We compute the $(1,1)$-Aeppli cohomology of compact simply-connected Lie groups. From this, we deduce that the Bismut flat metrics on the compact Bismut flat manifolds with finite fundamental group are globally stable for the pluriclosed flow. This prevents the existence of non-flat homogeneous Bismut Hermitian Einstein (hence also pluriclosed Calabi–Yau with torsion) metrics on C-spaces.

§ INTRODUCTION

Hermitian geometry lies at the intersection of complex and Riemannian geometry.
Indeed, a Hermitian manifold $(M,J,g)$ is endowed with a complex structure $J$ and a compatible metric $g$, i. e. such that $g$ is $J$-invariant and $\omega:=g(J\cdot,\cdot)$ is a non-degenerate $(1,1)$-form.
When the fundamental 2-form $\omega$ is closed (hence symplectic) the manifold is called Kähler.
There are several reasons to study Hermitian non-Kähler geometry; we distinguish two of them which are central to our dissertation:
to investigate non-Kähler Calabi–Yau geometries arising from theoretical physics, and completing the classification of complex surfaces.

Several natural connections have been introduced to deal with the non-Kähler case.
As a matter of fact, on a Hermitian manifold $(M,J,g)$ the Levi–Civita connection $\Na^{LC}$ preserves the complex structure, i. e. $\Na^{LC}J=0$, if and only if the metric is Kähler.
Henceforth, one considers other connections, possibly with torsion, that preserve the Hermitian structure.
Precisely, these connections $\Na$ satisfy $\Na J=\Na g=0$ and are called Hermitian connections.
For what concerns this article, we focus on the Bismut connection $\Na^B$ (also known as Strominger or Strominger–Bismut connection), since it appears in the context of classification of complex surfaces through the pluriclosed flow <cit.>, and it is used in describing models of heterotic string theory <cit.> like Calabi–Yau with torsion geometries.

The Bismut connection first appeared in a work of Bismut <cit.> in the context of non-Kähler index theory, and it is defined on a Hermitian manifold $(M,J,g)$ with respect to the Levi–Civita connection as

\begin{equation*}
\end{equation*}

for $x,y,z$ vector fields on $M$.
The crucial peculiarity of the Bismut connection is that it has a skew-symmetric torsion chosen to adapt to the Hermitian context.
It is indeed the only Hermitian connection with skew-symmetric torsion, which is given by

\begin{equation*}%\label{eq: int torsion}
g\left(T^B(\cdot,\cdot),\cdot\right) = d\omega(J\cdot,J\cdot,J\cdot) = -Jd\omega.
\end{equation*}

When the torsion of the Bismut connection is closed, that is $dJd\omega=0$, the metric is said pluriclosed (or strong Kähler with torsion, SKT in short).

Let $(M,J,g)$ be a Hermitian manifold.
The Hermitian metric $g$ is said to be Calabi–Yau with torsion (CYT in short) if the associated Bismut–Ricci tensor vanishes, i. e. $Ric^B(g)=0$, where $Ric^B(g)$ is given by tracing the Bismut curvature tensor $R^B(g)$ in the endomorphism components.
Calabi–Yau with torsion geometry plays a role in Physics after the works of Strominger <cit.> and Hull <cit.>.
Moreover, CYT structures on non-Kähler manifolds recently attracted attention as models for string compactifications, see e.g. <cit.>.

Being a Hermitian connection, $\Na^B$ determines a representative of the first Chern class $c_1(M)$ in de Rham cohomology via its Ricci form.
Therefore, the existence of Calabi–Yau with torsion structures is obstructed by having vanishing first Chern class.
It turns out <cit.> that every compact complex homogeneous space with vanishing first Chern class, after an appropriate deformation of the complex structure, admits a homogeneous CYT metric, provided that the complex homogeneous space also has an invariant volume form.
Then, even if the CYT metrics are generally not unique, in <cit.> it was proved the uniquenes among the homogeneous metrics on class $\mathcal{C}$ manifolds in the sense of <cit.> (with the exception of some specific cases).
Finally, these geometries also arise as static points of a parabolic evolution of Hermitian metrics.

In <cit.> the authors introduced an evolution equation for Hermitian metrics that preserves the pluriclosed condition and is hence called pluriclosed flow.
It evolves a starting pluriclosed metric $\omega_0$ as

\begin{equation*}
\begin{cases}
\frac{\p }{\p t}\omega (t) = - \left(Ric^B(\omega(t))\right)^{1,1},\\
\omega(0)= \omega_0, %, & \p\bar\p\omega_0=0.
\end{cases}
\end{equation*}

where $\left(\cdot\right)^{1,1}$ denotes the $(1,1)$-component of the form, and it preserves the pluriclosed condition since $\p\bar\p\left(Ric^B(\omega(t))\right)^{1,1}=0 $ by (<ref>).
The pluriclosed flow is particularly well-suited for compact complex surfaces since there is a pluriclosed representative in any conformal class of Hermitian metrics <cit.>.
As a matter of fact, it has been introduced as an analytic tool to understand the topology and geometry of compact complex surfaces.
In particular, it is conjectured <cit.> that on a minimal class VII surface with $b_2>0$ it should detect a rational curve, leading to a parabolic proof of the Global Spherical Shell conjecture <cit.> for $b_2=1$.
In <cit.> further analysis of the conjectural behavior of the pluriclosed flow on class VII surfaces has been made; see also <cit.> and the references therein for a survey on its possible application to the classification of complex surfaces.

Up to now, there is no definitive understanding of the behaviour of the pluriclosed flow and the singularities it may encounter.
In fact, a complete description of the long-time evolution of this flow has been reached only in the locally homogeneous setting for complex surfaces <cit.>, and for metrics of non-positive, flat, or negative holomorphic bisectional curvature <cit.>.
In <cit.>, <cit.> and <cit.> it has been developed an analysis of the behavior of the pluriclosed flow acting on invariant metrics on almost-abelian Lie groups, Hopf manifolds, and Oeljeklaus–Toma manifolds respectively.
Finally, long-time existence and convergence of the pluriclosed flow have been achieved, under a natural cohomological ansatz, on Bismut flat manifolds  <cit.>, which are complex manifolds equipped with a Hermitian metric whose Bismut curvature tensor vanishes, i. e. $R^B(g)\equiv 0$.
More precisely, given a compact Bismut flat manifold $(M,J, \omega_{BF})$, for any pluriclosed metric  $\omega_0$ such that

\begin{equation}\tag{Tor-class}\label{eq: cohomological condition}
[\p\omega_0] = [\p\omega_{BF} ] \in H^{2,1}_{\bar\p}(M,J),
\end{equation}

Theorem 1.2 of <cit.> (stated here as Theorem <ref>) ensures that the solution of the pluriclosed flow with initial data $\omega_0$ exists on $[0, \infty)$ and converges to a Bismut flat metric $\omega_\infty$.

The cohomological condition (<ref>) is natural since along the pluriclosed flow, the torsion of the Bismut connection satisfies $\p\omega_t = \p\omega_0 + \bar\p\beta(t)$, where $\beta$ evolves as

\begin{equation*}\begin{cases}
\frac{\p}{\p t}\beta = - \left(Ric^{B}(\omega_t)\right)^{2,0},\\
\beta_{|_{t=0}} = 0.
\end{cases}\end{equation*}

Moreover, it is needed in order to compare the Bismut connection of the evolving metric with the Bimsut connection of the reference Bismut flat metric on the same holomorphic Courant algebroid, and to prove that the former converges to the latter.
This is why from the argument in <cit.> one obtains that the flow is converging to a Bismut flat metric, but there is no evident relation between the endpoint of the flow $\omega_\infty$ and the reference metric $\omega_{BF}$.
However, we will see in Theorem <ref> that, a posteriori, we can obtain a better understanding of the relation between $\omega_\infty$ and $\omega_{BF}$.

It is evident that Theorem <ref> reduces the problem of understanding the long-time behavior of the pluriclosed flow on Bismut flat manifolds to check whether (<ref>) is satisfied by any pluriclosed metric.
In particular, if that is the case, then the Bismut flat metrics are globally-stable static points of the pluriclosed flow.
As a consequence, Theorem <ref> is extremely powerful when combined with the knowledge of the cohomology of the manifold.
For example, with Theorem <ref> at hand, the stability of the flat metrics on the torus for the pluriclosed flow (achieved in <cit.>) can be obtained just by using the $\p\bar\p$-Lemma to check (<ref>).
Moreover, we will see in the following that for compact Samelson spaces (see Definition <ref>) the knowledge of the dimension of the $(1,1)$-Aeppli cohomology group is enough to derive the global stability of the Bismut flat metrics.
This idea was used already for the Hopf surface, the Calabi–Eckmann threefold, and all the compact simple Lie groups of rank two, respectively in <cit.> and <cit.>.
Therefore, in this article, we use a refined version of the argument from <cit.>, which applies to a large class of Bismut flat manifolds, to compute their $(1,1)$-Aeppli cohomology, and prove our main result.
In Remark <ref> we explain the difference with the previous more common approach to the problem.
[Theorem <ref>]
Let $(M,J,\omega_{BF})$ be a compact Bismut flat manifold with finite fundamental group.
Then for any pluriclosed metric $\omega_0$ on $\left(M,J\right)$ the solution to the pluriclosed flow with initial data $\omega_0$ exists on $[0, \infty)$ and converges to a Bismut flat metric $\omega_{\infty}$.

For compact Bismut flat manifolds in general, the problem of global stability remains open.
In Example <ref> we explicitly construct a compact Bismut flat manifold with infinite fundamental group to highlight the main difficulties in extending our argument.
As far as we know, that is also the first example of a compact complex manifold $(M,J)$ equipped with a family of Bismut flat metrics $\left\{\omega_u\right\}_{u\in\C}$ which are diffeomorphic one to each other but have different torsion classes, i. e. $[\p\omega_u]\neq[\p\omega_{u'}]$ in $H^{2,1}_{\bar\p}(M,J)$.

As for the Kähler–Ricci flow and the Chern–Ricci flow <cit.>, it is expected that also the pluriclosed flow should converge to its static points (or to its solitons), if any.
The static points of the pluriclosed flow are pluriclosed metrics $\omega$ which satisfy

\begin{equation}\label{1,1 Bismut Einstein}
\left(Ric^B(\omega)\right)^{1,1} = \lambda \omega \, ,\quad\lambda\in\R.
\end{equation}

It turns out <cit.>, that (<ref>) forces either the metric to be Kähler or the Einstein constant $\lambda$ to vanish.
Thus, the non-Kähler static points of the pluriclosed flow are pluriclosed metric such that the $(1,1)$-component of the Bismut–Ricci form vanishes, and they are called Bismut Hermitian Einstein metrics in the literature <cit.>.
Up to now, the only known Bismut Hermitian Einstein metrics are the Bismut flat metrics and (trivially) the Kähler Einstein metrics.
Therefore, there is interest in finding non-Kähler non-flat examples, and this also motivates the search for metrics that are both pluriclosed and Calabi–Yau with torsion.

The first natural attempt is to look for homogeneous examples, also because of Theorem 3 in <cit.> about the existence of homogeneous CYT structure.
Therefore, suppose $M$ is a C-space, which was defined by Wang <cit.> as a compact complex manifold admitting a transitive action by a compact Lie group $G$ of biholomorphisms and finite fundamental group.
Then by Theorem 6.1 in <cit.> and Theorem 3.2 in <cit.> $M$ is forced to be a Bismut flat manifold, see Section <ref> for details.
However, on Bismut flat manifolds the existence of non-flat Bismut Hermitian Einstein metrics is in contrast with the global stability of the Bismut flat metrics for the pluriclosed flow.
Thus, as a consequence of Theorem <ref> we get the following result generalizing Corollary 5.1 in <cit.>.
[Theorem <ref>]
Let $(M,J)$ be a C-space and suppose that $g$ is a left-invariant Bismut Hermitian Einstein metric on it.
Then, up to finite cover, $M$ is a Lie group and $g$ is a bi-invariant metric.
In particular, $g$ is Bismut flat.

The paper is organized as follows.
In Section <ref>, we present the characterizzation of Bismut flat manifolds following <cit.>; we thus recall the construction of the complex structures on them as in <cit.>.
Section <ref> is devoted to the computation of the $(1,1)$-Aeppli cohomology of the universal cover of the compact Bismut flat manifolds with finite fundamental group.
Then, in Section <ref>, we use this knowledge to prove the global stability of the pluriclosed flow on these manifolds.
Finally, in Section <ref>, we use this stability property to prove the non-existence of non-flat homogeneous Bismut Hermitian Einstein metrics on C-spaces.

§ ACKNOWLEDGEMENTS

I would like to thank Francesco Pediconi and Jonas Stelzig for the help provided during this work and for
useful suggestions on the topics of this paper.
I am also grateful to Daniele Angella for his constant support and encouragement.

§ COMPLEX STRUCTURE ON BISMUT FLAT MANIFOLDS

By an explicit construction (briefly described in Section <ref>), Samelson showed <cit.> that any even-dimensional compact Lie group admits a left-invariant complex structure compatible with the bi-invariant metric coming from the Killing form.
Moreover, Alexandrov and Ivanov <cit.> proved that any even dimensional connected Lie group equipped with a bi-invariant metric $g$ and a left-invariant complex structure which is compatible with $g$ is Bismut flat.
Afterward, Wang, Yang, and Zheng <cit.> showed that up to taking the universal cover, these are the only existing compact Bismut flat manifolds.
In other words, simply-connected compact Bismut flat manifolds have been characterized as Samelson spaces, whose definition is as follows.
A Samelson space is a Hermitian manifold $(G, g, J)$, where $G$ is a connected and simply-connected, even-dimensional Lie group, $g$ a bi-invariant metric on $G$, and $J$ a left-invariant complex structure on $G$ that is compatible with $g$.

By Milnor's Lemma <cit.>, a simply-connected Lie group $G'$ with a bi-invariant metric must be the product of a compact semisimple Lie group with an additive vector group.

Let $G$ be a simply-connected Lie group with a bi-invariant metric $\langle \cdot,\cdot \rangle$. Then $G$ is isomorphic and isometric to the product $G_1 \times \cdots \times G_r \times {\mathbb R}^k$ where each $G_i$ is a simply-connected compact simple Lie group and ${\mathbb R}^k$ is the additive vector group with the flat metric.

Taking quotients of these manifolds one obtains the local Samelson spaces.
Let $(G', g, J)$ be a Samelson space, where $G' = G \times \R^k$ with $G$ semisimple.
Let $\rho: \Z^k \rightarrow I(G)$ be a homomorphism into the isometry group of $G$.
Then $\Gamma_\rho \sim \Z^k$ acts on $G \times \R^k$ by $\gamma(x, y) = (\rho(\gamma)(x), y + \gamma)$ as isometries, and it acts freely and properly discontinuously, so one gets a compact quotient $M_\rho = (G \times \R^k)/\Gamma_\rho$.
If the complex structure of $G'$ is preserved by $\Gamma_\rho$, then it descends down to $M_\rho$ and makes it a complex manifold.
In this case, the compact Hermitian manifold $M_\rho$ is called local Samelson space.

Such Hermitian manifolds are Bismut flat since their universal cover is so.
Moreover, they are all the compact Bismut flat manifolds up to finite cover.
[Theorem 1 in <cit.>]
Let $(M,J, g)$ be a compact Hermitian manifold whose Bismut connection is flat.
Then there exists a finite cover $M'$ of $M$ such that $M'$ is a local Samelson space $M_\rho$ defined as above.
Also, $M_\rho$ is diffeomorphic to $G \times \left(\S^1\right)^k$.

In <cit.> Pittie gave a complete description of the moduli of left-invariant complex structures on even-dimensional compact Lie groups, proving that they all come from Samelson's construction in <cit.>, namely, from a choice of a maximal torus, a complex structure on the Lie algebra of the torus, and a choice of positive roots for the Cartan decomposition.
Let us now recall this construction in more detail.

§.§ Cartan decomposition

Let $G$ be an even-dimensional connected Lie group and denote by ${\mathfrak g}$ the Lie algebra of $G$ and by ${\mathfrak g}^\C $ its complexification.
We recall that the Killing form of $G$ is given by

\begin{equation*}\label{eq: killing}
B(X,Y) := \tr \left(\mbox{ad}_X \circ\mbox{ad}_Y\right),
\end{equation*}

where $\mbox{ad}_X(Y)=[X,Y]$ for $X,Y\in\g$.
This is a symmetric bilinear form, and thanks to the Cartan criterion <cit.> it is non-degenerate if and only if $\g$ is semisimple.
If this is the case, its opposite is a metric which we indicate with $\left< \cdot,\cdot\right>:=-B(\cdot,\cdot)$.
Moreover, if $G$ is a simple Lie group then any invariant symmetric bilinear form on it is a scalar multiple of the Killing form.

Given $K$ a maximal torus of $G$, we denote by $\k$ its Lie algebra.
The action of $ $̨ on $\g$ is simultaneously diagonalizable <cit.>, and the eigenvalues of $\mbox{ad}\,\k$ on $$ are called {\em roots} of $$ with respect to $$̨.
Then one has the $\mbox{ad}(K)$-invariant roots decomposition

\begin{equation*}
{\mathfrak g}^\C = {\mathfrak k}^\C \oplus \sum_{\alpha\in R} {\mathfrak g}_{\alpha}  ,
\end{equation*}

Where ${\mathfrak k}^\C $ denotes the complexification of ${\mathfrak k}$, $R$ is the roots space and

$$ {\mathfrak g}_{\alpha} := \left\{v \in {\mathfrak g}^\C \; \big| \; [H, v] = \alpha(H)v \; \forall\; H \in {\mathfrak k}\right\}.$$

The algebra $\k^\C$ is also known as Cartan subalgebra and it is unique, up to conjugation by an automorphism of $\g$ <cit.>.
Moreover, the root spaces $\g_\alpha$ are one dimensional, and it holds <cit.>

\begin{equation}\label{eq: B orthogonal}
\langle\g_\alpha,\g_\beta\rangle = 0 \quad \text{if } \alpha+\beta\neq0.
\end{equation}

The bracket relations between root spaces can be easily computed and are

\begin{equation}\label{eq: roots brackets relations}
[\g_\alpha,\g_\beta] \begin{cases}
=\g_{\alpha+\beta} & \text{if }\alpha+\beta \text{ is a non-zero-root} ;\\
=0 & \text{if } \alpha+\beta  \text{ is not a root} ;\\
\subset \k & \text{if } \alpha+\beta=0 .
\end{cases}
\end{equation}

Furthermore, if $\alpha$ is a root, then also $-\alpha$ is a root, and $\g_{-\alpha}=\bar{\g_\alpha}$.
Therefore, one fix an element $H\in\k$ such that $α(H)≠0$ for every $α∈R$ (which exists since $R$ is finite), and says that a root $α$ is {\em positive} if $α(H)>0$.
A choice of the positive root system leads to the {\em Cartan decomposition}, namely $^$ decomposes as
\begin{equation}\label{eq: cartan decomposition}
{\mathfrak g}^\C = {\mathfrak k}^\C \oplus \sum_{\alpha\in R^+} {\mathfrak g}_{\alpha} \oplus {\mathfrak g}_{-\alpha} ,
\end{equation}
where $R^+$ is the space of positive roots.

\subsection{Samelson construction}\label{subsec: sam constr}

%In the following let $G$ be an even dimensional connected Lie group equipped with a bi-invariant metric $\langle \cdot ,\cdot \rangle$ and denote by ${\mathfrak g}$ the Lie algebra of $G$ and by ${\mathfrak g}^\C $ its complexification. $\langle \cdot ,\cdot \rangle$ will also denote the induced inner product on ${\mathfrak g}$. %We have $G=G_1 \times \cdots \times G_r\times {\mathbb R}^k$ as above. Since any bi-invariant metric on a compact simple Lie group is a constant multiple of the Killing form, the bi-invariant metric on each $G_i$ is unique up to constant.

The left-invariant almost-complex structures $J$ on $G$ are uniquely determined by their restriction to $$, which we still indicate with the same symbol.
Hence we look at them as linear maps $J: 𝔤  →𝔤$ such that $J^2=-𝕀_$.
Equivalently, an almost-complex structure on $G$ is determined by the subspace $⊂^$ of $(1,0)$-vectors, which clearly satisfies
$ ∩=0,  and  ⊕= ^. $
Finally, the integrability condition becomes $[,]⊂$, and thus, the complex structures on $G$ are in one-to-one correspondence with the complex Lie subalgebras $⊂^$, such that
$$[\s,\s]\subset\s\,,\quad \s \cap \g=0\,, \quad\text{and}\quad \s \oplus \bar\s = \g^\C .$$
Such subspaces are called {\em Samelson subalgebras} of $^$ \cite{MR994129}.

Samelson \cite{MR59287} first constructed examples of left-invariant complex structures on compact Lie groups as follows.
Consider the Cartan decomposition \eqref{eq: cartan decomposition}.
Since $(G)$ is even, the abelian Lie algebra $𝔨$ is even-dimensional as well. Thus it is possible to choose a complex structure on $$̨.
As before, this is equivalent to choosing a complex subalgebra ${\mathfrak a} \subset {\mathfrak k}^\C$ such that

$$[ {\mathfrak a}, {\mathfrak a}]\subset\mathfrak{a}\,,\quad {\mathfrak a} \cap {\mathfrak k}=0\,,\quad\text{and}\quad {\mathfrak a} \oplus \overline{{\mathfrak a}} = {\mathfrak k}^\C .$$

Now one could simply take

\begin{equation*}
{\mathfrak s} = {\mathfrak a} \oplus \sum_{\alpha\in R^+} {\mathfrak g}_{\alpha}
\end{equation*}

to be a Samelson subalgebra of $\g$.
Thanks to the relation in (<ref>), the positive root spaces are orthogonal with respect to $\langle\cdot,\cdot\rangle$.
Therefore, for a suitable choice of Samelson subalgebra on the torus the above complex structure is Hermitian together with the Killing metric.

§.§ Bi-invariant metrics compatible with Samelson complex structures

Let us suppose that $G=G_1\times\cdots\times G_s$ is a semisimple Lie group.
The Lie algebra of the maximal torus splits as

\[
\k=\k_1\oplus\cdots\oplus\k_s
\]

with respect to the Killing metric.
We say that a Samelson complex structure $J$ is reducible if there exist indexes $i_1,\ldots,i_p$ with $1\leq p<s$ such that

\[ J(\k_{i_1}\oplus\cdots\oplus\k_{i_p})\subset\k_{i_1}\oplus\cdots\oplus\k_{i_p}.\]

It is irreducible if it is not reducible.
Notice that a semisimple Lie group $G$ equipped with a Samelson complex structure $J$ can be always written as a product of Lie groups $G=G_1\times\cdots\times G_s$ such that the restriction of $J$ on any of the $G_i$ is an irreducible Samelson complex structure.

Consider a bi-invariant metric $g$ on $G$ and the decomposition of the Lie algebra in simple pieces with respect to the Killing form

$$\g = \g_1 \oplus \cdots \oplus \g_k.$$

We claim that the $\g_i$ are $g$-orthogonal one to each other.
As a matter of fact, consider the $1$-form on $G$ given by

$$ q = g (x,\cdot) \quad \text{for an arbitrary } x\in\g_1. $$

Then since $g$ is $\ad(G)$-invariant for any pair of elements $v,w\in\g_2\oplus\cdots\oplus\g_k$ it holds

$$ q([v,w])=g(x,[v,w])=g([v,x],w) = 0. $$

This implies that $q_{|_{\g_2\oplus\cdots\oplus\g_k}}=0$ since $[\g_2\oplus\cdots\oplus\g_k,\g_2\oplus\cdots\oplus\g_k]=\g_2\oplus\cdots\oplus\g_k$, and the same holds for the other $\g_i$'s.
Moreover, the metric $g$ must be a multiple of the Killing form once restricted to any simple component $G_i$.
Therefore, we can conclude that any bi-invariant metric $g$ on $G = G_1\times\cdots\times G_k$ is

$$ g = - \lambda_1 B_1 - \lambda_2 B_2 - \cdots - \lambda_s B_k, $$

for positive coefficients $\lambda_i\in\R_{>0}$ and $B_i$ the Killing form on $G_i$.

To sum up, suppose that $G$ is a semisimple Lie group equipped with a bi-invariant metric $g$ and a compatible Samelson complex structure $J$ which decomposes in $s$ irreducible components.
Thus $G = G_1\times\cdots\times G_s$ and $\left(G_i,J_{|_{G_i}}\right)$ are complex manifolds with irreducible Samelson complex structures $J_{|_{G_i}}$.
Then the bi-invariant metrics on $G$ compatible with $J$ form an $s$-dimensional family since they are all of the form

$$ \widetilde{g} = - \lambda_1 \,g_{|_{G_1}} - \lambda_2\, g_{|_{G_2}} - \cdots - \lambda_s\, g_{|_{G_s}}, $$

for positive coefficients $\lambda_i\in\R_{>0}$.

§ COHOMOLOGY OF SEMISIMPLE LIE GROUPS

Let $(G,J)$ be a semisimple Lie group of rank $r$ with maximal torus $T= \left(\S^1\right)^r \subset G$, and equipped with a Samelson complex structure $J$.
The Tits fibration, defined as

$$ \phi : G \rightarrow G/T: g \mapsto g\cdot T, $$

gives a toric fibration

$$ T \hookrightarrow G \xrightarrow{\phi} G/T$$

over the generalized flag manifold $\left(G/T,J_{G/T}\right)$.
We recall that the generalized flag manifolds are Kähler <cit.>, and that they can be endowed with a (unique) invariant Kähler–Einstein metric <cit.>.
Therefore, using Chern–Weil theory following <cit.> it is possible to understand the Dolbeault cohomology of $(G,J)$.
We thus combine it with the results from <cit.> in order to compute the Aeppli cohomology of semisimple Lie groups.
As we will see in the proof of Theorem <ref>, and in Remark <ref>, the Aeppli cohomology is particularly well-suited for studying pluriclosed metrics.
As a matter of fact, the Aeppli cohomology groups of a complex manifold $(M,J)$ are defined as (see <cit.>)

$$ H^{p,q}_{A}(M,J)=\frac{\ker\left\{ \p\circ\bar\p:\A^{p,q}_J(M)\rightarrow\A^{p+1,q+1}_J(M)\right\}}{\imm\left\{\p:\A^{p-1,q}_J(M)\rightarrow\A^{p,q}_J(M)\right\}+\imm\left\{\bar\p:\A^{p,q-1}_J(M)\rightarrow\A^{p,q}_J(M)\right\}} . $$

Therefore, the pluriclosed metrics always give a (possibly zero) representative in the $H^{1,1}_A(M,J)$.

Consider the connection one-form $\theta\in\A^1(G,\k)$ associated to the Tits fibration $\phi$.
We indicate with $\omega\in\A^2(G/T,\k)$ the associated curvature form, $d\theta = \phi^*\omega$.
We then define the maps on the dual of the Lie algebra of the torus

$$\Phi : \k^* \rightarrow \A^1(G) : v \mapsto v \circ \theta$$

$$ D : \k^* \rightarrow  \A^2(G/T): v \mapsto  v\circ\omega,$$

contracting respectively the toric components of $\theta$ and $\omega$.
In particular, if we choose an orthonormal basis for $\k$ with respect to the Killing form
$$\k = \left<z_1,\ldots,z_r\right>$$
the forms $θ$ and $ω$ can be written as
$$\theta = (\theta_1,\ldots,\theta_r),\quad \text{and } \omega=(\omega_1,\ldots,\omega_r).$$
Taking the dual basis
$$\k^* = \left<\xi^1,\ldots,\xi^r\right>,$$
the maps $Φ$ and $D$ are such that
$$\Phi(\xi^i) = \theta_i, \quad\text{and}\quad D(\xi^i)=\omega_i.$$
Now suppose we have a model for the Dolbeault and conjugate Dolbeault cohomologies of $G/T$
$$ \Psi: \left(\mathcal{Y},\eth,\bar\eth\right) \xrightarrow{\sim} \left(\A^{\bullet,\bullet}(G/T),\p,\bar\p\right). $$
The Dolbeault cohomology of $G/T$ satisfies $H^2(G/T) = H^1,1_(G/T,J_G/T)$ \cite[Paragraph 14.10]{MR102800}.
Therefore, by applying Proposition 8 of \cite{MR1255937} we obtain a model for the Dolbeault and conjugate Dolbeault cohomology of $G$ as follows:
\begin{equation}\tag{\bf Tanré}\label{eq: Tanré model}
\Psi\otimes\Phi : \left(\mathcal{Y}\otimes \bigwedge\left(\k^{1,0}\oplus\k^{0,1}\right), \delta, \bar\delta \right) \xrightarrow{\sim} \left(\A^{\bullet,\bullet}(G),\p,\bar\p\right),
\end{equation}
where $(^̨*)^= ^̨1,0⊕^̨0,1$ is the decomposition with respect to $J$, and $δ,δ̅$ satisfy
\begin{equation*}
\delta v =
\begin{cases}
\eth v    & \text{if } v\in\mathcal{Y},\\
0         & \text{if } v\in\k^{1,0},\\
D(v)      & \text{if } v\in\k^{0,1};
\end{cases}
\quad
\bar\delta v =
\begin{cases}
\bar\eth v    & \text{if } v\in\mathcal{Y},\\
D(v)        & \text{if } v\in\k^{1,0},\\
0           & \text{if } v\in\k^{0,1}.
\end{cases}
\end{equation*}
As stated in \cite[Paragraph 14.10]{MR102800}, the Dolbeault cohomology of $G/T$ is all concentrated on bi-degrees $(p,p)$.
It then immediately follows by applying the above model for the Dolbeault cohomology of $G$ that $H^0,1_(G,J)=r$ and $H_^p,0(G,J) = 0$ for all $p>0$.
Notice that this argument was used in \cite[Lemma 6.1]{MR4032184} to prove that in particular the $(3,0)$-Dolbeault cohomology vanishes.

We now clearify what is known about the Dolbeault cohomology of $G/T$, in order to take as $Ψ$ the simplest possible model.
We start by saying that the cohomology $H^*(G/T)$ is all generated by the unit and its $2$-dimensional classes, see \cite[Section 26]{MR51508}.
These classes are precisely the ones of the $ω_i$ as described in \cite[Proposition 14.6]{MR102800}, and the only relations among them are given by the polynomials $[ω_1,…,ω_r]$ which are invariant with respect to the action of the {\em Weil group} $W_G:=N(T)/T$ (for $N(T)$ the normalizer of $T$ in $G$), see \cite[Proposition 27.1]{MR51508}.
Finally, comparing the result in \cite[Proposition 27.2]{MR51508} with the de Rham cohomology of compact simple Lie groups one deduces that the first non-trivial relation among these $2$-forms occurs in degree $2$, in other words, it is a quadratic relation in the $ω_i$.
Since the Killing form is bi-invariant, and hence in particular invariant by the action of the Weil group, this quadratic relation must correspond to it in the following sense:
$$\omega_1\wedge\omega_1 + \omega_2\wedge\omega_2 + \cdots + \omega_r\wedge\omega_r = 0 .$$
In particular, for a semisimple Lie group, we have as many quadratic relations as the number of simple components: let us first distinguish the coordinates on the maximal torus as
$$\k = \left<z_1^1,\ldots,z^1_{n_1},\ldots,z^s_1,\ldots,z^s_{n_s}\right>,$$
where the upper indexes indicate the simple component they belong;
if we arrange the characteristic classes accordingly, then the quadratic relations are all of the type
\begin{equation}\label{eq: quadratic relations}
\sum_{i=1}^s b_i\sum_{j=1}^{n_i}\omega^i_j\wedge\omega^i_j=0 \quad \text{for } b_i\in\R,
\end{equation}
i.\,e. corresponding to the bi-invarian metrics on $G=G_1×⋯×G_s$.
%    \begin{prop}
%       Let $G$ be a semisimple Lie group of rank $r$ equipped with a Samelson complex structure $J$.
%      Then
%     $$H_{\bar\p}^{1,1}(G,J)\cong \C^r.$$
%    \begin{proof}
%       Using the model for the Dolbeault cohomology of $(G,J)$ one sees that
%      $H_{\bar\p}^{1,1}(G,J)$ is generated by the $(1,1)$-forms $\omega_i$.
%     Then the result follows by the fact that there are no non-trivial relations of order one between the $\omega_i$.
%    \textcolor{red}{potrebbero essere esatte}

%    We remark that even if the forms $\omega_i$ in the model are not $\bar\delta$-exact they are in $\imm\left(\delta\right)\cup\imm\left(\bar\delta\right)$, hence they are not representatives in the $H^{1,1}_A$ cohomology.

We now compute the $(1,1)$-Aeppli cohomology of $(G,J)$.
We first restrict to the case of irreducible Samelson complex structure; then the general case will follow by a K\"unneth formula.

\begin{theo}\label{th: 1,1 A cohomology}
Let $G$ be a semisimple Lie group equipped with a bi-invariant metric $g$ and a compatible irreducible Samelson complex structure $J$.
$$H_{A}^{1,1}(G,J)\cong \C.$$
\end{theo}
\begin{proof}
First of all, fix $r$ the rank of $G$ and $s$ the number of simple components, $G=G_1\times\cdots\times G_s$.
Then, notice that since the metric $g$ is bi-invariant it equals
$$g = -\lambda_1 B_{|_{\g_1}} -\lambda_2 B_{|_{\g_2}} - \cdots -\lambda_s B_{|_{\g_s}} ,$$
where $B$ is the Killing form of $G$ (see Section \ref{subsec: bi-inv metrics}).
Therefore we can scale the coordinates $\left<z_j^i\right>$ %as $w^i_j=\frac{1}{\lambda_i}z^i_j$
in order to obtain an orthonormal basis for $g$.
Let us keep the same symbols for the coordinates $\left<z_i\right>$ their dual $\left<\xi_i\right>$ and the connection and curvature forms $\theta_i$ and $\omega_i$ for the sake of simplicity.

Since \eqref{eq: Tanré model} is a model for Dolbeault and conjugate Dolbeault cohomology, thanks to the result of Stelzig \cite[Theorem C]{stelzig23}, it is also a model for Aeppli and {\em Bott--Chern} cohomologies.
We thus can use this model to construct a bi-complex quasi-isomorphic to $\left(\A^{\bullet,\bullet}(G),\p,\bar\p\right)$ with respect to the Aeppli cohomology.
Starting from the bi-complex freely-generated by the $\omega_i$'s and $\k^{1,0},\k^{0,1}$, namely
$$ \left(\bigwedge\left(\omega_1,\ldots,\omega_r\right) \otimes \bigwedge\left(\k^{1,0}\oplus\k^{0,1}\right), \delta, \bar\delta \right) ,$$
we obtain our bi-complex computing all cohomologies by erasing dots in the bi-degrees $(p,p)$ corresponding to the $W_G$-invariant polynomials in $\R[\omega_1,\ldots,\omega_r]$.
This affects the lower degrees starting from bi-degree $(2,2)$ with the relations \eqref{eq: quadratic relations}.
Since we are interested in the $(1,1)$-Aeppli cohomology we only care about the {\em squares} of the preliminary bi-complex which become L-shaped {\em zig-zags} after erasing dots in the bi-degrees $(2,2)$ as in the following figure (we refer to \cite[Section 1.1.4]{stelzig23} for the notation of squares and zig-zags).% \ref{fig: square to zigzag}.
\begin{figure}[h!] \begin{center}\begin{tikzpicture}
\newcommand\unito{1}

\draw[help lines, step=\unito] (0,0) grid (3.3*\unito,3.3*\unito);

\foreach \x in {0,...,2}
\node at (\unito*0.5+\unito*\x,-0.3) {\x};
\foreach \y in {0,...,2}
\node at (-0.3,\unito*0.5+\unito*\y) {\y};
\node at (-0.3, \unito*3.2) {$\vdots$};
\node at (\unito*3.2, -0.3) {$\cdots$};

\draw[help lines, step=\unito] (5*\unito,0) grid (8.3*\unito,3.3*\unito);

\foreach \x in {0,...,2}
\node at (\unito*5.5+\unito*\x,-0.3) {\x};
\foreach \y in {0,...,2}
\node at (\unito*5-0.3,\unito*0.5+\unito*\y) {\y};
\node at (\unito*5-0.3, \unito*3.2) {$\vdots$};
\node at (\unito*8.2, -0.3) {$\cdots$};

\node at (\unito*4,\unito*1.6) {$\rightarrow$};

\coordinate (11) at (3/2*\unito,3/2*\unito);
\coordinate (12) at (3/2*\unito,5/2*\unito);
\coordinate (21) at (5/2*\unito,3/2*\unito);
\coordinate (22) at (5/2*\unito,5/2*\unito);

\fill (11) circle (2.2pt);
\fill (21) circle (2.2pt);
\fill (12) circle (2.2pt);
\fill (22) circle (2.2pt);

\draw (11) -- (12);
\draw (11) -- (21);
\draw (21) -- (22);
\draw (12) -- (22);

\coordinate (11b) at (13/2*\unito,3/2*\unito);
\coordinate (12b) at (13/2*\unito,5/2*\unito);
\coordinate (21b) at (15/2*\unito,3/2*\unito);

\fill (11b) circle (2.2pt);
\fill (21b) circle (2.2pt);
\fill (12b) circle (2.2pt);

\draw (11b) -- (12b);
\draw (11b) -- (21b);

\end{tikzpicture}
%\label{fig: square to zigzag}
\end{center}
\end{figure}

We remark that all the forms $\omega_i$ are in $\imm\left(\delta\right)\cup\imm\left(\bar\delta\right)$ as
\begin{equation}\label{eq: omega_i are aeppli-exact}
2\omega_i = \delta\left(\xi^i + \im J \xi^i \right) + \bar\delta\left(\xi^i - \im J \xi^i \right).
\end{equation}
On the other hand, all the $(1,1)$-forms in $\k^{1,0}\wedge\,\k^{0,1}$ belong to a square in the preliminary bi-complex.
More concretely, given an element
$$ \sum_{i,j=1}^r A_i^j \left(\xi^i -\im J \xi^i \right) \wedge \left(\xi^j +\im J \xi^j \right) \in \k^{1,0}\wedge\k^{0,1},$$
it is not $\delta$ nor $\bar\delta$ exact, and it holds
\begin{multline*}
-\bar\delta\delta\left( \sum_{i,j=1}^r A_i^j \left(\xi^i -\im J \xi^i \right) \wedge \left(\xi^j +\im J \xi^j \right) \right)
= \bar\delta\left(\sum_{i,j=1}^r A_i^j \left(\xi^i -\im J \xi^i \right) \wedge D\left(\xi^j +\im J \xi^j \right) \right)\\
= \bar\delta\left(\sum_{i,j=1}^r A_i^j \left(\xi^i -\im J \xi^i \right) \wedge \left(\omega^j +\im \sum_{q=1}^r J^j_q \omega^q  \right) \right)
= \bar\delta\left( \sum_{i,j=1}^r \left(A_i^j +\im \sum_{q=1}^r J_j^q A_i^q \right) \left(\xi^i -\im J \xi^i \right) \wedge \omega^j  \right)\\
= \sum_{i,j=1}^r \left(A_i^j +\im \sum_{q=1}^r J_j^q A_i^q \right) D\left(\xi^i -\im J \xi^i \right) \wedge \omega^j
= \sum_{i,j=1}^r \left(A_i^j +\im \sum_{q=1}^r J_j^q A_i^q \right) \left(\omega^i -\im \sum_{p=1}^r J^i_p \omega^p \right) \wedge \omega^j \\
= \sum_{i,j=1}^r \left(A_i^j +\im \sum_{q=1}^r J_j^q A_i^q -\im \sum_{p=1}^r J^p_i A_p^j +\sum_{p,q=1}^r J^p_i J^q_j A_p^q    \right) \omega^i \wedge \omega^j \\
= \sum_{i,j=1}^r \left(A -\im AJ -\im JA - JAJ \right)^j_i \omega^i \wedge \omega^j,
\end{multline*}
where $\left(J^j_k\right)$ is the matrix associated to the endomorphism $J$ in the coordinates $\left<z_i\right>$.
We thus obtain a quadratic relation in the $\omega_i$'s.
Hence we are now interested in understanding how many of these squares break in zig-zags giving representatives in $H^{1,1}_A(G,J)$.
Thanks to equation \eqref{eq: quadratic relations}, this is equivalent to computing how many solutions there are of
\begin{equation}\label{eq: crucial}
Q + Q^t =
\begin{pmatrix}
b_1 \id &         &        &         \\
& b_2 \id &        &         \\
&         & \ddots &         \\
&         &        & b_s \id
\end{pmatrix}
\end{equation}
where $Q:=A-\im JA - \im AJ - JAJ$.
Notice that $JQ = \im Q$ and $QJ = \im Q$.
\begin{equation}\label{eq: 1}
-J\left( Q + Q^t\right)J = Q + Q^t .
\end{equation}
This forces the coefficients $b_i$ in \eqref{eq: crucial} to be all equal.
%        \begin{equation}\label{eq: matrix}
%           -J \begin{pmatrix}
%              b_1 \id &         &        &         \\
%                 & b_2 \id &        &         \\
%                &         & \ddots &         \\
%               &         &        & b_s \id
%      \end{pmatrix} J =
%     \begin{pmatrix}
%        b_1 \id &         &        &         \\
%           & b_2 \id &        &         \\
%          &         & \ddots &         \\
%         &         &        & b_s \id
Indeed, by multiplying \eqref{eq: 1} on the left by $J$ and using \eqref{eq: crucial} we have
\[
\begin{pmatrix}
\cdots -b_1Jz_1^1 \cdots \\
\vdots \\
\cdots -b_kJz_k^l \cdots \\
\vdots \\
\cdots -b_sJz^s_{n_s} \cdots
\end{pmatrix} =
\begin{pmatrix}
\vdots & & \vdots & & \vdots \\
b_1Jz_1^1 & \cdots & b_kJz^k_l & \cdots & b_sJz^s_{n_s} \\
\vdots & & \vdots & & \vdots
\end{pmatrix}.
\]
Therefore, if there exist two pairs $k=1,\ldots,s$, $l=1,\ldots, n_k$ and $p=1,\ldots,s$, $q=1,\ldots, n_p$ with $p\neq k$ such that $g(Jz_l^k,z^p_q)\neq 0$ it must be $b_k=b_p$.
Finally, since the complex structure is irreducible we can guarantee this to happen and get $b_1=b_2=\cdots=b_s$.

A straightforward computation shows that there is no non-zero matrix $Q$ which is anti-symmetric and satisfies $JQ=\im Q=QJ$.
Thus, there is only one solution (up to scalar multiplication) which is given by taking $A = \id$.
It corresponds to the $(1,1)$-form in $\k^{1,0}\wedge\,\k^{0,1}$
$$ 2\im\sum_{j=1}^r \xi^j\wedge J\xi^j ,$$
which is in the same Aeppli cohomology class of the metric.
Consequently, there is only one $(1,1)$-representative for the Aeppli cohomology of $(G,J)$, that is% the bi-invariant metric $g$. Thus
$$H^{1,1}_{A}(G,J) = \C\left<[ g(J\cdot,\cdot)]\right>.$$
\end{proof}

We know that $h^0,1_(G,J)= r$ and $h^p,0_(G,J)=0$ for $p≥1$, where $h^∙,∙_··(G,J) = H^∙,∙_··(G,J)$.
Therefore, in the bi-complex of $(G,J)$, there are $r$ zig-zags starting (horizontally) from bi-degree $(0,1)$.
These zig-zags can not have odd length (one or three) because that will count in de Rham cohomology which we know is $H^1(G)=0$.
Moreover, they can not have length four because this would imply $H^2,0_(G,J)≠0$.
Therefore they must have length equal to two, which implies
$$h_{\bar\p}^{1,1}(G,J) \geq h_{\bar\p}^{0,1}(G,J) = r.$$
We recall that the $(1,1)$-forms coming from the torus belong to squares or L-shaped zig-zags, hence they do not contribute to Dolbeault cohomology in bi-degree $(1,1)$.
Consequently, the dimension of the $(1,1)$-Dolbeault cohomology equals the rank of $G$ since $H^1,1_(G,J)$ is generated by the forms $ω_i$ for $i=1,…,r$ which satisfy \eqref{eq: omega_i are aeppli-exact}.

We picture the information collected up to now in the diagram in Figure \ref{fig: doublecmplex G}, where we define $ν_i := ξ^i +J ξ^i$ for $i=1,…,r$.

\begin{figure}[h!] \begin{center}\begin{tikzpicture}
\newcommand\unito{2}

\draw[help lines, step=\unito] (0,0) grid (3.3*\unito,3.3*\unito);

\foreach \x in {0,...,2}
\node at (\unito*0.5+\unito*\x,-0.3) {\x};
\foreach \y in {0,...,2}
\node at (-0.3,\unito*0.5+\unito*\y) {\y};

\node at (-0.3, \unito*3.2) {$\vdots$};
\node at (\unito*3.2, -0.3) {$\cdots$};
\node at (2*\unito+1/2*\unito, 2*\unito+1/2*\unito) {$\iddots$};
\node at (5/2*\unito,1/2*\unito) {$\cdots$};
\node at (1/2*\unito,5/2*\unito) {$\vdots$};
\node at (5/2*\unito,3/2*\unito) {$\cdots$};
\node at (3/2*\unito,5/2*\unito) {$\vdots$};

\coordinate (00) at (1/2*\unito,1/2*\unito);

\coordinate (01) at (1/2*\unito,30/16*\unito);
\coordinate (11b) at (6/5*\unito,30/16*\unito);
\coordinate (01bis) at (1/2*\unito,27/16*\unito);
\coordinate (11bbis) at (6/5*\unito,27/16*\unito);
\coordinate (01tri) at (1/2*\unito,21/16*\unito);
\coordinate (11btri) at (6/5*\unito,21/16*\unito);

\coordinate (10) at (30/16*\unito,1/2*\unito);
\coordinate (11a) at (30/16*\unito,6/5*\unito);
\coordinate (10bis) at (24/16*\unito,1/2*\unito);
\coordinate (11abis) at (24/16*\unito,6/5*\unito);
\coordinate (10tri) at (21/16*\unito,1/2*\unito);
\coordinate (11atri) at (21/16*\unito,6/5*\unito);

\coordinate (11c) at (5/3*\unito,5/3*\unito);
\coordinate (21) at (5/2*\unito,5/3*\unito);
\coordinate (12) at (5/3*\unito,5/2*\unito);

\fill (00) circle (2.5pt);
\fill (01) circle (2.5pt);
\fill (01bis) circle (2.5pt);
\fill (01tri) circle (2.5pt);
\fill (10) circle (2.5pt);
\fill (10bis) circle (2.5pt);
\fill (10tri) circle (2.5pt);
\fill (11a) circle (2.5pt);
\fill (11abis) circle (2.5pt);
\fill (11atri) circle (2.5pt);
\fill (11b) circle (2.5pt);
\fill (11bbis) circle (2.5pt);
\fill (11btri) circle (2.5pt);
\fill (11c) circle (2.5pt);
\fill (12) circle (2.5pt);
\fill (21) circle (2.5pt);

\node at (1/2*\unito,1/2*\unito-.4) {$1$};
\node at (-.4+1/2*\unito,30/16*\unito) {$\nu_1$};
\node at (-.4+1/2*\unito,27/16*\unito) {$\nu_2$};
\node at (-.4+1/2*\unito,21/16*\unito) {$\nu_r$};
\node at (5/7*\unito,.1+3/2*\unito) {$\vdots$};
\node at (30/16*\unito,1/2*\unito-.4) {$\bar\nu_r$};
\node at (24/16*\unito,1/2*\unito-.4) {$\bar\nu_2$};
\node at (21/16*\unito,1/2*\unito-.4) {$\bar\nu_1$};
\node at (3/2*\unito+.4,5/7*\unito) {$\cdots$};

\draw (10) -- (11a);
\draw (01) -- (11b);
\draw (01bis) -- (11bbis);
\draw (01tri) -- (11btri);
\draw (10bis) -- (11abis);
\draw (10tri) -- (11atri);
\draw (21) -- (11c) -- (12);

\end{tikzpicture}
\caption{Zig-zags of the double complex of $(G,J)$ involving bi-degrees $(1,1)$ or lower, up to squares.}
\label{fig: doublecmplex G}
\end{center}
\end{figure}

In order to extend Theorem \ref{th: 1,1 A cohomology} to a semisimple Lie group equipped with a general Samelson complex structure we use the K\"unneth formula for the Aeppli cohomology computed in \cite{stelzig23}.
Notice that the following result can also be obtained with a direct computation as in Theorem \ref{th: 1,1 A cohomology} without using K\"unneth's formula.

\begin{cor}\label{cor: 1,1 A cohomology reducible case}
Let $G$ be a semisimple Lie group equipped with a bi-invariant metric $g$ and a compatible Samelson complex structure $J$.
Suppose that $J$ decomposes in $s$ irreducible components, then
$$H_{A}^{1,1}(G,J)\cong \C^s.$$
\end{cor}
\begin{proof}
The complex manifold $(G,J)$ decomposes as a product $G=G_1\times\cdots\times G_s$ of complex manifolds $\left(G_i,J_{|_{G_i}}\right)$ where $J_{|_{G_i}}$ is an irreduble Samelson complex structure.
Then the result follows from Theorem \ref{th: 1,1 A cohomology} using the K\"unneth formula for Aeppli cohomology \cite[Corollary 1.36]{stelzig23}.
Indeed, using the same notations as in \cite{stelzig23}, we have the following short exact sequence
$$ 0 \rightarrow L \rightarrow H_A\left(G_1\times\cdots\times G_s,J\right) \rightarrow \left(\ker\p\bar\p\right)\left(H_A\left(G_1,J_{|_{G_1}}\right)\otimes \cdots \otimes H_A\left(G_s,J_{|_{G_s}}\right)\right) \rightarrow 0 .$$
In bi-degree $(1,1)$, the bi-complex structure (partially pictured in Figure \ref{fig: doublecmplex G}) is such that $L$ vanishes and hence
\begin{multline*}
H^{1,1}_A\left(G_1\times\cdots\times G_s,J\right) \cong \left[H_A^{1,1}\left(G_1,J_{|_{G_1}}\right)\otimes H_A^{0,0}\left(G_2,J_{|_{G_2}}\right)\otimes \cdots \otimes H_A^{0,0}\left(G_s,J_{|_{G_s}}\right)\right] \oplus \cdots \\
\oplus \left[H_A^{0,0}\left(G_1,J_{|_{G_1}}\right)\otimes \cdots \otimes H_A^{0,0}\left(G_{s-1},J_{|_{G_{s-1}}}\right)\otimes  H_A^{1,1}\left(G_s,J_{|_{G_s}}\right)\right]
\end{multline*}

\end{proof}

%    \begin{rmk*}
%       The result in Corollary \ref{cor: 1,1 A cohomology reducible case} can be also obtained with a direct computation as in Theorem \ref{th: 1,1 A cohomology} without using the K\"unneth formula.
%  \end{rmk*}

\begin{rmk}\label{rmk: crucial}
Since the only non-zero representatives in $H^{1,1}_A(G,J)$ comes from {\rm L}-shaped zig-zags, by counting the bi-invariant metrics, it can be proved that $s\leq h^{1,1}_A(G,J)\leq h^{2,1}_{\bar\p}(G,J)$.
Therefore, the results of Theorem \ref{th: 1,1 A cohomology} and Corollary \ref{cor: 1,1 A cohomology reducible case} would follow by proving that $h^{2,1}_{\bar\p}(G,J)=s$.
However, unless some particular cases (for example for rank less or equal than $4$ or some specific complex structures) we are not able to prove it.
Somehow, for these manifolds, it is, in general, easier to explicitly compute the Aeppli cohomology than the Dolbeault cohomology.
More precisely, the structure of the bi-complex computing all the cohomologies remains a black box from which we are only able to recover information about the {\rm L}-shaped zig-zags coming from squares.
We want to highlight that this is unusual since we have much more tools to compute the de Rham and Dolbeault cohomologies, while one usually recovers information on the Aeppli and Bott--Chern cohomologies from the formers (as for example in \cite{MR3340173,barbaro_pcflow}).
%Following this basic principle, in ?? we expressed obstructions to the existence of pluriclosed metrics in terms of the Dolbeault cohomology, even if the $(1,1)$-Aeppli cohomology is clearly the most natural choice.
Also in \cite{MR4032184} the authors use the Tanré's model to get information on the $(2,1)$ and $(3,0)$ Dolbeault cohomology groups and from that deduce the (non-)existence of pluriclosed metrics.
However, the $(1,1)$-Aeppli cohomology is clearly better suited for studying pluriclosed metrics. It is therefore interesting that here we use Tanré model to directly compute the Aepply cohomology and by-pass the understanding of the Dolbeault cohomology.
\end{rmk}

\section{Global stability}\label{sec: stability}
%    In this section, we prove that on compact Bismut flat manifold with finite fundamental group the Bismut flat metrics are globally stable for the pluriclosed flow.
%   This follows by combining the recent developments in the understanding of the long-time behavior of the pluriclosed flow, achieved in \cite{Garcia-Fernandez:2021tq}, and the knowledge of the cohomologies of this special class of manifolds, that we obtained in the previous section.\\

Bismut flat metrics are attractive (in the sense of the following theorem) for the pluriclosed flow in their {\em torsion classes}, which, for a metric $ω$, is the class of $ω$ in the $(2,1)$-Dolbeault cohomology.% $[\p\omega]\in H^{2,1}_{\bar\p}$.
\begin{theo}[Theorem 1.2 of \cite{Garcia-Fernandez:2021tq}]\label{thm: GJS convergence}
Let $(M,J, \omega_{BF})$ be a compact Bismut flat manifold. Given $\omega_0$ a pluriclosed metric such that $[\p\omega_0] = [\p\omega_{BF} ] \in H^{2,1}_{\bar\p}(M,J)$, the solution of the pluriclosed flow with initial data $\omega_0$ exists on $[0, \infty)$ and converges to a Bismut flat metric $\omega_\infty$.
\end{theo}

%    \begin{rmk*}
%       A priori, there is no clear relation between the endpoint of the flow $\omega_\infty$ and the background metric $\omega_{BF}$.
%      However, something more can actually be said about $\omega_\infty$ (see Theorem \ref{th: global stability}).
% \end{rmk*}

Thanks to the knowledge of the $(1,1)$-Aeppli cohomology of compact Samelson spaces achieved in the previous section, we can now check that the cohomological condition in Theorem \ref{thm: GJS convergence} holds for all the pluriclosed metrics on these manifolds.
It follows that the Bismut flat metrics on compact Bismut flat manifolds with finite fundamental group are globally stable for the pluriclosed flow.
We use here a refined version of the argument from \cite{barbaro_pcflow} which applies to this much more general case, proving the following result.
\begin{theo}\label{th: global stability}
Let $(G,J,\omega_{BF})$ be a compact Samelson space, that is a compact simply-connected semisimple Lie group with a Bismut flat Hermitian structure coming from the Killing form (as described in Section \ref{sec: 2}).
Suppose it decomposes in $s$ irreducible components $\left(G_i,J_{|_{G_i}}\right)$ and define $\omega_i:=\left(\omega_{BF}\right)_{|_{G_i}}$.
Then for any pluriclosed metric $\omega_0$ on $\left(G,J\right)$ there exist positive constants $\gamma_i$'s such that the solution to the pluriclosed flow with initial data $\omega_0$ exists on $[0, \infty)$ and converges to $\sum_{i=1}^s\gamma_i\omega_i$ up to diffeomorphism.
\end{theo}
\begin{proof}
First of all, given a pluriclosed metric $\omega$ on $\left(G,J\right)$ we show that its class in the $(1,1)$-Aeppli cohomology can not be zero.
Hence, suppose that $\omega$ is $\left(\p \,\rm{or}\, \bar\p\right)$-exact, namely there exists $\alpha\in\A^{1,0}_JG$ such that $\omega = \p\bar\alpha + \bar\p\alpha$.
$$\Omega:=\omega+\p\alpha+\bar\p\bar\alpha = d(\alpha+\bar\alpha)$$
is a symplectic form since it is obviously a closed $2$-form, and for any non-zero vector field $x\in\Cinf(G;TG)$ it holds
$$ \Omega(x,Jx) = \omega(x,Jx)+\p\alpha(x,Jx)+\bar\p\bar\alpha(x,Jx) = g(x,x) \neq 0. $$
However, a result of Borel \cite[Theorem 1]{Borel} ensures that $G$ does not admit any symplectic structure.\\

Now, following Section \ref{subsec: bi-inv metrics}, the restriction of $\omega_{BF}$ to any $\left(G_i,J_{|_{G_i}}\right)$ is a Bismut flat metric $\omega_i=\left(\omega_{BF}\right)_{|_{G_i}}$, which thanks to the above considerations is a representative of a non-zero class in $H^{1,1}_A\left(G_i,J_{|_{G_i}}\right)$.
Moreover, Corollary \ref{cor: 1,1 A cohomology reducible case} ensures that $H^{1,1}_A(G,J)$ is generated by the pull-backs of these metrics.
Thus for any pluriclosed metric $\omega_0$ on $(G,J)$ there exist constants $\lambda_i$ for $i=1,\ldots,s$ such that
$$[\omega_0] = \sum_{i=1}^s \lambda_i [\omega_{i}] \quad\text{in } H^{1,1}_A \left(G,J\right).$$
Here the constants $\lambda_i$'s must be positive because the calss of $\left(\omega_0\right)_{|_{G_i}} - \lambda_i\omega_i$ is zero in $H^{1,1}_A\left(G_i,J_{|_{G_i}}\right)$ hence it can not be a pluriclosed metric on $\left(G_i,J_{|_{G_i}}\right)$.
Consequently, by applying the $\p$ operator one gets that
$$[\p\omega_0] = \left[ \p \sum_{i=1}^s \lambda_i \omega_{i}\right] \quad\text{in } H^{2,1}_{\bar\p} \left(G,J\right).$$
Finally, we notice that the metric $\sum_{i=1}^s \lambda_i \omega_{i}$ is a Bismut flat metric on $(G,J)$ since it is bi-invariant and compatible with $J$.
Therefore, Theorem \ref{thm: GJS convergence} applies to ensure the long-time existence of the pluriclosed flow with initial data $\omega_0$ and convergence to a Bismut flat metric $\omega_\infty \in \sum_{i=1}^s \lambda_i [\omega_{i}]_A$.\\

Thanks to the characterization in Section \ref{sec: 2}, Bismut flat metrics are bi-invariant with respect to some Lie group structure, and all the bi-invariant metrics with respect to the action of $G$ which are compatible with $J$ are of the type $\sum_{i=1}^s \gamma_i \omega_{i}$ for positive constants $\gamma_i$'s.
We notice that a priori there might be different Lie group structures on $G$ as manifold.
However, the Milnor result (Lemma \ref{lem: Milnor}) ensures that there is only one up to diffeomorphisms of the underlying manifold.
Consequently,  $\omega_\infty$ and $\omega_{BF}$ are bi-invariant with respect to two isomorphic Lie group structures on $G$.
In particular, there exists $\varphi\in\mbox{Diff}(G)$ such that $\varphi^*\omega_\infty=\sum_{i=1}^s \gamma_i \omega_{i}$ for some positive constants $\gamma_i$'s.
\end{proof}

As an immediate corollary of Theorem \ref{th: global stability} we obtain the global stability of the Bismut flat metrics for the pluriclosed flow on any manifold whose universal cover is a compact Samelson space.
Indeed, we can prove the global existence of the pluriclosed flow and its convergence on the universal cover.
Thanks to the classification in Theorem \ref{thm: Bismut flat manifolds} these manifolds are precisely the compact Bismut flat manifolds with finite fundamental group.

\begin{theo}\label{th: global stability bis}
Let $(M,J,\omega_{BF})$ be a compact Bismut flat manifold with finite fundamental group.
Then for any pluriclosed metric $\omega_0$ on $\left(M,J\right)$ the solution to the pluriclosed flow with initial data $\omega_0$ exists on $[0, \infty)$ and converges to a Bismut flat metric $\omega_{\infty}$.
\end{theo}

\medskip

The case of compact Bismut flat manifolds with infinite fundamental group (which by Theorem \ref{thm: Bismut flat manifolds} corresponds to manifolds diffeomorphic to $G×(^1)^k$ for a semisimple Lie group $G$ and positive $k$) remains open.
%{We remark that also in [Fino - Gran] the authors stick on the semisimple case.}
In fact, the main argument we used in proving global stability was to show that the subspace generated by the classes of the Bismut flat metrics fills up the whole $H^1,1_A(G,J)$.
However, for {\em local} Samelson space the subspace of $H^1,1_A(G,J)$ generated by the classes of the pluriclosed metrics grows in such a way that it makes more difficult to verify that it is generated by Bismut flat metrics.
%In the above case, we are able to do it since we are actually working on a compact Samelson space, hence we have good control of the Bismut flat metrics which are just bi-invariant metrics with respect to some Lie group structure in the given isomorphism class.
%On the other hand, in general, we should work on a {\em local} Samelson space where we have a worse understanding of the family of Bismut flat metrics.% \textcolor{red}{because we need to take into account the quotient of $G\times\R^k$ by isometries of $G$}.
We present here an example in order to clarify this difficulty.

\begin{ex}\label{ex: higher rank}
Consider the simple Lie group of rank two $\SU(3)$ equipped with the Bismut flat Hermitian structure $\left(J_{\SU(3)},\omega_{BF}\right)$ coming from the Killing form, and the complex torus with the standard complex structure
Define $J:=J_{\SU(3)}\times J_{St}$ the product complex structure on $M:=\SU(3)\times\S^1\times\S^1$.
The lower bi-degrees of the bi-complex associated to $\left(\SU(3),J_{\SU(3)}\right)$ are as in Figure \ref{fig: doublecmplex G}.
Therefore, by applying the K\"unneth formula for Aeppli cohomology \cite[Corollary 1.36]{stelzig23} one gets that the $(1,1)$-Aeppli cohomology of $(M,J)$ is of dimension $4$ generated by the classes of $\omega_{BF}, \psi\wedge\bar\psi, \nu_1\wedge\bar\psi, \psi\wedge\bar\nu_1$, for $\psi$ the $(1,0)$-form generating the cohomology of the torus.
Thus, given a generic pluriclosed metric $\omega$ on $(M,J)$, its class in the $(1,1)$ Aeppli cohomology group $H^{1,1}_A(M,J)$ is
\begin{equation}\label{eq: above}
[\omega] =  \alpha[\omega_{BF}] + \beta \frac{\im}{2}[\psi\wedge\bar\psi] + u [\nu_1\wedge\bar\psi] - \bar u [\psi\wedge\bar\nu_1],
\end{equation}
for coefficients $\alpha,\beta\in\R$, and $u\in\C$ such that
\begin{equation}\label{eq: bove}
\begin{cases}
\alpha>0, \quad \beta>0,\\
\alpha\beta>4|u|^2.
\end{cases}\end{equation}
The above equation \eqref{eq: above} defines the subset of $H^{1,1}_A(M,J)$ generated by the classes of the pluriclosed metrics on $(M,J)$.
Henceforth, to apply the argument in Theorem \ref{th: global stability} one should verify that it is generated by the classes of the Bismut flat metrics, however, we can not argue as before counting the bi-invariant metrics coming from the Killing forms of the simple components.
We thus define a family of pluriclosed metrics as
\begin{align*}
\omega_{\alpha,\beta,u} := \alpha\omega_{BF} + \beta \frac{\im}{2}\psi\wedge\bar\psi + u \nu_1\wedge\bar\psi - \bar u \psi\wedge\bar\nu_1, %+ v\, \bar\p\varphi^4 + \bar v\, \p\bar\varphi^4 \\
%=& \frac{\im}{2}\left\{ \left(\alpha+2\Re(v)\right)\varphi^1\wedge\varphi^{\bar 1} + \left(\alpha -\sqrt{3}\Im(v)+\Re(v)\right)\varphi^2\wedge\varphi^{\bar2} \right. \\
%& \left. + \left(\alpha + \sqrt{3}\Im(v) + \Re(v)\right)\varphi^3\wedge\varphi^{\bar3} + \alpha\varphi^{4\bar4} + \beta \psi\wedge\bar\psi \right\} + u \varphi^4\wedge\bar\psi - \bar u \psi\wedge\bar\varphi^4
\end{align*}
with coefficients $\alpha,\beta$ and $u$ as in \eqref{eq: bove}, and by straightforward computations, we verify that all these metrics are Bismut flat.

Note that the metrics $\omega_{\alpha,\beta,u}$ give Bismut flat metrics on the cover $\SU(3)\times\R^2$ which are not bi-invariant with respect to the product Lie group structure whenever $u\neq0$.
Indeed, being bi-invariant means that for any $x,y,z$ in the Lie algebra
$$\omega_{\alpha,\beta,u} ([x,y],Jz) + \omega_{\alpha,\beta,u} ([x,z],Jy) = 0.$$
However, taking $z=\psi^*$ the dual of $\psi$ we get
\begin{equation*}
\omega_{\alpha,\beta,u} ([\cdot,\cdot\cdot],J\psi^*) + \omega_{\alpha,\beta,u} ([\cdot,\psi^*],J\cdot\cdot) =
\im \bar u \,\bar\nu_1([\cdot,\cdot\cdot]) = - \im \bar u \,d\,\bar\nu_1 \neq 0.
\end{equation*}
A posteriori, thanks to the characterization of Bismut flat metrics of Theorem \ref{thm: Bismut flat manifolds}, we may assert that they are bi-invariant with respect to another Lie group structure on $\SU(3)\times\R^2$ (as manifold) isomorphic to the previous one.
More precisely, by Lemma \ref{lem: Milnor}, for any $u\in\C$ there exists an isometry $$\phi_u : \left(\SU(3)\times\R^2,\omega_{\alpha,\beta,u}\right) \xrightarrow{\sim} \left(\SU(3)\times\R^2,\omega_{\alpha',\beta',0}\right).  $$
%In fact, $\omega_{BF}$ and $\psi\wedge\bar\psi$ are both bi-invariant forms since they come from the Killing forms on $\SU(3)$ and $\R^2$ respectively;
%while, $\nu_1$ is left-invariant but not right-invariant on $\SU(3)$.
%Therefore, on $\SU(3)\times\S^1\times\S^1$ there exist Lie groups structure different from the product one such that the metrics $\omega_{\alpha,\beta,u}$ with $u\neq 0$ are bi-invariant with respect to them.
%This corresponds with taking a non-trivial action $\rho:\Z^2\rightarrow I(\SU(3))$ when performing the quotient of $\SU(3)\times\R^2$, see Definition \ref{def: loc sam space}.
\end{ex}

\section{Homogeneous Bismut Hermitian Einstein metrics}\label{sec: non-flat BHE}

The existence of non-flat Bismut Hermitian Einstein metrics on Bismut flat manifolds with finite fundamental group is obstructed by the global stability property for the pluriclosed flow proved in Theorem \ref{th: global stability bis}.
Indeed, we have the following simple corollary of Theorem \ref{th: global stability bis}.
\begin{cor}\label{cor: obstruction on Bf}
Let $(M,J,\omega_{BF})$ be a compact Bismut flat manifold with finite fundamental group.
Then any Bismut Hermitian Einstein metric is Bismut flat.
\end{cor}

Let $(M,J)$ be a C-space, which we recall is a compact complex manifold admitting a transitive action by a compact Lie group of biholomorphisms and finite fundamental group.
Thanks to \cite[Theorem C]{MR66011}, such a space admits a transitive action of a compact semisimple Lie group, hence it can be written as $M=G/H$ where $G$ is a compact semisimple Lie group and $H$ a closed connected subgroup whose semisimple part coincides with the semisimple part of the centraliser of a toral subgroup of $G$.
Consequently, $G/H$ fibers over a generalized flag manifold $G/T$ with toric fibers:
$$T/H \hookrightarrow G/H \rightarrow G/T.$$
It is then possible to use Chern--Weil theory and the Tanré model to understand the Dolbeault cohomology of these manifolds, as in Section \ref{sec: cohomology comp}.
The authors of \cite{MR4032184} did it in order to check a cohomological obstruction due to Cavalcanti on the existence of pluriclosed metrics \cite[Theorem 5.15]{MR4157573}, and proved that pluriclosed structures on a C-space can only appear if it is a product of a compact Lie group and a generalized flag manifold (up to finite cover) \cite[Theorem 6.1]{MR4032184}.
More precisely, we have that
\begin{theo*}[Theorem 5.15 in \cite{MR4157573}]
Let  $(M,J)$ be a compact complex manifold which admits a pluriclosed metric and is such that
$$ H^{2,1}_{\bar\p}(M,J) = H^{3,0}_{\bar\p}(M,J) = 0. $$
Then $(M,J)$ admits a symplectic structure.
\end{theo*}
Combining this result with the knowledge of Dolbeault cohomology and the fact that there are no symplectic structures on non-K\"ahler C-spaces \cite[Theorem 1]{Borel}, Fino, Grantcharov, and Vezzoni deduced the following characterization.
\begin{theo*}[Theorem 6.1 in \cite{MR4032184}]
Every pluriclosed C-space is (up to a finite cover) the product of a compact
Lie group and a generalized flag manifold.
\end{theo*}
Now, the existence of a pluriclosed CYT metric, and, more generally, a Bismut Hermitian Einstein metric, on $(M,J)$ would force the first Chern class to vanish in the $(1,1)$-Aeppli cohomology.
In fact, for a Hermitian manifold $(M,J,g)$ it holds \cite[(2.7)]{MR1836272}
\begin{equation}\label{eq: ricci bismut e chern}
\left(Ric^B(\omega)\right)^{1,1} = Ric^{Ch}(\omega) + \im\left(\p\theta^{0,1} - \bar\p\theta^{1,0} \right),
\end{equation}
where $θ^1,0$ and $θ^0,1$ are respectively the $(1,0)$ and $(0,1)$ components of the Lee form.
However, generalized flag manifolds are K\"ahler--Einstein Fano \cite[Section 5]{MR303478}, while it is known that the first Chern class of Lie groups equipped with Samelson complex structures vanishes in de Rham (hence also Aeppli) cohomology, see e.\,g. \cite[Theorem 2]{MR2764884}.
Therefore, in order to have a Bismut Hermitian Einstein metric on $(M,J)$ we can not have the flag component.
We sum up this argument in the following proposition.
\begin{prop}
Let $(M,J)$ be a C-space, and suppose that it can be equipped with a Bismut Hermitian Einstein metric.
Then, up to finite cover, it is a compact semisimple Lie group equipped with a Samelson complex structure.
\end{prop}
We remark that not any Samelson complex structure is compatible with the existence of a Bismut flat metric; indeed, only the ones which are Hermitian together with a bi-invariant metric are so.
However, the existence of a left-invariant pluriclosed metric on a compact semisimple Lie group endowed with a Samelson complex structure forces the manifold to be a Bismut flat manifold.
More precisely, we have the following result due to Fino and Grantcharov.
\begin{theo*}[Theorem 3.2 in \cite{Fino22}]
Let $G$ be a compact semisimple Lie group of even rank endowed with a Samelson
complex structure $J$ and let $g$ be a left-invariant pluriclosed $J$-Hermitian metric.
Then the complex structure $J$ has to be compatible with a bi-invariant metric.
\end{theo*}
To summarize, a C-space $(M,J)$ can be equipped with a left-invariant Bismut Hermitian Einstein metric (if and) only if it is a Bismut flat manifold.
The `if' implication trivially follows by the fact that Bismut flat metric are bi-invariant and Bismut Hermitian Einstein.
We thus reduced to the setting where we can apply Corollary \ref{cor: obstruction on Bf} to get the following result.
\begin{theo}\label{thm: non-existence on C-spaces}
Let $(M, J)$ be a C-space and suppose that $g$ is a left-invariant Bismut Hermitian Einstein metric on it. Then, up to finite cover, $M$ is a Lie group and $g$ is a bi-invariant metric. In particular, $g$ is Bismut flat.
\end{theo}

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[bor] A. Borel, K¨ahlerian coset spaces of semisimple Lie groups, Proc. Nat. Acad. Sci. 40 (1954),

%%%%%%%%%%%%%%%%%%%%% Pezzo rimosso su (2,1)_Dolbeault %%%%%%%%%%%%%%%%%%%%%%

Let $G$ be a semisimple Lie group of rank $r$ with maximal torus $T⊂G$, and equipped with a Samelson complex structure $J$.
Given a semisimple Lie group $G$ of rank $r$ with maximal torus $T⊂G$, the Tits fibration, defined as
$$ \phi : G \rightarrow G/T: g \mapsto g\cdot T, $$
gives a toric fibration
$$ \left(\S^1\right)^r \hookrightarrow G \xrightarrow{\phi} G/T$$
over the generalized flag manifold $G/T$.
We recall that the generalized flag manifolds are K\"ahler \cite{Borel}, and that they can be endowed with a (unique) invariant K\"ahler–Einstein metric \cite{MR303478}.
We thus can use Chern--Weil theory following \cite{MR1255937} to compute the Dolbeault cohomology of $G$.\\

Consider the connection one-form $θ∈^1(G,)̨$ associated to the Tits fibration $ϕ$.
We indicate with $ω∈^2(G/T,)̨$ the associated curvature form, $dθ= ϕ^*ω$.
We then define the maps on the dual of the Lie algebra of the torus
$$\Phi : \k^* \rightarrow \A^1(G) : v \mapsto \iota_v\theta$$
$$ D : \k^* \rightarrow  \A^2(G/T): v \mapsto  \iota_v\omega,$$
contracting respectively the toric components of $θ$ and $ω$.
In particular, if we choose an orthonormal basis for $$̨ with respect to the Killing form

$$\k = \left<z_1,\ldots,z_r\right>$$

the forms $\theta$ and $\omega$ can be written as

$$\theta = (\theta_1,\ldots,\theta_r),\quad \text{and } \omega=(\omega_1,\ldots,\omega_r).$$

Taking the dual basis

$$\k^* = \left<\xi^1,\ldots,\xi^r\right>,$$

the maps $\Phi$ and $D$ are such that

$$\Phi(\xi^i) = \theta_i, \quad\text{and } D(\xi^i)=\omega_i.$$

Now suppose we have a model for the Dolbeault cohomology of $G/T$

$$ \Psi: \left(\mathcal{Y},\eth\right) \xrightarrow{\sim} \left(\A^{\bullet,\bullet}(G/T),\bar\p\right). $$

Since the Dolbeault cohomology $H^2(G/T) = H^{1,1}_{\bar\p}(G/T)$ by <cit.> by applying Proposition 8 of <cit.> we obtain a model for the Dolbeault cohomology of $G$ as follows

$$\Psi\otimes\Phi : \left(\mathcal{Y}\otimes \bigwedge^{\bullet,\bullet}\left(\k^{1,0}\oplus\k^{0,1}\right), \delta \right) \xrightarrow{\sim} \left(\A^{\bullet,\bullet}(G),\bar\p\right)$$

where $\left(\k^*\right)^\C = \k^{1,0}\oplus\k^{0,1}$ is the decomposition with respect to $J$, and $\delta$ satisfies

\begin{equation*}
\delta v =
\begin{cases}
\eth v    & \text{if } v\in\mathcal{Y};\\
D(v)        & \text{if } v\in\k^{1,0};\\
0           & \text{if } v\in\k^{0,1}.
\end{cases}
\end{equation*}

As stated in <cit.>, the Dolbeault cohomology of $G/T$ is all concentrated on bi-degrees $(p,p)$.
It then follows immediately by applying the above model for the Dolbeault cohomology of $G$ that $H^{0,1}_{\bar\p}(G)=r$ and $H_{\bar\p}^{p,0}(G) = 0$ for all $p>0$.
Notice that this argument was used in <cit.> to prove that in particular the $(3,0)$-Dolbeault cohomology vanishes.

The cohomology $H^*\left(G/T\right)$ is all generated by the unit and its $2$-dimensional classes, see <cit.>.
These classes are precisely the ones of the $\omega_i$ as described in <cit.>, and the only relations among them are given by the polynomials $\R[\omega_1,\ldots,\omega_r]$ which are invariant with respect to the action of the Weil group $W_G:=N(T)/T$, see <cit.>.
Finally, comparing the result in <cit.> with the de Rham cohomology of compact simple Lie groups one deduces that the first non-trivial relation among these $2$-forms occurs in degree $2$, in other words, it is a quadratic relation in the $\omega_i$.
Since the Killing form is bi-invariant, and hence in particular invariant by the action of the Weil group, this quadratic relation must correspond to it in the following sense:

$$\omega_1\wedge\omega_1 + \omega_2\wedge\omega_2 + \cdots + \omega_r\wedge\omega_r = 0 .$$

In particular, for a semisimple Lie group, we have as many quadratic relations as the number of simple components.
Let us distinguish the coordinates on the maximal torus as

$$\k = \left<z_1^1,\ldots,z^1_{n_1},\ldots,z^s_1,\ldots,z^s_{n_s}\right>,$$

where the upper indexes indicate the simple component they belong.
If we arrange the characteristic classes accordingly, then the quadratic relations are all of the type

\begin{equation}\label{eq: quadratic relations}
\sum_{i=1}^s b_i\sum_{j=1}^{n_i}\omega^i_j\wedge\omega^i_j=0 \quad \text{for } b_i\in\R.
\end{equation}

Let $G$ be a semisimple Lie group of rank $r$ equipped with a Samelson complex structure $J$.

$$H_{\bar\p}^{1,1}(G,J)\cong \C^r.$$

Using the model for the Dolbeault cohomology of $(G,J)$ one sees that
$H_{\bar\p}^{1,1}(G,J)$ is generated by the $(1,1)$-forms $\omega_i$.
Then the result follows by the fact that there are no non-trivial relations of order one between the $\omega_i$.

We now compute the $(2,1)$-Dolbeault cohomology of $(G,J)$.
We first restrict to the case of irreducible Samelson complex structure; then the general case will follow by Künneth formula.

Let $G$ be a semisimple Lie group equipped with a bi-invariant metric $g$ and a compatible irreducible Samelson complex structure $J$.

$$H_{\bar\p}^{2,1}(G,J)\cong \C.$$

First of all, notice now that since the metric $g$ is bi-invariant it equals

$$g = -\lambda_1 B_{|_{\g_1}} -\lambda_2 B_{|_{\g_2}} - \cdots -\lambda_s B_{|_{\g_s}} ,$$

where $B$ is the Killing form of $G$.
Therefore we can scale the coordinates $\left<z_j^i\right>$
in order to obtain an orthonormal basis for $g$.
Let us keep the same symbols for the coordinates $\left<z_i\right>$ their dual $\left<\xi_i\right>$ and the connection and curvature forms $\theta_i$ and $\omega_i$ for the sake of simplicity.

Using the model for the Dolbeault cohomology of $(G,J)$ one sees that
$H_{\bar\p}^{2,1}(G,J)$ is generated by the forms

$$ \sum_{i=1}^r v_i\wedge\omega_i, \quad \text{for } \quad v_i = \sum_{j=1}^r a^i_j\left(\xi^j -\im J \xi^j \right) \in \k^{0,1}.$$

Suppose that one of these gives a non-zero representative in Dolbeault cohomology.
Then it must be

\begin{align*}
0 &= \delta\left( \sum_{i=1}^r v_i\wedge\omega_i \right)
= \sum_{i=1}^r D(v_i)\wedge\omega_i
= \sum_{i=1}^r D\left(\sum_{j=1}^r a^i_j (\xi^j -\im J \xi^j )\right)\wedge\omega_i\\
&= \sum_{i=1}^r\sum_{j=1}^r a^i_j D\left(\xi^j - \im \sum_{k=1}^r J^j_k \xi^k \right)\wedge\omega_i
= \sum_{i=1}^r\sum_{j=1}^r \left(a_j^i - \im \sum_{k=1}^r J_j^k a^i_k \right)\omega_j\wedge\omega_i,
\end{align*}

where $\left(J^j_k\right)$ is the matrix associated to the endomorphism $J$ in the coordinates $\left<z_i\right>$.
In other words, we obtain a quadratic relation in the $\omega_i$.
Thus, if we define the matrix $A:=\left(a^i_j\right)_{i,j}$, equation (<ref>) gives

\begin{equation}\label{eq: crucial}
A - \im JA + \left(A - \im JA\right)^t =
\begin{pmatrix}
b_1 \id &         &        &         \\
& b_2 \id &        &         \\
&         & \ddots &         \\
&         &        & b_s \id
\end{pmatrix}
\end{equation}

Define the matrix $Q:=A-\im JA$.
By using (<ref>) we see that

\begin{align*}
Q^i_j &= A^i_j -\im\sum_{q=1}^r J^q_jA_q^i   \\
&= A^i_j -\im J_j^iA_i^i -\im\sum_{q=1, q\neq i}^r J^q_j\left(-A_i^q + \im\sum_{p=1}^rJ_i^pA_p^q + \im\sum_{p=1}^rJ_q^pA_p^i\right)\\
&= A^i_j +2\im J_i^jA_i^i -\im\sum_{q=1}^rJ_q^jA_i^q + 2 \sum_{p=1}^r J^j_iJ_i^pA^i_p - \sum_{q=1}^r\sum_{p=1}^r J^j_qJ_i^pA^q_p -\sum_{p=1}^r\delta_j^pA_p^i\\
&= -\im\sum_{q=1}^rA_i^qJ_q^j - \sum_{q=1}^r\sum_{p=1}^rJ_i^pA^q_pJ^j_q +2\im J^j_i\left(A^i_i - \im\sum_{p=1}^r J^p_iA^i_p\right)\\
&= \left(-JAJ-\im AJ\right)_i^j +2\im J^j_i Q_i^i\\
&= -\im\left(QJ\right)_i^j + \im\left(\left(Q+Q^t\right)J\right)_i^j\\
&= \im(Q^tJ)^j_i,
\end{align*}

where we used that the matrix $J=-J^t$ in the coordinates $\left<z_i\right>$.
Similarly, we also have

\begin{align*}
Q^i_i &= A^i_i -\im\sum_{q=1}^r J^q_iA_q^i   \\
&= A^i_i -\im\sum_{q=1}^r J^q_i\left(-A_i^q + \im\sum_{p=1}^rJ_i^pA_p^q + \im\sum_{p=1}^rJ_q^pA_p^i\right)\\
&= A^i_i -\im\sum_{q=1}^rJ_q^iA_i^q - \sum_{q=1}^r\sum_{p=1}^r J^i_qJ_i^pA^q_p -\sum_{p=1}^r\delta_i^pA_p^i\\
&= -\im\sum_{q=1}^rA_i^qJ_q^i - \sum_{q=1}^r\sum_{p=1}^rJ_i^pA^q_pJ^i_q \\
&= -\im\left(AJ-\im JAJ\right)^i_i\\
&= -\im (QJ)^i_i + \im\left(\left(Q+Q^t\right)J\right)^i_i
\end{align*}

In other words

$$Q = -(JQJ)^t = -JQ^tJ. $$

$$ -J\left( Q + Q^t\right)J = Q + Q^t .$$

Then using again (<ref>) we obtain

\begin{equation}\label{eq: matrix}
-J \begin{pmatrix}
b_1 \id &         &        &         \\
& b_2 \id &        &         \\
&         & \ddots &         \\
&         &        & b_s \id
\end{pmatrix} J =
\begin{pmatrix}
b_1 \id &         &        &         \\
& b_2 \id &        &         \\
&         & \ddots &         \\
&         &        & b_s \id
\end{pmatrix}
\end{equation}

which forces the coefficients $b_i$ to be all equal.
Indeed, by multiplying (<ref>) on the left by $J$ we have

\[
\begin{pmatrix}
\cdots -b_1Jz_1^1 \cdots \\
\vdots \\
\cdots -b_kJz_k^l \cdots \\
\vdots \\
\cdots -b_sJz^s_{n_s} \cdots
\end{pmatrix} =
\begin{pmatrix}
\vdots & & \vdots & & \vdots \\
b_1Jz_1^1 & \cdots & b_kJz^k_l & \cdots & b_sJz^s_{n_s} \\
\vdots & & \vdots & & \vdots
\end{pmatrix}.
\]

Therefore, if there exist two pairs $k=1,\ldots,s$, $l=1,\ldots, n_k$ and $p=1,\ldots,s$, $q=1,\ldots, n_p$ with $p\neq k$ such that $g(Jz_l^k,z^p_q)\neq 0$ it must be $b_k=b_p$.
Finally, since the complex structure is irreducible we can guarantee this to happen and get $b_1=b_2=\cdots=b_s$.

We have proved that there is only one possible $(2,1)$-representative for the Dolbeault cohomology of $(G,J)$ up to scalar multiplication.
It actually corresponds to the bi-invariant metric $g$.

$$H^{2,1}_{\bar\p}(G,J) = \C\left<[\p g(J\cdot,\cdot)]\right>.$$

Let $G$ be a semisimple Lie group equipped with a bi-invariant metric $g$ and a compatible Samelson complex structure $J$.
Suppose that $J$ decomposes in $s$ irreducible components, then

$$H_{\bar\p}^{2,1}(G,J)\cong \C^s.$$

The complex manifold $(G,J)$ decomposes as a product $G=G_1\times\cdots\times G_s$ of complex manifolds $\left(G_i,J_{|_{G_i}}\right)$ where $J_{|_{G_i}}$ is an irreduble Samelson complex structure.
Then the result follows using the Künneth formula for Dolbeault cohomology <cit.>

$$ H^{\bullet,\bullet}_{\bar\p}(G,J) \cong H^{\bullet,\bullet}_{\bar\p}\left(G_1,J_{|_{G_1}}\right)\otimes_\C \cdots \otimes_\C H^{\bullet,\bullet}_{\bar\p}\left(G_s,J_{|_{G_s}}\right) ,$$

and the fact that $H_{\bar\p}^{p,0}\left(G_i,J_{|_{G_i}}\right)=0$ for any $i=1,\ldots,s$ and $p>0$.

The result in Corollary <ref> can be also obtained with a direct computation as in Theorem <ref> without using the Künneth formula.