$\mu_{2}$, and then for $\mu_{k}$ ($k>2$) supposing $\mu_{2}=...=\mu_{k-1}=0$
which simplifies the computations. From Proposition 5.6, we know that the
infinitesimal action of a Hamiltonian $H=v_{2}p+...+v_{n}p^{n-1}$ on the
higher Beltrami differentials is given by

$\delta\mu_{2}=(\bar{\partial}-\mu_{2}\partial+\partial\mu_{2})v_{2}$

for $\mu_{2}$ and for $\mu_{k}$, supposing $\mu_{2}=...=\mu_{k-1}=0$, we
simply have

$\delta\mu_{k}=\bar{\partial}v_{k}.$

###### Proof.

First, we compute the variation of $\mu_{2}$ using equation (12.11):

$\delta\bm{\hat{}}{\mu}_{1}+\delta\bm{\hat{}}{\mu}_{2}\nabla+...+\delta\bm{\hat{}}{\mu}_{n}\nabla^{n-1}=[\bm{\hat{}}{v}_{1}+\bm{\hat{}}{v}_{2}\nabla+...+\bm{\hat{}}{v}_{n}\nabla^{n-1},-\bar{\nabla}\\!+\\!\bm{\hat{}}{\mu}_{1}\\!+\\!\bm{\hat{}}{\mu}_{2}\nabla+...+\bm{\hat{}}{\mu}_{n}\nabla^{n-1}]\mod\bm{\hat{}}{I}.$

Since the highest $\lambda$-term of $\bm{\hat{}}{\mu}_{2}$ is of degree 0, we
are interested in the part of degree $0$ of the coefficient of $\nabla$ in
$[\bm{\hat{}}{v}_{1}+\bm{\hat{}}{v}_{2}\nabla+...+\bm{\hat{}}{v}_{n}\nabla^{n-1},-\bar{\nabla}+\bm{\hat{}}{\mu}_{1}+\bm{\hat{}}{\mu}_{2}\nabla+...+\bm{\hat{}}{\mu}_{n}\nabla^{n-1}]\mod\bm{\hat{}}{I}$.

We first look on contributions coming from
$[\bm{\hat{}}{v}_{k}\nabla^{k-1},\bm{\hat{}}{\mu}_{l}\nabla^{l-1}]$ for
$k,l\geq 2$: If $k+l-3<n$ then we do not reduce modulo $\bm{\hat{}}{I}$, so
the highest term in $\lambda$ is of degree $4-(k+l)$. Since we have $k,l\geq
2$, the highest term comes from $k=l=2$, which gives
$v_{2}\partial\mu_{2}-\mu_{2}\partial v_{2}$. If $k+l-3\geq n$, we can have
terms with $\nabla^{m}$ with $n\leq m\leq k+l-3$. So we have to use
$\bm{\hat{}}{I}$ to reduce it. This reduction gives
$\nabla^{m}=c(\lambda)\nabla+\text{ other terms}$, and the highest term of
$c(\lambda)$ is of degree $m-2\leq k+l-5$. Hence, the highest term for
$[\bm{\hat{}}{v}_{k}\nabla^{k-1},\bm{\hat{}}{\mu}_{l}\nabla^{l-1}]$ is
$4-(k+l)+k+l-5=-1$.

The contributions from $\bm{\hat{}}{\mu}_{1}$ and $\bm{\hat{}}{v}_{1}$ also
have degree at most -1. There is one more contribution in degree 0 coming from
$[\bm{\hat{}}{v}_{2}\nabla,-\bar{\nabla}]$, which gives $\bar{\partial}v_{2}$.
Therefore, we have

$\delta\mu_{2}=(\bar{\partial}-\mu_{2}\partial+\partial\mu_{2})v_{2}.$

Now, suppose $\mu_{2}=...=\mu_{k-1}=0$ and compute the variation
$\delta\mu_{k}$ under an action generated by
$\bm{\hat{}}{v}_{k}\nabla^{k-1}+...+\bm{\hat{}}{v}_{n}\nabla^{n-1}$. From

$\delta\bm{\hat{}}{\mu}_{k}\nabla^{k-1}+...+\delta\bm{\hat{}}{\mu}_{n}\nabla^{n-1}=[\bm{\hat{}}{v}_{k}\nabla^{k-1}+...+\bm{\hat{}}{v}_{n}\nabla^{n-1},-\bar{\nabla}+\bm{\hat{}}{\mu}_{1}+\bm{\hat{}}{\mu}_{2}\nabla+...+\bm{\hat{}}{\mu}_{n}\nabla^{n-1}]\mod\bm{\hat{}}{I}$

we can analyze as above the contribution to the term of degree $2-k$ of the
coefficient of $\nabla^{k-1}$. Since $\bm{\hat{}}{v}_{l}$ is of degree at most
$2-l$ and $\bm{\hat{}}{\mu}_{l}$ of degree at most $1-l$ for $l<k$ (since we
suppose that $\mu_{l}=0$), we can see that
$[\bm{\hat{}}{v}_{l}\nabla^{l-1},\bm{\hat{}}{\mu}_{m}\nabla^{m-1}]$ cannot
contribute to the highest degree. The only contribution comes from the term
with $-\bar{\nabla}$. Thus,

$\delta\mu_{k}=\bar{\partial}v_{k}.$

This concludes the proof since the action of higher diffeomorphisms on the
$n$-complex structure has the same expression. ∎

######  Corollary 13.5.

With the action of higher diffeomorphisms, we can locally render $\Phi_{2}=0$.

The corollary directly follows from the previous theorem and the fact that the
higher complex structure can be locally trivialized (Theorem 5.7), i.e. we can
render $\mu_{2}=...=\mu_{n}=0$ locally and since
$\Phi_{2}=\mu_{2}\Phi_{1}+...+\mu_{n}\Phi_{1}^{n-1}$ this implies
$\Phi_{2}=0$.

######  Remark.

We see that a term $\bm{\hat{}}{v}_{k}\nabla^{k-1}$ can influence
$\bm{\hat{}}{\mu}_{i}$ with $i<k$ (unlike the case higher complex structures
where the simplification lemma 5.5 holds), but it does not influence the
highest term $\mu_{i}$. In the same vein, a term
$\bm{\hat{}}{v}_{k}\nabla^{k-1}$ with $k>n$ acts on parabolic connections, but
not on the highest terms. $\triangle$

######  Remark.

We can also compute the action of $\operatorname{Symp}_{0}(T^{*}\Sigma)$ on
the coordinates $\bm{\hat{}}{t}_{k}$ and their highest terms $t_{k}$. The
computation gives that $t_{k}$ transforms as a cotangent vector to the
$n$-complex structure, i.e. like
$I=\langle-p^{n}+t_{2}p^{n-2}+...+t_{n},-\bar{p}+\mu_{2}p+...+\mu_{n}p^{n-1}\rangle$
under the action of higher diffeomorphisms, modulo $t^{2}$. $\triangle$

#####  13.3.2 Global analysis

We show that the highest term in $\lambda$ in the zero-curvature condition
relates $(\mu_{k},t_{k})$ to the cotangent bundle
$T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$.

We know that the moment map of the hamiltonian action of
$\operatorname{Symp}_{0}(T^{*}\Sigma)$ on $\mathcal{A}\sslash\mathcal{P}$ is
given by $\xi_{k}=0$, i.e. the remaining curvature of a parabolic connection
has to vanish. For connections with parameter $\lambda$, this gives
$\xi_{k}(\lambda)=0$.

######  Theorem 13.6.

The highest term in $\lambda$ of $\xi_{k}(\lambda)=0$ gives the condition
$(\mathcal{C})$ of the cotangent bundle $T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$
(see Theorem 7.1).

The proof strategy is to reduce the analysis of the highest term in the
parabolic curvature to the expression
$\xi_{k}\mod\bm{\hat{}}{t}^{2}\mod\partial^{2}$. The following lemma shows
that we then get condition $(\mathcal{C})$. The hurried reader may skip the
proof of this technical lemma.

######  Lemma 13.7.

The parabolic curvature modulo $\bm{\hat{}}{t}^{2}$ and $\partial^{2}$ gives
condition $(\mathcal{C})$ on $T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$:

$\xi_{k}=(\bar{\partial}\\!-\\!\bm{\hat{}}{\mu}_{2}\partial\\!-\\!k\partial\bm{\hat{}}{\mu}_{k})\bm{\hat{}}{t}_{k}-\sum_{l=1}^{n-k}\left((l\\!+\\!k)\partial\bm{\hat{}}{\mu}_{l+2}+(l\\!+\\!1)\bm{\hat{}}{\mu}_{l+2}\partial\right)\bm{\hat{}}{t}_{k+l}\mod\bm{\hat{}}{t}^{2}\mod\partial^{2}.$

###### Proof.

The proof is a combination of several formulas:

  1. 1.

Proposition 12.2 together with the expression of the differential operators
(see (12.5)) give

$[\partial^{n}-\bm{\hat{}}{t}_{2}\partial^{n-2}-...-\bm{\hat{}}{t}_{n},-\bar{\partial}+\bm{\hat{}}{\alpha}_{nn}+\bm{\hat{}}{\alpha}_{n,n-1}\partial+...+\bm{\hat{}}{\alpha}_{n,1}\partial^{n-1}]=\sum_{k=2}^{n}\xi_{k}\partial^{n-k}\mod\bm{\hat{}}{I}$

where $\bm{\hat{}}{\alpha}_{ij}$ are the entries of the matrix $A_{2}$.

  2. 2.

Link between Poisson bracket and commutator:

$\displaystyle\\{p^{n}-\bm{\hat{}}{t}_{2}p^{n-2}-...-\bm{\hat{}}{t}_{n},-\bar{p}+\bm{\hat{}}{\mu}_{1}+\bm{\hat{}}{\mu}_{2}p+...+\bm{\hat{}}{\mu}_{n}p^{n-1}\\}$
$\displaystyle=\lim_{h\rightarrow
0}\frac{1}{h}[h^{n}\partial^{n}-\bm{\hat{}}{t}_{2}h^{n-2}\partial^{n-2}-...-\bm{\hat{}}{t}_{n},-h\bar{\partial}+\bm{\hat{}}{\mu}_{1}+\bm{\hat{}}{\mu}_{2}h\partial+...+\bm{\hat{}}{\mu}_{n}h^{n-1}\partial^{n-1}]\Big{|}_{\begin{subarray}{c}h\partial\mapsto
p\\\ h\bar{\partial}\mapsto\bar{p}\end{subarray}}.$

  3. 3.

Link between $\mod\partial^{2}$ and brackets: for $h$-connections $D_{1}(h)$
and $D_{2}(h)$, we have

$\lim_{h\rightarrow
0}\frac{1}{h}[D_{1}(h),D_{2}(h)]=\frac{1}{h}[D_{1}(h),D_{2}(h)]\mod\partial^{2}.$

  4. 4.

The formula from Proposition 7.3 linking the Poisson bracket to condition
$(\mathcal{C})$ (asserting that the spectral curve is Lagrangian):

$\displaystyle\\{p^{n}-\bm{\hat{}}{t}_{2}p^{n-2}-...-\bm{\hat{}}{t}_{n},-\bar{p}+\bm{\hat{}}{\mu}_{1}+\bm{\hat{}}{\mu}_{2}p+...+\bm{\hat{}}{\mu}_{n}p^{n-1}\\}$
$\displaystyle=$
$\displaystyle\sum_{k=2}^{n}\left((\bar{\partial}\\!-\\!\bm{\hat{}}{\mu}_{2}\partial\\!-\\!k\partial\bm{\hat{}}{\mu}_{k})\bm{\hat{}}{t}_{k}-\sum_{l=1}^{n-k}\left((l\\!+\\!k)\partial\bm{\hat{}}{\mu}_{l+2}+(l\\!+\\!1)\bm{\hat{}}{\mu}_{l+2}\partial\right)\bm{\hat{}}{t}_{k+l}\right)p^{n-k}\mod\bm{\hat{}}{t}^{2},I.$

Now, we are ready to conclude. By a direct computation, we can see that modulo
$\bm{\hat{}}{t}^{2},\partial^{2}$ we can replace
$\bm{\hat{}}{\alpha}_{n,n+1-l}$ by $\bm{\hat{}}{\mu}_{l}$ in point 2. Define
$D_{1}(h)=h^{n}\partial^{n}-\bm{\hat{}}{t}_{2}h^{n-2}\partial^{n-2}-...-\bm{\hat{}}{t}_{n}$
and
$D_{2}(h)=-h\bar{\partial}+\bm{\hat{}}{\mu}_{1}+\bm{\hat{}}{\mu}_{2}h\partial+...+\bm{\hat{}}{\mu}_{n}h^{n-1}\partial^{n-1}$.
Then using 1. to 4. and computing modulo $\bm{\hat{}}{t}^{2}$ and
$\partial^{2}$, we get:

$\displaystyle\sum_{k=2}^{n}\xi_{k}p^{n-k}$
$\displaystyle=\sum_{k=2}^{n}\xi_{k}(h\partial)^{n-k}\Big{|}_{h\partial\mapsto
p}$
$\displaystyle=\left(\frac{1}{h}[D_{1}(h),D_{2}(h)]\right)\Big{|}_{\begin{subarray}{c}h\partial\mapsto
p\\\ h\bar{\partial}\mapsto\bar{\partial}\end{subarray}}\mod\bm{\hat{}}{I}$
$\displaystyle=\lim_{h\rightarrow
0}\frac{1}{h}[D_{1}(h),D_{2}(h)]\Big{|}_{\begin{subarray}{c}h\partial\mapsto
p\\\ h\bar{\partial}\mapsto\bar{\partial}\end{subarray}}\mod\bm{\hat{}}{I}$
$\displaystyle=\\{p^{n}-\bm{\hat{}}{t}_{2}p^{n-2}-...-\bm{\hat{}}{t}_{n},-\bar{p}+\bm{\hat{}}{\mu}_{1}+\bm{\hat{}}{\mu}_{2}p+...+\bm{\hat{}}{\mu}_{n}p^{n-1}\\}\mod
I$
$\displaystyle=\sum_{k=2}^{n}\left((\bar{\partial}\\!-\\!\bm{\hat{}}{\mu}_{2}\partial\\!-\\!k\partial\bm{\hat{}}{\mu}_{k})\bm{\hat{}}{t}_{k}-\sum_{l=1}^{n-k}\left((l\\!+\\!k)\partial\bm{\hat{}}{\mu}_{l+2}+(l\\!+\\!1)\bm{\hat{}}{\mu}_{l+2}\partial\right)\bm{\hat{}}{t}_{k+l}\right)p^{n-k}$

Comparing coefficients, we get the lemma. ∎

We can now give the proof of Theorem 13.6:

###### Proof.

From the explicit expression of $\xi_{k}(\lambda)$, we know that only
derivatives, $\bm{\hat{}}{t}_{k}$’s and $\bm{\hat{}}{\mu}_{k}$’s appear. Since
we are only interested in the highest term, we can replace
$\bm{\hat{}}{t}_{k}$ by $\lambda^{k-1}t_{k}$ and $\bm{\hat{}}{\mu}_{k}$ by
$\lambda^{2-k}\mu_{k}$. Hence, we get an expression which is a tensor, since
both $t_{k}$ and $\mu_{k}$ are tensors (by Proposition 13.3). Since one term
is $\bar{\partial}t_{k}$, we know that the highest term of $\xi_{k}(\lambda)$
is a section of $K^{k}\otimes\bar{K}$ and is of degree $k-1$ in $\lambda$.

In addition, we know that every term in $\xi_{k}$, apart from
$\bar{\partial}t_{k}$, has at least one partial derivative $\partial$, which
adds a $K$-factor to the tensor. The rest is thus at most of type
$K^{k-1}\otimes\bar{K}$. The $\bar{K}$-factor comes from a unique $\mu_{m}$ in
each term. Once this $\mu_{m}$ fixed, only partial derivatives $\partial$ and
$t_{k}$’s contribute to the $K$-factor.

Since $t_{k}$ comes with a factor $\lambda^{k-1}$, we see that whenever there
is a term with a factor $t_{i}t_{j}$, the contribution in $\lambda$ is
$\lambda^{i+k-2}$ which is not optimal, since $t_{i+j}$ would contribute with
$\lambda^{i+j-1}$. In the same vein, whenever there is a term with at least
two $\partial$, so that the rest is a tensor of type at most
$K^{k-2}\otimes\bar{K}$, this term does not have an optimal contribution in
$\lambda$.

Therefore, the highest term in $\xi_{k}(\lambda)$ is the same as in
$\xi_{k}(\lambda)\mod\bm{\hat{}}{t}^{2}\mod\partial^{2}$. Finally, the
statement of the previous Lemma 13.7 concludes the proof of Theorem 13.6. ∎

With the previous theorem, we now understand the global meaning of the highest
terms $(\mu_{k},t_{k})$: the $\mu_{k}$ are the higher Beltrami differentials
coming from the higher complex structure, whereas the $t_{k}$ are a cotangent
vector to that higher complex structure. We can say that the semi-classical
limit of $\mathcal{A}\sslash\mathcal{P}\sslash\operatorname{Symp}_{0}$ is
$T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$, which confirms the twistor space picture
10.1.

The question remains how to determine the coefficients of lower degree in
$\bm{\hat{}}{\mu}_{k}$ and $\bm{\hat{}}{t}_{k}$. This will be discussed in the
next section.

###  14 Conjectural geometric approach to Hitchin components

In this section, we try to construct an analog to the non-abelian Hodge
correspondence in our setting: the existence and uniqueness of real twistor
lines. We give partial results and conjectures. Assuming the existence of real
twistor lines, we prove a canonical diffeomorphism between higher complex
structures and Hitchin components.

Consider $\mathcal{A}(\lambda)=\lambda\Phi+A+\lambda^{-1}\Phi^{*}$ where
$\Phi=\Phi_{1}+\Phi_{2}$ is given by an $n$-complex structure. Now, we look at
$\mathcal{A}(\lambda)$ as a twistor line, i.e. a section of the twistor space.
We impose the reality condition

$-\mathcal{A}(-1/\bar{\lambda})^{*}=\mathcal{A}(\lambda).$

Notice that $-1/\bar{\lambda}$ is the diametrically opposed point of $\lambda$
in $\mathbb{C}P^{1}$. For trivial $n$-complex structure the $*$-operator is
the hermitian conjugate $M^{*}=M^{\dagger}=\bar{M}^{\top}$. Intrinsically, the
operation $A\mapsto-A^{*}$ is an antiholomorphic involution which corresponds
to the compact real form of $\mathfrak{sl}_{n}$.

######  Remark.

For general higher complex structure, the real structure $*$ has to be defined
in such a way that $\operatorname{Symp}_{0}$ preserves it. It might be
necessary to have a hermitian structure on the bundle. $\triangle$

####  14.1 Case $n=2$ and $n=3$

Let us study the examples of smallest rank, those with $n=2$ and $n=3$. We
work locally, so we can suppose that the $n$-complex structure is trivial,
i.e. $\mu_{k}=0$ for $k=2,3$. We use the standard form from subsection 11.1.

For $n=2$, write $\Phi_{1}=\left(\begin{smallmatrix}0&0\\\
e^{\varphi}&0\end{smallmatrix}\right)$,
$A_{1}=\left(\begin{smallmatrix}a_{0}&a_{1}\\\
a_{2}&-a_{0}\end{smallmatrix}\right)$ and $A_{2}=-A_{1}^{\dagger}$. So we have

$\mathcal{A}(\lambda)=\begin{pmatrix}a_{0}&a_{1}\\\ a_{2}+\lambda
e^{\varphi}&-a_{0}\end{pmatrix}dz+\begin{pmatrix}-\bar{a}_{0}&-\bar{a}_{2}+\lambda^{-1}e^{\varphi}\\\
-\bar{a}_{1}&\bar{a}_{0}\end{pmatrix}d\bar{z}.$

Notice that this is example 13.1 with $\mu_{2}=0$ and $b_{1}=e^{\varphi}$. The
flatness equation gives

$\left\\{\begin{array}[]{cl}a_{2}e^{\varphi}&=\;0\\\
\bar{\partial}\varphi&=\;-2\bar{a}_{0}\\\
\bar{\partial}a_{1}&=\;2\bar{a}_{0}a_{1}\\\
\partial\bar{a}_{0}+\bar{\partial}a_{0}&=\;-a_{1}\bar{a}_{1}-e^{2\varphi}.\end{array}\right.$

The first equation gives $a_{2}=0$, the second
$a_{0}=-\frac{\partial\varphi}{2}$, the third is automatic once we write
$a_{1}=t_{2}e^{-\varphi}$, where $t_{2}$ is the holomorphic quadratic
differential from the affine connection (see section 11). Finally, the last
equation gives

$\partial\bar{\partial}\varphi=e^{2\varphi}+t_{2}\bar{t}_{2}e^{-2\varphi}$

which is the so-called $\mathbf{\cosh}$-Gordon equation, which is elliptic for
small $t_{2}$. So we see that the flat connection is uniquely determined by
$\mu_{2}=0,t_{2}$ and a solution to the $\cosh$-Gordon equation. More details
for this case can be found in [Fo08], in particular a link to minimal surface
sections in $\Sigma\times\mathbb{R}$.

For $n=3$, take $\Phi_{1}=\left(\begin{smallmatrix}&&\\\ c_{1}&&\\\
b_{2}&c_{2}&\end{smallmatrix}\right)$. As for $n=2$ the matrix $A_{1}$ is
upper triangular. Thus, we get

$\mathcal{A}(\lambda)=\begin{pmatrix}a_{0}&b_{0}&c_{0}\\\ \lambda
c_{1}&a_{1}&b_{1}\\\ \lambda b_{2}&\lambda
c_{2}&a_{2}\end{pmatrix}dz+\begin{pmatrix}-\bar{a}_{0}&\lambda^{-1}\bar{c}_{1}&\lambda^{-1}\bar{b}_{2}\\\
-\bar{b}_{0}&-\bar{a}_{1}&\lambda^{-1}\bar{c}_{2}\\\
-\bar{c}_{0}&-\bar{b}_{1}&-\bar{a}_{2}\end{pmatrix}d\bar{z}.$

Notice that we get the same expression as in example 11.3. With a diagonal
gauge, we can suppose
$c_{1}=e^{\varphi_{1}},c_{2}=e^{\varphi_{2}}\in\mathbb{R}_{+}$. Further, we
know the expressions for the holomorphic differentials to be
$t_{3}=c_{0}c_{1}c_{2}$ and $t_{2}=b_{0}c_{1}+b_{1}c_{2}+b_{2}c_{0}$, hence
$c_{0}=t_{3}e^{-\varphi_{1}-\varphi_{2}}$ and
$b_{1}=-e^{\varphi_{1}-\varphi_{2}}b_{0}-b_{2}t_{3}e^{-2\varphi_{2}-\varphi_{1}}$.

The flatness condition and the zero trace condition then give
$a_{0}=-\frac{2}{3}\partial\varphi_{1}-\frac{1}{3}\partial\varphi_{2}$,
$a_{1}=\frac{1}{3}\partial\varphi_{1}-\frac{1}{3}\partial\varphi_{2}$ and
$a_{2}=-a_{0}-a_{1}$.

Let us consider the case where $t_{2}=t_{3}=0$. Then $c_{0}=0$ and
$b_{1}=-e^{\varphi_{1}-\varphi_{2}}b_{0}$. The remaining equations of the
flatness are

$\left\\{\begin{array}[]{cl}\bar{\partial}b_{2}&=\;b_{2}(\bar{\partial}\varphi_{1}+\bar{\partial}\varphi_{2})-\bar{b}_{0}(e^{\varphi_{2}}+e^{2\varphi_{1}-\varphi_{2}})\\\
-\bar{\partial}b_{0}&=\;b_{0}\bar{\partial}\varphi_{1}+\bar{b}_{2}e^{\varphi_{2}}\\\
2\partial\bar{\partial}\varphi_{1}&=\;2e^{2\varphi_{1}}-e^{2\varphi_{2}}+b_{2}\bar{b}_{2}+b_{0}\bar{b}_{0}(2-e^{2\varphi_{1}-2\varphi_{2}})\\\
2\partial\bar{\partial}\varphi_{2}&=\;2e^{2\varphi_{2}}-e^{2\varphi_{1}}+b_{2}\bar{b}_{2}+b_{0}\bar{b}_{0}(-1+2e^{2\varphi_{1}-2\varphi_{2}}).\end{array}\right.$

For $b_{0}=b_{2}=0$ we get the Toda integrable system for $\mathfrak{sl}_{3}$.
This is the same solution as the one obtained from the non-abelian Hodge
correspondence applied to the principal nilpotent Higgs field. We see that we
need some extra data in order to impose $b_{0}=b_{2}=0$. The two variables
$b_{0}$ and $b_{2}$ are solutions to a system of differential equations. Thus,
we only need some initial conditions.

For $t_{2}=0$ and $t_{3}\neq 0$, if we impose $b_{0}=b_{1}=b_{2}=0$ and
$\varphi_{1}=\varphi_{2}=\varphi$, the flatness becomes Ţiţeica’s equation

$2\partial\bar{\partial}\varphi=e^{2\varphi}+t_{3}\bar{t}_{3}e^{-4\varphi}.$
(14.1)

From [LMc16], we know that Ţiţeica’s equation is linked to affine spheres,
minimal embeddings and Hitchin representations.

Before going to the general case, we push the similarity to Higgs bundles
further by choosing a special gauge.

####  14.2 Higgs gauge

Up to now, we have seen the flat connection $\mathcal{A}(\lambda)$ in two
gauges. The first, which we call symmetric gauge, is the form
$\mathcal{A}(\lambda)=\lambda\Phi+A+\lambda^{-1}\Phi^{*}$ where
$A_{2}=-A_{1}^{*}$ and the $*$-operator is the hermitian conjugate. The
second, which we call parabolic gauge and which in the literature is sometimes
called $W$-gauge or Drinfeld–Sokolov gauge, is the form described in equation
(13.1) where our parameters $\tilde{t}_{k}(\lambda)$ and
$\tilde{\mu}_{k}(\lambda)$ appear. The existence of parabolic gauge (see
subsection 12.2.3) assures that one can go from the symmetric to the parabolic
gauge. In Higgs theory, there is a third gauge used, which we call Higgs
gauge, characterized by $A_{2}=0$ and by the fact that $\Phi_{1}$ is a
companion matrix. Here we show that for trivial higher complex structure,
there exists the Higgs gauge in our setting. This allows to compare even
closer the Higgs bundle setting and our setting.

We start with the existence of the Higgs gauge for trivial higher complex
structure. We denote by $\mathcal{E}_{-}$ the sum of the negative simple
roots, i.e. $\mathcal{E}_{-}=\left(\begin{smallmatrix}0&&&\\\ 1&0&&\\\
&\ddots&\ddots&\\\ &&1&0\end{smallmatrix}\right)$.

######  Proposition 14.1.

For $\mu=0$ and a flat connection $\lambda\Phi+A+\lambda^{-1}\Phi^{*}$ in
symmetric gauge, there is a gauge $P$ which is lower triangular transforming
$\Phi_{1}$ to $\mathcal{E}_{-}$ and $A_{2}$ to 0.

###### Proof.

The statement is equivalent to the following two equations:

$P\Phi_{1}=\mathcal{E}_{-}P\;\text{ and }\;PA_{2}-\bar{\partial}P=0.$

The first matrix equation allows to express all entries $p_{i,j}$ of $P$ in
terms of the last row $(p_{n,k})_{1\leq k\leq n}$.

We then put $\Phi_{1}=P^{-1}\mathcal{E}_{-}P$ into the flatness equation
$0=\bar{\partial}\Phi_{1}+[A_{2},\Phi_{1}]$. After some manipulation, we get

$0=[\mathcal{E}_{-},(\bar{\partial}P)P^{-1}-PA_{2}P^{-1}].$

We know that the centralizer of $\mathcal{E}_{-}$ are polynomials in
$\mathcal{E}_{-}$. Hence we get

$\bar{\partial}P-PA_{2}=\begin{pmatrix}0&&&\\\ w_{2}&0&&\\\
\vdots&\ddots&\ddots&\\\ w_{n}&\cdots&w_{2}&0\end{pmatrix}P.$

Looking at the $n$ equations given by the last row, we can choose
$(p_{n,k})_{1\leq k\leq n}$ such that $w_{2}=...=w_{n}=0$. Therefore
$\bar{\partial}P=PA_{2}$, i.e. $A_{2}$ is transformed to 0. ∎

In the Higgs gauge, our flat connection takes the following form:

######  Proposition 14.2.

We suppose $\mu=0$. The flat connection $\mathcal{A}(\lambda)$ in Higgs gauge
is locally given by

$(\lambda\mathcal{E}_{-}+A)dz+\lambda^{-1}\mathcal{E}_{-}^{*}d\bar{z}$

where the $*$-operation is given by $M^{*}=HM^{\dagger}H^{-1}$ for some
hermitian matrix $H$. Further, we have
$\operatorname{tr}\mathcal{E}_{-}^{k}A=t_{k+1}$ and $A=-(\partial H)H^{-1}$.

###### Proof.

From the existence of Higgs gauge, we know that $\Phi_{1}=\mathcal{E}_{-}$ and
$A_{2}=0$. Since $\mu=0$, we also have $\Phi_{2}=0$. A direct computation
shows that if $P$ denotes the matrix from the Higgs gauge, the matrix
$\Phi_{1}^{*}$ transforms to
$PP^{\dagger}\mathcal{E}_{-}^{\dagger}(PP^{\dagger})^{-1}$. So
$H=PP^{\dagger}$ which is indeed a hermitian matrix.

Since $P$ is lower triangular, $t_{k+1}=\operatorname{tr}\Phi_{1}^{k}A_{1}$
transforms to $t_{k+1}=\operatorname{tr}\mathcal{E}_{-}^{k}A$.

Finally, since $A_{2}=P^{-1}\bar{\partial}P$ and $A_{2}=-A_{1}^{\dagger}$, we
get $A_{1}=-(\partial P^{\dagger})P^{\dagger\;-1}$ which transforms under $P$
to $A=-\partial(PP^{\dagger})(PP^{\dagger})^{-1}=-(\partial H)H^{-1}$. ∎

We see that $\mathcal{A}(\lambda)$ in the Higgs gauge becomes close to a Higgs
bundle. But in our setting the holomorphic differentials are in $A$, and not
in the Higgs field. We illustrate the similarity for $n=2$.

######  Example 14.3.

For $n=2$ and $\mu=0$, we have seen in the previous subsection 14.1 that in
symmetric gauge, our connection reads

$\mathcal{A}(\lambda)=\begin{pmatrix}-\frac{\partial\varphi}{2}&t_{2}e^{-\varphi}\\\
\lambda
e^{\varphi}&\frac{\partial\varphi}{2}\end{pmatrix}dz+\begin{pmatrix}\frac{\bar{\partial}\varphi}{2}&\lambda^{-1}e^{\varphi}\\\
-\bar{t}_{2}e^{-\varphi}&-\frac{\bar{\partial}\varphi}{2}\end{pmatrix}d\bar{z}.$

The flatness condition is equivalent to the $\cosh$-Gordon equation
$\partial\bar{\partial}\varphi=e^{2\varphi}+t_{2}\bar{t}_{2}e^{-2\varphi}$ and
$\bar{\partial}t_{2}=0$.

A direct computation gives the form in parabolic gauge:

$\mathcal{A}(\lambda)=\begin{pmatrix}0&\bm{\hat{}}{t}_{2}(\lambda)\\\
1&0\end{pmatrix}dz+\begin{pmatrix}-\frac{1}{2}\partial\bm{\hat{}}{\mu}_{2}&-\frac{1}{2}\partial^{2}\bm{\hat{}}{\mu}_{2}+\bm{\hat{}}{t}_{2}\bm{\hat{}}{\mu}_{2}\\\
\bm{\hat{}}{\mu}_{2}(\lambda)&\frac{1}{2}\partial\bm{\hat{}}{\mu}_{2}\end{pmatrix}d\bar{z}$

where $\bm{\hat{}}{t}_{2}(\lambda)=\lambda
t_{2}+(\partial\varphi)^{2}-\partial^{2}\varphi$ and
$\bm{\hat{}}{\mu}_{2}(\lambda)=-\lambda^{-1}\bar{t}_{2}e^{-2\varphi}$.

In Higgs gauge, we get

$\mathcal{A}(\lambda)=\begin{pmatrix}-\partial\varphi-
t_{2}p_{2}e^{-\varphi/2}&t_{2}\\\ \lambda-
a_{1}&\partial\varphi+t_{2}p_{2}e^{-\varphi/2}\end{pmatrix}dz+\begin{pmatrix}-\lambda^{-1}p_{2}e^{3\varphi/2}&\lambda^{-1}e^{2\varphi}\\\
-\lambda^{-1}p_{2}^{2}e^{\varphi}&\lambda^{-1}p_{2}e^{3\varphi/2}\end{pmatrix}d\bar{z}$

where $a_{1}=(\partial
p_{2}+\frac{3}{2}p_{2}\partial\varphi+t_{2}p_{2}^{2}e^{-\varphi/2})e^{-\varphi/2}$
and $p_{2}$ comes from the matrix of the Higgs gauge and satisfies
$\bar{\partial}p_{2}=-\bar{t}_{2}e^{-3\varphi/2}+p_{2}\frac{\bar{\partial}\varphi}{2}$.

Finally, we can compare to the non-abelian Hodge correspondence which gives

$\mathcal{A}(\lambda)=\begin{pmatrix}-\partial\varphi&\lambda t_{2}\\\
\lambda&\partial\varphi\end{pmatrix}dz+\begin{pmatrix}0&\lambda^{-1}e^{2\varphi}\\\
\lambda^{-1}\bar{t}_{2}e^{-2\varphi}\end{pmatrix}d\bar{z}.$ (14.2)

The flatness condition is equivalent to the $\sinh$-Gordon equation
$\partial\bar{\partial}\varphi=e^{2\varphi}-t_{2}\bar{t}_{2}e^{-2\varphi}$,
which is elliptic, and $\bar{\partial}t_{2}=0$. Notice that for $t_{2}=0$ (in
both approaches), the connection (14.2) is the same as our connection in the
Higgs gauge (since $p_{2}=0$ for $t_{2}=0$). $\triangle$

For non-trivial $n$-complex structure $\mu\neq 0$, there is no Higgs gauge.
Even for $n=2$, one can check that there is no $P$ satisfying
$P\Phi_{1}=\mathcal{E}_{-}P$ and $PA_{2}-\bar{\partial}P=0$.

####  14.3 General case

Set $t=(t_{2},...,t_{n})$ and $\mu=(\mu_{2},...,\mu_{n})$. To examine the
existence of an analog to the non-abelian Hodge correspondence, we discuss the
cases when $t=0$ or $\mu=0$.

Case $t=0$ and $\mu=0$. For the trivial structure we find the following
result, generalizing the observations for $n=2$ and $n=3$ from the previous
subsection 14.1.

######  Proposition 14.4.

For $\Phi_{2}=0$ and $t=0$, the flat connection $\mathcal{A}(\lambda)$ is
uniquely determined up to some finite initial data. There is a choice of
initial data such that the flatness equations are equivalent to the Toda
integrable system. In particular $\mathcal{A}(\lambda)$ is the same as the
connection given by the non-abelian Hodge correspondence applied to a
principal nilpotent Higgs field.

###### Proof.

Using Lemmas 11.1 and 11.2, we can write $\mathcal{A}_{1}(\lambda)$ in the
following form:

$\mathcal{A}_{1}(\lambda)=a_{0}+a_{1}T+...+a_{n}T^{n}$

where $a_{i}$ are diagonal matrices and $T$ is given by

$T=\begin{pmatrix}&1&&\\\ &&\ddots&\\\ &&&1\\\ \lambda&&&\end{pmatrix}.$
(14.3)

We denote by $a_{i,j}$ the $j$-th entry of the diagonal matrix $a_{i}$ and
$a_{i}^{\prime}$ the shifted matrix with $a^{\prime}_{i,j}=a_{i,j+1}$. We
write $a^{(k)}$ for the shift applied $k$ times. Notice that $aT=Ta^{(n-1)}$.
We can then write

$\mathcal{A}_{2}(\lambda)=a_{0}^{*}+T^{-1}a_{1}^{*}+...+T^{-n}a_{n}^{*}$

where $a^{*}_{i,j}=\pm\bar{a}_{i,j}$, the sign depends on whether the
coefficient comes with a $\lambda$ or not in $\mathcal{A}_{2}(\lambda)$.

By the standard form (Lemma 11.1) we can further impose
$a_{n,i}=e^{\varphi_{i}}$ for $i=1,...,n-1$ and $a_{n,0}=0$ since
$0=t_{n}=\prod_{i}a_{n,i}$. One of the flatness equations gives
$\bar{\partial}a_{n}=a_{n}(a_{0}^{(n-1)}-a_{0})$. Together with the condition
that the trace is 0, we can compute $a_{0}$. We get

$a_{0,i}=\sum_{k=1}^{i-1}\frac{k}{n}\partial\varphi_{k}-\sum_{k=i}^{n-1}\frac{n-k}{k}\partial\varphi_{k}.$
(14.4)

The other equations give a system of differential equations in
$a_{1},...,a_{n-1}$ which is quadratic. It allows the solution $a_{i}=0$ for
all $i=1,...,n-1$. In that case, using a diagonal gauge
$\operatorname{diag}(1,\lambda,...,\lambda^{n-1})$ the connection
$\mathcal{A}(\lambda)$ becomes

$\mathcal{A}(\lambda)=\begin{pmatrix}*&&&\\\ e^{\varphi_{1}}&*&&\\\
&\ddots&*&\\\
&&e^{\varphi_{n-1}}&*\end{pmatrix}dz+\begin{pmatrix}*&e^{\varphi_{1}}&&\\\
&*&\ddots&\\\ &&*&e^{\varphi_{n-1}}\\\ &&&*\end{pmatrix}d\bar{z}$ (14.5)

where on the diagonals are the $a_{0,i}$ and $-\bar{a}_{0,i}$ given by
equation (14.4). This is precisely the form of the Toda system. It is known
that the Hitchin equations for a principal nilpotent Higgs field are the Toda
equations for $\mathfrak{sl}_{n}$ (see [AF95], proposition 3.1). ∎

Notice that in particular the gauge class of the connection
$\mathcal{A}(\lambda)$ is independent of $\lambda\in\mathbb{C}^{*}$ (i.e. we
have a variation of Hodge structure). This is an intrinsic property which
might be used to fix the initial data.

Putting (14.5) in parabolic gauge, we get the following explicit formula for
our coordinates $\tilde{t}(\lambda)$ and $\tilde{\mu}(\lambda)$ (see also
proposition 3.1 and 4.4 in [AF95]):

######  Proposition 14.5.

For $\mu=0$ and $t=0$, one can choose initial conditions such that
$\tilde{\mu}_{k}(\lambda)=0$ and $\tilde{t}_{k}(\lambda)=w_{k}$ for all $k$,
where the $w_{k}$ are given by $\det(\partial-A_{1})=\prod_{i}(\partial-
a_{0,i})=\partial^{n}+w_{2}\partial^{n-2}+...+w_{n}$ (a “Miura transform”).
Furthermore, $A_{1}$ is diagonal given by equation (14.4) and the parabolic
gauge is upper triangular.

Case $t=0$. We get the following result:

######  Proposition 14.6.

For $t=0$, the connection $\mathcal{A}(\lambda)$ is determined by the flatness
condition and by some initial conditions. Its monodromy is in
$\operatorname{PSL}_{n}(\mathbb{R})$.

The idea of the proof is the following: locally, one can trivialize the higher
complex structure, so we are led to $\mu=0$ and $t=0$. Thus
$\mathcal{A}(\lambda)$ is given by the non-abelian Hodge correspondence and we
can apply Hitchin’s strategy to prove real monodromy, which is a local
argument.

###### Proof.

By Theorem 13.4, we know that we can locally render $\Phi_{2}=0$ by
trivializing the $n$-complex structure. Thus we can choose the initial
conditions such that $\mathcal{A}(\lambda)$ is given by the non-abelian Hodge
correspondence applied to the nilpotent Higgs field $\Phi_{1}$ (see
Proposition 14.4).

In [Hi92], Hitchin constructs a real form $\tau$, associated with the split
real form, which for $\mathfrak{sl}_{n}$ is given by a rotation of the matrix
by $180$ degrees composed with complex conjugation. He shows that
$\tau^{*}\Phi_{1}=\Phi_{1}^{*}$ and that
$\tau^{*}\mathcal{A}(\lambda)=\mathcal{A}(\lambda)$. This is a local
statement, therefore the monodromy of $\mathcal{A}(\lambda)$ has to be in the
fixed point set of $\tau$, so in $\operatorname{PSL}_{n}(\mathbb{R})$. ∎

Case $\mu=0$. For trivial $n$-complex structure, the standard form from Lemmas
11.1 and 11.2 allow to consider $\mathcal{A}(\lambda)$ as an affine connection
with special properties. We denote by $\mathcal{L}(\mathfrak{sl}_{n})$ the
loop algebra of $\mathfrak{sl}_{n}$. It is defined by
$\mathcal{L}(\mathfrak{sl}_{n})=\mathfrak{sl}_{n}\otimes\mathbb{C}[\lambda,\lambda^{-1}]$,
the space of Laurent polynomials with matrix coefficients. There is another
way to think of elements of $\mathcal{L}(\mathfrak{sl}_{n})$: as an infinite
periodic matrix $(M_{i,j})_{i,j\in\mathbb{Z}}$ with $M_{i,j}=M_{i+n,j+n}$ and
finite width (i.e. $M_{i,j}=0$ for all $\left|i+j\right|$ big enough). The
isomorphism is given as follows: to $\sum_{i=-N}^{N}N_{i}\lambda^{i}$ we
associate $M_{i,j}=(N_{k_{j}-k_{i}})_{r_{i},r_{j}}$ where $i=k_{i}n+r_{i}$ and
$j=k_{j}n+r_{j}$ are the Euclidean divisions of $i$ and $j$ by $n$ (so $0\leq
r_{i},r_{j}<n$), see also figure 14.1.

$\ldots$$\ldots$$\vdots$$\vdots$$\ddots$$\ddots$$N_{0}$$N_{0}$$N_{0}$$N_{1}$$N_{1}$$N_{2}$$N_{-1}$$N_{-1}$$N_{-2}$
Figure 14.1: Affine matrix as infinite periodic matrix

In the second viewpoint, a connection $\lambda\Phi+A+\lambda^{-1}\Phi^{*}$
with $\Phi_{1}$ lower triangular, $\Phi_{2}=0$ and thus $A_{1}$ upper
triangular, is precisely an infinite matrix with period $n$ and width $n$
(shown in figure 14.1 by dashed lines). The $(1,0)$-part
$\mathcal{A}_{1}(\lambda)$ is upper triangular ($\Phi_{1}$ is lower triangular
but $\lambda\Phi_{1}$ is upper triangular in the infinite matrix) and the
$(0,1)$-part $\mathcal{A}_{2}(\lambda)$ is lower triangular.

Thus, the flatness of $\mathcal{A}(\lambda)$ is a generalized Toda system,
replacing the tridiagonal property by “width equal to periodicity”. For
$t_{i}=0$ for $i=2,...,n-1$ but $t_{n}\neq 0$, we should get the usual affine
Toda system for $\mathcal{L}(\mathfrak{sl}_{n})$ as described in [Ba15].

######  Remark.

In order to describe $h$-connections, we can include parameters into
$\mathcal{L}(\mathfrak{sl}_{n})$ by considering its central extension
$\widehat{\mathfrak{sl}}_{n}$ or central coextension. $\triangle$

Since for $t=0$ we get an elliptic system, the system stays elliptic for at
least small $t\neq 0$, since ellipticity is an open condition (Cauchy-
Kowalewskaya theorem). So the generalized Toda system can be solved for small
$t$.

The study of this generalized Toda system is subject of future research.

General case. For $\mu\neq 0$ and $t\neq 0$, the system is still elliptic at
least for small $t$, since it is for $t=0$. We should get a generalized Toda
system with differentials $t_{k}$ satisfying the higher holomorphicity
condition $(\mathcal{C})$.

We conjecture that the connection
$\mathcal{A}(\lambda)=\lambda\Phi+A+\lambda^{-1}\Phi^{*}$ is uniquely
determined by $\mu$ and $t$. To be more precise:

######  Conjecture 14.7.

Given an element $[(\mu_{k},t_{k})]\in T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$ and
some finite extra data (initial conditions to differential equations), there
is a unique (up to unitary gauge) flat connection
$\mathcal{A}(\lambda)=\lambda\Phi+A+\lambda^{-1}\Phi^{*}$ satisfying

  1. 1.

Locally, $\Phi=\Phi_{1}dz+\Phi_{2}d\bar{z}$ with $\Phi_{1}$ principal
nilpotent and $\Phi_{2}=\mu_{2}\Phi_{1}+...+\mu_{n}\Phi_{1}^{n-1}$

  2. 2.

$-\mathcal{A}(-1/\bar{\lambda})^{*}=\mathcal{A}(\lambda)$ (reality condition)

  3. 3.

$t_{k}=\operatorname{tr}\Phi_{1}^{k-1}A_{1}$.

In addition, if $t_{k}=0$ for all $k$, then the monodromy of
$\mathcal{A}(\lambda)$ is in $\operatorname{PSL}_{n}(\mathbb{R})$.

Another formulation for the conjecture, which would generalize lemma 3 in
[Fo08], is the following:

######  Conjecture 14.8.

Let $\mathcal{A}(\lambda)$ be a family of
$\operatorname{PSL}_{n}(\mathbb{C})$-connections on $\Sigma$ satisfying

  1. 1.

Flatness:
$d\mathcal{A}(\lambda)+\mathcal{A}(\lambda)\wedge\mathcal{A}(\lambda)=0$

  2. 2.

Polynomiality: $\mathcal{A}(\lambda)=\lambda\Phi+A+\lambda^{-1}\Phi^{*}$

  3. 3.

Non-degeneracy: $\Phi_{1}$ is principal nilpotent

  4. 4.

Reality: $-\mathcal{A}(-1/\bar{\lambda})^{*}=\mathcal{A}(\lambda)$.

Then there is an open dense subset of such connections, which are, modulo a
choice of unitary gauge and a choice of a higher complex structure on
$\Sigma$, parameterized by holomorphic differentials $t_{k}\in H^{0}(K^{k})$
for $k=2,...,n$ and some finite data.

The open subset consists of those connections where $\Phi_{1}$ is principal
nilpotent and such that $\mu_{2}\bar{\mu}_{2}\neq 1$.

####  14.4 Link to Hitchin’s component

Assuming the existence and uniqueness of real twistor lines, we get the
desired link to Hitchin’s component:

######  Theorem 14.9.

If Conjecture 14.7 holds true, there is a canonical diffeomorphism between our
moduli space $\bm{\hat{}}{\mathcal{T}}^{n}$ and Hitchin’s component
$\mathcal{T}^{n}$.

###### Proof.

With Conjecture 14.7 we get a canonical way to associate a flat connection
$\mathcal{A}(\lambda=1)$ to a point in $T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$. By
Proposition 14.6 the monodromy of $\mathcal{A}(\lambda)$ for $t=0$ is in
$\operatorname{PSL}_{n}(\mathbb{R})$. Following Hitchin’s argument from
theorem 7.5 in [Hi92], we prove that the zero-section in
$T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$ where $t=0$ describes a connected
component of
$\operatorname{Rep}(\pi_{1}(\Sigma),\operatorname{PSL}_{n}(\mathbb{R}))$.

Since $\bm{\hat{}}{\mathcal{T}}^{n}$ is closed in
$T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$, the image of the map
$s:\bm{\hat{}}{\mathcal{T}}^{n}\rightarrow\operatorname{Rep}(\pi_{1}(\Sigma),\operatorname{PSL}_{n}(\mathbb{R}))$
is a closed submanifold. Furthermore both spaces have the same dimension by
Theorem 5.9. Therefore the image of $s$ is an open and closed submanifold,
i.e. a connected component.

Finally, for $\mu=0$ we get the same connection $\mathcal{A}(\lambda)$ as by
the non-abelian Hodge correspondence of the principal nilpotent Higgs field.
So the component described by $\bm{\hat{}}{\mathcal{T}}^{n}$ and Hitchin’s
component $\mathcal{T}^{n}$ coincide. ∎

Notice that the map between $\mathcal{T}^{n}$ and
$\bm{\hat{}}{\mathcal{T}}^{n}$ is something like an exponential map. For
$n=2$, Hitchin’s description of Teichmüller space is exactly via the
exponential map identifying a fiber of the cotangent bundle
$T^{*}_{\mu}\mathcal{T}^{2}$ to $\mathcal{T}^{2}$.

For $n=2$ we can explicitly give the correspondence between Hitchin’s
description of Teichmüller space and the description with the Beltrami
differential. In [Hi87a] Hitchin parameterizes $\mathcal{T}^{2}$ with
holomorphic quadratic differentials as follows: given a hyperbolic metrix
$\rho_{0}$ and $t\in H^{0}(K^{2})$, one associates the metric
$\rho:=\rho_{0}+t\bar{t}/\rho_{0}+t+\bar{t}$. Locally write
$\rho_{0}=e^{\varphi}dzd\bar{z}$. Then $\rho$ is of constant curvature -1 if
$\varphi$ satisfies the $\sinh$-Gordon equation

$\partial\bar{\partial}\varphi=e^{\varphi}-t\bar{t}e^{-\varphi}$

which is elliptic for all $t$. To find $\mu$, we have to write $\rho$ as
$fdwd\bar{w}$ with $f$ some positive real function and $dw=dz+\mu d\bar{z}$.
After some computation, we get a quadratic relation:

$t\mu^{2}-(e^{\varphi}+t\bar{t}e^{-\varphi})\mu+\bar{t}=0.$

For $t=0$ and $\mu_{k}=0$ for $k=3,...,n$ (but $\mu_{2}\neq 0$), we conjecture
that the monodromy of $\mathcal{A}(\lambda)$ is $n$-fuchsian, i.e. the
composition of a fuchsian map with the principal map
$\pi_{1}(\Sigma)\rightarrow\operatorname{PSL}_{2}(\mathbb{R})\rightarrow\operatorname{PSL}_{n}(\mathbb{R})$.
It is known that in the Hitchin parametrization, the fuchsian locus of
$\mathcal{T}^{n}$ is described by $t_{k}=0$ for $k=3,...,n$.

######  Corollary 14.10.

Hitchin’s component has a natural complex structure. Further, there is a
natural action of the mapping class group on it, preserving the complex
structure.

The first statement follows from Theorem 5.9 since we explicitly know the
cotangent space at a point. The second simply follows by the description of
Hitchin’s component as moduli space of some geometric structure on the
surface. Labourie describes this action in [La08] and shows that it is
properly discontinuous using cross ratios.

##  Part III Generalization to other Lie groups

En un mot, pour tirer la loi de l’expérience, il faut généraliser ;

c’est une nécessité qui s’impose à l’observateur le plus circonspect.

Henri Poincaré, La valeur de la science

Nigel Hitchin’s construction for components in character varieties works for
any adjoint group $G$ associated with a split form of a simple complex Lie
algebra. The construction of $n$-complex structures gives an approach to
$\operatorname{PSL}_{n}(\mathbb{R})$-Hitchin components. In this part, we
describe a generalization of the concepts from part I which might give a
geometric approach to $G$-Hitchin components. We define a so-called
$\mathfrak{g}$-complex structure using a generalization of the punctual
Hilbert scheme associated with a simple complex Lie algebra $\mathfrak{g}$. At
the beginning we treat an arbitrary simple Lie algebra $\mathfrak{g}$, but at
some point we restrict attention to classical Lie algebras.

To read this part, you only need to know basic facts about the punctual
Hilbert scheme (subsections 4.1 and 4.4). In order to see the analogy between
the $\mathfrak{g}$-complex structure and the $n$-complex structure (for which
$\mathfrak{g}=\mathfrak{sl}_{n}$), you should know the construction of the
higher complex structure, its moduli space and the spectral curve (section 5
and subsection 7.2).

Our inspiration how to define these new objects is twofold: on the one hand we
use the various descriptions of the punctual Hilbert scheme, especially the
matrix viewpoint, in order to generalize to an arbitrary $\mathfrak{g}$. On
the other, we got inspiration from Hitchin’s original paper [Hi92] (section 5)
where he starts with a principal nilpotent element and deforms it into an
element of a principal slice, a generalized companion matrix.

We signal to the reader that our definition of the $\mathfrak{g}$-Hilbert
scheme might get changed in the future, since as it is defined now, it is a
non-Hausdorff space. There should be a way to get a nice topological space,
using some adapted GIT quotient. This possible modification will not affect
the $\mathfrak{g}$-complex structure since only the regular part of the
$\mathfrak{g}$-Hilbert scheme plays a role in its construction.

We first introduce the $\mathfrak{g}$-Hilbert scheme and some interesting
subspaces and analyze their properties. We then proceed to the construction of
the $\mathfrak{g}$-complex structure and define higher diffeomorphisms of type
$\mathfrak{g}$. The local theory is trivial, like for the $n$-complex
structure (with a subtlety for $\mathfrak{g}$ of type $D_{n}$). The moduli
space of $\mathfrak{g}$-complex structures
$\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ enjoys similar properties as in the
case $\mathfrak{g}=\mathfrak{sl}_{n}$, and so it shares several properties
with Hitchin’s component. In particular, both are contractible and there is a
copy of Teichmüller space inside them. We then define a spectral curve which
is equipped with some extra structure depending on $\mathfrak{g}$ giving the
spectral data described in [Hi87b].

Most of the material of this part is published in the preprint [Th19]. A
collection of facts on regular elements in semisimple Lie algebras which we
need in this part can be found in appendix A.

###  15 Generalized punctual Hilbert scheme

In this section, we generalize the punctual Hilbert scheme to a
$\mathfrak{g}$-Hilbert scheme and explore the properties of the new object. We
analyze in detail its regular part in the case of a classical Lie algebra. We
use lots of analogies to the punctual Hilbert scheme and its properties from
sections 4 and 6.

####  15.1 Definitions and first properties

The punctual Hilbert scheme $\operatorname{Hilb}^{n}(\mathbb{C}^{2})$ has
several descriptions:

  * •

as a space of ideals (the idealic viewpoint)

  * •

as a desingularization of the configuration space $\mathfrak{h}^{2}/W$ for
$\mathfrak{g}=\mathfrak{gl}_{n}$

  * •

as a space of commuting matrices (the matrix viewpoint).

It is the matrix viewpoint which will be generalized. So let us quickly recall
it here:

$\operatorname{Hilb}^{n}(\mathbb{C}^{2})\cong\\{(A,B)\in\mathfrak{gl}_{n}^{2}\mid[A,B]=0,(A,B)\text{
admits a cyclic vector}\\}/GL_{n}.$

The main difficulty is to find an intrinsic condition which generalizes the
existence of a cyclic vector. Here is our proposal:

######  Definition 15.1.

The generalized punctual Hilbert scheme, or $\mathbf{\mathfrak{g}}$-Hilbert
scheme, denoted by $\operatorname{Hilb}(\mathfrak{g})$, is defined by

$\operatorname{Hilb}(\mathfrak{g})=\\{(A,B)\in\mathfrak{g}^{2}\mid[A,B]=0,\dim
Z(A,B)=\operatorname{rk}\mathfrak{g}\\}/G$

where $Z(A,B)$ denotes the common centralizer of $A$ and $B$, i.e. the set of
elements $C\in\mathfrak{g}$ which commute with both $A$ and $B$.

The condition on the dimension of the common centralizer does not come from
nowhere: Proposition A.9 of Appendix A shows that
$\operatorname{rk}\mathfrak{g}$ is the minimal possible dimension for the
centralizer of a commuting pair. Define the commuting variety by

$\operatorname{Comm}(\mathfrak{g})=\\{(A,B)\in\mathfrak{g}^{2}\mid[A,B]=0\\}.$

The $\mathfrak{g}$-Hilbert scheme is the set of all regular points of
$\operatorname{Comm}(\mathfrak{g})$ modulo $G$.

######  Remark.

Ginzburg has defined the notion of a principal nilpotent pair in [Gi00], which
is more restrictive than ours. He calls “nil-pairs” elements of our
$\mathfrak{g}$-Hilbert scheme, but it seems that he does not investigate them.
$\triangle$

Let us give two examples of elements in the $\mathfrak{g}$-Hilbert scheme:

######  Example 15.2.

Let $A\in\mathfrak{g}$ be a regular element. Then by a theorem of Kostant (see
A.6), its centralizer $Z(A)$ is abelian. So for any $B\in Z(A)$, we have
$Z(A)\subset Z(B)$, thus $Z(A,B)=Z(A)\cap Z(B)=Z(A)$ is of dimension
$\operatorname{rk}\mathfrak{g}$. Therefore
$[(A,B)]\in\operatorname{Hilb}(\mathfrak{g})$.

If $A$ is principal nilpotent, then $B\in Z(A)$ is also nilpotent. So
$[(A,B)]\in\operatorname{Hilb}_{0}(\mathfrak{g})$, the zero-fiber defined
below.

If $B=0$ then $[(A,0)]$ is in $\operatorname{Hilb}(\mathfrak{g})$ iff $A$ is
regular. $\triangle$

######  Example 15.3.

Let $(A,B)$ be a commuting pair of matrices in $\mathfrak{sl}_{n}$ admitting a
cyclic vector, i.e. an element of the reduced Hilbert scheme. One way to get
such a pair is the following construction: take a Young diagram (our
convention is to put the origin in the upper left corner as for matrices) with
$n$ boxes (see figure 15.1). Associate to each box a vector of a basis of
$\mathbb{C}^{n}$. Define $A$ to be the matrix which translates to the right,
i.e. sends a vector to the vector in the box to the right or to 0 if there is
none. Let $B$ be the matrix which translates to the bottom. Then $A$ and $B$
clearly commute and are nilpotent. In Proposition 15.5 below, we show that
$Z(A,B)$ is of minimal dimension in that case. $\triangle$

Figure 15.1: Young diagram and commuting nilpotent matrices

Guided by these examples, we define several subsets of the
$\mathfrak{g}$-Hilbert scheme and explore their relations. First, we define
the zero-fiber and the regular part which both play a mayor role in the
definition of a $\mathfrak{g}$-complex structure. We also define the cyclic
part, which is not intrinsically defined since it uses a representation of
$\mathfrak{g}$. The cyclic part will be used to define a map to a space of
ideals, getting a generalization of the original description of the punctual
Hilbert scheme by ideals.

######  Definition 15.4.

The zero-fiber of the $\mathfrak{g}$-Hilbert scheme is defined by

$\operatorname{Hilb}_{0}(\mathfrak{g})=\\{[(A,B)]\in\operatorname{Hilb}(\mathfrak{g})\mid
A\text{ and }B\text{ nilpotent}\\}.$

We define the regular part of the $\mathfrak{g}$-Hilbert scheme, denoted by
$\operatorname{Hilb}^{reg}(\mathfrak{g})$, to be those conjugacy classes
$[(A,B)]$ in which $A$ or $B$ is a regular element of $\mathfrak{g}$.

Finally for classical $\mathfrak{g}$, let $\rho$ denote the natural
representation of $\mathfrak{g}$ (i.e.
$\mathfrak{sl}_{n}\subset\mathfrak{gl}_{n},\mathfrak{so}_{n}\subset\mathfrak{gl}_{n}$
and $\mathfrak{sp}_{2n}\subset\mathfrak{gl}_{2n}$). Define the cyclic part of
the $\mathfrak{g}$-Hilbert scheme by

$\operatorname{Hilb}^{cycl}(\mathfrak{g})=\\{(A,B)\in\mathfrak{g}^{2}\mid[A,B]=0,(\rho(A),\rho(B))\text{
admits a cyclic vector}\\}/G.$

######  Remark.

In the definition of the cyclic part, it would be more natural to consider the
adjoint representation, but even in the case of $\mathfrak{sl}_{2}$, this
would give a map to a space of ideals, which is not the one of
$\operatorname{Hilb}^{2}_{red}(\mathbb{C}^{2})$.

Instead of the standard representation, one could also use a non-trivial
representation of minimal dimension, which for classical $\mathfrak{g}$ is
always the standard representation, apart from types $D_{3}$ and $D_{4}$. For
$D_{3}$, the two spin representations are of minimal dimension, and they give
the exceptional isomorphism between $\mathfrak{so}_{6}$ and
$\mathfrak{sl}_{4}$. For type $D_{4}$, there are three representations of
minimal dimension, the standard one and the two spin representations. All of
them are linked by outer automorphisms coming from the symmetry of the Dynkin
diagram. Thus the cyclic part of the $\mathfrak{so}_{8}$-Hilbert scheme is the
same for all three representations.

Taking the representation of minimal dimension has the additional advantage to
be well-defined for all $\mathfrak{g}$. This possibility has to be explored
further. $\triangle$

The first relation between the various Hilbert schemes is the inclusion of the
cyclic part in the $\mathfrak{g}$-Hilbert scheme, which justifies the name
“cyclic part”:

######  Proposition 15.5.

For $\mathfrak{g}$ of classical type, we have
$\operatorname{Hilb}^{cycl}(\mathfrak{g})\subset\operatorname{Hilb}(\mathfrak{g})$.

###### Proof.

Recall $\rho$ the natural representation of $\mathfrak{g}$ on
$\mathbb{C}^{m}$. For simplicity, we write $A$ instead of $\rho(A)$ here.

Let $(A,B)\in\mathfrak{g}^{2}$ admitting a cyclic vector $v$. Let $C\in
Z(A,B)$. Then $C$ is a polynomial in $A$ and $B$. Indeed, there is
$P\in\mathbb{C}[x,y]$ such that $Cv=P(A,B)v$. Since $C$ commutes with $A$ and
$B$, we get for any polynomial $Q$ that
$CQ(A,B)v=Q(A,B)Cv=Q(A,B)P(A,B)v=P(A,B)Q(A,B)v$. Thus $C=P(A,B)$.

Therefore the common centralizer of $(A,B)$ in $\mathfrak{gl}_{m}$ is
$\mathbb{C}[A,B]/I$ where $I=\\{P\in\mathbb{C}[x,y]\mid P(A,B)=0\\}$. We know
from section 4 that $I$ is of codimension $m$ since $(A,B)$ admits a cyclic
vector. Further, we have $Z(A,B)=Z_{\mathfrak{gl}_{m}}(A,B)\cap\mathfrak{g}$.
One can easily check that for $\mathfrak{g}$ of type $A_{n}$, a polynomial
$P(A,B)$ is in $\mathfrak{g}$ iff its constant term has a specific form, given
by the other coefficients (to ensure trace zero). For type $B_{n},C_{n}$ and
$D_{n}$, $P(A,B)$ is in $\mathfrak{g}$ iff $P$ is odd. One checks in each case
that the dimension of $Z(A,B)$ equals the rank of $\mathfrak{g}$. ∎

In general, the inclusion of the cyclic Hilbert scheme is strict as shows the
following example:

######  Example 15.6.

Consider $A=\left(\begin{smallmatrix}0&1&0\\\ 0&0&0\\\
0&0&0\end{smallmatrix}\right)$ and $B=\left(\begin{smallmatrix}0&0&1\\\
0&0&0\\\ 0&0&0\end{smallmatrix}\right)$ in $\mathfrak{sl}_{3}$. One easily
checks that the pair $(A,B)$ does not admit any cyclic vector, but that their
common centralizer is of dimension 2. So
$[(A,B)]\in\operatorname{Hilb}(\mathfrak{sl}_{3})\backslash\operatorname{Hilb}^{cycl}(\mathfrak{sl}_{3})$.
$\triangle$

This example will be used in subsection 15.9 to show that
$\operatorname{Hilb}(\mathfrak{g})$ is not Hausdorff.

In general, there is no link between regular and cyclic part. Example 15.3
shows that cyclic elements are not always regular and the following example
shows that regular element are not always cyclic:

######  Example 15.7.

For $\mathfrak{g}$ of type $D_{n}$, let $f$ be a principal nilpotent element.
Then one checks that $[(f,0)]\in\operatorname{Hilb}(\mathfrak{so}_{2n})$ is
regular but not cyclic (see also subsection 15.8). $\triangle$

Let us turn to the regular part. It turns out that if one fixes a principal
slice $f+Z(e)$ in $\mathfrak{g}$ (see Appendix A), there is a preferred
representative for regular classes:

######  Proposition 15.8.

Any class $[(A,B)]\in\operatorname{Hilb}^{reg}(\mathfrak{g})$ where $A$ is
regular can uniquely be conjugated to $(A\in f+Z(e),B\in Z(A))$.

###### Proof.

By the property of the principal slice, there is a unique conjugate of $A$
which is in the principal slice $f+Z(e)$. Denote still by $A$ and $B$ these
conjugates. The only thing to show is that $B$ is unique which is done in the
next lemma. ∎

######  Lemma 15.9.

If $A\in\mathfrak{g}$ is regular, $g\in G$ such that $Ad_{g}(A)=A$ and $B\in
Z(A)$, then $Ad_{g}(B)=B$.

###### Proof.

By Kostant’s theorem A.6, we know that $Z(A)$ is abelian. So the infinitesimal
version of the lemma is true. We conclude by the connectedness of the
stabilizer of $A$, given by the next lemma. ∎

######  Lemma 15.10.

For a regular element $A\in\mathfrak{g}$, its stabilizer
$\operatorname{Stab}(A)=\\{g\in G\mid Ad_{g}(A)=A\\}$ in the adjoint group $G$
is connected.

###### Proof.

Decompose $A$ into Jordan form: $A=A_{s}+A_{n}$ with $A_{s}$ semisimple,
$A_{n}$ nilpotent and $[A_{s},A_{n}]=0$. So $A_{n}\in Z(A_{s})$. The structure
of the centralizer $Z(A_{s})$ is well-known: it is a direct sum of a Cartan
$\mathfrak{h}$ containing $A_{s}$ with all root spaces $\mathfrak{g}_{\alpha}$
where $\alpha$ is a root such that $\alpha(A_{s})=0$. It is also known that
$Z(A_{s})$ is reductive, so a direct sum
$Z(A_{s})=\mathfrak{c}\oplus\mathfrak{g}_{s}$ where $\mathfrak{c}$ is the
center and $\mathfrak{g}_{s}$ is the semisimple part of $Z(A_{s})$. In
particular the center $\mathfrak{c}$ is included in $\mathfrak{h}$. So
$A_{n}\in\mathfrak{g}_{s}$ since $A_{n}$ is nilpotent. Denote by $G_{s}$ the
Lie group with trivial center with Lie algebra $\mathfrak{g}_{s}$.

We know that $A$ is regular iff $A_{n}$ is principal nilpotent in
$\mathfrak{g}_{s}$ (see [Ko63], proposition 0.4). We also know that the
$G$-equivariant fundamental group of the orbit of $A$ (which is the space of
connected components of $\operatorname{Stab}(A)$) is the same as the
$\operatorname{Stab}(A_{s})$-equivariant fundamental group of the
$\operatorname{Stab}(A_{s})$-orbit of $A_{n}$ (see Proposition 6.1.8. of
[CM93] adapted to the adjoint group). In other words, the connected components
of $\operatorname{Stab}_{G}(A)$ are the same as the connected components of
$\operatorname{Stab}_{G_{s}}(A_{n})$ since the
$\operatorname{Stab}(A_{s})$-orbit of $A_{n}$ is equal to the $G_{s}$-orbit of
$A_{n}$.

So we are reduced to the principal nilpotent case. Using the classification of
simple Lie algebras, one can explicitly check in Collingwood-McGovern’s book
[CM93] the tables 6.1.6. for classical $\mathfrak{g}$ and the tables at the
end of chapter 8 for exceptional $\mathfrak{g}$ that the stabilizer of a
principal nilpotent element is always connected. ∎

######  Remark.

It is surprising that the last lemma has never been stated (at least not to
our knowledge). It would be interesting to find a direct argument, without
using the classification of simple Lie algebras. $\triangle$

######  Corollary 15.11.

The regular zero-fiber
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})=\operatorname{Hilb}^{reg}(\mathfrak{g})\cap\operatorname{Hilb}_{0}(\mathfrak{g})$
is an affine variety of dimension $\operatorname{rk}\mathfrak{g}$.

###### Proof.

This directly follows from the previous proposition 15.8 using the fact that
$A\in f+Z(e)$ is nilpotent iff $A=f$. So
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})$ is described by $Z(f)$ which is
a vector space of dimension $\operatorname{rk}\mathfrak{g}$. ∎

We know that both the regular and the cyclic part are in general strictly
included in the $\mathfrak{g}$-Hilbert scheme. But they are dense subspaces:

######  Proposition 15.12.

The regular part $\operatorname{Hilb}^{reg}(\mathfrak{g})$ is dense in
$\operatorname{Hilb}(\mathfrak{g})$. For classical $\mathfrak{g}$, the cyclic
part is also dense in $\operatorname{Hilb}(\mathfrak{g})$.

###### Proof.

By a theorem of Richardson (see A.8), the set of semisimple commuting pairs is
dense in the commuting variety $\operatorname{Comm}(\mathfrak{g})$. So the set
of semisimple regular elements is also dense in
$\operatorname{Comm}(\mathfrak{g})$. Passing to the quotient by $G$, we get
that the classes of semisimple regular pairs are dense in
$\operatorname{Hilb}(\mathfrak{g})$ since
$\operatorname{Hilb}(\mathfrak{g})\subset\operatorname{Comm}(\mathfrak{g})/G$
and all semisimple regular pairs are in $\operatorname{Hilb}(\mathfrak{g})$.
Since the semisimple regular pairs are in the regular part, we get the density
of $\operatorname{Hilb}^{reg}(\mathfrak{g})$ in
$\operatorname{Hilb}(\mathfrak{g})$.

For classical $\mathfrak{g}$, we have the same argument for the cyclic part
since semisimple regular pairs are cyclic. ∎

To end the section, we state an analog of Kostant’s theorem about abelian
subalgebras of centralizers:

######  Proposition 15.13.

For any commuting pair $(A,B)\in\operatorname{Comm}(\mathfrak{g})$, there is
an abelian subspace of dimension $\operatorname{rk}\mathfrak{g}$ in the common
centralizer $Z(A,B)$.

###### Proof.

The proof is completely analogous to Kostant’s proof for Theorem A.6: we use a
limit argument. Let $(A_{n},B_{n})$ be a sequence of regular semisimple pairs
converging to $(A,B)$ (exists since regular semisimple pairs are dense). We
know that $Z(A_{n},B_{n})$ is a $\operatorname{rk}\mathfrak{g}$-dimensional
abelian subspace of $\mathfrak{g}$. Since the Grassmannian
$Gr(\operatorname{rk}\mathfrak{g},\dim\mathfrak{g})$ is compact, there is a
subsequence of $Z(A_{n},B_{n})$ which converges. It is easy to prove that the
limit is included in $Z(A,B)$ and is commutative. ∎

######  Corollary 15.14.

For $[(A,B)]\in\operatorname{Hilb}(\mathfrak{g})$, the common centralizer
$Z(A,B)$ is abelian.

In the following sections, we generalize as far as possible the other
viewpoints of the usual Hilbert scheme (resolution of configuration space and
idealic viewpoint) to our setting.

####  15.2 Chow map

We want to generalize the Chow map, which goes from
$\operatorname{Hilb}^{n}(\mathbb{C}^{2})$ to the configuration space (see
subsection 4.3).

Fix a Cartan subalgebra $\mathfrak{h}$ in $\mathfrak{g}$. Recall the Jordan
decomposition in a semisimple Lie algebra: for $x\in\mathfrak{g}$, there is a
unique pair $(x_{s},x_{n})$ with $x=x_{s}+x_{n}$, $x_{s}$ semisimple, $x_{n}$
nilpotent and $[x_{s},x_{n}]=0$. For a semisimple element $x$, denote by
$x^{*}$ a conjugate of $x$ in the Cartan $\mathfrak{h}$ (unique up to
$W$-action).

The Chow map
$ch:\operatorname{Hilb}(\mathfrak{g})\rightarrow\mathfrak{h}^{2}/W$ is defined
by

$ch([(A,B)])=[(A_{s}^{*},B_{s}^{*})]$

where the brackets $[.]$ denotes the equivalence class. For semisimple regular
pairs, this map corresponds to a simultaneous diagonalization.

######  Proposition 15.15.

The Chow map $ch$ is well-defined and continuous.

###### Proof.

Since $[A,B]=0$, we also have $[A_{s},B_{s}]=0$ by a simultaneous Jordan
decomposition in a faithful representation. Hence there is a conjugate of the
pair $(A_{s},B_{s})$ which lies in $\mathfrak{h}^{2}$. Since the adjoint
action of $G$ on $\mathfrak{g}$ restricts to the $W$-action on $\mathfrak{h}$,
the map $ch$ is well-defined.

The map $x\mapsto x_{s}^{*}$ is continuous which simply follows from the
continuity of eigenvalues. Hence the Chow map is continuous as well. ∎

######  Remark.

The Jordan decomposition $x\mapsto(x_{s},x_{n})$ is not continuous at all,
since semisimple elements are dense in $\mathfrak{g}$ for which we have
$x_{n}=0$ and for all non-semisimple elements we have $x_{n}\neq 0$. But the
map $x\mapsto x_{s}$ is continuous. $\triangle$

This map permits to think of a generic element of
$\operatorname{Hilb}(\mathfrak{g})$ as a point in $\mathfrak{h}^{2}/W$, or via
a representation of $\mathfrak{g}$ on $\mathbb{C}^{m}$, as a set of $m$ points
in $\mathbb{C}^{2}$ with a certain symmetry. For
$\mathfrak{g}=\mathfrak{sl}_{n}$ for example, these are $n$ points with
barycenter 0.

Since $\operatorname{Hilb}(\mathfrak{g})$ as a topological space is not
Hausdorff (see subsection 15.9), it cannot be a non-singular variety.
Nevertheless we conjecture the following:

######  Conjecture 15.16.

There is a modified version of $\operatorname{Hilb}(\mathfrak{g})$,
identifying some points, which is a smooth projective variety such that the
Chow morphism is a resolution of singularities.

####  15.3 Idealic map

For $\mathfrak{g}$ of classical type, we can associate to any regular element
of the $\mathfrak{g}$-Hilbert scheme an ideal, which we call idealic map. In
this subsection, $\mathfrak{g}$ is a classical Lie algebra. Recall the natural
representation $\rho$ of $\mathfrak{g}$ on $\mathbb{C}^{m}$ (see Definition
15.4). We write $A$ instead of $\rho(A)$.

We wish to define a map like in Proposition 4.4 between commuting matrices and
ideals:

$[(A,B)]\mapsto I(A,B)=\\{P\in\mathbb{C}[x,y]\mid P(A,B)=0\\}.$ (15.1)

If $[(A,B)]\in\operatorname{Hilb}^{cycl}(\mathfrak{g})$ is cyclic, this ideal
is of codimension $m$. But if the pair is not cyclic, there is no reason why
the codimension should be $m$. In fact, there are examples for $\mathfrak{g}$
of type $D_{n}$ where the codimension is smaller.

We wish the idealic map to be continuous, so $I$ has to be of constant
codimension. A strategy would be to define the idealic map $I$ on the cyclic
part $\operatorname{Hilb}^{cycl}(\mathfrak{g})$ (which is dense by Proposition
15.12) and to extend it by continuity. Unfortunately, the map can not be
extended in a continuous way as the following example shows:

######  Example 15.17.

Take $\mathfrak{g}$ of type $D_{n}$. Denote by $f$ a principal nilpotent
element. The pair $[(f,0)]\in\operatorname{Hilb}(\mathfrak{so}_{2n})$ is not
cyclic (see example 15.7). Using the matrix $S$ defined in equation (15.4), we
can approach $(f,0)$ by $(f,tS)$ or by $(f+tS^{\top},0)$ for
$t\in\mathbb{C}^{\times}$ going to 0. These pairs are all cyclic. In the first
case, the ideal is $I=\langle x^{2n-1},xy,y^{2}=t^{2}x^{2n-2}\rangle$ which
converges as $t$ goes to 0 to $\langle x^{2n-1},xy,y^{2}\rangle$. In the
second case, the ideal is $I=\langle x^{2n}+t^{2},y\rangle$ converging to
$\langle x^{2n},y\rangle$. $\triangle$

Because of this difficulty, our strategy is to define a space of ideals
$I_{\mathfrak{g}}(\mathbb{C}^{2})$, then a map
$\operatorname{Hilb}^{cycl}(\mathfrak{g})\rightarrow
I_{\mathfrak{g}}(\mathbb{C}^{2})$ and to extend it over the regular part
$\operatorname{Hilb}^{reg}(\mathfrak{g})$ (in a non-continuous way). The last
step is only necessary for $\mathfrak{g}$ of type $D_{n}$ since for the other
classical types the regular part is included in the cyclic part as we will see
in the sequel. The extension for $D_{n}$ will be defined ad hoc in subsection
15.8.

The previous section taught us to think of a generic element of
$\operatorname{Hilb}(\mathfrak{g})$ as a $m$-tuple of points in
$\mathbb{C}^{2}$ invariant under the Weyl group $W$. For type $A_{n}$ this
means that the barycenter of the points is the origin. For the other classical
types, this means that the set of points is symmetric with respect to the
origin. Thus the defining ideal of these points is also invariant under the
action of $W$. Hence the following definition.

######  Definition 15.18.

We define the space of ideals of type $\mathfrak{g}$, denoted by
$I_{\mathfrak{g}}(\mathbb{C}^{2})$, to be the set of ideals in
$\mathbb{C}[x,y]$ which are of codimension $m$ and $W$-invariant. For type
$B_{n},C_{n}$ and $D_{n}$ this means that $I$ is invariant under
$(x,y)\mapsto(-x,-y)$.

The map $I:\operatorname{Hilb}^{cycl}(\mathfrak{g})\rightarrow
I_{\mathfrak{g}}(\mathbb{C}^{2})$ given by equation (15.1) above is well-
defined. Indeed, the codimension is $m$ by cyclicity and the ideal is
$W$-invariant since this is a closed condition and it is true on the dense
subset of regular semisimple pairs.

Notice that $I_{\mathfrak{g}}(\mathbb{C}^{2})$ is the same for $\mathfrak{g}$
of type $C_{n}$ or $D_{n}$. But we will see that the idealic map $I$ has not
the same image in the two cases. We will also see that for $\mathfrak{g}$ of
type $A_{n},B_{n}$ or $C_{n}$ the idealic map is injective. But for type
$D_{n}$ it is not (it is generically 2 to 1). This comes from the fact that
the Weyl group acting on the generic $2n$ points, coming in $n$ pairs
$(P_{i},P_{i+1}=-P_{i})$, cannot exchange $P_{1}$ and $P_{2}$ while leaving
all other points fixed.

As for the usual Hilbert scheme, there is a direct link between the idealic
map and the Chow morphism:

######  Proposition 15.19.

The Chow map $ch$ is the composition of the idealic map with the map which
associates to an ideal its support, seen as an element of
$\mathfrak{h}^{2}/W$:

$ch([(A,B)])=\operatorname{supp}I(A,B).$

###### Proof.

The statement is true on regular semisimple pairs which is a dense subset. For
$\mathfrak{g}$ of type $A_{n}$, $B_{n}$ and $C_{n}$, it follows by continuity
of both the Chow map and the idealic map. For $D_{n}$, our definition of the
idealic map is to pick one of the various possible limits. In particular, the
support of the ideal is still given by the Chow map. ∎

####  15.4 Morphisms

In this subsection, we analyze the functorial behavior of the
$\mathfrak{g}$-Hilbert scheme. In particular we construct two maps linked to
the zero-fiber of the Hilbert scheme of $\mathfrak{sl}_{2}$ which will lead in
the construction of the moduli space $\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$
of $\mathfrak{g}$-complex structures to maps from and to Teichmüller space.

Let $\psi:\mathfrak{g}_{1}\rightarrow\mathfrak{g}_{2}$ be a morphism of Lie
algebras. For $[(A,B)]\in\operatorname{Hilb}(\mathfrak{g}_{1})$, we can
associate $[(\psi(A),\psi(B))]$ which is a well-defined map to
$\operatorname{Comm}(\mathfrak{g}_{2})/G_{2}$. But there is no reason why
$\dim Z(\psi(A),\psi(B))$ should be minimal.

If we accept Conjecture 15.16, that there is a modified version of the
$\mathfrak{g}$-Hilbert scheme which is a resolution of $\mathfrak{h}^{2}/W$,
we have a functorial behavior:

######  Proposition 15.20.

Assuming Conjecture 15.16, there is an induced map
$\operatorname{Hilb}(\mathfrak{g}_{1})\rightarrow\operatorname{Hilb}(\mathfrak{g}_{2})$.

###### Proof.

Choose Cartan subalgebras $\mathfrak{h}_{1}$ and $\mathfrak{h}_{2}$ such that
$\psi(\mathfrak{h}_{1})=\mathfrak{h}_{2}$. Consider the composition
$\mathfrak{h}_{1}^{2}\rightarrow\mathfrak{h}_{2}^{2}\rightarrow\mathfrak{h}_{2}^{2}/W_{2}$
using $\psi$ for the first arrow. Since $\psi$ induces a homomorphism between
the Weyl groups, we can factor the composition to get a map
$\mathfrak{h}_{1}^{2}/W_{1}\rightarrow\mathfrak{h}_{2}^{2}/W_{2}$. Finally,
consider the composition
$\operatorname{Hilb}(g_{1})\rightarrow\mathfrak{h}_{1}^{2}/W_{1}\rightarrow\mathfrak{h}_{2}^{2}/W_{2}$
where the first arrow comes from the minimal resolution. This is a continuous
map and by the universal property of a minimal resolution, the map lifts to
$\operatorname{Hilb}(\mathfrak{g}_{1})\rightarrow\operatorname{Hilb}(\mathfrak{g}_{2})$.
∎

Let us study this induced map in the case of the reduced Hilbert scheme
$\operatorname{Hilb}^{n}_{red}(\mathbb{C}^{2})$, which is a minimal resolution
(see subsection 4.3). Take
$\psi:\mathfrak{sl}_{m}\rightarrow\mathfrak{sl}_{n}$ inducing a map
$\operatorname{Hilb}^{m}_{red}(\mathbb{C}^{2})\rightarrow\operatorname{Hilb}^{n}_{red}(\mathbb{C}^{2})$.
In the matrix viewpoint, this map is not given by $[(\psi(A),\psi(B))]$.
Consider for example the map
$\psi:\mathfrak{sl}_{2}\rightarrow\mathfrak{sl}_{4}$ given on the standard
generators $(e,f,h)$ of $\mathfrak{sl}_{2}$ by

$\psi(e)=\left(\begin{smallmatrix}0&&&1\\\ &0&&\\\ &&0&\\\
&&&0\end{smallmatrix}\right),\psi(f)=\left(\begin{smallmatrix}0&&&\\\ &0&&\\\
&&0&\\\ 1&&&0\end{smallmatrix}\right)\text{ and
}\psi(h)=\left(\begin{smallmatrix}1&&&\\\ &0&&\\\ &&0&\\\
&&&-1\end{smallmatrix}\right).$

The element $[(h,0)]\in\operatorname{Hilb}^{2}_{red}(\mathbb{C}^{2})$
corresponds to the ideal $I=\langle x^{2}-1,y\rangle$ which through $\psi$
goes to $\langle x^{4}-x^{2},y\rangle$ which in turn gives the matrices
$[(M,0)]$ where $M=\left(\begin{smallmatrix}1&&&\\\ &0&1&\\\ &0&0&\\\
&&&-1\end{smallmatrix}\right)$. This is not $[(\psi(h),\psi(0))]$. It would be
interesting to describe the induced map in the matrix viewpoint.

Despite this complication, there are two cases where a map between
$\mathfrak{g}$-Hilbert schemes exists naturally.

The first one is linked to the principal map
$\psi:\mathfrak{sl}_{2}\rightarrow\mathfrak{g}$ which induces a map

$\operatorname{Hilb}(\mathfrak{sl}_{2})\rightarrow\operatorname{Hilb}^{reg}(\mathfrak{g}).$
(15.2)

Indeed, any non-zero element of $\mathfrak{sl_{2}}$ is regular and cyclic. So
if $[(A,B)]\in\operatorname{Hilb}(\mathfrak{sl}_{2})$ such that $A$ is non-
zero, there is by Proposition 15.8 a unique representative $(f+te,B\in
Z(e+tf))$ where $(e,f,h)$ denotes the standard generators of
$\mathfrak{sl}_{2}$ and $t\in\mathbb{C}$. So the image is
$[(\psi(f)+t\psi(e),\psi(B))]$. Since $(\psi(e),\psi(f),\psi(h))$ is a
principal $\mathfrak{sl}_{2}$-triple (property of the principal map), we know
that $\psi(f)+t\psi(e)$ is in the principal slice, thus it is regular, so we
land in $\operatorname{Hilb}^{reg}(\mathfrak{g})$.

The second one is a sort of inverse map to the first one, but only on the
level of the zero-fiber. Given
$[(A,B)]\in\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})$ where $A$ is regular,
there is a principal $\mathfrak{sl}_{2}$-subalgebra $\mathcal{S}$ with $A$ as
nilpotent element. There is no reason why $B$ should be in $\mathcal{S}$ but
there is a “best approximation” in the following sense:

######  Proposition 15.21.

Let $A$ be a principal nilpotent element and $B\in Z(A)$. Then there is a
unique $\mu_{2}\in\mathbb{C}$ such that $B-\mu_{2}A$ is not regular.

###### Proof.

The strategy of the proof is to use Proposition A.5 of the appendix which
characterizes principal nilpotent elements $x$ as those nilpotent elements
whose values $\alpha(x)$ for all simple roots $\alpha$ are non-zero. So the
proposition is equivalent to the statement that $\alpha_{1}(B)=\alpha_{2}(B)$
for all simple roots $\alpha_{1}$ and $\alpha_{2}$.

Let $R$ be a root system in $\mathfrak{h}^{*}$ and denote by $R_{+}$ and
$R_{s}$ the positive and respectively the simple roots. We can conjugate $A$
to the element given by $\alpha(A)=1$ if $\alpha\in R_{s}$ and $\alpha(A)=0$
otherwise.

For two simple roots $\alpha_{1}$ and $\alpha_{2}$ such that
$\alpha_{1}+\alpha_{2}\in R$, using $[A,B]=0$ we get:

$0=(\alpha_{1}+\alpha_{2})([A,B])=\alpha_{1}(A)\alpha_{2}(B)-\alpha_{2}(A)\alpha_{1}(B)=(\alpha_{2}-\alpha_{1})(B).$

Since $\mathfrak{g}$ is simple, its Dynkin diagram is connected, so
$\alpha_{1}(B)=\alpha_{2}(B)$ for all simple roots. The common value $\mu_{2}$
gives the unique complex number such that $B-\mu_{2}A$ is not regular. ∎

With this proposition, we can now define a map

$\mu:\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})\rightarrow\operatorname{Hilb}_{0}(\mathfrak{sl}_{2})$
(15.3)

given by $\mu([(A,B)])=[(e,\mu_{2}e)]$ or $[(\mu_{2}e,e)]$ depending whether
$A$ or $B$ is regular.

An equivalent way to define the map $\mu$ is the following: we can use the
previous proposition 15.21 to show that the centralizer $Z(A)$ of a principal
nilpotent element is a direct product

$Z(A)=\operatorname{Span}(A)\times Z(A)^{irreg}$

where $Z(A)^{irreg}$ denotes the irregular elements of $Z(A)$. The map $\mu$
is nothing but the projection to the first factor.

######  Remark.

We can describe the regular part of the $\mathfrak{g}$-Hilbert scheme
$\operatorname{Hilb}^{reg}(\mathfrak{g})$ as those classes $[(A,B)]$ such that
$\operatorname{Span}(A,B)$ intersects the regular part $\mathfrak{g}^{reg}$
non-trivially. This description is more symmetric since it does not prefer $A$
or $B$. From Proposition 15.21 we see that the intersection of
$\operatorname{Span}(A,B)$ with $\mathfrak{g}^{reg}$ is the whole two-
dimensional $\operatorname{Span}(A,B)$ from which we have to take out a line.
Hence, the intersection has two components. $\triangle$

In the following subsections, we study the regular part
$\operatorname{Hilb}^{reg}(\mathfrak{g})$ and its zero-fiber case by case for
classical $\mathfrak{g}$.

####  15.5 Case $A_{n}$

Consider $\mathfrak{g}=\mathfrak{sl}_{n}$ (of type $A_{n-1}$). We describe
first $\operatorname{Hilb}^{reg}_{0}(\mathfrak{sl}_{n})$, its idealic map and
then $\operatorname{Hilb}^{reg}(\mathfrak{sl}_{n})$ using Proposition 15.8.

Fix the following principal nilpotent element (with 1 on the diagonal line
just under the main diagonal):

$f=\begin{pmatrix}&&&\\\ 1&&&\\\ &\ddots&&\\\ &&1&\end{pmatrix}.$

This element $f$ is cyclic, so we know from 15.5 that the centralizer is given
by polynomials: $Z(f)=\\{\mu_{2}f+\mu_{3}f^{2}+...+\mu_{n}f^{n-1}\\}$. So an
element of $\operatorname{Hilb}^{reg}_{0}(\mathfrak{sl}_{n})$ can be
represented by $(f,Q(f))$ where $Q$ is a polynomial without constant term of
degree at most $n-1$. The coefficients $\mu_{i}$ are called higher Beltrami
coefficients.

Since here we have
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{sl}_{n})\subset\operatorname{Hilb}^{cycl}(\mathfrak{sl}_{n})$
(already $f$ is cyclic), the idealic map is given by

$I(f,Q(f))=\\{P\in\mathbb{C}[x,y]\mid P(f,Q(f))=0\\}=\langle
x^{n},-y+Q(x)\rangle.$

We recognize the big cell of the zero-fiber of the punctual Hilbert scheme.

To describe the whole regular part
$\operatorname{Hilb}^{reg}(\mathfrak{sl}_{n})$, we take the following
principal slice given by companion matrices:

$\begin{pmatrix}&&&t_{n}\\\ 1&&&\vdots\\\ &\ddots&&t_{2}\\\
&&1&\end{pmatrix}.$

Let $A$ be a matrix of companion type. Notice that the characteristic
polynomial of a companion matrix is given by $x^{n}+t_{2}x^{n-2}+...+t_{n}$.
Since $A$ is still cyclic, its centralizer consists of polynomials in $A$ with
constant term determined by the other coefficients (in order to ensure trace
zero). Thus, a representative of
$\operatorname{Hilb}^{reg}(\mathfrak{sl}_{n})$ is given by $(A,B=Q(A))$.

The idealic map is thus given by

$I(A,B)=\langle
x^{n}+t_{2}x^{n-2}+...+t_{n},-y+\mu_{1}+\mu_{2}x+...+\mu_{n}x^{n-1}\rangle$

where $\mu_{1}$ is given by
$\mu_{1}=\sum_{k=2}^{n-1}\frac{k}{n}t_{k}\mu_{k+1}\mod t^{2}$ (see equation
(6.2)). One recognizes the big cell of the reduced punctual Hilbert scheme.
Notice that the idealic map is injective here.

####  15.6 Case $B_{n}$

Consider $\mathfrak{g}=\mathfrak{so}_{2n+1}$. Represent $\mathfrak{g}$ on
$\mathbb{C}^{2n+1}$ using the metric given by $g(e_{i},e_{j})=\delta_{i,n-j}$
(where $e_{i}$ are standard vectors), i.e. $g=\left(\begin{smallmatrix}&&1\\\
&\udots&\\\ 1&&\end{smallmatrix}\right)$. A matrix $A$ is in $\mathfrak{g}$
iff $\sigma(A)=-A$ where $\sigma$ is the involution given by a reflection
along the anti-diagonal. In other words $A\in\mathfrak{g}$ iff
$A_{i,j}=A_{n+1-j,n+1-i}$ for all $i,j$.

We fix the following principal nilpotent element:

$f=\begin{pmatrix}&&&&&&\\\ 1&&&&&&\\\ &\ddots&&&&&\\\ &&1&&&&\\\ &&&-1&&&\\\
&&&&\ddots&&\\\ &&&&&-1&\end{pmatrix}.$

This element is cyclic, so its centralizer by 15.5 consists of all odd
polynomials: $Z(f)=\\{\mu_{2}f+\mu_{4}f^{3}+...+\mu_{2n}f^{2n-1}\\}$. A
representative of $\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})$ is thus given
by $(f,Q(f))$ where $Q$ is an odd polynomial of degree at most $2n-1$. The
coefficients $\mu_{2i}$ are called the higher Beltrami coefficients for
$B_{n}$.

A principal slice is given by

$\begin{pmatrix}&&&&&t_{2n}&\\\ 1&&&&\udots&&-t_{2n}\\\
&\ddots&&t_{2}&&\udots&\\\ &&1&&-t_{2}&&\\\ &&&-1&&&\\\ &&&&\ddots&&\\\
&&&&&-1&\end{pmatrix}.$

Let $A$ be a matrix of this type. Its characteristic polynomial is given by
$x^{2n+1}-2t_{2}x^{2n-1}+2t_{4}x^{2n-3}\pm...+(-1)^{n}\times 2t_{2n}x$. So we
can really think of the principal slice as a generalized companion matrix.
Changing slightly $t_{2i}$ we can get rid of signs and the factor 2 in the
characteristic polynomial, which we will do in the sequel.

The matrix $A$ is still cyclic, so we have
$\operatorname{Hilb}^{reg}(\mathfrak{g})\subset\operatorname{Hilb}^{cycl}(\mathfrak{g})$.
A representative of $\operatorname{Hilb}^{reg}(\mathfrak{g})$ is given by
$(A,B=Q(A))$ where $Q$ is still an odd polynomial of degree at most $2n-1$.
The idealic map is then given by

$I(A,B)=\langle
x^{2n+1}+t_{2}x^{2n-1}+t_{4}x^{2n-3}+...+t_{2n}x,-y+\mu_{2}x+\mu_{4}x^{3}+...+\mu_{2n}x^{2n-1}\rangle.$

This ideal is invariant under the map $(x,y)\mapsto(-x,-y)$. This is not
surprising since a generic element of the $\mathfrak{g}$-Hilbert scheme is a
pair of two diagonal matrices which for $\mathfrak{so}_{2n+1}$ are of the form
$\operatorname{diag}(x_{1},...,x_{n},0,-x_{n},...,-x_{1})$ and
$\operatorname{diag}(y_{1},...,y_{n},0,-y_{n},...,-y_{1})$. So they can be
thought of as $2n+1$ points in $\mathbb{C}^{2}$ with one point being the
origin and the other points being symmetric with respect to the origin. This
set is invariant under the map $-\operatorname{id}$, so is its defining ideal.

The next type, $C_{n}$, is quite similar to $B_{n}$.

####  15.7 Case $C_{n}$

Let $\mathfrak{g}=\mathfrak{sp}_{2n}$. We use the symplectic structure
$\omega=\sum_{i}e_{i}\wedge e_{n+i}$ of $\mathbb{C}^{2n}$ to represent
$\mathfrak{g}$. So a matrix

$\left(\begin{array}[]{c|c}A&B\\\ \hline\cr C&D\end{array}\right)$

is in $\mathfrak{g}$ iff $D=-A^{\top}$ and $B$ and $C$ are symmetric matrices.

Fix the principal nilpotent by

$f=\left(\begin{array}[]{cccc|cccc}&&&&&&&\\\ 1&&&&&&&\\\ &\ddots&&&&&&\\\
&&1&&&&&\\\ \hline\cr&&&&&-1&&\\\ &&&&&&\ddots&\\\ &&&&&&&-1\\\
&&&1&&&&\end{array}\right).$

This element is cyclic, so its centralizer is given by odd polynomials:
$Z(f)=\\{\mu_{2}f+\mu_{4}f^{3}+...+\mu_{2n}f^{2n-1}\\}$. As for $B_{n}$ we
call the $\mu_{2i}$ higher Beltrami coefficients.

A principal slice is given by

$\left(\begin{array}[]{cccc|cccc}&&&&t_{2n}&&&\\\ 1&&&&&t_{2n-2}&&\\\
&\ddots&&&&&\ddots&\\\ &&1&&&&&t_{2}\\\ \hline\cr&&&&&-1&&\\\ &&&&&&\ddots&\\\
&&&&&&&-1\\\ &&&1&&&&\end{array}\right).$

Let $A$ be an element of this form. Its characteristic polynomial is given by
$x^{2n}-t_{2}x^{2n-2}+t_{4}x^{2n-4}\pm...+(-1)^{n}t_{2n}$. By changing signs
in the $t_{2i}$ we can omit the minus signs in the characteristic polynomial.

The matrix $A$ is still cyclic so a representative of
$\operatorname{Hilb}^{reg}(\mathfrak{sp}_{2n})$ is given by $(A,B=Q(A))$ where
$Q$ is an odd polynomial of degree at most $2n-1$.

The idealic map reads

$I(A,B)=\langle
x^{2n}+t_{2}x^{2n-2}+t_{4}x^{2n-4}+...+t_{2n},-y+\mu_{2}x+\mu_{4}x^{3}+...+\mu_{2n}x^{2n-1}\rangle.$

As for $B_{n}$, this ideal is invariant under $-\operatorname{id}$ which comes
from the fact that two diagonal matrices in $\mathfrak{sp}_{2n}$ give $2n$
points in $\mathbb{C}^{2}$ which are symmetric with respect to the origin.

The last classical type, $D_{n}$, has some surprises.

####  15.8 Case $D_{n}$

Let $\mathfrak{g}=\mathfrak{so}_{2n}$. We use the same representation as for
$B_{n}$.

Fix the following principal nilpotent element:

$f=\begin{pmatrix}&&&&&&&\\\ 1&&&&&&&\\\ &\ddots&&&&&&\\\ &&1&&&&&\\\
&&1&0&&&&\\\ &&&-1&-1&&&\\\ &&&&&\ddots&&\\\ &&&&&&-1&\\\ \end{pmatrix}.$

This elements is not cyclic, since $f^{2n-1}=0$. A direct computation shows
that
$Z(f)=\\{\mu_{2}f+\mu_{4}f^{3}+...+\mu_{2n-2}f^{2n-3}\\}\cup\\{\sigma_{n}S\\}$
where $S$ is the matrix

$S=\left(\begin{array}[]{@{}ccc|ccc@{}}&&&&&\\\ &&&&&\\\ 1&&&&&\\\
\hline\cr-1&&&&&\\\ &&&&&\\\ &&1&-1&&\end{array}\right).$ (15.4)

We can give an intrinsic definition of the matrix $S$: let $R$ be a root
system and $v_{\alpha}$ be a root vector in $\mathfrak{g}$ for the root
$\alpha\in R$. Choose a basis $\alpha_{1},...,\alpha_{n}$ of $R$ (the simple
roots) such that $\alpha_{n-1}$ and $\alpha_{n}$ correspond to the two non-
adjacent vertices in the Dynkin diagram of $D_{n}$ (see figure 15.2). We can
choose $f$ to be $\sum_{i}v_{\alpha_{i}}$. The matrix $S$ is then given by

$S=v_{\alpha_{1}+...+\alpha_{n-1}}\pm
v_{\alpha_{1}+...+\alpha_{n-2}+\alpha_{n}}$

where the sign depends on the choice of the root vectors.

Figure 15.2: Dynkin diagram for $D_{n}$

A representative of $\operatorname{Hilb}^{reg}_{0}(\mathfrak{so}_{2n})$ is
given by $(A=f,B=Q(f)+\sigma_{n}S)$ where $Q$ is an odd polynomial of degree
at most $2n-3$. Such a pair is cyclic iff $\sigma_{n}\neq 0$.

Let us compute the ideal in the cyclic case. One easily checks that $fS=Sf$
and that $S^{2}=2f^{2n-2}$. Hence for
$B=\mu_{2}f+...+\mu_{2n-2}f^{2n-2}+\sigma_{n}S$, we get
$AB=fB=\mu_{2}f^{2}+...+\mu_{2n-2}f^{2n-2}$ and
$B^{2}=(\mu_{2}f+...+\mu_{2n-2}f^{2n-3})^{2}+2\sigma_{n}^{2}f^{2n-2}$. Hence,
the idealic map is given by

$I(A,B)=\langle
x^{2n-1},xy=\mu_{2}x^{2}+\mu_{4}x^{4}+...+\mu_{2n-2}x^{2n-2},y^{2}=\nu_{2}x^{2}+\nu_{4}x^{4}+...+\nu_{2n-2}x^{2n-2}\rangle$

where $\nu_{2k}=\sum_{i=1}^{k}\mu_{2i}\mu_{2k+2-2i}$ for $k=1,...,n-2$ and
$\nu_{2n-2}=2\sigma_{n}^{2}+\sum_{i=1}^{n-1}\mu_{2i}\mu_{2n-2i}$. So we see
that $(\mu_{2},\mu_{4},...,\mu_{2n-2},\nu_{2n-2})$ is a set of independent
variables which we call higher Beltrami differentials for $D_{n}$. We call
$\sigma_{n}$ a higher Beltrami differential as well. If $\sigma_{n}=0$, we
define the idealic map to be the continuous extension of the above ideal which
is still of the same form.

######  Remark.

We have seen in example 15.17 that inside
$\operatorname{Hilb}^{cycl}(\mathfrak{so}_{2n})$ there is no well-defined
continuous extension of the idealic map. But inside the zero-fiber, the limit
is unique. $\triangle$

The Hilbert scheme is covered with charts indexed by partitions (see
subsection 4.2). The chart in which $I$ is written corresponds to the
partition $2n=(2n-1)+1$ which we write also $[2n-1,1]$. In fact, this is the
highest partition of $2n$ of type $D_{n}$ (see [CM93], chapter 5 for special
types of partitions).

A principal slice is given by

$\left(\begin{array}[]{@{}cccc|cccc@{}}&&&\tau_{n}&-\tau_{n}&&t_{2n-2}&\\\
1&&&&&\udots&&-t_{2n-2}\\\ &\ddots&&t_{2}&t_{2}&&\udots&\\\
&&1&&&-t_{2}&&\tau_{n}\\\ \hline\cr&&1&0&&-t_{2}&&-\tau_{n}\\\ &&&-1&-1&&&\\\
&&&&&\ddots&&\\\ &&&&&&-1&\end{array}\right).$

Notice that the matrix for $\tau_{n}$ is $S^{\top}$. Let $A$ be a matrix of
this type. Its characteristic polynomial is given by

$\chi(A)=x^{2n}-4t_{2}x^{2n-2}+4t_{4}x^{2n-4}\pm...+(-1)^{n-1}\times
4t_{2n-2}x^{2}+(-1)^{n}\tau_{n}^{2}.$

By changing signs and factors in $t_{2i}$ and $\tau_{n}$, we can omit signs
and the factor 4 in the characteristic polynomial.

One can compute that the minimal polynomial of $A$ is equal to the
characteristic polynomial iff $\tau_{n}\neq 0$. So $A$ is cyclic iff
$\tau_{n}\neq 0$ (by Proposition A.3). In that case, the centralizer consists
of all odd polynomials in $A$ of degree at most $2n-1$. If $\tau_{n}=0$, the
centralizer is given by

$Z(A)=\\{\mu_{2}A+\mu_{4}A^{3}+...+\mu_{2n-2}A^{2n-3}\\}\cup\\{\sigma_{n}S_{t}\\}$

where the matrix $S_{t}$ is given by $S_{t}=S+t_{2n-2}S^{\top}$. The minimal
polynomial is given by $\chi(x)/x$ (which is a polynomial since $\tau_{n}=0$).

The pair $(A,B)$ is cyclic iff either $\tau_{n}\neq 0$ or $\tau_{n}=0$ and
$\sigma_{n}\neq 0$. In the first case, the idealic map is given by

$I=\langle
x^{2n}+t_{2}x^{2n-2}+t_{4}x^{2n-4}+...+t_{2n-2}x^{2}+\tau_{n}^{2},-y+\mu_{2}x+\mu_{4}x^{3}+...+\mu_{2n}x^{2n-1}\rangle.$

In the second case, we need three generators for the ideal, like for the zero-
fiber. We can compute that

$\displaystyle I(A,B)=\;\langle x^{2n-1}$
$\displaystyle=u_{2}x+u_{4}x^{3}+...+u_{2n-2}x^{2n-3}+uy,$ $\displaystyle xy$
$\displaystyle=v_{0}+v_{2}x^{2}+...+v_{2n-2}x^{2n-2},$ $\displaystyle y^{2}$
$\displaystyle=w_{0}+w_{2}x^{2}+...+w_{2n-2}x^{2n-2}\rangle$

where the coordinates can be chosen to be

$(u_{2},u_{4},...,u_{2n-2},u,v_{2},...,v_{2n-2},w_{2n-2})$

i.e. all the other variables are functions of these. These are Haiman
coordinates as explained in figure 6.1. In particular, we can see that the
coordinates $u$ and $\nu_{2n-2}$ are canonically conjugated.

The second ideal is in the chart corresponding to the partition $[2n-1,1]$
whereas the first corresponds to the trivial partition $[2n]$. If $u\neq 0$ we
can write the second ideal in the first chart, i.e. perform a coordinate
change in the Hilbert scheme. The link between the coordinates is given by

$\left\\{\begin{array}[]{cl}\tau_{n}^{2}=uv_{0}\\\ \mu_{2n}=\frac{1}{u}\\\
\mu_{2k}=-\frac{u_{2k}}{u}&\text{ for }1\leq k<n\\\
t_{2k}=u_{2n-2k}+uv_{2n-2k}&\text{ for }1\leq k\leq n-1.\end{array}\right.$

A regular pair $[(A,B)]$ which is not cyclic has both $\tau_{n}$ and
$\sigma_{n}$ equal to 0. In that case, we define the idealic map $I(A,B)$ to
be the limit of $I(A,B+tS_{t})$ for $t\in\mathbb{C}$ goes to 0. So we stay in
a chart associated to the partition $[2n-1,1]$.

Notice that the map from $\operatorname{Hilb}^{reg}(\mathfrak{g})$ to the
space of ideals $I_{\mathfrak{g}}(\mathbb{C}^{2})$ is not injective, since for
$\tau_{n}$ and $-\tau_{n}$ we get the same ideal. Even in the zero-fiber the
map is not injective, since $\sigma_{n}$ and $-\sigma_{n}$ give the same
ideal. In addition, the map is not surjective either. Indeed the ideal
$I=\langle x^{5}-y,xy,y^{2}\rangle\in I_{\mathfrak{g}}(\mathbb{C}^{2})$ is not
in the image since with the notation above we have $v_{0}=0$ and $u\neq 0$.
Changing the chart, we can compute that $\tau_{n}^{2}=uv_{0}=0$. But for a
matrix in $\operatorname{Hilb}^{reg}(\mathfrak{g})$ with $\tau_{n}=0$ we get
$u=0$.

######  Remark.

In the usual Hilbert scheme, there is only one cell of maximal dimension.
Comparing type $C_{n}$ and type $D_{n}$, we see that the zero-fiber of

$\\{I\text{ ideal of }\mathbb{C}[x,y]\mid\operatorname{codim}I=2n,I\text{
invariant under }-\operatorname{id}\\}$

has two components of maximal dimension,
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{sp}_{2n})$ and
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{so}_{2n})$. $\triangle$

######  Remark.

We notice the following analog to Higgs bundles: the pair
$[(f,0)]\in\operatorname{Hilb}(\mathfrak{so}_{2n})$ corresponds to the Higgs
field given by $\Phi=f$ on the bundle $V=K^{2}\oplus K\oplus K^{0}\oplus
K^{-2}\oplus K^{-1}\oplus K^{0}$. This Higgs bundle $(V,\Phi)$ is not stable,
only polystable. This could explain why the idealic map can not be
continuously extended to $[(f,0)]$. $\triangle$

####  15.9 Topology of $\mathfrak{g}$-Hilbert schemes*

It is clear that $\operatorname{Hilb}(\mathfrak{g})$ is a topological space,
as a quotient of a subset of $\mathfrak{g}^{2}$. In this section, we explore
this topology of $\operatorname{Hilb}(\mathfrak{g})$, especially for
$\mathfrak{g}=\mathfrak{sl}_{n}$. We then formulate some conjectures on its
general structure.

For $\mathfrak{g}=\mathfrak{sl}_{2}$, every non-zero element
$A\in\mathfrak{g}$ is regular and cyclic. Since the centralizer of the pair
$(0,0)$ is all of $\mathfrak{sl}_{2}$, this pair is not in
$\operatorname{Hilb}(\mathfrak{sl}_{2})$. Thus we have
$\operatorname{Hilb}(\mathfrak{sl}_{2})=\operatorname{Hilb}^{cycl}(\mathfrak{sl}_{2})=\operatorname{Hilb}^{2}_{red}(\mathbb{C}^{2})$
which is a smooth projective variety.

For $\mathfrak{g}=\mathfrak{sl}_{3}$, a detailed analysis, putting $A$ into
Jordan normal form, shows that $(A,B)$ has minimal centralizer and is not
cyclic iff it is conjugated to a pair
$P_{1}(b)=\left(\left(\begin{smallmatrix}0&1&0\\\ 0&0&0\\\
0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&b&1\\\ 0&0&0\\\
0&0&0\end{smallmatrix}\right)\right)$. So

$\operatorname{Hilb}(\mathfrak{sl}_{3})=\operatorname{Hilb}^{3}_{red}(\mathbb{C}^{2})\cup\\{P_{1}(b)\mid
b\in\mathbb{C}\\}.$

At first sight, the topology seems to be a smooth variety (the reduced Hilbert
scheme) and a complex line. But a closer look shows that each point of the
extra line is infinitesimally close to a point in the variety, meaning that
these two points cannot be separated by open sets, infringing the Hausdorff
property. The pair $P_{1}(b)$ is infinitesimally close to
$P_{2}(b)=\left(\left(\begin{smallmatrix}0&1&0\\\ 0&0&0\\\
0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&b&0\\\ 0&0&0\\\
0&1&0\end{smallmatrix}\right)\right)$. Indeed any neighborhood of the first
pair $P_{1}(b)$ contains $\left(\left(\begin{smallmatrix}0&1&0\\\ 0&0&0\\\
0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&b&1\\\ 0&0&0\\\
0&s&0\end{smallmatrix}\right)\right)$ for some small $s\in\mathbb{C}$ which is
conjugated to $\left(\left(\begin{smallmatrix}0&1&0\\\ 0&0&0\\\
0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&b&s\\\ 0&0&0\\\
0&1&0\end{smallmatrix}\right)\right)$ which lies in a neighborhood of the
second pair$P_{2}(b)$. Since the idealic map is continuous and for
$\mathfrak{sl}_{n}$ injective on the cyclic part, there cannot be another
point of the cyclic part which is infinitesimally close to the first pair
$P_{1}(b)$. Finally, two elements of the extra line can be separated by open
sets. Hence, the space $\operatorname{Hilb}(\mathfrak{sl}_{3})$ is obtained
from a smooth variety by adding “double points” (here in the sense of
infinitesimally close points) along a complex line.

Since the idealic map is injective on the cyclic part
$\operatorname{Hilb}^{cycl}(\mathfrak{sl}_{n})$, the same analysis holds for
$\mathfrak{sl}_{n}$, i.e. $\operatorname{Hilb}(\mathfrak{sl}_{n})$ is obtained
from a smooth variety (the reduced Hilbert scheme) by adding double points.

There should exist a procedure, like a GIT quotient, giving a modified
$\mathfrak{g}$-Hilbert scheme which is a Hausdorff space. The GIT quotient
does not apply here since $\\{(A,B)\in\mathfrak{g}^{2}\mid[A,B]=0,\dim
Z(A,B)=\operatorname{rk}\mathfrak{g}\\}$ is not a closed variety. In the
language of GIT quotients, the pairs $P_{1}$ and $P_{2}$ above are both
semistable, but there is no polystable element in their closure.

To give a feeling about what happens, consider the action of $\mathbb{R}_{>0}$
on $\mathbb{R}^{2}\backslash\\{(0,0)\\}$ given by
$\lambda.(x_{1},x_{2})=(\lambda x_{1},\lambda^{-1}x_{2})$. The orbits are
drawn in figure 15.3. The quotient space is a set of two lines $L_{1}$ and
$L_{2}$ with origins $O_{1}$ and $O_{2}$ together with two extra points
$O_{3}$ and $O_{4}$ such that the pairs
$(O_{1},O_{3}),(O_{1},O_{4}),(O_{2},O_{3})$ and $(O_{2},O_{4})$ are
infinitesimally close points (the four points $O_{i}$ correspond to the four
half-axis). In the figure, the dashed lines indicate infinitesimally close
points. From the GIT perspective, all points are semistable (take the constant
function 1), the four half-axis are semistable and all other orbits are
stable. The orbits of the half-axis are closed in
$\mathbb{R}^{2}\backslash\\{(0,0)\\}$ so they should be polystable, but in the
quotient the points are still infinitesimally close.

Figure 15.3: Non-Hausdorff quotient

We conjecture the following:

######  Conjecture 15.22.

There is a generalized GIT quotient procedure identifying infinitesimally
close points in $\operatorname{Hilb}(\mathfrak{g})$, giving a modified
$\mathfrak{g}$-Hilbert scheme which is Hausdorff.

In particular one should find the reduced Hilbert scheme for
$\mathfrak{g}=\mathfrak{sl}_{n}$. See also Conjecture 15.16 for a modified
$\mathfrak{g}$-Hilbert scheme as a resolution of $\mathfrak{h}^{2}/W$.

Assume a smooth version of the $\mathfrak{g}$-Hilbert scheme exists. In the
$\mathfrak{sl}_{n}$-case the reduced Hilbert scheme is covered with charts
parameterized by partitions of $n$, which also parameterizes nilpotent orbits
of $\mathfrak{sl}_{n}$. For $\mathfrak{g}$ of classical type, the nilpotent
orbits are parameterized by special partitions (see [CM93], chapter 5). In
general, we conjecture the following for the zero-fiber of the
$\mathfrak{g}$-Hilbert scheme:

######  Conjecture 15.23.

The smooth version of $\operatorname{Hilb}_{0}(\mathfrak{g})$ is covered with
charts parameterized by nilpotent orbits and all these charts are necessary to
cover $\operatorname{Hilb}_{0}(\mathfrak{g})$. In particular for classical
$\mathfrak{g}$, we conjecture that the modified version of
$\operatorname{Hilb}_{0}(\mathfrak{g})$ is isomorphic to the space of ideals
of $\mathbb{C}[x,y]$ which are of codimension $m$, $W$-invariant, supported at
0 and which lie in a chart associated to a partition of type $\mathfrak{g}$.

###  16 $\mathfrak{g}$-complex structures

Using the $\mathfrak{g}$-Hilbert scheme we are able to construct a new
geometric structure on a smooth surface, generalizing both complex and higher
complex structures.

####  16.1 Definition

Recall from section 5 that the higher complex structure is defined as a
section $I$ of $\operatorname{Hilb}^{n}_{0}(T^{*\mathbb{C}}\Sigma)$ satisfying
$I(z)\oplus\overline{I}(z)=\langle p,\bar{p}\rangle$ at every point
$z\in\Sigma$ (see Definition 5.1). We call the condition on $I$ the reality
constraint. In this definition we exclusively use the idealic viewpoint of the
punctual Hilbert scheme. Since the $\mathfrak{g}$-Hilbert scheme uses the
matrix viewpoint, we have to think of $n$-complex structures as special
1-forms. More precisely, an $n$-complex structure is a gauge class of a
$\mathfrak{sl}_{n}$-valued 1-form which can be locally written as
$\Phi_{1}dz+\Phi_{2}d\bar{z}$ where $[(\Phi_{1},\Phi_{2})]$ is in the zero-
fiber of the Hilbert scheme.

We are now ready to give the definition of a $\mathfrak{g}$-complex structure,
but one difficulty stays: we have to incorporate the reality constraint in the
matrix viewpoint. Recall from Proposition 15.21 the map
$\mu_{2}:\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})\rightarrow\mathbb{C}$
associating to $[(A,B)]$ the unique $\mu_{2}\in\mathbb{C}$ such that
$B-\mu_{2}A$ is irregular.

######  Definition 16.1.

A $\mathbf{\mathfrak{g}}$-complex structure is a gauge class of elements
locally of the form

$A(z)dz+B(z)d\bar{z}\in\Omega^{1}(\Sigma,\mathfrak{g})=\Omega^{1}(\Sigma,\mathbb{C})\otimes\mathfrak{g}$

such that

$[(A(z),B(z))]\in\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})$

and $\mu_{2}(z)\bar{\mu}_{2}(z)\neq 1$ for all $z\in\Sigma$.

Notice that for complex structures, the map $\mu_{2}(z)$ is nothing but the
Beltrami differential. So our reality constraint coincides with the one for
complex structures. In particular, for $\mathfrak{g}=\mathfrak{sl}_{2}$, we
get a usual complex structure. In the general case, we have the following:

######  Proposition 16.2.

A $\mathfrak{g}$-complex structure induces a complex structure on $\Sigma$.

###### Proof.

Recall from 15.4 equation (15.3) the map
$\mu:\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})\rightarrow\operatorname{Hilb}_{0}(\mathfrak{sl}_{2})$
given by $\mu([(A,B)])=[(e,\mu_{2}e)]$ or $[(\mu_{2}e,e)]$ depending on
whether $A$ or $B$ is regular. Since a $\mathfrak{sl}_{2}$-complex structure
is a complex structure, the map $\mu$ induces a map from
$\mathfrak{g}$-complex structures to complex structures. ∎

######  Remark.

To define the map $\mu$ in 15.4, we really need $\mathfrak{g}$ to be simple.
Thus, only for $\mathfrak{g}$ simple we get a unique complex structure out of
a $\mathfrak{g}$-complex structure. $\triangle$

In the definition of a higher complex structure from section 5, we use the
zero-fiber $\operatorname{Hilb}^{n}_{0}(\mathbb{C}^{2})$, without imposing to
be in the regular part. The fact that we actually are in the regular part
follows from the reality constraint $I\oplus\overline{I}=\langle
p,\bar{p}\rangle$. The same can be obtained for $\mathfrak{g}$ of classical
type, where we can reformulate the definition of $\mathfrak{g}$-complex
structures in a nicer way using the idealic map.

####  16.2 Idealic viewpoint

Recall the space of ideals $I_{\mathfrak{g}}(\mathbb{C}^{2})$ constructed in
15.3. Denote by $I_{\mathfrak{g},0}(\mathbb{C}^{2})$ the set of those ideals
of $I_{\mathfrak{g}}(\mathbb{C}^{2})$ which are supported at the origin (the
zero-fiber). We can rewrite the definition of a $\mathfrak{g}$-complex
structure in the following way:

######  Definition 16.3.

For classical $\mathfrak{g}$, a $\mathbf{\mathfrak{g}}$-complex structure is a
section $I$ of $I_{\mathfrak{g},0}(T^{*\mathbb{C}}\Sigma)$ such that for all
$z\in\Sigma$:

$I(z)\oplus\overline{I}(z)=\left\\{\begin{array}[]{cl}\langle
p,\bar{p}\rangle&\text{ if }\mathfrak{g}\text{ of type }A_{n},B_{n},C_{n}\\\
\langle p,\bar{p}\rangle^{2}&\text{ if }\mathfrak{g}\text{ of type
}D_{n}.\end{array}\right.$

Notice that the condition on the ideals does not depend on coordinates since
$\langle p,\bar{p}\rangle$ is the maximal ideal associated to the origin.

We prove the equivalence of both definitions. For that recall that to an ideal
$I$ one can associate a class of commuting matrices $[(A,B)]$ (see subsection
4.4).

######  Proposition 16.4.

For classical $\mathfrak{g}$, the condition on $I\oplus\overline{I}$ given in
Definition 16.3 is equivalent to $[(A(z),B(z))]$ being in the regular part
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})$ and having
$\mu_{2}\bar{\mu}_{2}\neq 1$, i.e. the condition in Definition 16.1.

###### Proof.

The backwards direction is a direct computation using the preferred
representatives for $\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})$ from
Proposition 15.8. So we concentrate on the direct implication.

Case $A_{n}$. The case $\mathfrak{g}$ of type $A_{n}$ has been treated in
Proposition 5.2.

Case $B_{n}$. For $\mathfrak{g}$ of type $B_{n}$ the standard representation
gives $\mathfrak{so}_{2n+1}\hookrightarrow\mathfrak{sl}_{2n+1}$. By virtue of
the case $A_{n}$, we know that $I\oplus\overline{I}=\langle p,\bar{p}\rangle$
implies $\mu_{2}\bar{\mu}_{2}\neq 1$ and $(A,B)$ regular for
$\mathfrak{sl}_{2n+1}$, i.e.

$I(A,B)=\langle
p^{2n+1},-\bar{p}+\mu_{2}p+\mu_{3}p^{2}+...+\mu_{2n}p^{2n}\rangle.$

Since we know that in case $B_{n}$, the ideal $I$ is invariant under the map
$-\operatorname{id}$, we get $\mu_{2k+1}=0$ for all $k=1,...,n-1$. So $I$
corresponds to a pair $(f,Q(f))$ for $Q$ an odd polynomial of degree at most
$2n-1$, which is precisely a representative of
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{so}_{2n+1})$ (see subsection 15.6).

Case $C_{n}$. This case is exactly analogous to $B_{n}$ via the injection
$\mathfrak{sp}_{2n}\hookrightarrow\mathfrak{sl}_{2n}$.

Case $D_{n}$. We imitate the strategy of the proof for case $A_{n}$ (see
Proposition 5.2) with only difference that we have to go further in the
analysis, needing some computations. The main argument is an iteration process
which always ends since $p^{k}\bar{p}^{l}=0\mod I$ for $k+l\geq 2n$.

Put $I_{1}=(I\mod\langle p,\bar{p}\rangle^{2})$, i.e. the set of all terms of
degree at most 1 appearing in $I$. If $I_{1}$ is of dimension 2, then
$I=\langle p,\bar{p}\rangle$ since both $p$ and $\bar{p}$ can be expressed by
higher terms which by iteration become 0. If $I_{1}$ is of dimension 1, then
we have a relation of the form $\bar{p}=\mu_{2}p+p^{2}R(p,\bar{p})$ where $R$
is a polynomial, which gives $\bar{p}$ as a polynomial in $p$ by iteration. We
can then explicitly check that $I\oplus\overline{I}$ is either $\langle
p,\bar{p}\rangle$ or $\langle p=\bar{p},p\bar{p},p^{2}\rangle$. Since we
suppose $I=\langle p,\bar{p}\rangle^{2}$, we get $\dim I_{1}=0$, i.e.
$I_{1}=\\{0\\}.$

Put $I_{2}=(I\mod\langle p,\bar{p}\rangle^{3})$. We have
$I_{2}\oplus\overline{I}_{2}=(I\oplus\overline{I})_{2}=\langle
p^{2},p\bar{p},\bar{p}^{2}\rangle$ by assumption on $I$. If $I_{2}$ is of
dimension 3, then all of $p^{2},p\bar{p}$ and $\bar{p}^{2}$ can be expressed
by higher terms. By iteration, we get $I=\langle
p^{2},p\bar{p},\bar{p}^{2}\rangle$ which is not of type $D_{n}$. If $\dim
I_{2}\leq 1$, then we also have $\dim\overline{I}_{2}\leq 1$, so $2\geq\dim
I_{2}+\dim\overline{I}_{2}=\dim\langle
p^{2},p\bar{p},\bar{p}^{2}\rangle_{2}=3$, a contradiction. Hence $\dim
I_{2}=2.$

There is a term containing $p\bar{p}$ in $I_{2}$ since if not, no such term
would neither exist in $\overline{I}_{2}$, so neither in
$I_{2}\oplus\overline{I}_{2}=\langle p^{2},p\bar{p},\bar{p}^{2}\rangle$, a
contradiction. Without loss of generality, we can assume that there is another
term containing $\bar{p}^{2}$ (if not change the role of $I$ and
$\overline{I}$).

So there exist $\alpha,\beta,\gamma,\delta\in\mathbb{C}$ such that

$\left\\{\begin{array}[]{cl}\bar{p}^{2}=\alpha p^{2}+\beta p\bar{p}&\mod
I_{2}\\\ p\bar{p}=\gamma p^{2}+\delta\bar{p}^{2}&\mod
I_{2}.\end{array}\right.$

If $\beta\gamma\neq 1$, we can simplify by substitution one into the other to

$\left\\{\begin{array}[]{cl}\bar{p}^{2}=\alpha^{\prime}p^{2}&\mod I_{2}\\\
p\bar{p}=\gamma^{\prime}p^{2}&\mod I_{2}.\end{array}\right.$

If $\beta\gamma=1$, we have $p^{2}\in I_{2}$, so
$p\bar{p}=\delta\bar{p}^{2}\mod I_{2}$, so changing $I$ to $\overline{I}$ we
are in the previous situation.

Iterating the substitution process we get that $\bar{p}^{2}$ and $p\bar{p}$
are polynomials in $p$. Using the invariance of $I$ under
$-\operatorname{id}$, we see that these are polynomials in $p^{2}$, i.e. even
polynomials. So the most generic ideal is given by

$I=\langle
p^{2n-1},p\bar{p}=\mu_{2}p^{2}+\mu_{4}p^{4}+...+\mu_{2n-2}p^{2n-2},\bar{p}^{2}=\nu_{2}p^{2}+\nu_{4}p^{4}+...+\nu_{2n-2}p^{2n-2}\rangle$

which corresponds to a regular element of
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{so}_{2n})$. One checks that
$I\oplus\overline{I}$ with $I$ of the form above equals $\langle
p,\bar{p}\rangle^{2}$ iff $\mu_{2}\bar{\mu_{2}}\neq 1$. ∎

To end this section, we determine the geometric nature of the various higher
Beltrami coefficients. Since $p$ and $\bar{p}$ are linear coordinates on
$T^{*\mathbb{C}}\Sigma$, we can identify $p=\frac{\partial}{\partial
z}=\partial$ and $\bar{p}=\frac{\partial}{\partial\bar{z}}=\bar{\partial}$.
Denote by $K$ the canonical bundle, i.e. $K=T^{*(1,0)}\Sigma$, and by
$\Gamma(B)$ the space of sections of a bundle $B$.

Analyzing the behavior under a coordinate change $z\mapsto w(z,\bar{z})$
analogous to the computation in subsection 5.1, we get

$\mu_{i}\in\Gamma(K^{1-i}\otimes\bar{K})\text{ and
}\nu_{2i}\in\Gamma(K^{-2i}\otimes\bar{K}^{2}).$ (16.1)

Since $\sigma_{n}^{2}$ has the same nature as $\nu_{2n-2}$ (see 15.8), we get
$\sigma_{n}\in\Gamma(K^{1-n}\otimes\bar{K})$.

###  17 Moduli space

In this section, we define the moduli space of $\mathfrak{g}$-complex
structures and explore its properties. In the whole section we suppose
$\mathfrak{g}$ of classical type. We first have to define an equivalence
relation on $\mathfrak{g}$-complex structures, which is accomplished by the
notion of higher diffeomorphisms of type $\mathfrak{g}$.

####  17.1 Higher diffeomorphisms

For higher complex structures, we defined in subsection 5.2 higher
diffeomorphisms to be hamiltonian diffeomorphisms of $T^{*}\Sigma$ preserving
the zero-section $\Sigma\subset T^{*}\Sigma$. This gives the higher
diffeomorphisms for type $A_{n}$. We generalize this idea to other classical
$\mathfrak{g}$. We use the standard representation of $\mathfrak{g}$ on
$\mathbb{C}^{m}$, i.e.
$\mathfrak{sl}_{n}\subset\mathfrak{gl}_{n},\mathfrak{so}_{n}\subset\mathfrak{gl}_{n}$
and $\mathfrak{sp}_{2n}\subset\mathfrak{gl}_{2n}$.

One way to think of a $\mathfrak{g}$-complex structure is as a $m$-tuple of
1-forms with some symmetry, which collapses all to the zero-section. The space
of higher diffeomorphisms which we are looking for has to preserve this
symmetry. For example for $\mathfrak{sl}_{n}$, we have $n$ sections whose
barycenter at every fiber is the origin, i.e. their sum gives the zero-
section. That is why we have to impose that the hamiltonian diffeomorphisms of
$T^{*}\Sigma$ preserve the zero-section.

For $\mathfrak{g}$ of type $B_{n},C_{n}$ or $D_{n}$, the set of $m$ points is
symmetric with respect to the origin. Thus we define:

######  Definition 17.1.

A higher diffeomorphism of type $B_{n},C_{n}$ or $D_{n}$ is a hamiltonian
diffeomorphism of $T^{*}\Sigma$ invariant under the map
$(z,p,\bar{p})\mapsto(z,-p,-\bar{p})$. We denote by
$\operatorname{Symp}(\mathfrak{g},\Sigma)$ the space of higher diffeomorphisms
of type $\mathfrak{g}$.

This group is the same for types $B_{n},C_{n}$ and $D_{n}$, but differs from
$\operatorname{Symp}_{0}(T^{*}\Sigma)$. In coordinates a hamiltonian
diffeomorphism is generated by a function $H(z,\bar{z},p,\bar{p})$ which can
be Taylor developed to $\sum_{k,l}w_{k,l}(z,\bar{z})p^{k}\bar{p}^{l}$. It is
invariant under $-\operatorname{id}$ iff it has only odd terms, i.e.
$w_{k,l}=0$ for all $k+l$ even. Notice that the associated flow automatically
preserves the zero-section since $w_{0,0}=0$.

####  17.2 Action on $\mathfrak{g}$-complex structures

We can now analyze how higher diffeomorphisms act on $\mathfrak{g}$-complex
structures.

Intuitively, hamiltonian diffeomorphisms of $T^{*}\Sigma$ act on the space of
1-forms, so also on $m$-tuples of them. The invariance condition implies that
the symmetry of the collection of 1-forms is preserved. This action persists
at the limit when the $m$-tuple of 1-forms is collapsed to the zero-section.

To compute the action, it is better to work in the idealic viewpoint. We
imitate the steps from the higher complex structure (see section 5.2).

Let $I$ be an ideal representing a $\mathfrak{g}$-complex structure, with
generators $\langle f_{1},...,f_{r}\rangle$. Each $f_{k}$ can be considered as
a function on $T^{*\mathbb{C}}\Sigma$, so its variation under a Hamiltonian
$H$ is given by the Poisson bracket $\\{H,f_{k}\\}$. The tangent space at $I$
in the space of all ideals of codimension $m$ is the set of all ring
homomorphisms from $I$ to $A/I$ (see Proposition 5.4). Thus a Hamiltonian $H$
changes $I$ to $\langle f_{1}+\varepsilon\\{H,f_{1}\\}\mod
I,...,f_{r}+\varepsilon\\{H,f_{r}\\}\mod I\rangle$.

For $n$-complex structures, the simplification lemma 5.5 allowed us to reduce
$H$ modulo $I$. This is still possible:

######  Proposition 17.2.

For classical $\mathfrak{g}$, the generators $f_{i}$ of an ideal
$I\in\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})$ satisfy
$\\{f_{i},f_{j}\\}=0\mod I$.

###### Proof.

For $A_{n}$, we have $I=\langle
p^{n},\bar{p}=\mu_{2}p+...+\mu_{n}p^{n-1}=Q(p)\rangle$. We compute
$\\{p^{n},-\bar{p}+Q(p)\\}=np^{n-1}\partial Q=0\mod I$ since there is no
constant term in $Q$. The same argument holds for $B_{n}$ and $C_{n}$ since
their ideals are special cases of the ideal of type $A_{n}$.

For $D_{n}$, the ideal $I$ is given by

$\displaystyle\langle p^{2n-1},p\bar{p}$
$\displaystyle=\mu_{2}p^{2}+\mu_{4}p^{4}+...+\mu_{2n-2}p^{2n-2}=Q(p)+\mu_{2n-2}p^{2n-2},$
$\displaystyle\bar{p}^{2}$
$\displaystyle=\nu_{2}p^{2}+\nu_{4}p^{4}+...+\nu_{2n-2}p^{2n-2}=R(p)+\nu_{2n-2}p^{2n-2}\rangle.$

As before the Poisson brackets with the first generator $p^{2n-1}$ vanishes
modulo $I$ since $Q$ and $R$ have no constant terms. To compute the last
Poisson bracket, define $\tilde{Q}=Q/p$. By the relations in $I$, we have
$R=\tilde{Q}^{2}+p^{2n-2}\tilde{R}$ for some polynomial $\tilde{R}$ (see
subsection 15.8). Remark further that
$\\{a(z,\bar{z})p^{k}\bar{p}^{l},b(z,\bar{z})p^{k^{\prime}}\bar{p}^{l^{\prime}}\\}=0\mod
I$ whenever $k+l+k^{\prime}+l^{\prime}>n-1$ since any term of degree $n-1$ in
$p$ and $\bar{p}$ is in $I$ and the Poisson bracket lowers this degree by 1.
With all this, we compute

$\displaystyle\\{-p\bar{p}+Q+\mu_{2n-2}p^{2n-2},-\bar{p}^{2}+R+\nu_{2n-2}p^{2n-2}\\}$
$\displaystyle=$
$\displaystyle\;\\{-p\bar{p}+p\tilde{Q}+\mu_{2n-2}p^{2n-2},-\bar{p}^{2}+\tilde{Q}^{2}+p^{2n-2}(\tilde{R}+\nu_{2n-2})\\}$
$\displaystyle=$
$\displaystyle\;\\{-p\bar{p}+p\tilde{Q},-\bar{p}^{2}+\tilde{Q}^{2}\\}$ by
degree argument $\displaystyle=$
$\displaystyle\;2(p\bar{p}-p\tilde{Q})\bar{\partial}\tilde{Q}-2(\bar{p}-\tilde{Q})\tilde{Q}\partial\tilde{Q}$
$\displaystyle=$
$\displaystyle\;2(p\bar{p}-Q)(\bar{\partial}\tilde{Q}-\frac{\tilde{Q}}{p}\partial\tilde{Q})$
$\displaystyle=$
$\displaystyle\;2\mu_{2n-2}p^{2n-2}(\bar{\partial}\tilde{Q}-\frac{\tilde{Q}}{p}\partial\tilde{Q})$
$\displaystyle\mod I$ $\displaystyle=$ $\displaystyle\;0$ $\displaystyle\mod
I$

where the last line comes from the fact that $p$ divides the polynomial
$\bar{\partial}\tilde{Q}-\frac{\tilde{Q}}{p}\partial\tilde{Q}$. ∎

As a consequence, when computing the action of a Hamiltonian $H$ on a
$\mathfrak{g}$-complex structure, we can reduce it modulo $I$. In particular
if $H\mod I=0$, the higher diffeomorphism generated by $H$ does not act at
all. For $\mathfrak{g}$ of type $A_{n},B_{n}$ or $C_{n}$ we can reduce $H$ to
a polynomial in $p$, and for $D_{n}$ we can reduce it to
$H=w_{-}\bar{p}+\sum_{k=0}^{n-2}w_{2k+1}p^{2k+1}$.

####  17.3 Local theory

Now, we can study the local theory of $\mathfrak{g}$-complex structures. Let
$z_{0}$ be a point on $\Sigma$ and take a small chart around it with image the
unit disk $\Delta$ in the complex plane (with $z_{0}$ the origin).

######  Theorem 17.3 (Local theory).

For $\mathfrak{g}$ of type $A_{n}$, $B_{n}$ or $C_{n}$, the
$\mathfrak{g}$-complex structure can be locally trivialized, i.e. there is a
higher diffeomorphism of type $\mathfrak{g}$ which sends all higher Beltrami
differentials to 0 for all small $z\in\mathbb{C}$.

For $\mathfrak{g}$ of type $D_{n}$, all $\mathfrak{g}$-complex structures with
non-vanishing $\sigma_{n}$ on the unit disk $\Delta$ are equivalent under
higher diffeomorphisms. However, the zero locus of $\sigma_{n}$ on $\Delta$ is
an invariant.

###### Proof.

The proof for $\mathfrak{g}$ of type $A_{n}$ was done in Theorem 5.7, using a
method in the spirit of the proof of Darboux’s theorem in symplectic geometry.

If $\mathfrak{g}$ is of type $B_{n}$ or $C_{n}$, the standard representation
realizes the $\mathfrak{g}$-complex structure as a substructure of type
$A_{n}$. Since the last is trivializable, so is the $\mathfrak{g}$-complex
structure in that case.

For $\mathfrak{g}$ of type $D_{n}$, we use the same method as for type $A_{n}$
by a hamiltonian flow argument. We start with an ideal $I$ determined by
$(\mu_{2},\mu_{4},...,\mu_{2n-2},\nu_{2n-2})$, the higher Beltrami
differentials. The action on $\mu_{2i}$ is the same as for
$\mathfrak{g}=\mathfrak{sl}_{2n}$ so we can trivialize them using a
Hamiltonian $H$ which is a polynomial in $p$. So we are left with

$I=\langle p^{2n-1},p\bar{p},-\bar{p}^{2}+\nu_{2n-2}p^{2n-2}\rangle.$ (17.1)

We have seen at the end of subsection 17.2 that in the case $D_{n}$, any
Hamiltonian can be reduced to
$H=w_{-}\bar{p}+\sum_{k=0}^{n-2}w_{2k+1}p^{2k+1}$. The only part of this
Hamiltonian acting on $\nu_{2n-2}$ is $H=w_{-}\bar{p}$, which also changes
$\mu_{2n-2}$. So in order to assure that $\mu_{2n-2}$ stays zero, we use

$H=w_{-}\bar{p}+w_{2n-3}p^{2n-3}.$

We compute the action of this Hamiltonian on the ideal $I$. For the second
generator of $I$ we get:

$\\{w_{-}\bar{p}+w_{2n-3}p^{2n-3},-p\bar{p}\\}=p^{2n-2}(\bar{\partial}w_{2n-3}+\nu_{2n-2}\partial
w_{-})\mod I.$

For the third generator of $I$ we get

$\\{w_{-}\bar{p}+w_{2n-3}p^{2n-3},-\bar{p}^{2}+\nu_{2n-2}p^{2n-2}\\}=p^{2n-2}(w_{-}\bar{\partial}\nu_{2n-2}+2\nu_{2n-2}\bar{\partial}w_{-})\mod
I.$

Denote by $\mu_{2-2}^{t}$ and $\nu_{2n-2}^{t}$ the image of $\mu_{2n-2}$ and
$\nu_{2n-2}$ under the flow generated by $H$ at time $t$. From the above
computation we get

$\left\\{\begin{array}[]{cl}\frac{d}{dt}\mu_{2n-2}^{t}&=\bar{\partial}w_{2n-3}+\nu_{2n-2}^{t}\partial
w_{-}\\\
\frac{d}{dt}\nu_{2n-2}^{t}&=(w_{-}\bar{\partial}+2\bar{\partial}w_{-})\nu_{2n-2}^{t}.\end{array}\right.$

Instead of keeping $\nu_{2n-2}$, we work with the higher Beltrami differential
$\sigma_{n}$. Since all the $\mu_{2i}$ are zero in $I$, we have
$\nu_{2n-2}=\sigma_{n}^{2}$. Therefore we get from the second equation above
$\frac{d}{dt}(\sigma_{n}^{2})=(w_{-}\bar{\partial}+2\bar{\partial}w_{-})(\sigma_{n}^{2})$
which gives

$\frac{d}{dt}\sigma_{n}^{t}=\bar{\partial}(w_{-}^{t}\sigma_{n}^{t}).$ (17.2)

We wish to have $\frac{d}{dt}\mu_{2n-2}^{t}=0$ to keep $\mu_{2n-2}=0$. For
$\sigma_{n}$, we show that we can deform it to the constant function 1 on the
unit disk, assuming $\sigma_{n}$ vanishes nowhere on $\Delta$. We choose the
path $\sigma_{n}^{t}=(1-t)\sigma_{n}^{0}+t$ from the initial $\sigma_{n}^{0}$
to the constant function 1. If $\sigma_{n}^{t}=0$ for some $t$, we have to
slightly modify the path. We get
$\frac{d}{dt}\sigma_{n}^{t}=1-\sigma_{n}^{0}$.

Denote by $T$ the local inverse of the $\bar{\partial}$-operator, i.e.
$\bar{\partial}(Tf)=f=T\bar{\partial}f$ for all $f\in L^{2}(\Delta)$. The
operator $T$ is a pseudo-differential operator given by

$Tf(z)=\frac{1}{2\pi i}\int_{\mathbb{C}}\frac{f(\zeta)}{\zeta-z}d\zeta\wedge
d\bar{\zeta}.$

We can solve equation (17.2) using $T$:

$w_{-}^{t}=\frac{1}{\sigma_{n}^{t}}T(1-\sigma_{n}^{0}).$

Notice that the Hamiltonian we need is time-dependent. Putting this solution
into the equation for $\frac{d}{dt}\mu_{2n-2}^{t}$, we can solve for
$w_{2n-3}$:

$w_{2n-3}^{t}=-T(\partial w_{-}^{t}\nu_{2n-2}^{t}).$

Finally, we multiply $H$ by a bump function, a function on $\Delta$ which is 1
in a neighborhood of the origin and 0 outside a bigger neighborhood of the
origin, which ensures that the hamiltonian vector field is compactly
supported, so it can be integrated to all times. In particular for $t=1$ we
get $\sigma_{n}(z)=1$ for all $z$ near the origin.

To show that the zero locus of $\sigma_{n}$ can not be changed by a higher
diffeomorphism, consider the singularity defined by

$f(p,\bar{p})=-\frac{\nu_{2n-2}}{2n-1}p^{2n-1}+p\bar{p}^{2}=0$

which is a Kleinian singularity of type $D_{2n}$ if $\nu_{2n-2}\neq 0$. Its
deformation ideal $\left\langle\frac{\partial f}{\partial p},\frac{\partial
f}{\partial\bar{p}}\right\rangle$ is directly linked to our ideal $I$ from
equation (17.1) by

$\left\langle\frac{\partial f}{\partial p},\frac{\partial
f}{\partial\bar{p}}\right\rangle\oplus\langle p,\bar{p}\rangle^{n-1}=\langle
p^{2n-1},p\bar{p},-\bar{p}^{2}+\nu_{2n-2}p^{2n-2}\rangle.$

Since the type of a singularity is invariant under diffeomorphisms, so is its
deformation ideal. This is why we cannot change $\nu_{2n-2}=0$ to
$\nu_{2n-2}\neq 0$ by higher diffeomorphisms. ∎

######  Remark.

It is interesting to notice the appearance of Kleinian singularities, which
have an $ADE$-classification. The fact that for $\mathfrak{g}$ of type $D_{n}$
the singularity is of type $D_{2n}$ is linked to the representation of
$\mathfrak{so}_{2n}$ on $\mathbb{C}^{2n}$. There should be a more intrinsic
way to link $\mathfrak{g}$-complex structures to singularities of type
$\mathfrak{g}$.

An idea in this direction is the following: the singularity of type
$\mathfrak{g}$ appears inside the Lie algebra $\mathfrak{g}$, more precisely
inside the nilpotent variety along the subregular locus (see [St74]). A
minimal resolution of this singularity is given by the Springer resolution.
There should be a link between $\mathfrak{g}$-Hilbert schemes and the Springer
resolution. $\triangle$

Since there are no local invariants for $\mathfrak{g}$-complex structures,
only their global geometry is non-trivial.

####  17.4 Definition of the moduli space

To define the moduli space of $\mathfrak{g}$-complex structures, there is one
more subtlety: in order to get one component, we have to fix an orientation on
$\Sigma$. We then call a complex structure compatible if the induced
orientation coincides with the given orientation on $\Sigma$. We call a
$\mathfrak{g}$-complex structure compatible if the induced complex structure
is.

######  Definition 17.4.

The moduli space $\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ is the space of all
compatible $\mathfrak{g}$-complex structures modulo the action of higher
diffeomorphisms of type $\mathfrak{g}$.

Notice that a $\mathfrak{g}$-complex structure is compatible iff
$\mu_{2}(z,\bar{z})\bar{\mu}_{2}(z,\bar{z})<1$. Reverting the orientation on
$\Sigma$ we get another copy of $\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$
corresponding to those $\mathfrak{g}$-complex structures with
$\mu_{2}(z,\bar{z})\bar{\mu}_{2}(z,\bar{z})>1$.

For $\mathfrak{g}=\mathfrak{sl}_{2}$ we get Teichmüller space since we can
reduce any Hamiltonian to $H=w(z,\bar{z})p$ which generates a linear
diffeomorphism of $T^{*}\Sigma$, coming from a diffeomorphism on $\Sigma$
isotopic to the identity.

The moduli space has the following properties:

######  Theorem 17.5 (Global theory).

For $\mathfrak{g}$ of type $A_{n},B_{n}$ or $C_{n}$, and a surface $\Sigma$ of
genus $g\geq 2$, the moduli space $\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ is
a contractible manifold of complex dimension $(g-1)\dim\mathfrak{g}$. In
addition, its cotangent space at any point $I$ is given by

$T^{*}_{I}\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}=\bigoplus_{m=1}^{r}H^{0}(K^{m_{i}+1})$

where $(m_{1},...,m_{r})$ are the exponents of $\mathfrak{g}$ and
$r=\operatorname{rk}\mathfrak{g}$ denotes the rank of $\mathfrak{g}$.

For type $D_{n}$, the moduli space $\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$
is a contractible topological space. The locus where the zero-set of the
higher Beltrami differential $\sigma_{n}$ is a discrete set on $\Sigma$ is a
smooth manifold with the same properties as above (dimension and cotangent
space).

Notice that the differentials in $H^{0}(K^{m_{i}+1})$ are holomorphic with
respect to the complex structure induced from the $\mathfrak{g}$-complex
structure (see Proposition 16.2).

For the case $D_{n}$, we conjecture that the moduli space
$\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ is a topological manifold
everywhere. The points where the zero-set of $\sigma_{n}$ is not discrete can
have a cotangent space which is strictly bigger than the space of holomorphic
differentials. One can think for example of the curve in $\mathbb{R}^{2}$
given by $t\mapsto(t^{3},t^{2})$, shown in figure 17.1, which has a cusp at
the origin, but is still a topological manifold.

Figure 17.1: Curve with cusp

###### Proof.

The case for $A_{n}$ has been treated in Theorem 5.9. The cases $B_{n}$ and
$C_{n}$ are exactly analogous:

One shows that at every point, the cotangent space exists and is of the form
stated in the theorem. From this follows that
$\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ is a manifold. We have to check the
appearance of the exponents of the Lie algebra. Since $\mu_{2i}$ is a section
of $K^{1-2i}\otimes\bar{K}$ (see equation (16.1)) its dual $t_{2i}$ is a
section of $K^{2i}$. Since the exponents for $B_{n}$ and $C_{n}$ are the same
and equal to $(1,3,...,2n-1)$, we get the desired form stated in the theorem.

For $\mathfrak{g}$ of type $D_{n}$ we consider the subset on
$\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ where the zero-locus of $\sigma_{n}$
is discrete in $\Sigma$. By the local theory, we know that there is a
coordinate system in which $\mu_{2i}=0$ for all $i=1,...,n-1$. In that case,
we know that the variation of $\mu_{2i}$ under a higher diffeomorphism
generated by $H=w_{-}\bar{p}+\sum_{k=0}^{n-2}w_{2k+1}p^{2k+1}$ is given by
$\delta\mu_{2i}=\bar{\partial}w_{2i-1}$ and equation (17.2) gives
$\delta\sigma_{n}=\bar{\partial}(w_{-}\sigma_{n})$. The variation of
$\mu_{2i}$ is the same as in the case of type $A_{n}$, so we know that these
contribute to the cotangent bundle by a term $H^{0}(K^{2i})$. For the term
$\sigma_{n}$ we use the pairing between differential of type $(1-n,1)$ and of
type $(n,0)$ given by integration over the surface. We get

$\displaystyle(\\{\delta\sigma_{n}\\}/\bar{\partial}(w_{-}\sigma_{n}))^{*}$
$\displaystyle=\\{t_{n}\in\Gamma(K^{n})\mid\textstyle\int
t_{n}\bar{\partial}(w_{-}\sigma_{n})=0\;\forall\,w_{-}\in\Gamma(\bar{K})\\}$
$\displaystyle=\\{t_{n}\in\Gamma(K^{n})\mid\textstyle\int\bar{\partial}t_{n}w_{-}\sigma_{n}=0\;\forall\,w_{-}\in\Gamma(\bar{K})\\}$
$\displaystyle=\\{t_{n}\in\Gamma(K^{n})\mid\bar{\partial}t_{n}=0\\}$
$\displaystyle=H^{0}(K^{n})$

where we used that $\sigma_{n}$ vanishes only on a discrete set.

Hence the cotangent bundle is given by

$T^{*}_{I}\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}=\bigoplus_{m=1}^{n-1}H^{0}(K^{2m})\oplus
H^{0}(K^{n}).$

The exponents of $\mathfrak{so}_{2n}$ are precisely $(1,3,...,2n-3,n-1)$, so
the cotangent bundle is of the form stated in the theorem.

For the dimension of $\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$, we use $\dim
H^{0}(K^{m_{i}+1})=(g-1)(2m_{i}+1)$ by Riemann-Roch (using $g\geq 2$). We get

$\dim\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}=(g-1)\sum_{i=1}^{r}(2m_{i}+1)=(g-1)\dim\mathfrak{g}$

using a well-known formula coming from the decomposition of $\mathfrak{g}$ as
$\mathfrak{sl}_{2}$-module using the principal $\mathfrak{sl}_{2}$-triple.

Contractibility for all types is analogous to the case $A_{n}$. ∎

The previous theorem gives lots of common features between our moduli space
$\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ and the $G$-Hitchin component, in
particular the contractibility and the dimension. There is another common
property to notice:

######  Proposition 17.6.

There is an injection of Teichmüller space into the moduli space
$\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$.

###### Proof.

The proposition follows from the map
$\psi:\operatorname{Hilb}(\mathfrak{sl}_{2})\rightarrow\operatorname{Hilb}^{reg}(\mathfrak{g})$
constructed in equation (15.2). This map restricts to a map between the zero-
fibers and extends over the surface $\Sigma$. Finally the map descends to the
quotient by higher diffeomorphisms since for $\mathfrak{sl}_{2}$ we only
quotient by diffeomorphisms of $\Sigma$.

For injectivity, it is possible to check for classical $\mathfrak{g}$ that a
Hamiltonian of degree at least 2 does not act on $\mu_{2}$. We then exactly
reason as in the proof of Proposition 5.10. ∎

The same property holds for $G$-Hitchin components which can be defined as the
deformation space of representations of the form
$\pi_{1}(\Sigma)\rightarrow\operatorname{PSL}_{2}(\mathbb{R})\rightarrow G$
where the first map is a fuchsian representation and the second one is the
principal map. So inside the $G$-Hitchin component sits a copy of Teichmüller
space.

We conjecture the equivalence of Hitchin’s component and the moduli space of
$\mathfrak{g}$-complex structures:

######  Conjecture 17.7.

The moduli space $\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ is canonically
homeomorphic to the Hitchin component in the character variety
$\operatorname{Rep}(\pi_{1}(\Sigma),G)$ where $G$ is the real split Lie group
associated to $\mathfrak{g}$.

####  17.5 Spectral curve

In this subsection, we construct a spectral curve in $T^{*\mathbb{C}}\Sigma$
associated to a cotangent vector to $\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$,
i.e. a $\mathfrak{g}$-complex structure and a set of holomorphic
differentials. The case for $\mathfrak{g}$ of type $A_{n}$ was treated in
section 7.

In Proposition 6.4 we proved that the zero-fiber
$\operatorname{Hilb}^{n}_{0}(\mathbb{C}^{2})$ is Lagrangian in the reduced
Hilbert scheme $\operatorname{Hilb}^{n}_{red}(\mathbb{C}^{2})$. This stays
true for all classical $\mathfrak{g}$:

######  Proposition 17.8.

For classical $\mathfrak{g}$, the regular zero-fiber
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})$ is a Lagrangian subspace of
$\operatorname{Hilb}^{reg}(\mathfrak{g})$.

###### Proof.

Since we are in the regular part, Proposition 15.8 gives a parametrization.
For classical $\mathfrak{g}$, the standard representation allows to consider
$\operatorname{Hilb}^{reg}(\mathfrak{g})$ as subset of
$\operatorname{Hilb}^{m}_{red}(\mathbb{C}^{2})$ which remains symplectic and
we can explicitly check that the zero-fiber
$\operatorname{Hilb}^{reg}_{0}(\mathfrak{g})$ is Lagrangian. ∎

For general $\mathfrak{g}$ we conjecture the following:

######  Conjecture 17.9.

The modified smooth version of the $\mathfrak{g}$-Hilbert scheme is symplectic
and the zero-fiber is a Lagrangian subspace.

If we assume Conjecture 15.16, stating that the modified version of the
$\mathfrak{g}$-Hilbert scheme is a minimal resolution of $\mathfrak{h}^{2}/W$,
we get a symplectic structure. Indeed $\mathfrak{h}^{2}=T^{*}\mathfrak{h}$ has
a canonical symplectic structure, which is invariant under the action of $W$.
Hence it lifts to the minimal resolution.

Now we construct the spectral curve. First, we look at $\mathfrak{g}$ of type
$A_{n}$, $B_{n}$ or $C_{n}$. We can write a point in
$T^{*}\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ as an equivalence class of
higher Beltrami differentials $\mu_{i}$ and holomorphic differentials $t_{i}$.
To write in a uniform way, set $\mu_{i}$ or $t_{i}$ to 0 whenever it does not
appear for $\mathfrak{g}$. For example for type $B_{n}$ or $C_{n}$ all
variables with odd index are 0.

Associate polynomials $P(p)=\sum_{i}t_{i}p^{m-i}$ and
$Q(p)=\sum_{i}\mu_{i}p^{i-1}$ (where $m$ is the dimension of the standard
representation of $\mathfrak{g}$). Put $I=\langle-p^{m}+P,-\bar{p}+Q\rangle$.
Define the spectral curve $\bm{\tilde{}}{\Sigma}\subset T^{*\mathbb{C}}\Sigma$
by the zero set of the two generators of $I$. It is a ramified cover over
$\Sigma$ with $m$ sheets.

For $\mathfrak{g}$ of type $D_{n}$, a generic point in the cotangent bundle
$T^{*}\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ corresponds to the ideal

$I=\langle
p^{2n}+t_{2}p^{2n-2}+...+t_{2n-2}p^{2}+\tau_{n}^{2},-\bar{p}+\mu_{2}p+...+\mu_{2n}p^{2n-1}\rangle$

which can be seen as a special case of $A_{n}$. Thus we can proceed as above.
In the case where $\tau_{n}=0$ we have seen in 15.8 that the ideal changes to
an ideal with three generators. The zero-set of these generators still define
a spectral curve in $T^{*\mathbb{C}}\Sigma$. It is the limit of the curve when
$\tau_{n}\rightarrow 0$.

######  Theorem 17.10.

The spectral curve $\bm{\tilde{}}{\Sigma}$ is Lagrangian modulo $t^{2}$.

This is the precise analog of Theorem 7.3.

###### Proof.

In the case where the ideal has two generators $-p^{m}+P$ and $-\bar{p}+Q$,
this is equivalent to $\\{-p^{m}+P,-\bar{p}+Q\\}=0\mod I\mod t^{2}$ for $I\in
T^{*}\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$. For $A_{n}$, the proof is given
in Theorem 7.3. For $B_{n}$ and $C_{n}$ it is completely analogous since the
$\mathfrak{g}$-complex structure can be seen as a special case of $A_{n}$.

For $\mathfrak{g}$ of type $D_{n}$, a generic ideal has still two generators,
so we have a special case of $A_{n}$. If the ideal has three generators, the
spectral curve is still Lagrangian since it can be obtained as a limit of
Lagrangian curves, and the property of being Lagrangian is closed. ∎

Since the spectral curve is Lagrangian to order 1, the periods are well-
defined up to this order. The ratios of these periods should give coordinates
on $T^{*}\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$ and also on
$\bm{\hat{}}{\mathcal{T}}_{\mathfrak{g}}$. For the trivial
$\mathfrak{g}$-complex structure (where all higher Beltrami differentials are
0) we recover Hitchin’s spectral curve.

Finally, we can recover the same spectral data as Hitchin in his paper on
stable bundles [Hi87b]. From a $\mathfrak{g}$-complex structure we get a
bundle $V$ over the surface $\Sigma$ whose fiber at a point $z\in\Sigma$ is
$\mathbb{C}[p,\bar{p}]/I(z)$ where we use the idealic viewpoint. We also get a
line bundle $L$ on $\bm{\tilde{}}{\Sigma}$ whose fiber is the eigenspace of
$M_{p}$, the multiplication operator by $p$ in the quotient
$\mathbb{C}[p,\bar{p}]/I$. This gives the spectral data for type $A_{n}$.

For $\mathfrak{g}$ of type $C_{n}$, we get in addition an involution $\sigma$
on the spectral curve $\bm{\tilde{}}{\Sigma}$ given by
$(p,\bar{p})\mapsto(-p,-\bar{p})$. For $\mathfrak{g}$ of type $D_{n}$, the
spectral curve is singular, having a double point. The spectral data is given
by a desingularization of $\bm{\tilde{}}{\Sigma}$, the involution $\sigma$ as
for $C_{n}$ and the line bundle $L$. For $\mathfrak{g}$ of type $B_{n}$, there
is a canonical subbundle $V_{0}\subset V=\mathbb{C}[p,\bar{p}]/I$ generated by
the span of the image of $1\in\mathbb{C}[p,\bar{p}]$ in the quotient
$\mathbb{C}[p,\bar{p}]/I$ (since for $B_{n}$, we have $I\subset\langle
p,\bar{p}\rangle$). Thus the vector bundle $V$ is an extension
$V_{0}\rightarrow V\rightarrow V_{1}$. The spectral data is given by
$(V_{0},V_{1},\sigma,L,\bm{\tilde{}}{\Sigma})$.

##  Part IV Perspectives

„Ich habe bemerkt, sagte Herr K., das wir viele abschrecken von unserer Lehre
dadurch, dass wir auf alles eine Antwort wissen. Könnten wir nicht im
Interesse der Propaganda eine Liste der Fragen aufstellen, die uns ganz
ungelöst erscheinen?“ 333“I have noticed,” said Mr. K., “that we put many
people off our teaching because we have an answer to everything. Could we not,
in the interests of propaganda, draw up a list of the questions that appear to
us completely unsolved?”, Translation by Martin Chalmers

Bertolt Brecht, Geschichten vom Herrn Keuner

In this last part of the thesis, we present open questions, conjectures and
several links between higher complex structures and other domains.

In previous chapters we already pointed at several questions and conjectures
which remain open. We present them in a uniform way and discuss possible
approaches and answers. In particular we discuss the conjectural hyperkähler
structure of $T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$ with its twistor space given
in figure 10.1, geometric interpretations of
$T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$ as almost-Fuchsian representations, period
coordinates on $T^{*}\bm{\hat{}}{\mathcal{T}}^{n}$, the fiber of the forgetful
map $\bm{\hat{}}{\mathcal{T}}^{n}\rightarrow\bm{\hat{}}{\mathcal{T}}^{n-1}$
which would partially answer a conjecture of Labourie from [La06], and
generalizations to Lie algebras other than $\mathfrak{sl}_{n}$.

The field of higher Teichmüller theory is vast and studies numerous
mathematical objects. We give an outlook on possible connections between
higher complex structures and various fields. In particular we discuss links
to other components in the character variety, spectral networks, cluster
varieties, opers, W-algebras and SYZ-mirror symmetry.

###  18 Open questions and conjectures

We have already encountered several open questions and conjectures in the