot^{+,z}_n
\end{equation}
defines then a system of positive roots in $\Rgot$. We notice that
$\Rgot^{+,z}_n$ is the set of roots $\beta\in \Rgot$ satisfying $\langle\beta, z\rangle=1$. Hence $\Rgot^{+,z}_n$ is invariant relatively to the action of the
Weyl group $W=N(T)/T$.

Let us recall the following classical fact concerning the parametrization of the $G$-orbits in $\Ccal_{G/K}^0$ via the holomorphic chamber
$$
\chol:=\{\xi\in\tgot^*_{\geq 0},(\xi,\beta) >0,\ \forall \beta\in \Rgot^{+,z}_{n}\}.
$$

\medskip

\begin{prop}\label{prop:Gz}
\begin{itemize}
\item The set $\Ccal_{G/K}^0\cap \tgot^*_{\geq 0}$ is contained in $\chol$.
\item  For any compact subset $\Kcal$ of $\chol$, there exists $c_\Kcal>0$ such that
$\langle \xi,g z\rangle \geq c_\Kcal \|g z\|$,  $\forall g\in G$, $\forall \xi\in\Kcal$.
\item The map $\Ocal\mapsto \Ocal\cap \tgot^*_{\geq 0}$ defines a bijective map between the set of $G$-orbits in $\Ccal_{G/K}^0$ and the holomorphic chamber
$\chol$.
\end{itemize}
\end{prop}
{\em Proof :}
Let $\xi\in\tgot^*_{\geq 0}\cap\Ccal_{G/K}^0$. The first point is proved if we check that  $(\xi,\beta) >0$  for any $\beta\in\Rgot^{+,z}_n$.
Let $X_\beta,Y_\beta\in \pgot$ such that $X_\beta+i Y_\beta\in  (\pgot\otimes\Cbb)_\beta$. We choose the following normalization~:
the vector $h_\beta:=[X_\beta,Y_\beta]$ satisfies $\langle\beta,h_\beta\rangle=1$. We see then that $(\xi,\beta)=\frac{1}{\|h_\beta\|^2}\langle \xi,h_\beta\rangle$
for any $\xi\in \ggot^*$. Standard computation \cite{Paneitz83} gives
$$
e^{t\,{\rm ad}(X_\beta)}z=z +({\rm cosh}(t)-1)h_\beta +{\rm sinh}(t)Y_\beta,\quad \forall t\in\Rbb.
$$
By definition, we must have $\langle \xi+\eta,e^{t\,{\rm ad}(X_\beta)} z\rangle\geq 0, \forall t\in\Rbb$, for any $\eta\in \tgot^*$ small enough. It imposes that
$\langle \xi,h_\beta\rangle>0$. The first point is settled.

Now choose some maximal strongly orthogonal system $\Sigma\subset \Rgot^{+,z}_n$. The real span $\agot$ of the $X_\beta,\beta\in\Sigma$ is a maximal abelian subspace of $\pgot$. Hence any element $g\in G$ can be written $g=k e^{X(t)} k'$ with $X(t)=\sum_{\beta\in\Sigma}t_\beta X_\beta$ and $k,k'\in K$. We get
\begin{equation}\label{eq:Gz}
gz=k\left(z +\sum_{\beta\in\Sigma}({\rm cosh}(t_\beta)-1)h_\beta +\sum_{\beta\in\Sigma}{\rm sinh}(t_\beta)Y_\beta\right)
\end{equation}
and
$$
\langle \xi,g z\rangle=\langle k^{-1}\xi,z\rangle + \sum_{\beta\in\Sigma}({\rm cosh}(t_\beta)-1) \langle k^{-1}\xi,h_\beta\rangle.
$$

For any $\xi\in \chol$, we define $c_\xi := \min_{\beta\in \Rgot^{+,z}_n}\langle \xi,h_\beta\rangle>0$. Let $\pi:\kgot^*\to \tgot^*$ be the projection. We have
$\langle k^{-1}\xi,z\rangle=\langle \pi(k^{-1}\xi),z\rangle$ and $\langle k^{-1}\xi,h_\beta\rangle=\langle \pi(k^{-1}\xi),h_\beta\rangle$.
The convexity theorem of Kostant tell us that $\pi(k^{-1}\xi)$ belongs to the convex hull of $\{w\xi, w\in W\}$. It follows that
$\langle k^{-1}\xi,z\rangle\geq \langle \xi,z\rangle>0 \quad {\rm and}\quad \langle k^{-1}\xi,h_\beta\rangle\geq c_\xi>0$.
We obtain then that $\langle \xi,g z\rangle \geq \frac{1}{2}\min (\langle \xi,z\rangle, c_\xi) e^{\|X(t)\|}$ for any $\xi\in\chol$, where $\|X(t)\|=\sup |t_\beta|$.
From (\ref{eq:Gz}), it is not difficult to see that there exists $C>0$ such that $\|gz\|\leq C e^{\|X(t)\|}$ for any $g=k e^{X(t)} k'$.

Let $\Kcal$ be a compact subset of $\chol$. Take $c_\Kcal=\frac{1}{2C}\min (\min_{\xi\in\Kcal}\langle \xi,z\rangle, \min_{\xi\in\Kcal}c_\xi)>0$. The previous computations show that $\langle \xi,g z\rangle \geq c_\Kcal \|g z\|,\quad \forall g\in G,\ \forall \xi\in\Kcal$. The second point is proved.

The first two points show that $\tgot^*_{\geq 0}\cap\Ccal_{G/K}^0=\chol$. The third point follows then from the fact that $\Ccal_{G/K}^0$
is contained in $\ggot^*_{e}$ (see Lemma \ref{lem:C-se}).
$\Box$

\medskip

We have a canonical map ${\rm p}: \Ccal_{G/K}^0\to \chol$ defined by the relations $G\xi\cap \tgot^*_{\geq 0}=\{ {\rm p}(\xi)\}$, $\forall \xi\in \Ccal_{G/K}^0$. The following result is needed in \S \ref{sec:proof-theo-A}.
\begin{lem}\label{lem:p-continu}
The map ${\rm p}$ is continuous.
\end{lem}
{\em Proof :} Let $(\xi_n)$ be a sequence of $\Ccal_{G/K}^0$ converging to $\xi_\infty\in \Ccal_{G/K}^0$. Let $\xi'_n ={\rm p}(\xi_n)$ and
$\xi'_\infty ={\rm p}(\xi_\infty)$: we have to prove that the sequence $(\xi'_n)$ converges to $\xi'_\infty$. We choose elements $g_n,g_\infty\in G$ such that
$\xi_n=g_n\xi'_n,\forall n$, and $\xi_\infty=g_\infty\xi'_\infty$.

First we notice that $-b(\xi_n,\xi_n)=\|\xi'_n\|^2$, hence the sequence $(\xi'_n)$ is bounded. We will now prove that the sequence $(g_n)$ is bounded.

 Let $\epsilon>0$ such that
$\langle\xi_\infty+\eta,gz\rangle \geq 0$, $\forall g\in G$, $\forall \|\eta\|\leq \epsilon$. If $\|\xi-\xi_\infty\|\leq \epsilon/2$, we write
$\xi=\frac{1}{2}(\xi_\infty+ 2(\xi-\xi_\infty))+\frac{1}{2}\xi_\infty$, and then
$$
\langle\xi,gz\rangle= \frac{1}{2}\langle\xi_\infty+ 2(\xi-\xi_\infty),gz\rangle +\frac{1}{2}\langle\xi_\infty,gz\rangle\geq \frac{1}{2}\langle\xi_\infty,gz\rangle,
\quad \forall g\in G.
$$
Now we have $\langle\xi'_n,z\rangle=\langle\xi_n,g_nz\rangle\geq \frac{1}{2}\langle\xi_\infty,g_nz\rangle$ if $n$ is large enough. This shows that the sequence
$\langle\xi_\infty,g_nz\rangle$ is bounded. If we use the second point of Proposition \ref{prop:Gz},
we can conclude that the sequence $(g_n)$ is bounded.

Let $(\xi'_{\phi(n)})$ be a subsequence converging to $\ell\in\tgot^*_{\geq 0}$. Since  $(g_{\phi(n)})$ is bounded, there exists a
subsequence $(g_{\phi\circ\psi(n)})$ converging to $h\in G$. From the relations $\xi_{\phi\circ\psi(n)}=$ \break
$g_{\phi\circ\psi(n)}\xi'_{\phi\circ\psi(n)},\forall n\in\Nbb$, we obtain $\xi_\infty=h\ell$. Then $\ell={\rm p}(\xi_\infty)=
\xi_\infty'$. Since every subsequence of $(\xi'_{n})$ has a subsequential limit to $\xi_\infty'$, then the sequence $(\xi'_n)$ converges to $\xi'_\infty$.
$\Box$

\subsection{The cone $\Pi_{\rm hol}(\tilde{G},G)$ is closed}\label{sec:cone-closed}

We suppose that $G/K$ is a complex submanifold of a Hermitian symmetric space $\wG/\wK$. In other words,  $\tilde{G}$ is a reductive real Lie group
such that $G\subset \tilde{G}$ is a closed subgroup preserved by the Cartan involution, and $\tilde{K}$ is a maximal compact subgroup of $\tilde{G}$
containing $K$. We denote by $\tilde{\ggot}$ and $\tilde{\kgot}$ the Lie algebras of $\tilde{G}$ and $\tilde{K}$ respectively. We suppose that there exists a $\tilde{K}$-invariant element $z\in\kgot$ such that $\ad(z)\vert_{\tilde{\pgot}}$ defines a complex structure on $\tilde{\pgot}$ : $(\ad(z)\vert_{\tilde{\pgot}})^2= -Id_{\tilde{\pgot}}$.

Let $\Ccal_{\wG/\wK}\subset \wg^*$ be the closed invariant cone associated to the Hermitian symmetric space $\wG/\wK$. We start with the following key fact.
\begin{lem}
The projection $\pi_{\ggot,\wg}: \wg^*\to\ggot^*$ sends $\Ccal_{\wG/\wK}^0$ into $\Ccal_{G/K}^0$.
\end{lem}
{\em Proof :}
Let $\tilde{\xi}\in \Ccal_{\wG/\wK}^0$ and $\xi=\pi_{\ggot,\wg}(\tilde{\xi})$. Then $\langle \tilde{\xi}+\tilde{\eta},\tilde{g} z\rangle \geq 0,\ \forall \tilde{g}\in \wG$
if $\tilde{\eta}\in\wg^*$ is small enough. It follows that $\langle \xi+\pi_{\ggot,\wg}(\tilde{\eta}),g z\rangle=\langle \tilde{\xi}+\tilde{\eta},g z\rangle \geq 0,\ \forall g\in G$
 if $\tilde{\eta}$ is small enough. Since $\pi_{\ggot,\wg}$ is an open map, we can conclude that $\xi\in\Ccal_{G/K}^0$.
$\Box$

\medskip

Let $\wT$ be a maximal torus of $\wK$, with Lie algebra $\wt$. The $\wG$-orbits in the interior of $\Ccal_{\wG/\wK}$ are parametrized by the holomorphic chamber
$\wchol\subset\wt^*$.
The previous Lemma says that the projection $\pi_{\ggot,\wg}(\tilde{\Ocal})$ of any $\wG$-orbit $\tilde{\Ocal}\subset \Ccal_{\wG/\wK}^0$ is the union of $G$-orbits
$\Ocal\subset \Ccal^0_{G/K}$.  So it is natural to study the following object
\begin{equation}
\Pi_{\rm hol}(\tilde{G},G):=\left\{(\tilde{\xi},\xi)\in \wchol\times\chol ; \ G\xi\,\subset \,\pi_{\ggot,\tilde{\ggot}}\big(\tilde{G}\tilde{\xi}\,\big)\right\}.
\end{equation}

Here is a first result.

\begin{prop}\label{prop:pi-z-compact}
$\Pi_{\rm hol}(\tilde{G},G)$ is a closed cone of $\wchol\times\chol$.
\end{prop}

{\em Proof :}
Suppose that a sequence $(\tilde{\xi}_n,\xi_n)\in \Pi_{\rm hol}(\tilde{G},G)$ converge to $(\tilde{\xi}_\infty,\xi_\infty)\in\wchol\times\chol$.
By definition, there exists a sequence $(\tilde{g}_n,g_n)\in \tilde{G}\times G$ such that
$g_n \xi_n=\pi_{\ggot,\tilde{\ggot}}(\tilde{g}_n\tilde{\xi}_n)$. Let $\tilde{h}_n:=g_n^{-1}\tilde{g}_n$, so that $\xi_n=\pi_{\ggot,\tilde{\ggot}}(\tilde{h}_n\tilde{\xi}_n)$ and
$\langle \tilde{h}_n\tilde{\xi}_n, z\rangle=\langle \xi_n, z\rangle$. We use now that the sequence $\langle \xi_n, z\rangle$ is bounded, and that the sequence
$\tilde{\xi}_n$ belongs to a compact subset of $\wchol$. Thanks to the second point of Proposition \ref{prop:Gz}, these facts imply that $\|\tilde{h}_n^{-1}z\|$ is a bounded sequence. Hence $\tilde{h}_n$ admits a subsequence converging to $\tilde{h}_\infty$. So we get $\xi_\infty=\pi_{\ggot,\tilde{\ggot}}(\tilde{h}_\infty\tilde{\xi}_\infty)$, and that proves that $(\tilde{\xi}_\infty,\xi_\infty)\in\Pi_{\rm hol}(\tilde{G},G)$.
$\Box$

\subsection{Rational and weakly regular points}\label{sec:rational}

Let $(M,\Omega)$ be a symplectic manifold. We suppose that there exists a line bundle $\Lcal$ with connection $\nabla$ that prequantizes the $2$-form $\Omega$: in other words, $\nabla^2=- i\, \Omega$. Let $K$ be a compact connected Lie group acting on $\Lcal\to M$, and leaving the connection  invariant. Let $\Phi_K:M\to\kgot^*$ be the moment map defined by Kostant's relations
\begin{equation}\label{eq:kostant}
L_X-\nabla_X=i\langle \Phi_K,X\rangle,\quad \forall X\in\kgot.
\end{equation}
Here $L_X$ is the Lie derivative acting on the sections of $\Lcal\to M$.

Remark that relations (\ref{eq:kostant}) imply, via the equivariant Bianchi formula, the relations
\begin{equation}\label{eq:symplectic-moment-map}
\iota(X_M)\Omega=- d\langle \Phi_K,X\rangle,\quad \forall X\in\kgot,
\end{equation}
where $X_M(m) :=\frac{d}{dt}\vert_{t=0}e^{-tX}m$ is the vector field on $M$ generated by $X\in\kgot$.

\begin{defi}\label{defi:weakly-regular}
Let $\dim_K(M):=\min_{m\in M}\dim\kgot_m$. An element $\xi\in\kgot^*$ is a weakly-regular value of $\Phi_K$ if for all $m\in\Phi_K^{-1}(\xi)$ we have
$\dim\kgot_m=\dim_K(M)$.
\end{defi}

When $\xi\in\kgot^*$ is a weakly-regular value of $\Phi_K$, the constant rank theorem tells us that $\Phi_K^{-1}(\xi)$ is a submanifold of $M$ stable under the action of the stabilizer subgroup
$K_\xi$. We see then that the reduced space
\begin{equation}\label{eq:symplectic-reduction}
M_\xi:=\Phi_K^{-1}(\xi)/K_\xi
\end{equation}
is a symplectic orbifold.

Let $T\subset K$ be a maximal torus with Lie algebra $\tgot$. We consider the lattice $\wedge:=\frac{1}{2\pi}\ker(\exp:\tgot\to T)$ and the dual lattice
$\wedge^*\subset \tgot^*$ defined by $\wedge^*=\hom(\wedge,\Zbb)$. We remark that $i\eta$ is a differential of a character of $T$ if and only if $\eta\in\wedge^*$.
The $\Qbb$-vector space generated by the lattice $\wedge^*$ is denoted by $\tgot^*_\Qbb$: the vectors belonging to
$\tgot^*_\Qbb$ are designed as rational. An affine subspace $V\subset \tgot^*$ is called rational if it is affinely generated by its rational points.

We also fix a closed positive Weyl chamber $\tgot^*_{\geq 0}$. For each relatively open face $\sigma\subset \tgot^*_{\geq 0}$,
the stabilizer $K_\xi$ of points $\xi\in\sigma$ under the coadjoint action, does not depend on $\xi$, and will be denoted $K_\sigma$.
The Lie algebra $\ggot_\sigma$ decomposes into its semi-simple and central parts $\kgot_\sigma=[\kgot_\sigma,\kgot_\sigma]\oplus\zgot_\sigma$.
The subspace $\zgot_\sigma^*$ is defined to be the annihilator of $[\kgot_\sigma,\kgot_\sigma]$, or equivalently the fixed point set of the coadjoint
$K_\sigma$ action. Notice that $\zgot_\sigma^*$ is a rational subspace of  $\tgot^*$, and that the face $\sigma$ is an open cone of $\zgot_\sigma^*$,

We suppose that the moment map $\Phi_K$ is {\em proper}. The Convexity Theorem \cite{Atiyah82,Guillemin-Sternberg82.bis,Kirwan.84.bis,sjamaar98,L-M-T-W} tells
us that  $\Delta_K(M):={\rm Image}(\Phi_K)\, \bigcap\, \tgot_{\geq 0}^*$ is a closed, convex, locally polyhedral set.

\begin{defi}
We denote by $\Delta_K(M)^0$ the subset of $\Delta_K(M)$ formed by the {\em weakly-regular values} of the moment map
$\Phi_K$ contained in $\Delta_K(M)$.
\end{defi}

We will use the following remark in the next sections.

\begin{lem}\label{lem:rational-dense}
The subset $\Delta_K(M)^0\cap \tgot_\Qbb^*$ is dense in $\Delta_K(M)$.
\end{lem}
{\em Proof :}
Let us first explain why $\Delta_K(M)^0$ is a dense open subset of $\Delta_K(M)$. There exists a unique open face $\tau$ of the Weyl chamber $\tgot^*_{\geq 0}$ such as
$\Delta_K(M)\cap\tau$ is dense in $\Delta_K(M)$~: $\tau$ is called the {\em principal} face in \cite{L-M-T-W}.
The Principal-cross-section Theorem \cite{L-M-T-W} tells us that $Y_\tau :=\Phi^{-1}(\tau)$ is a symplectic  $K_\tau$-manifold, with a trivial action of
$[K_\tau,K_\tau]$. The line bundle $\Lcal_\tau:=\Lcal\vert_{Y_\tau}$ prequantizes the symplectic structure on $Y_\tau$, and relations
(\ref{eq:symplectic-moment-map}) show that $[K_\tau,K_\tau]$ acts trivially on $\Lcal_\tau$. Moreover the restriction of $\Phi_K$ on
$Y_\tau$ is the moment map $\Phi_\tau: Y_\tau\to \zgot_\tau^*$ associated to the action of the torus $Z_\tau=\exp(\zgot_\tau)$ on $\Lcal_\tau$.

Let $I\subset \zgot_\tau^*$ be the smallest affine subspace containing $\Delta_K(M)$. Let $\zgot_I\subset\zgot_\tau$ be the annihilator of
the subspace parallel to $I$ : relations (\ref{eq:symplectic-moment-map}) show that $\zgot_I$ is the generic infinitesimal stabilizer
of the $\zgot_\tau$-action  on $Y_\tau$. Hence $\zgot_I$ is the Lie algebra of the torus $Z_I:=\exp(\zgot_I)$.

We see then that any regular value of $\Phi_\tau: Y_\tau\to I$, viewed as a map with codomain $I$, is a weakly-regular value of the moment map $\Phi_K$. It explains
why $\Delta_K(M)^0$ is a dense open subset of $\Delta_K(M)$.

The convex set $\Delta_K(M)\cap\tau$ is equal to $\Delta_{Z_\tau}(Y_\tau):={\rm Image}(\Phi_\tau)$, it is sufficient to check that
$\Delta_{Z_\tau}(Y_\tau)^0\cap \tgot_\Qbb^*$ is dense in  $\Delta_{Z_\tau}(Y_\tau)$. The subtorus $Z_I\subset Z_\tau$ acts trivially on $Y_\tau$, and it acts on the line bundle $\Lcal_\tau$
through a character $\chi$. Let $\eta\in \wedge^*\cap\tgot^*_\tau$ such that $d\chi=i\eta\vert_{\zgot_I}$. The affine subspace $I$ which is equal to $\eta+(\zgot_I)^{\perp}$ is rational.
Since the open subset $\Delta_{Z_\tau}(Y_\tau)^0$ generates the rational affine subspace $I$, we can conclude that $\Delta_{Z_\tau}(Y_\tau)^0\cap \tgot_\Qbb^*$
is dense in  $\Delta_{Z_\tau}(Y_\tau)$.
$\Box$

\subsection{Weinstein's theorem}\label{sec:weinstein}

Let $\tilde{a}\in \wchol$. Consider the Hamiltonian action of the group $G$ on the coadjoint orbit $\wG\tilde{a}$. The moment map
$\Phi_G^{\tilde{a}}:\wG\tilde{a}\to\ggot^*$ corresponds to the restriction of the projection $\pi_{\ggot,\tilde{\ggot}}$ to $\wG\tilde{a}$. In this setting,
the following conditions holds :
\begin{enumerate}
\item The action of $G$ on $\wG\tilde{a}$ is proper.
\item  The moment map $\Phi_G^{\tilde{a}}$ is a proper map since the map $\langle\Phi_G^{\tilde{a}},z\rangle$ is proper (see the second point of Proposition
\ref{prop:Gz}).
\end{enumerate}

Conditions 1. and 2. imposes that the image of $\Phi_G^{\tilde{a}}$ is contained in the open subset
$\ggot^*_{se}$ of strongly elliptic elements \cite{pep-jems}. Thus, the $G$-orbits contained in the image of $\Phi_G^{\tilde{a}}$ are parametrized by
the following subset of the holomorphic chamber $\chol$ :
$$
\Delta_G(\wG\tilde{a}):={\rm Image}(\Phi_G^{\tilde{a}})\, \bigcap \, \tgot^*_{\geq 0}.
$$
We notice that $\Pi_{\rm hol}(\tilde{G},G)=\bigcup_{\tilde{a}\in \wchol}\{\tilde{a}\}\times\Delta_G(\wG\tilde{a})$.

Like in Definition \ref{defi:weakly-regular}, an element $\xi\in\ggot^*$ is a {\em weakly-regular} value of $\Phi_G^{\tilde{a}}$ if for all
$m\in(\Phi_G^{\tilde{a}})^{-1}(\xi)$ we have
$\dim\ggot_m=\min_{x\in \wG\tilde{a}}\dim(\ggot_x)$. We denote by  $\Delta_G(\wG\tilde{a})^0$ the set of elements $\xi\in\Delta_G(\wG\tilde{a})$
that are weakly-regular for $\Phi_G^{\tilde{a}}$.

\begin{theo}[Weinstein]\label{theo:weinstein}
For any $\tilde{a}\in \wchol$, $\Delta_G(\wG\tilde{a})$ is a closed convex subset contained in $\chol$.
\end{theo}
{\em Proof :}
We recall briefly the arguments of the proof (see \cite{Weinstein01} or \cite{pep-jems}[\S 2]). Under Conditions 1. and 2., one checks easily that $Y_{\tilde{a}}:=(\Phi_G^{\tilde{a}})^{-1}(\kgot^*)$ is a smooth
$K$-invariant symplectic submanifold of $\wG\tilde{a}$ such that
\begin{equation}\label{eq:slice-Y}
\wG\tilde{a}\simeq G\times_K Y_{\tilde{a}}.
\end{equation}
The moment map $\Phi_K^{\tilde{a}}:Y_{\tilde{a}}\to\kgot^*$, which corresponds to the restriction of  the map $\Phi_G^{\tilde{a}}$ to $Y_{\tilde{a}}$,
is a proper map. Hence the Convexity Theorem tells us that $\Delta_K(Y_{\tilde{a}}):={\rm Image}(\Phi_K^{\tilde{a}})\bigcap \tgot^*_{\geq 0}$ is a closed, convex, locally polyhedral set. Thanks to the isomorphism (\ref{eq:slice-Y}) we see that $\Delta_G(\wG\tilde{a})$ coincides with the
closed convex subset $\Delta_K(Y_{\tilde{a}})$. The proof is completed.
$\Box$

\medskip

The next Lemma is used in \S \ref{sec:proof-theo-A}.

\begin{lem}\label{lem:rational-dense-coadjoint}
Let $\tilde{a}\in \wchol$ be a rational element. Then $\Delta_G(\wG\tilde{a})^0\cap \tgot_\Qbb^*$ is dense in $\Delta_G(\wG\tilde{a})$.
\end{lem}
{\em Proof :}
Thanks to (\ref{eq:slice-Y}), we know that $\Delta_G(\wG\tilde{a})=\Delta_K(Y_{\tilde{a}})$. Relation (\ref{eq:slice-Y}) shows also that
$\Delta_G(\wG\tilde{a})^0=\Delta_K(Y_{\tilde{a}})^0$. Let $N\geq 1$ such that $\tilde{\mu}=N\tilde{a}\in \wedge^*\cap \chol$. The stabilizer subgroup $\wG_{\tilde{\mu}}$ is compact, equal to $\wK_{\tilde{\mu}}$. Let us denote by $\Cbb_{\tilde{\mu}}$ the one-dimensional representation of $\wK_{\tilde{\mu}}$ associated to  $\tilde{\mu}$.
The convex set $\Delta_G(\wG\tilde{a})$ is equal to $\frac{1}{N}\Delta_G(\wG\tilde{\mu})$, so it is sufficient to check that
$\Delta_G(\wG\tilde{\mu})^0\cap \tgot_\Qbb^*=\Delta_K(Y_{\tilde{\mu}})^0\cap \tgot_\Qbb^*$ is dense in $\Delta_G(\wG\tilde{\mu})=\Delta_K(Y_{\tilde{\mu}})$. The coadjoint orbit $\wG\tilde{\mu}$ is prequantized by the line bundle $\wG\times_{K_{\tilde{\mu}}}\Cbb_{\tilde{\mu}}$ and the symplectic slice $Y_{\tilde{\mu}}$ is prequantized by the line bundle $\Lcal_{\tilde{\mu}}:=\wG\times_{K_{\tilde{\mu}}}\Cbb_{\tilde{\mu}}\vert_{Y_{\tilde{\mu}}}$.
Thanks to Lemma \ref{lem:rational-dense}, we know that $\Delta_K(Y_{\tilde{\mu}})^0\cap \tgot_\Qbb^*$ is dense in $\Delta_K(Y_{\tilde{\mu}})$ : the
proof is complete. $\Box$

\section{Holomorphic discrete series}\label{sec:holo-series}

\subsection{Definition}\label{sec:holo-series-def}

We return to the framework of \S \ref{sec:cone-elliptic}. We recall the notion of holomorphic discrete series representations associated to a
Hermitian symmetric spaces $G/K$. Let us introduce
$$
\chol^\rho:=\left\{ \xi\in\tgot^*_{\geq 0} \vert\ (\xi,\beta)\geq (2\rho_n,\beta) , \  \forall \beta\in \Rgot_n^{+,z}\right\}
$$
where  $2\rho_n=\sum_{\beta\in\Rgot_n^{+,z}}\beta$ is $W$-invariant.

\begin{lem}
\begin{enumerate}
\item We have $\chol^\rho\subset\chol$.
\item For any $\xi\in\chol$ there exists $N\geq 1$ such that $N\xi\in\chol^\rho$.
\end{enumerate}
\end{lem}
{\em Proof :}
The first point is due to the fact that $(\beta_0,\beta_1)\geq 0$ for any $\beta_0,\beta_1\in \Rgot_n^{+,z}$. The second point is obvious.
$\Box$

We will be interested in the following subset of dominants weights :
$$
\Ghol:=\wedge^*_+\bigcap \chol^\rho.
$$
Let ${\rm Sym}(\pgot)$ be the symmetric algebra of the complex $K$-module $(\pgot,\mathrm{ad}(z))$.

\begin{theo}[Harish-Chandra]
For any $\lambda\in\Ghol$, there exists an irreducible unitary representation of $G$, denoted $V^G_\lambda$, such that
the vector space of $K$-finite vectors is $V^G_\lambda\vert_K:=V_\lambda^K\otimes {\rm Sym}(\pgot)$.
\end{theo}

The set $V^G_\lambda,\lambda\in\Ghol$ corresponds to the holomorphic discrete series representations associated to the complex structure ${\rm ad}(z)$.

\subsection{Restriction}\label{sec:holo-series-restriction}

We come back to the framework of \S \ref{sec:cone-closed}. We consider two compatible Hermitian symmetric spaces $G/K\subset \wG/\wK$, and we look after the restriction of holomorphic discrete series representations of $\wG$ to the subgroup $G$.

Let $\tilde{\lambda}\in \wGhol$. Since the representation $V^{\wG}_{\tilde{\lambda}}$ is discretely admissible relatively the circle group $\exp(\Rbb z)$, it is also discretely admissible relatively to $G$. We can be more precise  \cite{Jakobsen-Vergne, Martens,Toshi-JFA98} :

\begin{prop}\label{prop:restriction-V-lambda-G-prime}
We have an Hilbertian direct sum
$$
V^{\wG}_{\tilde{\lambda}}\vert_{G}=\bigoplus_{\lambda\in \Ghol} m_{\tilde{\lambda}}^{\lambda} \ V^{G}_\lambda,
$$
where the multiplicity $m_{\tilde{\lambda}}^{\lambda} :=[V^{G}_\lambda:V^{\wG}_{\tilde{\lambda}}]$ is finite for any $\lambda$.
\end{prop}

The Hermitian $\wK$-vector space $\tilde{\pgot}$, when restricted to the $K$-action, admits an orthogonal decomposition $\tilde{\pgot}=\pgot\oplus \qgot$. Notice that the symmetric algebra ${\rm Sym}(\qgot)$ is an admissible $K$-module.

In \cite{Jakobsen-Vergne}, H.P. Jakobsen and M. Vergne obtained the following nice characterization of the multiplicities $[V^{G}_\lambda:V^{\wG}_{\tilde{\lambda}}]$.
Another proof is given in \cite{pep-jems}, \S 4.4.
\begin{theo}[Jakobsen-Vergne]\label{theo:JV}
Let $(\tilde{\lambda},\lambda)\in \wGhol\times\Ghol$. The multiplicity $[V^{G}_\lambda:V^{\wG}_{\tilde{\lambda}}]$ is equal to the multiplicity of the
representation $V^{K}_\lambda$ in ${\rm Sym}(\qgot)\otimes V^{\wK}_{\tilde{\lambda}}\vert_{K} $.
\end{theo}

\subsection{Discrete analogues of $\Pi_{\rm hol}(\tilde{G},G)$ }\label{sec:LR}

We define the following discrete analogues of the cone $\Pi_{\rm hol}(\tilde{G},G)$:

\begin{equation}
\Pi_{\rm hol}^\Zbb(\tilde{G},G):=\left\{(\tilde{\lambda},\lambda)\in \wGhol\times\Ghol\ ; \ [V^{G}_\lambda:V^{\wG}_{\tilde{\lambda}}]\neq 0 \right\}.
\end{equation}
and
\begin{equation}
\Pi_{\rm hol}^\Qbb(\tilde{G},G):=\left\{(\tilde{\xi},\xi)\in \wchol\times\chol\ ; \ \exists N\geq 1,\ (N\xi,N\tilde{\xi})\in \Pi_{\rm hol}^\Zbb(\tilde{G},G)\right\}.
\end{equation}

We have the following key fact.

\begin{prop}\label{prop:monoid}
${}$
\begin{itemize}
\item $\Pi_{\rm hol}^\Zbb(\tilde{G},G)$ is a subset of $ \tilde{\wedge}^* \times \wedge^*$ stable under the addition.
\item $\Pi_{\rm hol}^\Qbb(\tilde{G},G)$ is a $\Qbb$-convex cone of the $\Qbb$-vector space $\tilde{\tgot}^*_\Qbb \times \tgot^*_\Qbb$.
\end{itemize}
\end{prop}
{\em Proof :}
Suppose that $a_1:=(\tilde{\lambda}_1,\lambda_1)$ and $a_2:=(\tilde{\lambda}_2, \lambda_2)$ belongs to
$\Pi_{\rm hol}^\Zbb(\tilde{G},G)$. Thanks to Theorem \ref{theo:JV}, we know that the $K$-modules ${\rm Sym}(\qgot)\otimes (V^K_{\lambda_j})^*\otimes V^{\wK}_{\tilde{\lambda}_j}\vert_{K}$ possess a non-zero invariant vector $\phi_j$, for $j=1,2$.

Let $\mathbb{X}:=\overline{K/T}\times \wK/\wT$ be the product of flag manifolds. The complex structure is normalized so that
$\T _{([e],[\tilde{e}])}\mathbb{X}\simeq \ngot_-\oplus \tilde{\ngot}_+$, where $\ngot_-=\sum_{\alpha<0}(\kgot_\Cbb)_\alpha$ and
$\tilde{\ngot}_+=\sum_{\tilde{\alpha}>0}(\tilde{\kgot}_\Cbb)_{\tilde{\alpha}}$.
We associate to each data $a_j$, the holomorphic line bundle $\Lcal_j:=K\times_T \Cbb_{-\lambda_j}\boxtimes \wK\times_{\wT} \Cbb_{-\tilde{\lambda}_j}$ on
$\mathbb{X}$. Let $H^0(\mathbb{X},\Lcal_j)$ be the space of holomorphic sections of the line bundle $\Lcal_j$. The Borel-Weil Theorem tell us that
$H^0(\mathbb{X},\Lcal_j)\simeq (V^K_{\lambda_j})^*\otimes V^{\wK}_{\tilde{\lambda}_j}\vert_{K}$, $\forall j\in\{1,2\}$.

We have $\phi_j\in \left[{\rm Sym}(\qgot)\otimes H^0(\mathbb{X},\Lcal_j)\right]^K$, $\forall j$, and then $\phi_1\phi_2\in {\rm Sym}(\qgot)\otimes H^0(\mathbb{X},\Lcal_1\otimes \Lcal_2)$ is a non-zero invariant vector. Hence
$[{\rm Sym}(\qgot)\otimes (V^K_{\lambda_1+\lambda_2})^*\otimes V^{\wK}_{\tilde{\lambda}_1+\tilde{\lambda}_2}\vert_{K}]^K\neq 0$.
Thanks to Theorem \ref{theo:JV}, we can conclude that $a_1+a_2=(\tilde{\lambda}_1+\tilde{\lambda}_2,\lambda_1+\lambda_2)$ belongs to
$\Pi_{\rm hol}^\Zbb(\tilde{G},G)$. The first point is proved. From the first point, one checks easily that
\begin{itemize}
\item[-] $\Pi_{\rm hol}^\Qbb(\tilde{G},G)$ is stable under addition,
\item[-] $\Pi_{\rm hol}^\Qbb(\tilde{G},G)$ is stable by expansion by a non-negative rational number.
\end{itemize}
The second point is settled.
$\Box$

\subsection{Riemann-Roch numbers}\label{sec:RR-numbers}

 We come back to the framework of \S \ref{sec:rational}.

 We associate to a dominant weight $\mu\in\wedge^*_{+}$ the (possibly singular) symplectic reduced space
 $M_\mu:=\Phi_K^{-1}(\mu)/K_\mu$, and the (possibly singular) line bundle
 over $M_\mu$~:
 $$
 \Lcal_{\mu}:=\left(\Lcal\vert_{\Phi_K^{-1}(\mu)}\otimes \Cbb_{-\mu}\right)/K_\mu.
 $$

 \underline{Suppose first that $\mu$ is a weakly-regular value of $\Phi_K$.} Then $M_\mu$ is an orbifold equipped with a symplectic structure
 $\Omega_\mu$, and $\Lcal_{\mu}$ is a line orbi-bundle over $M_\mu$ that prequantizes the symplectic structure.
 By choosing an almost complex structure on $M_\mu$ compatible with $\Omega_\mu$, we get a decomposition
 $\wedge \T^* M_\mu\otimes \Cbb = \oplus_{i,j}\wedge^{i,j} \T^* M_\mu$ of the bundle of differential forms.
 Using Hermitian structure in the tangent bundle $\T M_\mu$ of $M_\mu$, and in the fibers of $\Lcal_{\mu}$, we define a
Dolbeaut-Dirac operator
$$
 D^+_{\mu} : \Acal^{0,+}(M_\mu,\Lcal_{\mu})\longrightarrow \Acal^{0,-}(M_\mu,\Lcal_{\mu})
$$
where $\Acal^{i,j}(M_\mu,\Lcal_{\mu})=\Gamma(M_\mu,\wedge^{i,j} \T^* M_\mu\otimes \Lcal_{\mu})$.

\begin{defi}
Let $\mu\in\wedge^*_{+}$ be a weakly-regular value of the moment map $\Phi_K$. The Riemann-Roch number
$RR(M_\mu,\Lcal_{\mu})\in\Zbb$ is defined as the index of the elliptic operator $D^+_{\mu}$:
$RR(M_\mu,\Lcal_{\mu})= \dim(ker(D^+_{\mu})) -\dim(coker(D^+_{\mu}))$.
\end{defi}

\medskip

\underline{Suppose that $\mu\notin \Delta_K(M)$.} Then $M_\mu=\emptyset$ and we take $RR(M_\mu,\Lcal_{\mu})=0$.

\medskip

\underline{Suppose now that $\mu\in \Delta_K(M)$ is not (necessarily) a weakly-regular value of $\Phi_K$.}
Take a small element $\epsilon\in\tgot^*$ such that $\mu+\epsilon$ is a weakly-regular value of $\Phi_K$ belonging to $\Delta_K(M)$.
We consider the symplectic orbifold $M_{\mu+\epsilon}$: if $\epsilon$ is small enough,
$$
\Lcal_{\mu,\epsilon}:=\left(\Lcal\vert_{\Phi_K^{-1}(\mu+\epsilon)}\otimes \Cbb_{-\mu}\right)/K_{\mu+\epsilon}.
$$
is a line orbi-bundle over $M_{\mu+\epsilon}$ .

We have the following important result (see \S 3.4.3 in \cite{pep-vergne-ams}).
\begin{prop}Let $\mu\in \Delta_K(M)\cap \wedge^*$.
The Riemann-Roch number  $RR(M_{\mu+\epsilon},\Lcal_{\mu,\epsilon})$ do not depend on the choice of $\epsilon$ small enough so that
$\mu+\epsilon\in \Delta_K(M)$ is a weakly-regular value of $\Phi_K$.
\end{prop}

We can now introduce the following definition.

\begin{defi}\label{def:RR-numbers}
Let $\mu\in\wedge^*_{+}$.  We define
$$
\Qcal(M_\mu,\Omega_\mu)=
\begin{cases}
0 \hspace{29mm} {\rm if }\quad  \mu\notin \Delta_K(M),\\
RR(M_{\mu+\epsilon},\Lcal_{\mu,\epsilon})\hspace{4mm}  {\rm if }\quad \mu\in \Delta_K(M).
\end{cases}
$$
Above, $\epsilon$ is chosen small enough so that $\mu+\epsilon\in \Delta_K(M)$ is a weakly-regular value of $\Phi_K$.
\end{defi}

Let $n\geq 1$. The manifold $M$, equipped with the symplectic structure $n\Omega$,  is prequantized by the line bundle
$\Lcal^{\otimes n}$: the corresponding moment map is $n\Phi_K$. For any dominant weight $\mu \in  \wedge^*_{+}$, the symplectic reduction of
$(M,n\Omega)$ relatively to the weight $n\mu$ is $(M_\mu,n\Omega_\mu)$. Like in definition \ref{def:RR-numbers}, we consider the following Riemann-Roch numbers
$$
\Qcal(M_\mu,n\Omega_\mu)=
\begin{cases}
0 \hspace{36mm} {\rm if }\quad  \mu\notin \Delta_K(M),\\
RR(M_{\mu+\epsilon},(\Lcal_{\mu,\epsilon})^{\otimes n})\hspace{4mm}  {\rm if }\quad \mu\in \Delta_K(M)\ {\rm and} \  \|\epsilon\|<<1.
\end{cases}
$$

Kawasaki-Riemann-Roch formula shows that $n\geq 1\mapsto \Qcal(M_\mu,n\Omega_\mu)$ is a quasi-polynomial map
\cite{vergne96,Loizides19}. When $\mu$ is a weakly-regular value of $\Phi_K$, we denote by
${\rm vol}(M_\mu):=\frac{1}{d_\mu}\int_{M_\mu}(\frac{\Omega_\mu}{2\pi})^{\frac{\dim M_\mu}{2}}$ the symplectic volume of the symplectic orbifold
$(M_\mu,\Omega_\mu)$. Here $d_\mu$ is the generic value of the map $m\in \Phi_K^{-1}(\mu)\mapsto {\rm cardinal}(K_m/K^0_m)$.

The following proposition is a direct consequence of Kawasaki-Riemann-Roch formula (see \cite{Loizides19} or \S 1.3.4 in \cite{pep-duistermaat}).

\begin{prop}\label{prop:quasi-polynomial-non-zero}
Let $\mu\in\Delta_K(M)\cap \wedge^*_{+}$  be a weakly-regular value of $\Phi_K$. Then we have
$\Qcal(M_\mu,n\Omega_\mu)\ \sim\ {\rm vol}(M_\mu) \,n^{\frac{\dim M_\mu}{2}}$, when $n\to\infty$.
In particular the map $n\geq 1\mapsto \Qcal(M_\mu,n\Omega_\mu)$ is non-zero.
\end{prop}

\medskip

\subsection{Quantization commutes with reduction}\label{sec:QR=0}

Let us explain the  ``{\em Quantization commutes with Reduction}'' Theorem proved in \cite{pep-jems}.

We fix $\tilde{\lambda}\in \wGhol$. The coadjoint orbit $\wG\tilde{\lambda}$ is prequantized by the line bundle
 $\wG\times_{K_{\tilde{\lambda}}}\Cbb_{\tilde{\lambda}}$, and the moment map $\Phi_G^{\tilde{\lambda}}:\wG\tilde{\lambda}\to\ggot^*$
 corresponding to the $G$-action on
 $\wG\times_{K_{\tilde{\lambda}}}\Cbb_{\tilde{\lambda}}$ is equal to the restriction of the map $\pi_{\ggot,\tilde{\ggot}}$ to $\wG\tilde{\lambda}$.

The symplectic slice $Y_{\tilde{\lambda}}=(\Phi_G^{\tilde{\lambda}})^{-1}(\kgot^*)$ is prequantized by the line bundle
$\Lcal_{\tilde{\lambda}}:=\wG\times_{K_{\tilde{\lambda}}}\Cbb_{\tilde{\lambda}}\vert_{Y_{\tilde{\lambda}}}$. The moment map
$\Phi_K^{\tilde{\lambda}}:Y_{\tilde{\lambda}}\to\kgot^*$ corresponding to the $K$-action  is equal to the restriction of
$\Phi_G^{\tilde{\lambda}}$ to $Y_{\tilde{\lambda}}$.

For any $\lambda\in \Ghol$, we consider the (possibly singular) symplectic reduced space
$$
\mathbb{X}_{\tilde{\lambda},\lambda}:=(\Phi_K^{\tilde{\lambda}})^{-1}(\lambda)/K_\lambda,
$$
equipped with the reduced symplectic form $\Omega_{\tilde{\lambda},\lambda}$, and the (possibly singular) line bundle
$$
\mathbb{L}_{\tilde{\lambda},\lambda}:=\left(\Lcal_{\tilde{\lambda}}\vert_{(\Phi_K^{\tilde{\lambda}})^{-1}(\lambda)}\otimes \Cbb_{-\lambda}\right)/K_\lambda.
$$

Thanks to Definition \ref{def:RR-numbers}, the geometric quantization
$\Qcal(\mathbb{X}_{\tilde{\lambda},\lambda},\Omega_{\tilde{\lambda},\lambda})\in\Zbb$ of those compact symplectic spaces
$(\mathbb{X}_{\tilde{\lambda},\lambda},\Omega_{\tilde{\lambda},\lambda})$ are well-defined even if they are singular.
In particular $\Qcal(\mathbb{X}_{\tilde{\lambda},\lambda},\Omega_{\tilde{\lambda},\lambda})=0$ when
$\mathbb{X}_{\tilde{\lambda},\lambda}:=\emptyset$.

The following Theorem is proved in \cite{pep-jems}.

\begin{theo}\label{theo:QR-jems}
Let $\tilde{\lambda}\in \wGhol$. We have an Hilbertian direct sum
$$
V^{\wG}_{\tilde{\lambda}}\vert_{G}=\bigoplus_{\lambda\in \Ghol} \Qcal(\mathbb{X}_{\tilde{\lambda},\lambda},\Omega_{\tilde{\lambda},\lambda}) \ V^{G}_\lambda,
$$
It means that for any $\lambda\in \Ghol$, the multiplicity of the representation $V^{G}_\lambda$ in
the restriction $V^{\wG}_{\tilde{\lambda}}\vert_{G}$ is equal to the geometric quantization
$\Qcal(\mathbb{X}_{\tilde{\lambda},\lambda},\Omega_{\tilde{\lambda},\lambda})$ of the (compact) reduced space $\mathbb{X}_{\tilde{\lambda},\lambda}$.
\end{theo}

\begin{rem}
Let $({\tilde{\lambda},\lambda})\in \wGhol\times\Ghol$. Theorem \ref{theo:QR-jems}. shows that
$$
\left[V^{G}_{n\lambda} : V^{\wG}_{n\tilde{\lambda}}\right]= \Qcal(\mathbb{X}_{\tilde{\lambda},\lambda},n\Omega_{\tilde{\lambda},\lambda})
$$
for any $n\geq 1$.
\end{rem}

\section{Proofs of the main results}\label{sec:proof}

We come back to the setting of \S \ref{sec:cone-closed}:  $G/K$ is a complex submanifold of a Hermitian symmetric space $\wG/\wK$.
It means that there exits a $\tilde{K}$-invariant element $z\in\kgot$ such that $\ad(z)$ defines complex structures on $\tilde{\pgot}$ and $\pgot$.
We consider the orthogonal decomposition $\tilde{\pgot}=\pgot\oplus\qgot$, and we denote by ${\rm Sym}(\qgot)$ the symmetric algebra of the complex $K$-module
$(\qgot,{\rm ad}(z))$.

\subsection{Proof of Theorem A}\label{sec:proof-theo-A}

The set $\Pi_{\rm hol}(\tilde{G},G)$ is equal to $\bigcup_{\tilde{a}\in \wchol}\{\tilde{a}\}\times\Delta_G(\wG\tilde{a})$. We define
$$
\Pi_{\rm hol}(\tilde{G},G)^0:=\bigcup_{\tilde{a}\in \wchol} \{\tilde{a}\} \times \Delta_G(\wG\tilde{a})^0.
$$

We start with the following result.

\begin{lem}\label{lem:dense-Pi}
The set $\Pi_{\rm hol}(\tilde{G},G)^0\bigcap \tilde{\tgot}^*_\Qbb \times \tgot^*_\Qbb$ is dense in $\Pi_{\rm hol}(\tilde{G},G)$.
\end{lem}
{\em Proof :}
Let $(\tilde{\xi},\xi)\in \Pi_{\rm hol}(\tilde{G},G)$: take $\tilde{g}\in\wG$ such that $\xi=\pi_{\ggot,\tilde{\ggot}}(\tilde{g}\tilde{\xi})$. We consider a sequence
$\tilde{\xi}_n\in \wchol\cap\tilde{\tgot}^*_\Qbb$ converging to $\tilde{\xi}$. Then $\xi_n:=\pi_{\ggot,\tilde{\ggot}}(\tilde{g}\tilde{\xi}_n)$ is a sequence of $\Ccal^0_{G/K}$ converging to
$\xi\in\chol$. Since the map ${\rm p}: \Ccal_{G/K}^0\to \chol$ is continuous (se Lemma \ref{lem:p-continu}), the sequence $\eta_n:={\rm p}(\xi_n)$ converges to
${\rm p}(\xi)=\xi$. By definition, we have $\eta_n\in \Delta_G(\wG\,\tilde{\xi}_n)$ for any $n\in\Nbb$. Since $\tilde{\xi}_n$ are rational, each subset
$\Delta_G(\wG\tilde{\xi}_n)^0\cap \tgot_\Qbb^*$ is dense in $\Delta_G(\wG\tilde{\xi}_n)$ (see Lemma \ref{lem:rational-dense-coadjoint}). Hence, $\forall n\in\Nbb$, there exists $\zeta_n\in \Delta_G(\wG\,\tilde{\xi}_n)^0\cap \tgot^*_\Qbb$ such that
$\|\zeta_n-\eta_n\|\leq 2^{-n}$. Finally, we see that $(\tilde{\xi}_n,\zeta_n)$ is a sequence of rational elements of $\Pi_{\rm hol}(\tilde{G},G)^0$ converging to
$(\xi,\tilde{\xi})$.
$\Box$

\medskip

The main purpose of this section is the proof of the following theorem.

\medskip

\begin{theo}\label{theo:rational-Pi}
The following inclusions hold
\begin{eqnarray*}
\Pi_{\rm hol}(\tilde{G},G)^0\, \bigcap\, \tilde{\tgot}^*_\Qbb \times \tgot^*_\Qbb
\quad \underset{(1)}{\mbox{\Large$\subset$}}\quad
\Pi_{\rm hol}^\Qbb(\tilde{G},G)
\quad \underset{(2)}{\mbox{\Large$\subset$}}\quad
\Pi_{\rm hol}(\tilde{G},G).
\end{eqnarray*}
\end{theo}

Lemma \ref{lem:dense-Pi} and Theorem \ref{theo:rational-Pi} gives the important Corollary.

\begin{coro}\label{coro: pi-hol-dense}
$\Pi_{\rm hol}^\Qbb(\tilde{G},G)$ is dense in $\Pi_{\rm hol}(\tilde{G},G)$.
\end{coro}

\medskip

{\em Proof of Theorem \ref{theo:rational-Pi} :}
Let $(\tilde{\mu},\mu)\in \Pi_{\rm hol}^\Qbb(\tilde{G},G)$: there exists $N\geq 1$ such that
$(N\tilde{\mu},N\mu)\in\Pi_{\rm hol}^\Zbb(\tilde{G},G)$. The multiplicity $[V^{G}_{N\mu}:V^{\wG}_{N\tilde{\mu}}]$ is non-zero, and thanks to Theorem
\ref{theo:QR-jems}, it implies that the reduced space $\mathbb{X}_{N\tilde{\mu},N\mu}$ is non-empty. In other words
$(N\tilde{\mu},N\mu)\in \Pi_{\rm hol}(\tilde{G},G)$. The inclusion $(2)$ is proven.

Let $(\tilde{\mu},\mu)\in \Pi_{\rm hol}(\tilde{G},G)^0\bigcap \tgot^*_\Qbb\times\tilde{\tgot}^*_\Qbb$. There exists $N_o\geq 1$ such that
$\lambda:=N_o\mu \in \Ghol$,  $\tilde{\lambda}:=N_o\tilde{\mu}\in\wGhol$ and $\lambda\in \Delta_G(\wG\tilde{\lambda})^0$: the element $\lambda$ is a weakly-regular value of the moment map $\wG\tilde{\lambda}\to\ggot^*$. Theorem \ref{theo:QR-jems} tell us that, for any $n\geq 1$,  the multiplicity
$[V^{G}_{n\lambda}:V^{\wG}_{n\tilde{\lambda}}]$ is equal to Riemann-Roch number
$\Qcal(\mathbb{X}_{\tilde{\lambda},\lambda}, n\Omega_{\tilde{\lambda},\lambda})$. Since the map
$n\mapsto \Qcal(\mathbb{X}_{\tilde{\lambda},\lambda}, n\Omega_{\tilde{\lambda},\lambda})$ is non-zero
(see Proposition \ref{prop:quasi-polynomial-non-zero}), we can conclude that there exists $n_o\geq 1$ such that
$[V^{G}_{n_o\lambda}:V^{\wG}_{n_o\tilde{\lambda}}]\neq 0$. In other words,  we obtain $n_oN_o(\tilde{\mu},\mu)\in \Pi_{\rm hol}^\Zbb(\tilde{G},G)$
and so $(\tilde{\mu},\mu)\in \Pi_{\rm hol}^\Qbb(\tilde{G},G)$. The inclusion $(1)$ is settled.
$\Box$

\medskip

\medskip

Now we can terminate the proof of Theorem A.

\medskip

Thanks to Proposition \ref{prop:monoid}, we know that $\Pi_{\rm hol}^\Qbb(\tilde{G},G)$ is a $\Qbb$-convex cone.  Since $\Pi_{\rm hol}(\tilde{G},G)$ is a closed subset of $\wchol\times\chol$ (see Proposition \ref{prop:pi-z-compact}),
we can conclude, by a density argument,  that $\Pi_{\rm hol}(\tilde{G},G)$ is a closed convex cone of $\wchol\times\chol$.

\subsection{The affine variety $\wK_\Cbb\times\qgot$}\label{sec:affine}

Let $\tilde{\kappa}$ be the Killing form on the Lie algebra $\tilde{\ggot}$. We consider the $\wK$-invariant symplectic structures $\Omega_{\tilde{\pgot}}$ on $\tilde{\pgot}$, defined by the relation
$$
\Omega_{\tilde{\pgot}}(\tilde{Y},\tilde{Y}')=\tilde{\kappa}(z,[\tilde{Y},\tilde{Y}']),\quad \forall \tilde{Y},\tilde{Y}'\in\tilde{\pgot}.
$$
We notice that the complex structure ${\rm ad}(z)$ is adapted to $\Omega_{\tilde{\pgot}}$ : $\Omega_{\tilde{\pgot}}(\tilde{Y},{\rm ad}(z)\tilde{Y})>0$ if $\tilde{Y}\neq 0$.

We denote by $\Omega_{\qgot}$ the restriction of
$\Omega_{\tilde{\pgot}}$ on the symplectic subspace $\qgot$. The moment map $\Phi_{\qgot}$ associated to the $K$-action on
$(\qgot, \Omega_{\qgot})$ is defined by the relations $\langle\Phi_{\qgot}(Y),X\rangle= \frac{-1}{2}\tilde{\kappa}([X,Y],[z,Y])$,
$\forall (X,Y) \in\pgot\times \qgot$.  In particular, $\langle\Phi_{\qgot}(Y),z\rangle = \frac{-1}{2}\|Y\|^2$, $\forall Y \in\qgot$,
so the map $\langle\Phi_{\qgot},z\rangle$ is proper.

The complex reductive group $\tilde{K}_\Cbb$ is equipped with the following action of
$\tilde{K}\times K$ : $(\tilde{k},k)\cdot a= \tilde{k} a k^{-1}$.   It has a canonical structure of a smooth affine variety.
There is a diffeomorphism of the cotangent bundle $\T^*\tilde{K}$ with $\tilde{K}_\Cbb$ defined as follows. We identify $\T^*\tilde{K}$ with $\tilde{K}\times \tilde{\kgot}^*$
by means of left-translation and then with $\tilde{K}\times \tilde{\kgot}$ by means of an invariant inner product on  $\tilde{\kgot}$. The map $\varphi:\tilde{K}\times \tilde{\kgot}\to \tilde{K}_\Cbb$ given by $\varphi(a,X)=a e^{iX}$ is a diffeomorphism. If we use $\varphi$ to transport the complex structure of $\tilde{K}_\Cbb$ to $\T^*\tilde{K}$, then the resulting complex structure on $\T^*\tilde{K}$ is compatible with the symplectic structure on $\T^*\tilde{K}$, so that $\T^*\tilde{K}$
becomes a K\"{a}hler Hamiltonian $\tilde{K}\times K$-manifold (see \cite{Hall97}, \S 3). The moment map relative to the $\tilde{K}\times K$-action is the proper
map $\Phi_{\tilde{K}} \oplus \Phi_K : \T^*\tilde{K} \to \tilde{\kgot}^*\oplus \kgot^*$ defined by
$\Phi_{\tilde{K}}(\tilde{a},\tilde{\eta})=-\tilde{a}\tilde{\eta}$ and $\Phi_K(\tilde{a},\tilde{\eta})=\pi_{\kgot,\wk}(\tilde{\eta})$.
Here $\pi_{\kgot,\wk}: \tilde{\kgot}^* \to \kgot^*$ is the projection dual to the inclusion $\kgot \croc \tilde{\kgot}$ of Lie algebras.

Finally we consider the K\"{a}hler Hamiltonian $\tilde{K}\times K$-manifold
$\T^*\tilde{K}\times \qgot$, where $\qgot$ is equipped with the symplectic structure $\Omega_\qgot$.
Let us denote by $\Phi: \T^*\tilde{K}\times \qgot\to \tilde{\kgot}^*\oplus \kgot^*$
the moment map relative to the $\tilde{K}\times K$-action~:
\begin{equation}\label{eq:momentcotangent}
\Phi(\tilde{a},\tilde{\eta},Y)=\left(-\tilde{a}\tilde{\eta},\pi_{\kgot,\wk}(\tilde{\eta})+\Phi_\qgot(Y)\right).
\end{equation}

Since $\Phi$ is proper map, the Convexity Theorem tell us that
$$
\Delta(\T^*\tilde{K}\times \qgot):={\rm Image}(\Phi)\bigcap \, \tilde{\tgot}^*_{\geq 0}\times \tgot^*_{\geq 0}
$$
is a closed convex locally polyhedral set.

We consider now the action of $\wK\times K$ on the affine variety $\tilde{K}_\Cbb\times \qgot$. The set of highest weights of
$\tilde{K}_\Cbb\times \overline{\qgot}$ is the semigroup
$$
\Delta^{\Zbb}(\tilde{K}_\Cbb\times\qgot)\subset \tilde{\wedge}^*_+\times\wedge^*_+
$$
consisting of all dominant weights $(\tilde{\lambda},\lambda)$ such that the irreducible $\wK\times K$-representation
$V_{\tilde{\lambda}}^{\wK}\otimes V_{\lambda}^{K}$ occurs in the coordinate ring $\Cbb[\tilde{K}_\Cbb\times \qgot]$.
A direct application of the Peter-Weyl Theorem gives the following characterization :
\begin{equation}\label{eq:delta-Z-affine}
(\tilde{\lambda},\lambda)\in\Delta^{\Zbb}(\tilde{K}_\Cbb\times \qgot)\Longleftrightarrow
\left[ V^{\wK}_{\tilde{\lambda}}\vert_K\otimes V^{K}_\lambda\otimes {\rm Sym}(\qgot) \right]^K\neq 0.
\end{equation}

We denote by $\Delta^{\Qbb}(\tilde{K}_\Cbb\times \qgot)$ the $\Qbb$-convex cone generated by the semigroup \break $\Delta^{\Zbb}(\tilde{K}_\Cbb\times \qgot)$:
$(\tilde{\xi},\xi)\in \Delta^{\Qbb}(\tilde{K}_\Cbb\times \qgot)$ if and only if $\exists N\geq 1$, $N(\tilde{\xi},\xi)\in \Delta^{\Zbb}(\tilde{K}_\Cbb\times \qgot)$.

\medskip

The following important fact is classical (see Theorem 4.9 in \cite{sjamaar98}).

\begin{prop}\label{prop:delta-affine-dense}
The Kirwan polyhedron $\Delta(\T^*\tilde{K}\times \qgot)$ is equal to the closure of the $\Qbb$-convex cone $\Delta^{\Qbb}(\tilde{K}_\Cbb\times \qgot)$.
\end{prop}

\subsection{Proof of Theorem B}

Consider the semigroup $\Pi^\Zbb(\tilde{G},G)$ of $\tilde{\wedge}^*_+ \times \wedge^*_+$ (see Definition \ref{defi:horn-cone-general}) and the $\Qbb$-convex cone
$\Pi^\Qbb(\tilde{G},G):=\{(\tilde{\xi},\xi)\in \tilde{\tgot}^*_{\geq 0}\times \tgot^*_{\geq 0} \ ; \ \exists N\geq 1, N(\tilde{\xi},\xi)\in \Pi^\Zbb(\tilde{G},G)\}$.

The Jakobsen-Vergne theorem says that $\Pi_{\rm hol}^\Zbb(\tilde{G},G)=\Pi^\Zbb(\tilde{G},G)\,\bigcap \, \wGhol\times\Ghol$. Hence, the convex cone $\Pi_{\rm hol}^\Qbb(\tilde{G},G)$ is equal to
$\Pi^\Qbb(\tilde{G},G)\cap \wchol\times\chol$. Thanks to (\ref{eq:delta-Z-affine}), we know that
$(\tilde{\xi},\xi)\in \Pi^\Qbb(\tilde{G},G)$ if and only if $(\tilde{\xi},\xi^*)\in \Delta^{\Qbb}(\tilde{K}_\Cbb\times \qgot)$.
The density results obtained in Proposition \ref{prop:delta-affine-dense} and Corollary \ref{coro: pi-hol-dense} gives finally Theorem B.

\begin{theo}\label{theo:theorem-B}
\begin{enumerate}
\item We have $\Pi_{\rm hol}(\tilde{G},G)=\Pi(\tilde{G},G)\,\bigcap \,\wchol\times\chol $.
\item An element $(\tilde{\xi},\xi)$ belongs to  $\Pi_{\rm hol}(\wG,G)$ if and only if
$$
(\tilde{\xi},\xi)\in\wchol\times\chol\quad {\rm and}\quad (\tilde{\xi},\xi^*)\in \Delta(\T^*\wK\times \qgot).
$$
\end{enumerate}
\end{theo}

\medskip

\subsection{Proof of Theorem C}

We denote by $\bar{\qgot}$ the $K$-vector space $\qgot$ equipped with the opposite symplectic form $-\Omega_{\qgot}$ and opposite complex structure
$-{\rm ad}(z)$. The moment map relative to the $K$-action on $\bar{\qgot}$ is denoted by $\Phi_{\bar{\qgot}}=-\Phi_{\qgot}$.

\begin{lem} Any element $(\tilde{\xi},\xi)\in\tilde{\tgot}^*_{\geq 0}\times\tgot^*_{\geq 0}$ satisfies the equivalence :
$$
(\tilde{\xi},\xi^*)\in \Delta(\T^*\wK\times \qgot) \Longleftrightarrow \xi\in \Delta_K(\wK\tilde{\xi}\times \overline{\qgot}).
$$
\end{lem}
{\em Proof :} Thanks to (\ref{eq:momentcotangent}), we see immediatly that $\exists (\tilde{a},\tilde{\eta},Y)\in\T ^*\wK\times \qgot$ such that
$(\tilde{\xi},\xi^*)=\Phi(\tilde{a},\tilde{\eta},Y)$ if and only if $\exists (\tilde{b},Z)\in \wK\times \qgot$ such that
$\xi=\pi_{\kgot,\wk}(\tilde{b}\tilde{\xi})+\Phi_{\bar{\qgot}}(Z)$. $\Box$

\medskip

At this stage we know that  $\Delta_G(\wG\tilde{\mu})=\Delta_K(\wK\tilde{\mu}\times \overline{\qgot})\cap \chol$. Hence, Theorem C will follows from the next result.

\begin{prop}\label{prop:delta-chol}
For any $\tilde{\mu}\in \wchol$, the Kirwan polyhedron $\Delta_K(\wK\tilde{\mu}\times \overline{\qgot})$ is contained in $\chol$.
\end{prop}

{\em Proof :}  By definition $\chol=\Ccal_{G/K}^0\cap\tgot^*_{\geq 0}$, so we have to prove that
$\pi_{\kgot,\tilde{\kgot}}(\wK\tilde{\mu})+ {\rm Image}( \Phi_{\bar{\qgot}})$ is contained in $\Ccal_{G/K}^0$.
By definition $\wK\tilde{\mu}\subset \Ccal_{\wG/\wK}^0$, and then $\pi_{\kgot,\tilde{\kgot}}(\wK\tilde{\mu})\subset \Ccal_{G/K}^0$. Since
$\Ccal_{G/K}^0 +\Ccal_{G/K}\subset \Ccal_{G/K}^0$, it is sufficient to check that ${\rm Image}( \Phi_{\bar{\qgot}})\subset \Ccal_{G/K}$.
Let $\Phi_{\tilde{\pgot}}$ be the moment map relative to the action of $\wK$ on $(\tilde{\pgot},\Omega_{\tilde{\pgot}})$. As
${\rm Image}( \Phi_{\bar{\qgot}})\subset \pi_{\kgot,\tilde{\kgot}}\left({\rm Image}( -\Phi_{\tilde{\pgot}})\right)$, the following lemma will terminate the proof of
Proposition \ref{prop:delta-chol}. $\Box$

\begin{lem}
The image of the moment map $-\Phi_{\tilde{\pgot}}$ is contained in $\Ccal_{\wG/\wK}$.
\end{lem}
{\em Proof :}
Let $z^*\in\tilde{\tgot}^*$ such that $\langle z^*,\tilde{X}\rangle= -\tilde{\kappa}(z,\tilde{X})$, $\forall\tilde{X}\in\tilde{\ggot}$. Consider the coadjoint orbit
$\tilde{\Ocal}=\wG z^*$ equipped with its canonical symplectic structure $\Omega_{\tilde{\Ocal}}$: the symplectic vector space $\T_{z^*}\tilde{\Ocal}$ is canonically
isomorphic to $(\tilde{\pgot},-\Omega_{\tilde{\pgot}})$. In \cite{McDuff}, McDuff proved that $(\tilde{\Ocal},\Omega_{\tilde{\Ocal}})$ is diffeomorphic, as a
$\wK$-symplectic manifold, to the symplectic vector space $(\tilde{\pgot},-\Omega_{\tilde{\pgot}})$ (see \cite{Deltour-these,Deltour-JDG} for a generalization of this fact).
McDuff's results shows in particular that ${\rm Image}(-\Phi_{\tilde{\pgot}})=\pi_{\tilde{\ggot},\tilde{\kgot}}(\tilde{\Ocal})$. Our proof is completed if we check that
$\pi_{\tilde{\ggot},\tilde{\kgot}}(\tilde{\Ocal})\subset\Ccal_{\wG/\wK}$: in other words, if $\langle\pi_{\tilde{\ggot},\tilde{\kgot}}(\tilde{g}_0\, z^*),\tilde{g}_1z\rangle\geq 0$,
$\forall \tilde{g}_0,\tilde{g}_1\in \wG$. But
\begin{eqnarray*}
2\langle\pi_{\tilde{\ggot},\tilde{\kgot}}(\tilde{g}_0\, z^*),\tilde{g}_1\,z\rangle &=& \langle\tilde{g}_0\, z^*,\tilde{g}_1z+\Theta(\tilde{g}_1)z\rangle\\
&=& -\tilde{\kappa}(z,\tilde{g}_0^{-1}\tilde{g}_1\, z)-\tilde{\kappa}(z,\tilde{g}_0^{-1}\Theta(\tilde{g}_1)z).
\end{eqnarray*}
With (\ref{eq:Gz}) in hand, it is not difficult to see that $-\tilde{\kappa}(z,\tilde{g}\, z)\geq 0$ for every $\tilde{g}\in\wG$. We thus verified that
$\pi_{\tilde{\ggot},\tilde{\kgot}}(\tilde{\Ocal})\subset\Ccal_{\wG/\wK}$. $\Box$

\section{Facets of the cone $\Pi_{\rm hol}(\tilde{G},G)$}\label{sec:cone:faces}

We come back to the framework of \S \ref{sec:affine}. We consider the K\"{a}hler Hamiltonian $\tilde{K}\times K$-manifold
$\T^*\tilde{K}\times \qgot$. The moment map, $\Phi: \T^*\tilde{K}\times \qgot\to \tilde{\kgot}^*\oplus \kgot^*$, relative to the $\tilde{K}\times K$-action is defined by (\ref{eq:momentcotangent}).

In this section, we adapt to our case the result of \S 6 of \cite{pep-ressayre-pair} concerning the parametrization of the facets of Kirwan polyhedrons in
terms of Ressayre's data.

\subsection{Admissible elements}

We choose maximal torus $\wT\subset\wK$ and $T\subset K$ such that $T\subset \wT$. Let $\Rgot_o$ and $\Rgot$ be respectively the set of roots for the action of $T$ on
$(\wg/\ggot)\otimes\Cbb$ and $\ggot\otimes\Cbb$. Let $\tilde{\Rgot}$ be the set of roots for the action of $\wT$ on $\wg\otimes\Cbb$.
Let $\Rgot^+\subset \Rgot$ and  $\tilde{\Rgot}^+\subset \tilde{\Rgot}$  be the systems of positive roots defined in (\ref{eq:R-plus}).
Let $W,\tilde{W}$ be the Weyl groups of $(T,K)$ and $(\wT,\wK)$. Let $w_o\in W$ be the longest element.

We start by introducing the notion of admissible elements. The group $\hom(U(1),T)$ admits a natural identification with the lattice
$\wedge:=\frac{1}{2\pi}\ker(\exp:\tgot\to T)$. A vector $\gamma\in\tgot$ is called rational if it belongs to the $\Qbb$-vector space
$\tgot_\Qbb$ generated by $\wedge$.

We consider the $\wK\times K$-action on $N:=\T^* \wK \times \qgot$. We associate to any subset $\Xcal\subset N$, the integer
$\dim_{\wK\times K}(\Xcal)$ (see (\ref{eq:dim-K})).

\begin{defi}
A non-zero element $(\tilde{\gamma},\gamma)\in\tilde{\tgot}\times \tgot$
is called {\em admissible} if the elements $\tilde{\gamma}$ and $\gamma$ are rational and if $\dim_{\wK\times K}(N^{(\tilde{\gamma},\gamma)})-\dim_{\wK\times K}(N)\in\{0,1\}$.
\end{defi}

If $\gamma\in\tgot$, we denote by $\Rgot_o\cap\gamma^\perp$ the subsets of weight vanishing against $\gamma$. We start with the
following lemma whose proof is left to the reader (see \S 6.1.1 of \cite{pep-ressayre-pair}).

\begin{lem}
\begin{enumerate}
\item $N^{(\tilde{\gamma},\gamma)}\neq \emptyset$ if and only if $\tilde{\gamma}\in \tilde{W}\gamma$.
\item $\dim_{\wK\times K}(N)=\dim_T(\tilde{\ggot}/\ggot)=\dim(\tgot)-\dim({\rm Vect}(\Rgot_o))$.
\item For any $\tilde{w}\in\tilde{W}$, $\dim_{\wK\times K}(N^{(\tilde{w}\gamma,\gamma)})=
\dim_T(\tilde{\ggot}^\gamma/\ggot^\gamma)=\dim(\tgot)-\dim({\rm Vect}(\Rgot_o\cap\gamma^\perp))$.
\end{enumerate}
\end{lem}

The next result is a direct consequence of the previous lemma.

\begin{lem}
The admissible elements relative to the $\wK\times K$-action on $\T^* \wK \times \qgot$ are of the form $(\tilde{w}\gamma,\gamma)$ where
$\tilde{w}\in \tilde{W}$ and $\gamma$ is
a non-zero rational element satisfying ${\rm Vect}(\Rgot_o)\cap \gamma^\perp={\rm Vect}(\Rgot_o\cap\gamma^\perp)$.
\end{lem}

\subsection{Ressayre's data}

\begin{defi}
\begin{enumerate}
\item Consider the linear action $\rho: G\to {\rm GL}_\Cbb(V)$ of a compact Lie group on a complex vector space $V$. For any $(\eta,a)\in\ggot\times\Rbb$, we define the vector subspace
$V^{\eta=a}=\{v\in V, d\rho(\eta)v=i av\}$.  Thus, for any $\eta\in\ggot$, we have the decomposition
$V=V^{\eta>0}\oplus V^{\eta=0}\oplus V^{\eta<0}$ where $V^{\eta>0}=\sum_{a>0}V^{\eta=a}$, and $V^{\eta<0}=\sum_{a<0}V^{\eta=a}$.
\item The real number $\tr_{\eta}(V^{\eta>0})$ is defined as the sum $\sum_{a>0}a\,\dim(V^{\eta=a})$.
\end{enumerate}
\end{defi}

We consider an admissible element $(\tilde{w}\gamma,\gamma)$. The submanifold of $N\simeq \wK_\Cbb \times \qgot$ fixed  of by $(\tilde{w}\gamma,\gamma)$
is $N^{(\tilde{w}\gamma,\gamma)}= \tilde{w} \wK_\Cbb^\gamma \times \qgot^{\gamma}$. There is a canonical isomorphism between the manifold
$N^{(\tilde{w}\gamma,\gamma)}$ equipped with the action of $\tilde{w}\wK^\gamma \tilde{w}^{-1}\times K^\gamma$
with the manifold $\wK_\Cbb^\gamma \times \qgot^{\gamma}$ equipped with the action of $\wK^\gamma \times K^\gamma$.
The tangent bundle $(\T N\vert_{N^{(\tilde{w}\gamma,\gamma)}})^{(\tilde{w}\gamma,\gamma)>0}$ is isomorphic to
$N^{\gamma_w}\times \tilde{\kgot}_\Cbb^{\gamma>0}\times \qgot^{\gamma>0}$.

The choice of positive roots $\Rgot^+$ (resp. $\tilde{\Rgot}^+$) induces a decomposition $\kgot_\Cbb=\ngot\oplus\tgot_\Cbb\oplus \overline{\ngot}$ (resp. $\wk_\Cbb=\tilde{\ngot}\oplus\tilde{\tgot}_\Cbb\oplus \overline{\tilde{\ngot}}$), where
$\ngot=\sum_{\alpha\in\Rgot^+}(\kgot\otimes\Cbb)_\alpha$ (resp. $\tilde{\ngot}=\sum_{\tilde{\alpha}\in\tilde{\Rgot}^+}(\wk\otimes\Cbb)_{\tilde{\alpha}}$).
 We consider the map
$$
\rho^{\tilde{w},\gamma}: \wK_\Cbb^\gamma \times \qgot^\gamma \longrightarrow
\hom \left(\tilde{\ngot}^{\tilde{w}\gamma>0}\times \ngot^{\gamma>0}\,,\, \tilde{\kgot}_\Cbb^{\gamma>0}\times\qgot^{\gamma>0}\right)
$$
defined by the relation
$$
\rho^{\tilde{w},\gamma}(\tilde{x},v): (\tilde{X},X)\longmapsto ((\tilde{w} \tilde{x})^{-1}\tilde{X}-X\, ;\, X\cdot v),
$$
for any $(\tilde{x},v)\in \wK_\Cbb^\gamma \times \qgot^\gamma$.
\medskip

\begin{defi}
$(\gamma,\tilde{w})\in\tgot\times \tilde{W}$ is a Ressayre's data if
\begin{enumerate}
\item $(\tilde{w}\gamma,\gamma)$ is admissible,
\item $\exists (\tilde{x},v)$ such that $\rho^{\tilde{w},\gamma}(\tilde{x},v)$ is bijective.
\end{enumerate}
\end{defi}

\begin{rem}
In \cite{pep-ressayre-pair}, the Ressayre's data were called {\em regular infinitesimal $B$-Ressayre's pairs}.
\end{rem}

Since the linear map $\rho^{\tilde{w},\gamma}(\tilde{x},v)$ commutes with the $\gamma$-actions, we obtain the following necessary conditions.

\begin{lem}\label{lem:relations-A-B}
If $(\gamma,\tilde{w})\in\tgot\times \tilde{W}$ is a Ressayre's data, then
\begin{itemize}
\item Relation (A) : $\dim(\tilde{\ngot}^{\tilde{w}\gamma>0})+\dim(\ngot^{\gamma>0})=\dim(\tilde{\kgot}_\Cbb^{\gamma>0})+\dim(\qgot^{\gamma>0})$.
\item Relation (B) : $\tr_{\tilde{w}\gamma}(\tilde{\ngot}^{\tilde{w}\gamma>0})+\tr_{\gamma}(\ngot^{\gamma>0})=
\tr_{\gamma}(\tilde{\kgot}_\Cbb^{\gamma>0})+\tr_{\gamma}(\qgot^{\gamma>0})$.
\end{itemize}
\end{lem}

\begin{lem}
Relation (B) is equivalent to
\begin{equation}\label{eq:trace-condition-w-tilde}
\sum_{\stackrel{\alpha\in\Rgot^+}{\langle\alpha,\gamma\rangle> 0}}\langle\alpha,\gamma\rangle=
\sum_{\stackrel{\tilde{\alpha}\in\tilde{\Rgot}^+}{\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle> 0}}\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle.
\end{equation}
\end{lem}

{\em Proof :} First one sees that
$\tr_{\gamma}(\qgot^{\gamma>0})=\tr_{\gamma}(\tilde{\pgot}^{\gamma>0})-\tr_{\gamma}(\pgot^{\gamma>0})=
\sum_{\stackrel{\tilde{\alpha}\in \tilde{\Rgot}^+_n}{\langle\tilde{\alpha},\gamma\rangle>0}}\langle\tilde{\alpha},\gamma\rangle
-\sum_{\stackrel{\alpha\in \Rgot^+_n}{\langle\alpha,\gamma\rangle>0}}\langle\alpha,\gamma\rangle$, and
$\tr_{\gamma}(\wk_\Cbb^{\gamma>0})=\tr_{\tilde{w}\gamma}(\wk_\Cbb^{\tilde{w}\gamma>0})=\tr_{\tilde{w}\gamma}(\tilde{\ngot}^{\tilde{w}\gamma>0})+
\sum_{\stackrel{\tilde{\alpha}\in \tilde{\Rgot}^+_c}{\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle>0}}\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle$.
Relation $(B)$ is equivalent to
\begin{equation}\label{eq:equivalent-relation-B}
\tr_{\gamma}(\ngot^{\gamma>0})+\sum_{\stackrel{\alpha\in \Rgot^+_n}{\langle\alpha,\gamma\rangle>0}}\langle\alpha,\gamma\rangle=
\sum_{\stackrel{\tilde{\alpha}\in \tilde{\Rgot}^+_n}{\langle\tilde{\alpha},\gamma\rangle>0}}\langle\tilde{\alpha},\gamma\rangle
+\sum_{\stackrel{\tilde{\alpha}\in \tilde{\Rgot}^+_c}{\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle>0}}\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle.
\end{equation}
Since $\tilde{\Rgot}^+_n$ is invariant under the action of the Weyl group $\tilde{W}$, the right hand side of (\ref{eq:equivalent-relation-B}) is equal to
$\sum_{\stackrel{\tilde{\alpha}\in\tilde{\Rgot}^+}{\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle>0}}\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle$.
Since the left hand side of (\ref{eq:equivalent-relation-B}) is equal to
$\sum_{\stackrel{\alpha\in \Rgot^+}{\langle\alpha,\gamma\rangle>0}}\langle\alpha,\gamma\rangle$, the proof of the Lemma is complete. $\Box$

\subsection{Cohomological characterization of Ressayre's data}

Let $\gamma\in\tgot$ be a non-zero rational element. We denote by $B\subset K_\Cbb$ and by $\tilde{B}\subset \wK_\Cbb$ the Borel subgroups with
Lie algebra $\bgot=\tgot_\Cbb\oplus \ngot$ and $\tilde{\bgot}=\tilde{\tgot}_\Cbb\oplus \tilde{\ngot}$. Consider the parabolic subgroup $P_\gamma\subset K_\Cbb$ defined by
\begin{equation}\label{eq:P-gamma}
P_\gamma=\{g\in K_\Cbb, \lim_{t\to\infty}\exp(-it\gamma)g\exp(it\gamma)\ {\rm exists}\}.
\end{equation}
Similarly, one defines a parabolic subgroup $\tilde{P}_\gamma\subset \wK_\Cbb$.

We work with the projective varieties $\Fcal_\gamma:=K_\Cbb/P_\gamma$, $\tilde{\Fcal}_\gamma:=\tilde{K}_\Cbb/\tilde{P}_\gamma$,
and the canonical embedding $\iota: \Fcal_\gamma\to  \tilde{\Fcal}_\gamma$. We associate to any $\tilde{w}\in\tilde{W}$, the Schubert cell
$$
\tilde{\Xgot}^o_{\tilde{w},\gamma}:= \tilde{B}[\tilde{w}]\subset \tilde{\Fcal}_\gamma.
$$
and the Schubert variety $\tilde{\Xgot}_{\tilde{w},\gamma}:=\overline{\tilde{\Xgot}^o_{\tilde{w},\gamma}}$. If $\tilde{W}^\gamma$ denotes the subgroup of $\tilde{W}$ that fixes
$\gamma$, we see that  the Schubert cell $\tilde{\Xgot}^o_{\tilde{w},\gamma}$ and the Schubert variety $\tilde{\Xgot}_{\tilde{w},\gamma}$ depends only of the class of $\tilde{w}$ in
$\tilde{W}/\tilde{W}^\gamma$.

On the variety $\Fcal_\gamma$, we consider the Schubert cell $\Xgot^o_{\gamma}:= B[e]$ and the Schubert variety $\Xgot_{\gamma}:=\overline{\Xgot^o_{\gamma}}$.

We consider the cohomology\footnote{Here, we use singular cohomology with integer coefficients.} ring $H^*(\tilde{\Fcal}_\gamma,\Zbb)$ of $\tilde{\Fcal}_\gamma$.
If Y is an irreducible closed subvariety of $\tilde{\Fcal}_\gamma$, we denote by $[Y]\in H^{2n_Y}(\tilde{\Fcal}_\gamma,\Zbb)$ its cycle class in cohomology : here $n_Y={\rm codim}_\Cbb(Y)$. Let $\iota^*:H^*(\tilde{\Fcal}_\gamma,\Zbb)\to H^*(\Fcal_\gamma,\Zbb)$ be the pull-back map in cohomology.
Recall that the cohomology class $[pt]$  associated to a singleton $Y=\{pt\}\subset \Fcal_\gamma$   is a basis of
$H^{\mathrm{max}}(\Fcal_\gamma,\Zbb)$.

\medskip

To an oriented real vector bundle $\Ecal\to N$ of rank $r$, we can associate its Euler class ${\rm Eul}(\Ecal)\in H^{2r}(N,\Zbb)$.
When $\Vcal\to N$ is a complex vector bundle, then ${\rm Eul}(\Vcal_\Rbb)$ corresponds to the top Chern class $c_{p}(\Vcal)$, where $p$ is the complex rank of $\Vcal$, and
$\Vcal_\Rbb$ means $\Vcal$ viewed as a real vector bundle oriented by its complex structure (see \cite{Bott-Tu}, \S 21).

The isomorphism $\qgot^{\gamma>0}\simeq \qgot/ \qgot^{\gamma\leq 0}$ shows that $\qgot^{\gamma>0}$ can be viewed as a $P_\gamma$-module. Let
$[\qgot^{\gamma>0}]= K_\Cbb\times_{P_\gamma}\qgot^{\gamma>0}$ be the corresponding complex vector bundle on $\Fcal_\gamma$.
We denote simply by ${\rm Eul}(\qgot^{\gamma>0})$ the Euler class ${\rm Eul}([\qgot^{\gamma>0}]_\Rbb)\in H^*(\Fcal_\gamma,\Zbb)$.

The following characterization of Ressayre's data was obtained in \cite{pep-ressayre-pair}, \S 6. Recall that $\Rgot_o$ denotes the set of weights relative to the $T$-action on
$(\wg/\ggot)\otimes\Cbb$.

\begin{prop}
An element $(\gamma,\tilde{w})\in\tgot\times \tilde{W}$ is a Ressayre's data if and only if the following conditions hold :
\begin{itemize}
\item $\gamma$ is non-zero and rational.
\item ${\rm Vect}(\Rgot_o\cap \gamma^\perp)={\rm Vect}(\Rgot_o)\cap \gamma^\perp$.
\item $[\Xgot_{\gamma}]\cdot \iota^*([\tilde{\Xgot}_{\tilde{w},\gamma}])\cdot {\rm Eul}(\qgot^{\gamma>0})= k[pt],\ k\geq 1$ in $H^*(\Fcal_\gamma,\Zbb)$.
\item $\sum_{\stackrel{\alpha\in\Rgot^+}{\langle\alpha,\gamma\rangle> 0}}\langle\alpha,\gamma\rangle=
\sum_{\stackrel{\tilde{\alpha}\in\tilde{\Rgot}^+}{\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle> 0}}\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle$.
\end{itemize}
\end{prop}

\subsection{Parametrization of the facets}\label{sec:facets}

We can finally describe the Kirwan polyhedron $\Delta(\T^*\tilde{K}\times \qgot)$ (see \cite{pep-ressayre-pair}, \S 6).

\begin{theo}\label{theo:delta}
An element $(\tilde{\xi},\xi)\in\tilde{\tgot}_{\geq 0}^*\times \tgot_{\geq 0}^*$ belongs to $\Delta(\T^*\tilde{K}\times  \qgot)$ if and only if
$$
\langle \tilde{\xi},\tilde{w}\gamma\rangle+\langle \xi,\gamma\rangle\geq 0
$$
for any Ressayre's data $(\gamma,\tilde{w})\in\tgot\times \tilde{W}$.
\end{theo}

Theorem \ref{theo:delta} and Theorem \ref{theo:theorem-B} permit us to give the following description of the convex cone
$\Pi_{\rm hol}(\tilde{G},G)$.

\begin{theo}
An element $(\tilde{\xi},\xi)$ belongs to  $\Pi_{\rm hol}(\wG,G)$ if and only if $(\tilde{\xi},\xi)\in\wchol\times\chol$ and
$$
\langle \tilde{\xi},\tilde{w}\gamma\rangle\geq \langle \xi,w_0\gamma\rangle
$$
for any $(\gamma,\tilde{w})\in\tgot\times \tilde{W}$ satisfying the following conditions:
\begin{itemize}
\item $\gamma$ is non-zero and rational.
\item ${\rm Vect}(\Rgot_o\cap \gamma^\perp)={\rm Vect}(\Rgot_o)\cap \gamma^\perp$.
\item $[\Xgot_{\gamma}]\cdot \iota^*([\tilde{\Xgot}_{\tilde{w},\gamma}])\cdot {\rm Eul}(\qgot^{\gamma>0})= k[pt],\ k\geq 1$ in $H^*(\Fcal_\gamma,\Zbb)$.
\item $\sum_{\stackrel{\alpha\in\Rgot^+}{\langle\alpha,\gamma\rangle> 0}}\langle\alpha,\gamma\rangle=
\sum_{\stackrel{\tilde{\alpha}\in\tilde{\Rgot}^+}{\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle> 0}}\langle\tilde{\alpha},\tilde{w}_0\tilde{w}\gamma\rangle$.\end{itemize}
\end{theo}

\section{Example: the holomorphic Horn cone $\Horn(p,q)$}\label{sec:horn-p-q}

Let  $p\geq q\geq 1$. We consider the pseudo-unitary group $G=U(p,q)\subset GL_{p+q}(\Cbb)$ defined by the relation : $g\in U(p,q)$ if and only if
$g{\rm Id}_{p,q}g^*={\rm Id}_{p,q}$ where ${\rm Id}_{p,q}$ is the diagonal matrice ${\rm Diag}({\rm Id}_p,-{\rm Id}_q)$.

We work with the maximal compact subgroup $K=U(p)\times U(q)\subset G$. We have the Cartan decomposition
$\ggot=\kgot\oplus \pgot$, where $\pgot$ is identified with the vector space $M_{p,q}$ of $p\times q$ matrices through the map
$$
X\in M_{p,q}\longmapsto\left(\begin{array}{cc} 0 & X \\ X^* & 0\\\end{array}\right).
$$
We work with the element $z_{p,q}=\frac{i}{2}{\rm Id}_{p,q}$ which belongs to the center of $\kgot$. The adjoint action of $z_{p,q}$ on  $\pgot$ corresponds to the standard complex structure on
$M_{p,q}$.

The trace on $\mathfrak{gl}_{p+q}(\Cbb)$ defines an identification $\ggot\simeq \ggot^*=\hom(\ggot,\Rbb)$:  to $X\in\ggot$ we associate
$\xi_X\in\ggot^*$ defined by $\langle \xi_X,Y\rangle=-{\rm Tr}(XY)$. Thus, the $G$-invariant cone $\Ccal_{G/K}$ defined by $z_{p,q}$ can be viewed as the following cone
of $\ggot$:
$$
\Ccal(p,q)=\left\{X\in\ggot,\ {\rm Im}\left(\tr( gXg^{-1} {\rm Id}_{p,q})\right) \geq 0,\ \forall g\in U(p,q)\right\}.
$$

Let $T\subset U(p)\times U(q)$ be the maximal torus formed by the diagonal matrices.
The Lie algebra $\tgot$ is identified with $\Rbb^p\times\Rbb^q$ through the map ${\bf d}:\Rbb^p\times\Rbb^q\to \ugot(p)\times\ugot(q)$ :
${\bf d}_{x}={\rm Diag}(ix_1,\cdots,ix_p,i x_{p+1},\cdots, i x_{p+q})$. The Weyl chamber is
$$
\tgot_{\geq 0}=\left\{x\in \Rbb^{p}\times\Rbb^q,\ x_1\geq \cdots \geq x_p \ {\rm and}\  x_{p+1}\geq \cdots \geq x_{p+q}\right\}.
$$
Proposition \ref{prop:Gz} tells us that the $U(p,q)$ adjoint orbits in the interior of $\Ccal(p,q)$ are parametrized by the holomorphic chamber
$$
\Ccal_{p,q}=\left\{x\in \Rbb^p\times\Rbb^q, x_1\geq \cdots \geq x_p > x_{p+1}\geq \cdots \geq x_{p+q}\right\} \,\subset\, \tgot_{\geq 0}.
$$

\begin{defi}
The holomorphic Horn cone $\Horn(p,q):=\Horn^2(U(p,q))$ is defined by the relations
$$
\Horn(p,q)=\left\{(A,B,C)\in(\Ccal_{p,q})^3,\ U(p,q){\bf d}_C\subset U(p,q){\bf d}_A+U(p,q){\bf d}_B\right\},
$$
\end{defi}

\medskip

Let us detail the description given of $\Horn(p,q)$ by Theorem B. For any $n\geq 1$, we consider the semigroup
$\wedge_n^+=\{(\lambda_1\geq \cdots\geq\lambda_n)\}\subset\Zbb^n$. If $\lambda=(\lambda',\lambda'')\in \wedge_p^+\times\wedge_q^+$, then
$V_\lambda:=V^{U(p)}_{\lambda'}\otimes V^{U(q)}_{\lambda''}$ denotes the irreducible representation of $U(p)\times U(q)$ with highest weight $\lambda$.
We denote by ${\rm Sym}(M_{p,q})$ the symmetric algebra of $M_{p,q}$.

\begin{defi}\label{defi:horn-cone}
\begin{enumerate}
\item $\horn^{\Zbb}(p,q)$ is the semigroup of $(\wedge_p^+\times\wedge_q^+)^3$ defined by the conditions:
$$
 (\lambda,\mu,\nu)\in \horn^{\Zbb}(p,q)\Longleftrightarrow \left[V_\nu\,:\,V_\lambda\otimes V_\mu\otimes {\rm Sym}(M_{p,q})\right]\neq 0.
$$
\item $\horn(p,q)$ is the convex cone of $(\tgot_{\geq 0})^3$ defined as the closure of
$\Qbb^{>0}\cdot \horn^{\Zbb}(p,q)$.
\end{enumerate}
\end{defi}

Theorem B asserts that
\begin{equation}
\Horn(p,q)=\horn(p,q)\,\bigcap \,(\Ccal_{p,q})^3.
\end{equation}

\medskip

In another article \cite{pep:horn-p-q}, we obtained a recursive description of the cones $\horn(p,q)$.
This allows us to give the following description of the holomorphic Horn cone $\Horn(2,2)$.

\begin{exam}
An element $(A,B,C)\in(\Rbb^4)^3$ belongs to $\Horn(2,2)$ if and only if the following conditions holds:

\begin{equation*}
\boxed{
\begin{array}{rcl}
a_1 \geq a_2 & > & a_3 \geq a_4\\
b_1 \geq b_2 & > & b_3 \geq b_4\\
c_1 \geq c_2 & > & c_3 \geq c_4
\end{array}
}
\end{equation*}

$$
\boxed{a_1+a_2+a_3+a_4+b_1+b_2+b_3+b_4=c_1+c_2+c_3+c_4}
$$

$$
\boxed{a_1+a_2+b_1+b_2\leq c_1+c_2}
$$

\begin{equation*}
\boxed{
\begin{array}{rcl}
a_2+b_2&\leq & c_2\\
a_2+b_1&\leq & c_1\\
a_1+b_2&\leq & c_1
\end{array}
}
\end{equation*}

\begin{equation*}
\boxed{
\begin{array}{rcl}
a_3+b_3 &\geq & c_3\\
a_3+b_4 &\geq & c_4\\
a_4+b_3 &\geq & c_4
\end{array}
}
\end{equation*}

\begin{equation*}
\boxed{
\begin{array}{rcl}
a_2+a_4+b_2+b_4 &\leq & c_1+c_4 \\
a_2+a_4+b_2+b_4 &\leq & c_2+c_3\\
a_2+a_4+b_1+b_4 &\leq & c_1+c_3\\
a_1+a_4+b_2+b_4 &\leq & c_1+c_3\\
a_2+a_4+b_2+b_3 &\leq & c_1+c_3\\
a_2+a_3+b_2+b_4 &\leq & c_1+c_3
\end{array}
}
\end{equation*}
\end{exam}

{\small


text:
\section{Consistent Systems}
We prove Theorem \ref{Theorem3.1}, the main result of this section, using a sequence of lemmas. We follow the argument presented in \cite{Hegde2019}, adapting those lemmas for our assumptions on the relaxation parameters. We also direct the reader to the original source \cite{Natterer2001}.

\begin{lemma} \label{genEig}
Let $\mathcal{H}$ be a Hilbert space and $\mathcal{K}$ be a closed subspace of $\mathcal{H}$. Let $U$ be a linear operator on $\mathcal{H}$ with the following properties:
\begin{enumerate}
    \item $U\vec{x}=\vec{x}$ for every $\vec{x}\in\mathcal{K}$,
    \item $\mathcal{K}^\perp$ is an invariant subspace for $U$ (i.e., $U(\mathcal{K}^\perp)\subseteq \mathcal{K}^\perp$),
    \item $\| U|_{\mathcal{K}^\perp}\| < 1$.
\end{enumerate}
Given a sequence $\{\vec{x}_k\}$ in $\mathcal{H}$ such that
\[
\lVert \vec{x}_k\rVert \le 1 \text{ and } \lim_{k\to \infty}\lVert U\vec{x}_k\rVert=1,
\]
it follows that
\[
\lim_{k\to\infty} (I-U)\vec{x}_k=\vec{0}.
\]
\end{lemma}

\begin{proof}
For convenience, we denote $\alpha = \|U|_{\mathcal{K}^\perp}\|$, and let $P$ be the orthogonal projection onto $\mathcal{K}^\perp$. We claim that $\|P\vec{x}_k\| \rightarrow 0$. Indeed, we have
\begin{align*}
1 &= \lim_{k\to\infty}\|U\vec{x}_k\|^2 \\
&= \lim_{k\to\infty}\|U(I - P)\vec{x}_k + UP\vec{x}_k\|^2 \\
&= \lim_{k\to\infty}\left(\|(I - P)\vec{x}_k\|^2 + \|UP\vec{x}_k\|^2\right) \\
&\leq \lim\inf\left(\|(I - P)\vec{x}_k\|^2 + \alpha^2\|P\vec{x}_k\|^2\right) \\
&= \lim\inf\left(\|\vec{x}_k\|^2 - (1 - \alpha^2)\|P\vec{x}_k\|^2\right) \\
&\leq \lim\inf\left(1 - (1 - \alpha^2)\|P\vec{x}_k\|^2\right) \\
&= 1 - (1 - \alpha^2)\lim\sup\|P\vec{x}_k\|^2 \leq 1.
\end{align*}
We therefore observe that $1 - (1 - \alpha^2)\lim\sup\|P\vec{x}_k\|^2 = 1$ so that $\lim\sup\|P\vec{x}_k\| = 0$, as desired. Hence $$\lim_{k\to\infty}(I-U)\vec{x}_k=\lim_{k\to\infty}(I-U)(P\vec{x}_k)=\vec{0}.$$

\end{proof}

\begin{lemma} \label{PGi} Fix an integer $i \in \{1,2,...,c\}$, an enumeration of the subnetworks. Suppose that $\{\vec{x}_k\}$ is a sequence in $\mathbb{C}^d$ such that
\[
\lVert\vec{x}_k \rVert \le 1 \text{ and } \lim _{k \to \infty} \lVert P_{G_i}^{\vec{\omega}} \vec{x}_k \rVert = 1.
\]
It follows that
\[ \lim _ {k \to \infty} (I-P_{G_i}^{\vec{\omega}})\vec{x}_k = \vec{0}.
\]
\end{lemma}

\begin{proof}
Let $\mathcal{K} = \left\{\vec{a}_u\,:\,u\in G_i\right\}^\perp$. The proof consists of simply verifying that $P_{G_i}^{\vec{\omega}}$ satisfies the conditions of Lemma \ref{genEig}.

As observed in Lemma \ref{contractions}, we have that $P^{\vec{\omega}}_{G_i}\vec{x} = \vec{x}$ for every $\vec{x} \in \mathcal{K}$ and that $\mathcal{K}^\perp$ is an invariant subspace for $P^{\vec{\omega}}_{G_i}$. Condition (3) of Lemma \ref{genEig} follows from the assumptions on the relaxation parameters, specifically $\|P^{\vec{\omega}}_{G_i}\vec{x}\| \leq \alpha_i\|\vec{x}\|$ for every $\vec{x} \in \mathcal{K}^\perp$.

\end{proof}

\begin{lemma}\label{PGir}
Fix an integer $i \in \{1,2,...,c\}$, an enumeration of the subnetworks. Suppose that $\{\vec{x}_k\}$ is a sequence in $\mathbb{C}^d$ such that
\[
\|\vec{x}_k\| \leq 1 \text{ and } \lim_{k \rightarrow\infty}\|P^{\vec{\omega}}_{G_i,r}\vec{x}_k\| = 1.
\]
It follows that
\[
\lim_{k\rightarrow\infty}(I - P^{\vec{\omega}}_{G_i,r})\vec{x}_k = \vec{0}.
\]
\end{lemma}

\begin{proof}
Note that
\begin{align*}
(I - P^{\vec{\omega}}_{G_i}P^{\vec{\omega}}_{g_i}\cdots P^{\vec{\omega}}_r)\vec{x}_k = (I - P^{\vec{\omega}}_{g_i}\cdots P^{\vec{\omega}}_r)\vec{x}_k + (I - P^{\vec{\omega}}_{G_i})P^{\vec{\omega}}_{g_i}\cdots P^{\vec{\omega}}_r\vec{x}_k.
\end{align*}
Since $\|P^{\vec{\omega}}_{g_i}\cdots P^{\vec{\omega}}_r\vec{x}_k\| \leq 1$, we have $(I - P^{\vec{\omega}}_{G_i})P^{\vec{\omega}}_{g_i}\cdots P^{\vec{\omega}}_r\vec{x}_k \rightarrow \vec{0}$ from Lemma \ref{PGi}. Hence it suffices to show $(I - P^{\vec{\omega}}_{g_i}\cdots P^{\vec{\omega}}_r)\vec{x}_k \rightarrow \vec{0}$. Consider the path from $r$ to $g_i$, say $r = u_1 \rightarrow u_2 \rightarrow ... \rightarrow u_n = g_i$, and let $\mathcal{K} = \{\vec{a}_{u_j}\,:\,1\leq j\leq n\}^\perp$. We check Lemma \ref{genEig}. Conditions (1) and (2) are straightforward to check, so we only show condition (3). Assume by way of contradiction that $\|P^{\vec{\omega}}_{g_i}\cdots P^{\vec{\omega}}_r|_{\mathcal{K}^\perp}\| = 1$. By continuity and compactness, there then exists a unit vector $\vec{x} \in \mathcal{K}^\perp$ such that $\|P^{\vec{\omega}}_{g_i}\cdots P^{\vec{\omega}}_r\vec{x}\| = 1$. From this observation and Lemma \ref{contraction}, it follows that $\vec{x} \in \mathcal{K}$ so that $\vec{x} = \vec{0}$, which is a contradiction.

\end{proof}

\begin{lemma} \label{Lemma3.6}
Suppose that $\{\vec{x}_k\}$ is a sequence in $\mathbb{C}^d$ such that
\[
\lVert \vec{x}_k \rVert \le 1 \text{ and } \lim_{k\to \infty} \lVert P^{\vec{\omega}}\vec{x}_k \rVert = 1.
\]
It follows that
\[
\lim_{k \to \infty} (I - P^{\vec{\omega}})\vec{x}_k = \vec{0}.
\]
\end{lemma}

\begin{proof}
Recalling Equation \ref{Pomega}, we note that
\[
(I-P^{\vec{\omega}})\vec{x}_k = \sum_{i=1}^c w(r,g_i)(I-P^{\vec{\omega}}_{G_i,r})\vec{x}_k.
\]
Therefore it suffices to show that the hypotheses of Lemma \ref{PGir} are satisfied. From Lemma \ref{contractions}, we have $\lVert P_{G_i,r}^{\vec{\omega}}\vec{x}_k \rVert \le 1$ and, thus,
\[
1 = \lim_{k\to\infty}\lVert P^{\vec{\omega}}\vec{x}_k \rVert \le \lim\inf\sum _{i=1}^c w(r,g_i) \lVert P_{G_i,r}^{\vec{\omega}}\vec{x}_k \rVert \le 1.
\]
It follows that, for each $i \in \{1,2,...,c\}$,
\[
\lim _{k \to \infty} \lVert P_{G_i,r}^{\vec{\omega}}\vec{x}_k \rVert = 1.
\]

\end{proof}

\begin{proposition}\label{inconsistentlemma1}
If $\|P^{\vec{\omega}}\vec{x}\| = \|\vec{x}\|$, then $\vec{x} \in \mathcal{R}(A^*)^\perp$.
\end{proposition}

\begin{proof}
Note that
\[
\|\vec{x}\| = \left\|\sum_i w(r,g_i)P^{\vec{\omega}}_{G_i}P^{\vec{\omega}}_{g_i}...P^{\vec{\omega}}_r\vec{x}\right\| \leq \sum_i w(r,g_i)\|P^{\vec{\omega}}_{G_i}P^{\vec{\omega}}_{g_i}...P^{\vec{\omega}}_r\vec{x}\| \leq \|\vec{x}\|.
\]
Therefore it follows that $\|P^{\vec{\omega}}_{G_i}P^{\vec{\omega}}_{g_i}...P^{\vec{\omega}}_r\vec{x}\| = \|\vec{x}\|$ for all $i$. Hence $\|P^{\vec{\omega}}_r\vec{x}\| = \|\vec{x}\|$ which, by Lemma \ref{contraction}, implies that $\vec{x} \in \mathcal{N}(S_r)$ and $P^{\vec{\omega}}_r\vec{x} = \vec{x}$. We then inductively find $\vec{x} \in \mathcal{N}(S_{g_i})\cap ...\cap\mathcal{N}(S_r)$,  $P^{\vec{\omega}}_{g_i}\vec{x} = ... = P^{\vec{\omega}}_r\vec{x} = \vec{x}$, and $\|P^{\vec{\omega}}_{G_i}\vec{x}\| = \|\vec{x}\|$. Now let $P$ be the orthogonal projection onto $\{\vec{a}_u\,:\,u\in G_i\}^\perp$. Then, as argued in Lemma \ref{contractions}, we find
\begin{align*}
\|\vec{x}\|^2 &= \|P^{\vec{\omega}}_{G_i}\vec{x}\|^2 \\
&= \|P^{\vec{\omega}}_{G_i}P\vec{x} + P^{\vec{\omega}}_{G_i}(I - P)\vec{x}\|^2 \\
&= \|P\vec{x} + P^{\vec{\omega}}_{G_i}(I - P)\vec{x}\|^2 \\
&= \|P\vec{x}\|^2 + \|P^{\vec{\omega}}_{G_i}(I - P)\vec{x}\|^2 \\
&\leq \|P\vec{x}\|^2 + \alpha_i^2\|(I - P)\vec{x}\|^2 \\
&\leq \|\vec{x}\|^2.
\end{align*}
Therefore $P\vec{x} = \vec{x}$ so that $\vec{x} \in \mathcal{N}(S_u)$ for every $u \in G_i$, which concludes the proof.

\end{proof}

The next lemma is an immediate consequence of Proposition \ref{inconsistentlemma1}.

\begin{lemma} \label{Lemma3.7}
Let $\mathcal{V}$ be the collection of all of the vertices in the network. Then
\[
\mathcal{N}(I-P^{\vec{\omega}}) = \bigcap_{v \in \mathcal{V}} \mathcal{N}(I-P_v).
\]
\end{lemma}

\begin{lemma}\label{Lemma3.8}
Let $\mathcal{V}$ be the collection of all of the vertices in the network. As $k \to \infty$, $(P^{\vec{\omega}})^k$ converges strongly to the orthogonal projection onto
\[
\bigcap_{v \in \mathcal{V}} \mathcal{N}(I-P_v) = \mathcal{N} (A).
\]
\end{lemma}

\begin{proof}
Using Lemmas \ref{Lemma3.6} and \ref{Lemma3.7} with the observation that $\mathcal{N}(S_v) = \mathcal{N}(I - P_v)$, the proof is identical to the proof of Lemma 3.5 in \cite{Natterer2001}.

\end{proof}

\begin{theorem} \label{Theorem3.1}
If the system of equations $A\vec{x} = \vec{b}$ is consistent, then the sequence of estimates $\{\vec{x}^{(n)}\}$ from the distributed Kaczmarz algorithm given by the recursion
\[
\vec{x}^{(n+1)} = Q^{\vec{\omega}}\vec{x}^{(n)} = \sum _ {\ell \in \mathcal{L}} w(r,\ell) Q^{\vec{\omega}}_\ell\cdots Q^{\vec{\omega}}_r \vec{x}^{(n)},
\]
with admissible relaxation parameters, converges to the solution of minimal norm provided that the initial estimate $\vec{x}^{(0)} \in \mathcal{R}(A^*)$.
\end{theorem}

\begin{proof}
Let $\vec{x}$ be a solution to the system of equations, and let $v$ be any vertex in the network. Then, from Equation \ref{QandP}, we have
\[
\vec{x} = Q_v^{\vec{\omega}}\vec{x} = P^{\vec{\omega}}_v\vec{x} + \omega_v\vec{h}_v.
\]
Let $\vec{y}$ be an arbitrary vector. From Equation \ref{QandP}, again, we find
\[
Q_v^{\vec{\omega}}\vec{y} = P_v^{\vec{\omega}}\vec{y} + \omega_v\vec{h}_v = P_v^{\vec{\omega}}(\vec{y} - \vec{x}) + \vec{x}.
\]
It then immediately follows from this last identity that
\begin{align*}
Q^{\vec{\omega}}\vec{y} &= \sum_{\ell \in \mathcal{L}}w(r,\ell)Q^{\vec{\omega}}_{\ell}\cdots Q^{\vec{\omega}}_r\vec{y} \\
&= \left(\sum_{\ell \in \mathcal{L}}w(r,\ell)P^{\vec{\omega}}_\ell\cdots P^{\vec{\omega}}_r(\vec{y} - \vec{x})\right) + \vec{x} \\
&= P^{\vec{\omega}}(\vec{y} - \vec{x}) + \vec{x}.
\end{align*}
Further, for every positive integer $k$,
\[
(Q^{\vec{\omega}})^k\vec{y} = (P^{\vec{\omega}})^k(\vec{y} - \vec{x}) + \vec{x}.
\]
Then, from Lemma \ref{Lemma3.8}, we have that
\[
(Q^{\vec{\omega}})^k \vec{y} \to T(\vec{y}-\vec{x}) + \vec{x}
\]
where $T$ is the orthogonal projection onto $\mathcal{N}(A)$. Now, if $\vec{y} = \vec{x}^{(0)} \in \mathcal{R}(A^*)$, then $T(\vec{y} - \vec{x}) + \vec{x} = (I - T)\vec{x}$ is the solution of minimal norm, which concludes the proof.

\end{proof}

config: arxiv


TITLE: If the annual inflation rate is 3.2%, how can I calculate the daily inflation rate??
QUESTION [1 upvotes]: I know that I cannot just take 3.2% and divide by 365.  Sorry I'm a math noob...

REPLY [0 votes]: In general:
Let the daily rate be $r$, $y$ the yearly rate, and $d$ the number of trading in a year. Then you have:
$$
\begin{align}
(1+r)^d &= (1+y) \\
 1+r &= (1+y)^{\frac{1}{d}}  \\
 1+r &= (1+y)^{\frac{1}{d}} \\
 r & =  (1+y)^{\frac{1}{d}} -1\\
\end{align}
$$
In your case, $r = (1+0.032)^{\frac{1}{252}} - 1 = 0.0001250025227173$


TITLE: Whence "Durchschnitt" and "Vereinigung"?
QUESTION [10 upvotes]: Today the set-theoretic operations of intersection $\cap$ [German: Durchschnitt] and union $\cup$ [German: Vereinigung] are standard.
The modern notations are present in the first edition of van der Waerden's Moderne Algebra (1930). However, the notation is mostly missing from Steinitz' Algebraische Theorie der Körper (1910), which was a precursor of van der Waerden. Instead of Menge (for set), Steinitz uses System. The term Durchschnitt is present in Steinitz, but the word Vereinigung and the symbols $\cap,\cup$ are missing.
Has anything been written on the history of set-theoretic language in algebra?

REPLY [15 votes]: An extensive discussion of the origin of "Menge" is given in Earliest Known Uses of Some of the Words of Mathematics (scroll down to "Set and Set Theory"). Cantor's (1880) Über unendliche linear Punktmannigfaltigkeiten is one of the earliest uses. It contains the notation $\{\cdots\}$ for a set and introduces the term "Vereinigung" for the union (symbol ${\cal M}$) and Durchschnitt for the intersection (symbol ${\cal D}$)

Schröder (1877) had previously used the notation $+$ and $\times$ for union and intersection. This was changed into the presently used symbols $\cup$ and $\cap$ by Peano (1888) to avoid confusion with addition and multiplication in algebra.
Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann (1888).

In this brief work by Schröder (37 pages) the mathematical logic is developed that forms the introduction of the present book. I found it useful to replace the logical symbols $\times,+,A_1,0,1$ used by Schröder by the symbols $\cap,\cup,-A$, ⚪, ⚫ in order to avoid a possible confusion between symbols from logic and from mathematics (a possible confusion noted by Schröder himself). I also introduced the logical symbols $\lt$ and $\gt$, although not strictly necessary...


TITLE: Pentagram Geometry
QUESTION [2 upvotes]: Consider a regular pentagon with vertices (in clockwise order) $A,B,C,D,E$. Let $A '$ be the point of intersection of $BD$
and $CE$, let $B '$ be the point of intersection of $CE$ and $DA$ and so on. If the triangle $AC 'D '$ has area $1$, what is the area of
pentagon $A 'B 'C 'D 'E '$?
I've found a few angles and lengths of sides using trig, but I can't see an answer in sight. Can anyone come with an answer with proof. Thanks.

REPLY [3 votes]: In a regular pentagon, the ratio between the length of a diagonal and the length of a side is $\frac{1+\sqrt{5}}{2}=\varphi$. It follows that, if the area of the darker triangle in the picture is $1$, the area of $BED$ is $\varphi^4$ and the area of the dotted triangle is $\varphi^2$.
By a straightforward decomposition, the area of the smaller pentagon is:
$$ \varphi^4 - \varphi^2 - 2 = (3\varphi+2)-(\varphi+1)-2 = 2\varphi-1=\color{red}{\sqrt{5}}.$$


TITLE: Let $f:\mathbb{R} \to \mathbb{R}_l$ be a continuous function then what is $f$?
QUESTION [1 upvotes]: Let $f:\mathbb{R} \to \mathbb{R}_l$ be a continuous function then what is $f$?

My conjecture is that $f$ will be all forms of increasing continuous monotone functions.
Let us pick a basis element of the form $[a,b)$ such that $f^{-1}[a,b)$ will be $[f^{-1}(a),f^{-1}b)$ .
Now, $[f^{-1}(a),f^{-1}(b))$ is an open set in $\mathbb{R}$ [as $\mathbb{R}_l$ is finer than $\mathbb{R}$].
I think this will hold.
The basis of $\mathbb{R}_l$ is of the form $[a,b)$[It is the lower limit topology on $\mathbb{R}$]
Will any continuous monotone function satisfying the above condition?

REPLY [1 votes]: Assume that $f:\mathbb{R}\to\mathbb{R}_l$ is continuous and takes (at least) two values $a$ and $b$ with $a<b.$ This implies that $f^{-1}[a,b)$ and $f^{-1}[b,\infty)$ are both nonempty.
Now note that $\{ (-\infty,a), [a,b), [b,\infty) \}$ is a partition of open sets in $\mathbb{R}_l.$ The properties of $f$ then implies that $\{ f^{-1}(-\infty,a), f^{-1}[a,b), f^{-1}[b,\infty) \}$ is a partition of open sets in $\mathbb{R}.$
But since $\mathbb{R}$ is connected, the only possibility is then that exactly one of these three sets equals $\mathbb{R}$ and that the other ones are empty. That contradicts that $f^{-1}[a,b)$ and $f^{-1}[b,\infty)$ are both nonempty.
Thus $f$ can only take one value.