# The companion section for classical groups

T. Hameister, B. C. Ngô

###### Abstract

We use the companion matrix construction for $\mathrm{GL}_{n}$ to build
canonical sections of the Chevalley map
$[\mathfrak{g}/G]\to\mathfrak{g}{/\\!\\!/}G$ for classical groups $G$ as well
as the group $G_{2}$. To do so, we construct canonical tensors on the
associated spectral covers. As an application, we make explicit lattice
descriptions of affine Springer fibers and Hitchin fibers for classical groups
and $G_{2}$.

##  1 Introduction

Let $G$ be a reductive group over $k$, and denote by $\mathfrak{g}$ its Lie
algebra. The Chevalley map

$\chi\colon\mathfrak{g}\to\mathfrak{g}{/\\!\\!/}G,$

where $\mathfrak{g}{/\\!\\!/}G:=\mathrm{Spec}(k[\mathfrak{g}]^{G})$ denotes
the invariant theoretic quotient of $\mathfrak{g}$ by the adjoint action of
$G$, is of fundamental importance in the construction of the Hitchin system
[Hit87]. In particular, for $\mathfrak{g}=\mathfrak{gl}_{n}$, $\chi$ sends a
matrix to its characteristic polynomial.

In [Kos63], Kostant exhibited a section of the Chevalley map for a general
reductive group $G$ under the assumption that the characteristic of $k$ does
not divide the order of the Weyl group. Kostant’s section was generalized in
[BČ22] and [AFV18], including the case of characteristics $p>2$ for classical
groups and the group $G_{2}$. As explained in [Ngo10], this section can be
used to construct sections of the Hitchin fibration and affine Springer
fibers. However, Kostant’s construction can be counter-intuitive for
computations. To illustrate this latter point, consider the case
$G=\mathrm{GL}_{3}(k)$, in which case
$\mathfrak{g}{/\\!\\!/}G={\mathbb{A}}^{3}$ is the 3-dimensional affine space.
The Kostant section is the map sending

$(a_{1},a_{2},a_{3})\in\mathfrak{g}{/\\!\\!/}G\quad\mapsto\quad\begin{pmatrix}\frac{a_{1}}{3}&\frac{a_{1}^{2}}{6}+\frac{a_{2}}{2}&-\frac{4a_{1}^{3}}{27}-\frac{a_{1}a_{2}}{3}-a_{3}\\\
1&\frac{a_{1}}{3}&\frac{a_{1}^{2}}{6}+\frac{a_{2}}{2}\\\
0&1&\frac{a_{1}}{3}\end{pmatrix}\in\mathfrak{g}$

If you introduced this problem to an undergraduate student of linear algebra,
of course, they would not give you the answer above; they might instead
suggest the map:

$(a_{1},a_{2},a_{3})\in\mathfrak{g}{/\\!\\!/}G\quad\mapsto\quad\begin{pmatrix}0&0&-a_{3}\\\
1&0&-a_{2}\\\ 0&1&-a_{1}\end{pmatrix}\in\mathfrak{g}$

sending a characteristic polynomial to its _companion matrix_. The section to
the Hitchin map that Hitchin constructed in [Hit87] is not strictly the same
as the one of [Ngo10] in the sense that he does not rely on the Kostant
section but another section that feels more like a generalization of the
companion matrix. Instead of the companion matrix, a map
$\mathfrak{g}{/\\!\\!/}G\to\mathfrak{g}$, we will construct a map
$\mathfrak{g}{/\\!\\!/}G\to[\mathfrak{g}/G]$, where $[\mathfrak{g}/G]$ is the
quotient of $\mathfrak{g}$ by the adjoint action of $G$ in the sense of
algebraic stack. This section will be called the companion section, which is
free of any choice. The present note aims to explicitly construct the
companion section for classical groups, including the symplectic and
orthogonal groups and $G_{2}$. As an application of the companion sections, we
will give elementary descriptions of affine Springer fibers and Hitchin fibers
for classical groups similar to the description of the Hitchin fibers in the
linear case due to Beauville-Narasimhan-Ramanan.

The emphasis of this work is on providing case-by-case explicit formulas for
the companion section for classical groups. It is also possible to construct
the companion section uniformly. This will be the subject of our subsequent
work.

## Acknowledgments

The second author was partially supported by NSF grant DMS 2201314. The
authors thank Alexis Bouthier for pointing out the literature on Kostant
sections in small characteristics. Both authors thank the anonymous reviewer
for helpful feedback.

##  2 Tensors defining classical groups

We will recall the standard definition of classical groups as the subgroup of
the linear groups fixing certain tensors. This is very well known for
symplectic and orthogonal groups but a bit less known for $G_{2}$, which in a
certain respect could qualify as a classical group as well.

Let $V$ be a $2n$-dimensional vector space over a base field $k$, $V^{*}$ its
dual vector space. The linear group $\mathrm{GL}(V)$ acts on the space
$\wedge^{2}V^{*}$ of alternating bilinear forms on $V$ with an open orbit. An
alternating bilinear form $\mu\in\wedge^{2}V^{*}$ is considered non-degenerate
if it lies in this open orbit. This is equivalent to requiring the induced map
$\mu:V\to V^{*}$ to be an isomorphism. The stabilizer of such a non-degenerate
alternating bilinear form is a symplectic group $G$. We note that
$\mu\in\wedge^{2}V^{*}$ is non-degenerate if
$\wedge^{n}\mu\in\wedge^{2n}V^{*}$ is a non-zero vector of the 1-dimensional
vector space $\wedge^{2n}V^{*}$ and as a result, $G$ is contained in the
special linear group $\mathrm{SL}(V)$. Then, a $G$-bundle over a $k$-scheme
$S$ consists of a locally free $\mathcal{O}_{S}$-module $\mathcal{V}$ of rank
$2n$ equipped with an alternating bilinear form
$\wedge^{2}_{S}\mathcal{V}\to\mathcal{O}_{S}$ which is non-degenerate
fiberwise. Although the embedding of $G=\mathrm{Sp}_{2n}$ into
$\mathrm{GL}_{2n}$ may differ by conjugation by an element of
$\mathrm{GL}_{2n}$, as we are more concerned with $G$-bundles than $G$ itself,
the specific choice of non-degenerate alternating form $\mu\in\wedge^{2}V^{*}$
is immaterial. We will write $G=\mathrm{Sp}_{2n}$.

Let $V$ be a $n$-dimensional vector space over a base field $k$, $V^{*}$ its
dual vector space. The linear group $\mathrm{GL}(V)$ acts on the space
$S^{2}V^{*}$ of symmetric bilinear forms on $V$ with an open orbit. A
symmetric bilinear form $\mu\in S^{2}V^{*}$ is considered non-degenerate if it
lies in this open orbit. This is also equivalent to the induced map $\mu:V\to
V^{*}$ being an isomorphism, which in turn is equivalent to the induced map
$\wedge^{n}\mu:\wedge^{n}V\to\wedge^{n}V^{*}$ being an isomorphism of
1-dimensional vector spaces. We note that $\wedge^{n}V$ and $\wedge^{n}V^{*}$
are dual as vector spaces so that for every choice of a basis vector
$\omega\in\wedge^{n}V$, we have a dual basis vector
$\omega^{*}\in\wedge^{n}V^{*}$. A basis vector $\omega\in\wedge^{n}V$ is said
to be compatible with $\mu$ if the equation $\wedge^{n}\mu(\omega)=\omega^{*}$
is satisfied. This equation has exactly two non-zero solutions
$\omega\in\wedge^{n}V$, which differ by a sign. The stabilizer of a non-
degenerate symmetric bilinear form $\mu\in S^{2}V^{*}$ is an orthogonal group
$O(\mu)$. The stabilizer of a pair $(\mu,\omega)$ consisting of a non-
degenerate symmetric bilinear form $\mu\in S^{2}V^{*}$ and a compatible basis
vector $\omega\in\wedge^{n}V$ is the special orthogonal group
$\mathrm{SO}(\mu,\omega)$ which is the neutral component of $O(\mu)$. We note
that $\mathrm{SO}(\mu,\omega)=O(\mu)\cap\mathrm{SL}(V)$ so that the special
orthogonal group can also be defined as the stabilizer of a pair
$(\mu,\omega)$ as above but without requiring $\omega$ being compatible with
$\mu$. The stabilizer of any such pair is a special orthogonal group $G$. A
$G$-bundle over a $k$-scheme $S$ consists then in a locally free
$\mathcal{O}_{S}$-module $\mathcal{V}$ of rank $n$ equipped a symmetric
bilinear form $\wedge^{2}_{S}\mathcal{V}\to\mathcal{O}_{S}$ which is non-
degenerate fiberwise. The embedding of $G=\mathrm{SO}_{n}$ into
$\mathrm{GL}_{n}$ depends on the form $\mu$ and is well defined only up to
conjugation by $\mathrm{GL}_{n}$. However, as we are more concerned with
$G$-bundles than $G$ itself, choosing a specific non-degenerate symmetric form
$\mu\in\wedge^{2}V^{*}$ is immaterial. We will write $G=\mathrm{SO}_{n}$.

There is a simple tensor definition of $G_{2}$ due to Engel [Eng00]. Let $V$
be a 7-dimensional vector space. The linear group $\mathrm{GL}(V)$ acts on the
space $\wedge^{3}V^{*}$ of non-degenerate trilinear forms on $V$ with an open
orbit. We will follow Hitchin’s [Hit00] in formulating the equation defining
this open orbit . We will denote the contraction $\wedge^{3}V^{*}\times
V\to\wedge^{2}V^{*}$ by $\mu_{v}$ for $\mu\in\wedge^{2}V^{*}$ and $v\in V$.
For $v_{1},v_{2}\in V$ and $\mu\in\wedge^{3}V^{*}$, we then have

$\mu_{v_{1}}\wedge\mu_{v_{2}}\wedge\mu\in\wedge^{7}V^{*}.$

By choosing a non-zero vector $\iota$ of the determinant $\wedge^{7}V$, $\mu$
gives rise to a symmetric bilinear form $\nu\in S^{2}V^{*}$

$\nu(v_{1},v_{2})=\langle\iota,\mu_{v_{1}}\wedge\mu_{v_{2}}\wedge\mu\rangle$
(1)

which is non-degenerate if and only if $\mu$ lies in the open orbit of
$\wedge^{3}V^{*}$. We will say that $\mu$ is a non-degenerate 3-form on $V$.
The stabilizer of a non-degenerate 3-form is a group $G_{2}\ltimes\mu_{3}(k)$
where $\mu_{3}(k)$ is the group of 3rd roots of unity in $k$; We obtain the
connected component, a group of type $G_{2}$, by taking the intersection with
$SL(V)$. A $G_{2}$-bundle over a $k$-scheme $S$ is thus a locally free
$\mathcal{O}_{S}$-module $\mathcal{V}$ of rank 7 equipped with an alternating
trilinear form $\mu\in\wedge^{3}\mathcal{V}^{*}$ which is non-degenerate
fiberwise together with a trivialization of the determinant. Again, a
different choice of nondegenerate 3-form $\mu$ may give a $\mathrm{GL}_{7}$
conjugate embedding of $G_{2}$ into $\mathrm{GL}_{7}$. However, such a choice
is immaterial for us.

##  3 Spectral cover and the companion matrix

For all groups $G$ discussed previously, including symplectic, special
orthogonal, and $G_{2}$, $G$ is defined as a subgroup of $\mathrm{GL}_{n}$
fixing certain tensors. We call the inclusion $G\to\mathrm{GL}_{n}$ the
standard representation of $G$. We also have the induced inclusion of Lie
algebras $\mathfrak{g}\to\mathfrak{gl}_{n}$ compatible with the adjoint
actions of $G$ and $\mathrm{GL}_{n}$. We derive a morphism between invariant
theoretic quotients

$\mathfrak{c}=\mathfrak{g}{/\\!\\!/}G\to\mathfrak{gl}_{n}{/\\!\\!/}\mathrm{GL}_{n}=\mathfrak{c}_{n}$

which is a closed embedding for symplectic groups, odd special orthogonal
groups, and $G_{2}$, but not for even orthogonal groups. For
$\mathrm{GL}_{n}$, we have a spectral cover
$\mathfrak{s}_{n}\to\mathfrak{c}_{n}$, defined in Section 3.1, which is a
finite flat morphism of degree $n$ so that $\mathcal{O}_{\mathfrak{s}_{n}}$ is
a locally free $\mathcal{O}_{\mathfrak{c}_{n}}$-module of rank $n$ given with
a canonical endomorphism $[x]$ which is the usual companion matrix. The main
result of this work can be formulated as follows:

######  Theorem 3.1.

Let $G$ be a symplectic group, odd special orthogonal group, or $G_{2}$ group
and $G\to\mathrm{GL}_{n}$ its standard representation. Let
$\mathfrak{c}\to\mathfrak{c}_{n}$ be the induced map of Chevalley quotients
which is a closed embedding in these cases. Then the restriction
$\mathcal{O}_{\mathfrak{s}_{n}}$ to $\mathfrak{c}$

$\mathcal{V}=\mathcal{O}_{\mathfrak{c}}\otimes_{\mathcal{O}_{\mathfrak{c}_{n}}}\mathcal{O}_{\mathfrak{s}_{n}}$

as locally free $\mathcal{O}_{\mathfrak{c}}$-module affords a canonical tensor
defining a $G$-reduction and the companion matrix for $\mathrm{GL}_{n}$
defines a canonical map $\mathfrak{g}{/\\!\\!/}G\to[\mathfrak{g}/G]$ which is
a section of the natural map $[\mathfrak{g}/G]\to\mathfrak{g}{/\\!\\!/}G$.
This statement remains valid for even orthogonal groups after replacing
$\mathfrak{c}\times_{\mathfrak{c}_{n}}\mathfrak{s}_{n}$ by its normalization.

We prove the theorem by a case-by-case analysis. In particular, we will
construct the explicit tensors required in each case.

###  3.1 Linear groups

We first recall how the companion matrix is connected to the universal
spectral cover in the case $\mathrm{GL}_{n}$. In this case, the Chevalley
quotient $\mathfrak{g}{/\\!\\!/}G$ is the $n$-dimensional affine space
$\mathbf{A}^{n}$ and the map $\chi:\mathfrak{g}\to\mathfrak{g}{/\\!\\!/}G$ is
given by the characteristic polynomial
$\chi(\gamma)=(a_{1}(\gamma),\ldots,a_{n}(\gamma))$ where
$\gamma\in\mathfrak{g}$ and
$a_{i}(\gamma)=(-1)^{i}\mathrm{tr}(\wedge^{i}\gamma)$. In this case we have
$\mathfrak{c}_{n}=\mathrm{Spec}(A_{n})$ where $A_{n}=k[a_{1},\ldots,a_{n}]$.
The spectral cover $\mathfrak{s}_{n}=\mathrm{Spec}(B_{n})$ where $B_{n}$ is
the $A_{n}$-algebra

$B_{n}=A_{n}[x]/(x^{n}+a_{1}x^{n-1}+\cdots+a_{n})$

which is a free $A_{n}$-module of rank $n$ as the images of
$1,x,\ldots,x^{n-1}$ form an $A_{n}$-basis of $B_{n}$. We also note that
$B_{n}$ is a regular $k$-algebra as it is isomorphic to the polynomial algebra
of variables $a_{1},\dots,a_{n-1},x$. On the other hand, $B_{n}$ is equipped
with an $A_{n}$-linear operator $[x]:B_{n}\to B_{n}$ given by $b\mapsto bx$.
To give a map $\mathfrak{c}_{n}\to[\mathfrak{gl}_{n}/\mathrm{GL}_{n}]$ is
equivalent to the data of a rank $n$ vector bundle
$\mathcal{E}\to\mathfrak{c}_{n}$ together with a an
$\mathcal{O}_{\mathfrak{c}_{n}}$-linear endomorphism of $\mathcal{E}$; that
is, at the level of modules, a free, rank $n$ $A_{n}$ module with an $A_{n}$
linear endomorphism. Hence, $B_{n}$ with the operator $[x]$ provides us with
an $A_{n}$-point of $[\mathfrak{gl}_{n}/\mathrm{GL}_{n}]$, and we have thus
constructed a map $[x]:\mathfrak{c}_{n}\to[\mathfrak{gl}_{n}/\mathrm{GL}_{n}]$
which is a section of
$\chi:[\mathfrak{gl}_{n}/\mathrm{GL}_{n}]\to\mathfrak{c}_{n}$. In term of
matrices with respect to the $A_{n}$-basis of $B_{n}$ given by
$1,x,\ldots,x^{n-1}$, $[x]$ is given by the usual companion matrix

$x_{\bullet}=\begin{pmatrix}0&&\cdots&-a_{n}\\\ 1&0&\cdots&-a_{n-1}\\\
&\cdots&0&\\\ 0&&1&-a_{1}\end{pmatrix}\in\mathfrak{gl}_{n}(A)$ (2)

The companion matrix thus gives us a map
$x_{\bullet}:\mathfrak{c}\to\mathfrak{g}$ in the case $G=\mathrm{GL}_{n}$
taking a point $a=(a_{1},\dots,a_{n})$ of $\mathfrak{c}$ to the matrix above.
This construction is a section to the characteristic polynomial map. However,
it is often more useful to think of $[x]$ as a map
$[x]:\mathfrak{c}\to[\mathfrak{g}/G]$ in the case $G=\mathrm{GL}_{n}$.

Let $\mathfrak{g}$ come equipped with the homothety action of
${\mathbb{G}}_{m}$ and $\mathfrak{c}$ with the induced action $t\cdot
a_{i}=t^{i}a_{i}$. There is an issue with using the companion matrix to
construct a section to the Hitchin map as the companion map
$x_{\bullet}:\mathfrak{c}\to\mathfrak{g}$ is not $\mathbb{G}_{m}$-equivariant.
We note, however, that the stack-valued map
$[x]:\mathfrak{c}\to[\mathfrak{g}/G]$ is almost $\mathbb{G}_{m}$-equivariant
in the sense that after a base change by the isogeny
$\mathbb{G}_{m}\to\mathbb{G}_{m}$ given by $t\mapsto t^{2}$, it becomes
equivariant because of the identity

$\mathrm{ad}(\mathrm{diag}(t^{n-1},t^{n-3},\ldots,t^{1-n}))(\gamma)=t^{-2}\begin{pmatrix}0&0&\cdots&-t^{2n}a_{n}\\\
1&0&\cdots&-t^{2n-2}a_{n-1}\\\ &\cdots&&\\\ 0&&1&-t^{2}a_{1}\end{pmatrix}.$
(3)

This explains why we have a section to the Hitchin map after choosing a square
root of the canonical bundle as in [Hit87].

As we intend to use the companion matrix (2) to construct a canonical section
to the Chevalley map $\chi:[\mathfrak{g}/G]\to\mathfrak{c}$ for classical
groups, it is useful to further investigate the linear algebraic structure of
$B_{n}$ as an $A_{n}$-module. We have a symmetric $A_{n}$-bilinear map
$\xi:B_{n}\otimes_{A_{n}}B_{n}\to A_{n}$ given by

$\xi(b_{1}\otimes_{A_{n}}b_{2})=\mathrm{tr}_{B_{n}/A_{n}}(b_{1}b_{2})$

thus an element $\xi\in S^{2}_{A_{n}}B_{n}^{*}$. Because this element induces
degenerate forms over the ramification locus of $B_{n}$ over $A_{n}$, we need
a correction term to get a symmetric bilinear form that is non-degenerate
fiberwise. We will describe this correction and the associated nondegenerate
form in Lemma 3.2.

The pairing $\xi$ defines an $A_{n}$-linear map $\mu:B_{n}\to B_{n}^{*}$ where
$B_{n}^{*}=\mathrm{Hom}_{A_{n}}(B_{n},A_{n})$ and
$\mu(b_{1})(b_{2})=\xi(b_{1},b_{2})$. We note that the $A_{n}$-module
$B_{n}^{*}$ is naturally a $B_{n}$-module and $\mu:B_{n}\to B_{n}^{*}$ is
$B_{n}$ linear; thus, it is uniquely determined by the image of $1\in B_{n}$
that we will also denote by $\mu\in B_{n}^{*}$. We will show that $B_{n}^{*}$
is a free $B_{n}$-module of rank 1, construct a generator of $B_{n}^{*}$ and
find an explicit formula for $\mu\in B_{n}^{*}$ as a multiple of this
generator.

######  Lemma 3.2.

Let us denote by $v_{0},\ldots,v_{n-1}$ the basis of $B_{n}$ given by the
images of $1,x,\ldots,x^{n-1}$ in $B_{n}$ and $v_{0}^{*},\ldots,v_{n-1}^{*}$
the dual basis of $B_{n}^{*}$. Then $\beta^{*}=v_{n-1}^{*}$ is a generator of
$B_{n}^{*}$ as a $B_{n}$-module. Let us denote $f^{\prime}\in
B_{n}=A_{n}[x]/(f)$ the image of the derivative

$nx^{n-1}+(n-1)a_{1}x^{n-2}+\cdots+a_{n-1}\in A_{n}[x]$

of the universal polynomial $f=x^{n}+a_{1}x^{n-1}+\cdots+a_{n}\in A_{n}[x]$.
Then we have $\mu=f^{\prime}\beta^{*}$.

###### Proof.

First, the discriminant $d$ of the universal polynomial $f$, defined as the
resultant between $f$ and its derivative, is a nonzero element of the
polynomial ring $A_{n}$. Indeed, $d$ defines the ramification divisor of the
finite flat covering $\mathfrak{s}\to\mathfrak{c}$, which is generically étale
for there exist separable polynomials in $k^{\prime}[x]$ of degree $n$ with
coefficients in any infinite field $k^{\prime}$ containing $k$. We denote
$A^{\prime}=A_{n}[d^{-1}]$ the localization of $A_{n}$ obtained by inverting
$d$, and $B^{\prime}=B_{n}\otimes_{A_{n}}A^{\prime}$. By construction,
$f^{\prime}$ is an invertible element of $B^{\prime}$. The trace map
$\mathrm{tr}_{B^{\prime}/A^{\prime}}:B^{\prime}\to A^{\prime}$ of $B^{\prime}$
as free $A^{\prime}$-module of rank $n$ is now given by the Euler formula (cf.
III.6, Lemma 2 in [Ser13])

$\mathrm{tr}_{B^{\prime}/A^{\prime}}\left(\frac{x^{k}}{f^{\prime}}\right)=\begin{cases}0&\text{if
}k<n-1\\\ 1&\text{if }k=n-1\end{cases}$

If $v_{0},\ldots,v_{n-1}$ denote the basis of $B^{\prime}$ given by the images
of $1,x,\ldots,x^{n-1}$ in $B_{n}$ and $v_{0}^{*},\ldots,v_{n-1}^{*}$ the dual
basis of $(B^{\prime})^{*}$, then we derive from the Euler formula that the
identities

$\mu(v_{i})=f^{\prime}\left(v^{*}_{n-1-i}+\sum_{j<i}a^{\prime}_{i,j}v^{*}_{n-1-j}\right)$
(4)

hold in $B_{n}^{*}\otimes_{A_{n}}A^{\prime}$ for some $a^{\prime}_{i,j}\in
A^{\prime}$. In particular, we have $\mu(v_{0})=f^{\prime}v^{*}_{n-1}$. As the
localization map $B_{n}^{*}\to B_{n}^{*}\otimes_{A_{n}}A^{\prime}$ is
injective, this identity also holds in $B_{n}^{*}$. It follows that
$\mu=f^{\prime}v_{n-1}^{*}$ as desired. ∎

As a consequence, we have a canonical nondegenerate bilinear form
$\beta^{*}:B_{n}\otimes_{A_{n}}B_{n}\to A_{n}$ which is symmetric with respect
to which the $A_{n}$-linear operator $[x]:B_{n}\to B_{n}$ is anti-self-
adjoint; that is, for all $v_{1},v_{2}\in B_{n}$, we have

$\beta^{*}(xv_{1},v_{2})+\beta^{*}(v_{1},xv_{2})=0.$

For $G=\mathrm{SL}_{n}$, the Lie algebra $\mathfrak{g}=\mathfrak{sl}_{n}$ is
the space of traceless matrices. We have $\mathfrak{c}=\mathrm{Spec}(A)$ where
$A=k[a_{2},\ldots,a_{n}]$. We note that for $a_{1}=0$, the companion matrix
(2) is traceless and thus gives rise to a $A$-point on $\mathfrak{sl}_{n}$.
The companion map $\gamma:\mathfrak{c}\to\mathfrak{g}$ induces a map
$[\gamma]:\mathfrak{c}\to[\mathfrak{g}/G]$. The latter lays over the point of
$BG$ with values in $A$ corresponding to the $\mathrm{SL}_{n}$-bundle
corresponding to rank $n$ vector bundle $B$ equipped with the trivialization
of the determinant given by the basis $1,x,\ldots,x^{n-1}$. The formula (3)
shows that the map $[\gamma]:\mathfrak{c}\to[\mathfrak{g}/G]$ is equivariant
with respect to the isogeny $\mathbb{G}_{m}\to\mathbb{G}_{m}$ given by
$t\mapsto t^{2}$ for the diagonal matrix
$\mathrm{diag}(t^{n-1},t^{n-3},\ldots,t^{1-n})$ belonging to
$\mathrm{SL}_{n}$.

###  3.2 Symplectic groups

In the case $G=\mathrm{Sp}_{2n}$, we have $\mathfrak{c}=\mathrm{Spec}(A)$ with
$A=k[a_{2},\ldots,a_{2n}]$. The spectral cover $\mathfrak{s}=\mathrm{Spec}(B)$
where

$B=A[x]/(x^{2n}+a_{2}x^{2n-2}+\cdots+a_{2n})$

is a free $A$-module of rank $2n$, is equipped with an involution $\tau:B\to
B$ given $\tau(x)=-x$. The companion matrix (2) gives a $A$-linear
endomorphism of $B$ as a free $A$-module. For the companion matrix to produce
a section to the Chevalley map $[\mathfrak{g}/G]\to\mathfrak{c}$ in the
symplectic case, we need to construct a canonical nondegenerate symplectic
form $\omega$ on the $A$-module $B$ for which $\gamma$ is anti-self-adjoint in
the sense that

$\omega(\gamma v_{1},v_{2})+\omega(v_{1},\gamma v_{2})=0$

for all $v_{1},v_{2}\in B$.

The standard representation $\mathrm{Sp}_{2n}\to\mathrm{GL}_{2n}$ induces a
map on Chevalley bases
$\mathfrak{c}\to\mathfrak{c}_{2n}=\mathrm{Spec}(A_{2n})$ where
$A_{2n}=k[a_{1},\ldots,a_{2n}]$ which identidies $\mathfrak{c}$ with the
closed subscheme of $\mathfrak{c}_{2n}$ defined by the ideal generated by
$a_{1},a_{3},\ldots,a_{2n-1}$. We have $B=A\otimes_{A_{2n}}B_{2n}$ where
$B_{2n}$ is the finite free $A_{2n}$-algebra defining the spectral covering of
$\mathfrak{c}_{2n}$. If we denote $B^{*}=\mathrm{Hom}_{A}(B,A)$ then we have
$B^{*}=A\otimes_{A_{2n}}B^{*}_{2n}$ where
$B^{*}_{2n}=\mathrm{Hom}_{A_{2n}}(B_{2n},A_{2n})$. The generator
$\beta_{2n}^{*}$ of the free $B_{2n}$-module $B^{*}_{2n}$ defined in Lemma 3.2
then induces a generator $\beta^{*}$ of $B^{*}$ as a free $B$-module of rank
one which can also be viewed as the bilinear form $\beta^{*}:B\otimes_{A}B\to
A$ given by
$b_{1}\otimes_{A}b_{2}=\mathrm{tr}_{B/A}({f^{\prime}}^{-1}b_{1}b_{2})$ after
localization.

The bilinear form $\omega:B\otimes_{A}B\to A$

$\omega(b_{1},b_{2})=\beta^{*}(b_{1},\tau(b_{2}))=\mathrm{tr}_{B/A}({f^{\prime}}^{-1}b_{1}\tau(b_{2}))$

with the second identity only making sense after localization of $A$ making
$f^{\prime}$ invertible, is a non-degenerate symplectic form for which $[x]$
is anti-self-adjoint. Indeed, we have

$\omega(b_{1},b_{2})=-\omega(b_{2},b_{1})$

because $\tau(f^{\prime})=-f^{\prime}$ for $f^{\prime}\in A[x]$ is an odd
polynomial as $f\in A[x]$ is an even polynomial. The equation
$\omega(xb_{1},b_{2})+\omega(b_{1},xb_{2})=0$ can be derived from
$\tau(x)=-x$.

It follows that we have a morphism

$[x]:\mathfrak{c}\to[\mathfrak{g}/G]$

which deserves to be called the companion map for the symplectic group. To
obtain a companion matrix $x_{\bullet}:\mathfrak{c}\to\mathfrak{g}$, it is
enough to find a trivialization of the $G$-bundle associated with the non-
degenerate symplectic form $\omega:B\otimes_{A}B\to A$. For most applications,
particularly the Hitchin fibration, we only need the section
$[x]:\mathfrak{c}\to[\mathfrak{g}/G]$.

###  3.3 Odd special orthogonal groups

In the case $G=\mathrm{SO}_{2n+1}$, we have $\mathfrak{c}=\mathrm{Spec}(A)$
with $A=k[a_{2},a_{4},\dots,a_{2n}]$. The spectral cover is defined as
$\mathfrak{s}=\mathrm{Spec}(B)$ where $B=A[x]/(f)$ with $f=xf_{0}$ and
$f_{0}=x^{2n}+a_{2}x^{2n-2}+\cdots+a_{2n}$. $B$ is a free $A$-module of rank
$2n+1$. As in the symplectic case, we will define a symmetric non-degenerate
bilinear form $B\otimes_{A}B\to A$ for which the multiplication by $x$ is
anti-self-adjoint.

The standard representation $\mathrm{SO}_{2n+1}\to\mathrm{GL}_{2n+1}$ gives
rise to a map
$\mathfrak{c}\to\mathfrak{c}_{2n+1}=\mathrm{Spec}(k[a_{1},\ldots,a_{2n+1}])$
which is a closed embedding defined by the ideal generated by
$a_{1},a_{3},\ldots,a_{2n+1}$. We have $B=A\otimes_{A_{2n+1}}B_{2n+1}$ where
$B_{2n+1}$ is the finite free $A_{2n+1}$-module of rank $2n+1$ defining the
spectral cover in the case $\mathrm{GL}_{2n+1}$. We also have
$B^{*}=A\otimes_{A_{2n+1}}B_{2n+1}^{*}$ where $B^{*}=\mathrm{Hom}_{A}(B,A)$
and $B_{2n+1}^{*}=\mathrm{Hom}_{A_{2n+1}}(B_{2n+1},A_{2n+1})$. Following the
discussion in the linear case $B_{2n+1}^{*}$ is a free $B_{2n+1}$ generated by
the element $\beta^{*}_{2n+1}=(f^{\prime})^{-1}\mu$ where $\mu$ is the trace
form $\mu(b_{1}\otimes_{A}b_{2})=\mathrm{tr}(b_{1}b_{2})$. It induces a
generator $\beta^{*}$ of $B^{*}$ as a $B$-module. We define the bilinear form
$\omega:B\otimes_{A}B\to A$ by

$\omega(b_{1},b_{2})=\beta^{*}(b_{1},\tau(b_{2}))=\mathrm{tr}({f^{\prime}}^{-1}b_{1}\tau(b_{2})).$
(5)

The bilinear form $\omega$ is a nondegenerate bilinear form because
$\beta^{*}$ is. It is symmetric because $\tau(f^{\prime})=f^{\prime}$ as
$f^{\prime}$ is an even polynomial. The equation
$\omega(xb_{1},b_{2})+\omega(b_{1},xb_{2})$ can be derived from the fact
$\tau(x)=-x$.

By choosing a trivialization of the determinant, we obtain a companion map
$[x]:\mathfrak{c}\to[\mathfrak{g}/G]$ for $G=\mathrm{SO}_{2n+1}$.

###  3.4 Even special orthogonal groups

The case $G=\mathrm{SO}_{2n}$ is slightly more difficult for the map
$\mathfrak{c}\to\mathfrak{c}_{2n}$ induced by the standard representation of
$\mathrm{SO}_{2n}$ is not a closed embedding. Indeed, we have
$\mathfrak{c}_{2n}=\mathrm{Spec}(A_{2n})$ where
$A_{2n}=k[a_{1},\ldots,a_{2n}]$ but $\mathfrak{c}=\mathrm{Spec}(A)$ where
$A=k[a_{2},\ldots,a_{2n-2},p_{n}]$ where $p_{n}$ is the Pfaffian satisfying
$p_{n}^{2}=a_{2n}$ does not lie in the image of $A_{2n}\to A$. If $B_{2n}$ is
the spectral cover of $A_{2n}$ and $B=A\otimes_{A_{2n}}B_{2n}$ then we have

$B=A[x]/(x^{2n}+a_{2}x^{2n-2}+\cdots+a_{2n-2}x^{2}+p_{n}^{2}).$

As indicated by Hitchin [Hit87], the true spectral cover for even special
orthogonal groups is not $B$ but its blowup $\widetilde{B}$ along the singular
locus defined by $x$. We have

$\widetilde{B}=A[x,p_{n-1}]/\big{(}xp_{n-1}-p_{n},\;x^{2n-2}+a_{2}x^{2n-4}+\cdots+a_{2n-2}+p_{n-1}^{2}\big{)}$

which is a free $A$-module of rank $2n$ and smooth as a $k$-algebra. We have
an involution $\tau$ on $B$ and $\widetilde{B}$ given by $\tau(x)=-x$ and
$\tau(p_{n-1})=-p_{n-1}$.

The dualizing sheaf $\omega_{\widetilde{B}/A}$ is a free rank-one
$\widetilde{B}$-module, canonically isomorphic to $\widetilde{B}$ away from
the ramification locus. As a $\widetilde{B}$-submodule of
$\mathrm{Fr}(\widetilde{B})$ it is generated by the inverse of the different
$\mathfrak{D}_{\widetilde{B}/A}$ which is given by the formula

$\displaystyle\mathfrak{D}_{\widetilde{B}/A}$ $\displaystyle=$
$\displaystyle\det\begin{pmatrix}-p_{n-1}&f^{\prime}\\\
-x&2p_{n-1}\end{pmatrix}$ $\displaystyle=$
$\displaystyle(n-1)x^{2(n-1)}+(n-2)a_{2}x^{2(n-2)}+\cdots+a_{2n-2}x^{2}+p_{n-1}^{2}.$

In other words, the bilinear form $\widetilde{B}\otimes_{A}\widetilde{B}\to A$
given by

$b_{1}\otimes_{A}b_{2}\mapsto\mathrm{tr}_{\widetilde{B}/A}(\mathfrak{D}_{\widetilde{B}/A}^{-1}b_{1}b_{2})$

is non-degenerate. As in the symplectic and odd special orthogonal cases, we
now consider the symmetric bilinear form

$\omega(b_{1},b_{2})=\mathrm{tr}_{\widetilde{B}/A}(\mathfrak{D}_{\widetilde{B}/A}^{-1}b_{1}\tau(b_{2}))$

Then $\omega$ is a non-degenerate symmetric bilinear form because
$\tau(\mathfrak{D}_{\widetilde{B}/A})=\mathfrak{D}_{\widetilde{B}/A}$, and it
satisfies

$\omega(xb_{1},b_{2})=-\omega(b_{1},xb_{2}).$

After a choice of trivialization of the determinant of $\widetilde{B}$ as a
free $A$-module of rank $n$, the multiplication by $x$ gives rise to the
companion section $[x]:\mathfrak{g}{/\\!\\!/}G\to[\mathfrak{g}/G]$ for the odd
special orthogonal group $G=\mathrm{SO}_{2n+1}$.

###  3.5 The group $G_{2}$

In the case $G_{2}$, the invariant quotient is $A=k[e,q]$ with $\deg(e)=2$ and
$\deg(q)=6$. The spectral cover $\mathfrak{s}=\mathrm{Spec}(B)$ of
$\mathfrak{c}=\mathrm{Spec}(A)$ given by

$B=A[x]/\left(xf_{0}\right)\quad\text{for
}f_{0}=x^{6}-ex^{4}+\frac{e^{2}}{4}x^{2}+q$

is a reducible cover of $A$ with two components corresponding to the quotient
maps

$B\to B^{\prime}=A[x]/(f_{0})\quad\text{and}\quad B\to A=A[x]/(x).$

The cover $\mathfrak{s}^{\prime}=\mathrm{Spec}(B^{\prime})$ of $A$ is finite,
flat of degree 6, and factors through two subcovers, of degrees 2 and 3,
corresponding to the sub-$A$-algebras

$A\subset A[y]/\left(y^{3}-ey^{2}+\frac{e^{2}}{4}y+q\right)\subset
B^{\prime}\quad\text{where }y=x^{2}$ $A\subset
A[z]/\left(z^{2}+q\right)\subset B^{\prime}\quad\text{where
}z=x\left(x^{2}-\frac{e}{2}\right)$

Let $\epsilon\in B[q^{-1}]^{*}:=\mathrm{Hom}_{A[q^{-1}]}(B[q^{-1}],A[q^{-1}])$
be dual to $f_{0}$; $\delta_{i}\in B[q^{-1}]^{*}$ be dual to $x^{i}$; and
$\eta_{i}\in B[q^{-1}]^{*}$ be dual to $x^{i}z$ for $i=1,2,3$. Let ${\rm
tr}_{z}$ denote the skew-symmetric bilinear form on $B$ given by

${\rm tr}_{z}(g,h)={\rm Tr}_{{\rm Frac}(B)/{\rm
Frac}(A)}\left(\frac{g(x)h(-x)z}{f(x)}\right)$

We will denote by $\rho$ the 3-form on $B[q^{-1}]$ given by

$\rho:=\delta_{1}\wedge\delta_{2}\wedge\eta_{3}+\delta_{1}\wedge\eta_{2}\wedge\delta_{3}+\eta_{1}\wedge\delta_{2}\wedge\delta_{3}-q\cdot\eta_{0}\wedge\eta_{1}\wedge\eta_{2}+\epsilon\wedge{\rm
tr}_{z}$ (6)

A priori, the 3-form above is valued in $A[q^{-1}]$. The next proposition
tells us that it restricts to an element of $\bigwedge^{3}_{A}B^{*}$.

######  Proposition 3.3.

Restricting the 3-form $\rho$ to $B\to B[q^{-1}]$ induces a 3-form
$\rho\in\bigwedge^{3}_{A}B^{*}$. In other words, $\rho$ takes values in $A$
when restricted to $B$.

###### Proof.

Consider the $A$-basis of $B$ given by

$\\{1,x^{i},x^{i}z\colon i=1,2,3\\}.$

This differs from the $A[q^{-1}]$-basis

$\\{f_{0},x^{i},x^{i}z\colon i=1,2,3\\}$

of $B[q^{-1}]$ only by scaling $f_{0}$. As $\rho$ is valued in $A$ on the
$A$-linear span of the latter basis, it suffices to check the contraction
$\iota_{1}\rho$ of $\rho$ along $1\in B$ is valued in $A$. We compute

$\displaystyle\iota_{1}\rho$
$\displaystyle=\eta_{1}\wedge\eta_{2}-\frac{e}{2}\eta_{2}\wedge\eta_{3}+\frac{1}{q}({\rm
tr}_{z}-\delta_{1}\wedge\delta_{2}+\frac{e}{2}\delta_{2}\wedge\delta_{3}-i_{x^{3}z}{\rm
tr}_{z}\wedge\epsilon+\frac{e}{2}i_{xz}{\rm tr}_{z}\wedge\epsilon)$
$\displaystyle=\eta_{1}\wedge\eta_{2}-\frac{e}{2}\eta_{2}\wedge\eta_{3}+\Big{[}\frac{1}{q}({\rm
tr}_{z}-\delta_{1}\wedge\delta_{2}+\frac{e}{2}\delta_{2}\wedge\delta_{3})-\iota_{1}{\rm
tr}_{z}\wedge\epsilon\Big{]}$

Rewriting the latter in terms of a dual basis $\xi_{i}$, $i=0,\dots,6$ for the
$A$-basis $\\{x^{i}\colon i=0,\dots,6\\}$ of $B$, we see that the expression
in square brackets above is

$\iota_{1}\rho=\mathrm{e}_{3}\wedge\mathrm{e}_{6}+\mathrm{e}_{4}\wedge\mathrm{e}_{5}-\frac{3e}{2}\mathrm{e}_{5}\wedge\mathrm{e}_{6}$

whose image lies in $A$. ∎

As the previous proposition illustrates, working with the form $\rho$ requires
significantly more computational effort. As such, Propositions 3.4 and 3.5
will be checked primarily with computer algebra packages. These computations
were done in Macaulay2; explicit code for each calculation is referred to in
Appendix A.

######  Proposition 3.4.

Let $\nu$ be the bilinear form associated to $\rho$ as in equation (1) and let
$\omega\in S^{2}_{A}B^{*}$ be the symmetric, nondegenerate form given by the
formula (5). Then, $\nu=-2^{4}3^{2}\omega$.

######  Proposition 3.5.

The form $\rho$ is compatible with the endomorphism $[x]$, in the sense that

$\rho(xb_{1},b_{2},b_{3})+\rho(b_{1},xb_{2},b_{3})+\rho(b_{1},b_{2},xb_{3})=0.$

As such, the form $\rho$ together with a trivialization of the determinant
gives a map $[x]\colon\mathfrak{c}\to[\mathfrak{g}_{2}/G_{2}]$.

##  4 Special components

In the previous section, we gave explicit formulas for the tensors defining
the reduction of the vector bundle
$\mathcal{O}_{\mathfrak{c}\times_{\mathfrak{c}_{n}}\mathfrak{s}_{n}}$ to $G$
so that the companion section for $\mathrm{GL}_{n}$ induces the companion
section for classical group $G$. These explicit formulas may feel like
miracles, especially in the $G_{2}$ case where a computer algebra system is
needed. In this section, we will derive them from the geometry of spectral
covers, which makes the construction more conceptual, especially in the
$G_{2}$ case. In subsequent work, we use this approach to construct the
companion section uniformly.

###  4.1 Special form and component associated with a subcover

Let $A$ be a $k$-algebra, $B$ a finite flat $A$-algebra of degree $n$
generated by one element $b\in B$, and $A^{\prime}\subset B$ an $A$-subalgebra
of $B$ such that $A^{\prime}$ is finite flat of degree $m$ over $A$ generated
by one element $a^{\prime}\in A^{\prime}$ and $B$ is a finite flat
$A^{\prime}$-algebra of degree $d$ generated by $b$. Under these assumptions,
we have $B\simeq A[x]/P(x)$ where $P(x)$ is the characteristic polynomial of
the $A$-linear $b:B\to B$ defined as the multiplication by $b$. Similarly we
have $A^{\prime}\simeq A[x]/(P_{1}(x))$ where $P_{1}(x)$ is the the
characteristic polynomial of the $A$-linear operator $a^{\prime}:A^{\prime}\to
A^{\prime}$, and $B\simeq A^{\prime}[x]/P_{2}(x)$ where $P_{2}(x)$ is the
characteristic polynomial of the $A^{\prime}$-linear operator $b:B\to B$.

Assuming that the characteristic of $k$ is greater than $d$, we want to
construct an alternating $d$-form

$\omega_{A^{\prime}}:\wedge^{d}_{A}B\to A$

supported on a special component of $\mathrm{Spec}(S^{d}_{A}B)$ isomorphic to
$\mathrm{Spec}(A^{\prime})$. We explain what this means. As far as we know,
the concept of non-degeneracy for $d$-forms is not yet defined for $d\geq 3$
and thus we can prove the it only for $d=1$ or $d=2$. However, we expect that
the form we construct is non-degenerate for a reasonable definition of this
concept. As to the special component, $\bigwedge^{d}_{A}B$ is a module over
the ring of symmetric tensors $(\bigotimes_{A}^{d}B)^{\mathfrak{S}_{d}}$. We
will construct a surjective homomomorphism of $A$-algebras
$(\bigotimes_{A}^{d}B)^{\mathfrak{S}_{d}}\to A^{\prime}$ which realizes
$\mathrm{Spec}(A^{\prime})$ as an irreducible component of
$\mathrm{Spec}((\bigotimes_{A}^{d}B)^{\mathfrak{S}_{d}})$ if $B$ is
generically étale over $A$ and $A^{\prime}$ is a domain.

The homomorphism of $A$-algebras $(\bigotimes_{A}^{d}B)^{\mathfrak{S}_{d}}\to
A^{\prime}$ is constructed as follows. Let
$P_{2}(x)=x^{d}+a^{\prime}_{1}x^{d-1}+\cdots+a^{\prime}_{d}$ be the
characteristic polynomial of the $A^{\prime}$-linear map $b:B\to B$. Then we
have

$B=A^{\prime}[x]/(x^{d}+a^{\prime}_{1}x^{d-1}+\cdots+a^{\prime}_{d}).$

We consider the polynomial ring $R=k[x_{1},\ldots,x_{d}]$ and the subring $S$
of invariant polynomials under the symmetric group $\mathfrak{S}_{d}$. We have

$S=k[x_{1},\ldots,x_{d}]^{\mathfrak{S}_{d}}=k[\alpha_{1},\ldots,\alpha_{d}]$

with

$\alpha_{i}=(-1)^{i}\sum_{1\leq j_{1}<\cdots<j_{i}\leq
d}\alpha_{j_{1}}\ldots\alpha_{j_{d}}.$

Since $R$ and $S$ are regular, and $R$ is a finite generated $S$-module, $R$
is a finite flat $S$-algebra of degree $d!$. We consider the homomorphism of
algebras $S\to A^{\prime}$ given by $\alpha_{i}\mapsto a^{\prime}_{i}$ and the
base change $A^{\prime}\otimes_{S}R$ which is a finite flat
$A^{\prime}$-algebra of degree $d!$ equipped with an action of
$\mathfrak{S}_{d}$. We have
$(A^{\prime}\otimes_{S}R)^{\mathfrak{S}_{d}}=A^{\prime}$. Moreover, for every
$i\in\\{1,\ldots,d\\}$ we have a homomorphism of $A^{\prime}$-algebras $B\to
R\otimes_{S}A^{\prime}$ given by $x\mapsto x_{i}$ which together give rise to
a surjective homomorphism of $A^{\prime}$-algebras
$\bigotimes_{A^{\prime}}^{d}B\to A^{\prime}\otimes_{S}R$, which is
$\mathfrak{S}_{d}$-equivariant. We derive a $\mathfrak{S}_{d}$-equivariant
surjective homomorphism of $A$-algebras

$\bigotimes_{A}^{d}B\to\bigotimes_{A^{\prime}}^{d}B\to
A^{\prime}\otimes_{S}R.$ (7)

By taking the $\mathfrak{S}_{d}$-invariant, we obtain the desired homomorphism
of algebras

$S^{d}_{A}B=(\bigotimes_{A}^{d}B)^{\mathfrak{S}_{d}}\to A^{\prime},$

which is surjective because taking $\mathfrak{S}_{d}$-invariants is an exact
functor under the characteristic assumption.

We will now construct a special $d$-form on $B$

$\omega_{A^{\prime}}:\bigwedge^{d}_{A}B\to A$

supported on the special component. As above, we have a surjective
homomorphism of algebras $\mathfrak{S}_{d}$-equivariant surjective
homomorphisms of $A$-algebras $\otimes^{d}_{A}B\to\otimes^{d}_{A^{\prime}}B\to
A^{\prime}\otimes_{S}R$ which induces a surjective $A$-linear maps of the
alternating parts $\bigwedge^{d}_{A}B\to\bigwedge^{d}_{A^{\prime}}B\to
A^{\prime}\otimes_{S}R^{\mathrm{sgn}}$ where $R^{\mathrm{sgn}}$ is the direct
factor of $R$ as $S$-module in which $\mathfrak{S}_{d}$ acts as the sign
character. It is known that $R^{sgn}$ is a free $S$-module generated by
$\prod_{1\leq i<j\leq d}(x_{i}-x_{j})$. We thus obtains a surjective
$A$-linear map $\bigwedge_{A}^{d}B\to A^{\prime}$. By composing it with the
generator of $\mathrm{Hom}_{A}(A^{\prime},A)$ constructed in 3.2 we obtain the
special $d$-form $\omega_{A^{\prime}}:\bigwedge^{d}_{A}B\to A$ which is
supported by the special by construction.

Let us discuss the non-degeneracy of the special $d$-form
$\omega_{A^{\prime}}:\bigwedge^{d}_{A}B\to A$. For $d=1$, this follows from
Lemma 3.2. We can check that it is also non-degenerate everywhere for $d=2$.
For $d\geq 3$, we don’t know a general definition of non-degeneracy but it
easy to see that the special form $\omega_{A^{\prime}}$ is everywhere non-
zero. In dimension 6 and 7 where the definition of non-degeneracy is
available, we will check that the special $d$-form is everywhere non-
degenerate by direct calculation.

###  4.2 $\mathrm{Sp}_{2n}$ case

We recall in the case $G=\mathrm{Sp}_{2n}$, we have
$\mathfrak{c}=\mathrm{Spec}(A)$ with $A=k[a_{2},\ldots,a_{2n}]$. The spectral
cover $\mathfrak{s}=\mathrm{Spec}(B)$ where

$B=A[x]/(x^{2n}+a_{2}x^{2n-2}+\cdots+a_{2n})$

is a free $A$-module of rank $2n$, is equipped with an involution $\tau:B\to
B$ given $\tau(x)=-x$. We consider the subalgebra $A^{\prime}$ of $B$
consisting of elements fixed under $\tau$

$A^{\prime}=A[y]/(y^{n}+a_{2}y^{n-1}+\cdots+a_{2n}).$

We then have $B=A^{\prime}[x]/(x^{2}-y)$.

The construction of the special form and special component in 4.1 gives rise
to an alternating form

$\omega_{A^{\prime}}:\wedge^{2}_{A}B\to A$

supported in the special component
$\mathfrak{c}^{\prime}=\mathrm{Spec}(A^{\prime})$ of
$(\mathfrak{s}\times_{\mathfrak{c}}\mathfrak{s}){/\\!\\!/}\mathfrak{S}_{2}$
where $\mathfrak{s}=\mathrm{Spec}(B)$ and $\mathfrak{c}=\mathrm{Spec}(A)$. The
homomorphism (7) $\mathrm{Sym}_{A}^{2}(B)\to A^{\prime}$ can be explicitly
computed elements of the form:

$b\otimes_{A}1+1\otimes_{A}b\mapsto\mathrm{tr}_{B/A^{\prime}}(b).$

In particular, $x\otimes_{A}1+1\otimes_{A}x$ be long to the kernel of
$\mathrm{Sym}_{A}^{2}(B)\to A^{\prime}$, and in fact on can verify that it is
a generator of the kernel. Since $x\otimes_{A}1+1\otimes_{A}x$ annihilates
$\omega_{A^{\prime}}$ we have

$\omega_{A^{\prime}}(xb_{1},b_{2})+\omega_{A^{\prime}}(b_{1},xb_{2})=0$

for every $b_{1},b_{2}\in B$. By Lemma 3.2, the 2-form $\omega_{A^{\prime}}$
is everywhere non-degenerate. We can also see by explicit calculation that the
form $\omega_{A^{\prime}}$ is the same as the 2-form we constructed in
subsection 3.2 by means of the Euler formula.

###  4.3 $G_{2}$ case

In the case $G_{2}$, the invariant quotient is $A=k[e,q]$ with $\deg(e)=2$ and
$\deg(q)=6$. The spectral cover $\mathfrak{s}=\mathrm{Spec}(B)$ with

$B=A[x]/\left(xf_{0}\right)\quad\text{for
}f_{0}=x^{6}-ex^{4}+\frac{e^{2}}{4}x^{2}+q$

is a reducible cover of $A$ with two components corresponding to the quotient
maps

$B\to B^{\prime}=A[x]/(f_{0})\quad\text{and}\quad B\to A$

We will define a canonical 3-form on $B$ out of a 3-form and a 2-form on
$B^{\prime}$ associated to subalgebras

$\displaystyle A\subset A^{\prime}=A[z]/\left(z^{2}+q\right)=k[e,y]$
$\displaystyle\subset$ $\displaystyle
B^{\prime}=A^{\prime}[x]/(x^{3}-\frac{e}{2}x-z)$ $\displaystyle A\subset
A^{\prime\prime}=A[y]/\left(y^{3}-ey^{2}+\frac{e^{2}}{4}y+q\right)=k[e,z]$
$\displaystyle\subset$ $\displaystyle
B^{\prime}=A^{\prime\prime}[x]/(x^{2}-y).$

Since both $A^{\prime}$ and $A^{\prime\prime}$ are regular algebras, they are
finite flat $A$-modules of rank 2 and 3, respectively, whereas $B$ are finite
flat $A^{\prime}$-module and $A^{\prime\prime}$-module of rank 3 and 2,
respectively. The construction of the special form associated with a subcover
gives rise to

$\omega_{A^{\prime}}:\wedge^{3}_{A}B^{\prime}\to A\mbox{ and
}\omega_{A^{\prime\prime}}:\wedge^{2}_{A}B^{\prime}\to A$

supported on the special components
$\mathfrak{c}^{\prime}=\mathrm{Spec}(A^{\prime})$ and
$\mathfrak{c}^{\prime\prime}=\mathrm{Spec}(A^{\prime\prime})$ of
$\mathfrak{s}^{\times_{\mathfrak{c}}^{3}}{/\\!\\!/}\mathfrak{S}_{3}$ and
$\mathfrak{s}^{\times_{\mathfrak{c}}^{2}}{/\\!\\!/}\mathfrak{S}_{2}$,
respectively. By arguing as in the symplectic case, we see that
$\omega_{A^{\prime}}$ is annihilated by
$x\otimes_{A}1\otimes_{A}1+1\otimes_{A}x\otimes_{A}1+1\otimes_{A}1\otimes_{A}x$
and $\omega_{A^{\prime}}$ by $x\otimes_{A}1+1\otimes_{A}x$. It follows that as
alternating forms, they satisfy the relations:

$\displaystyle\omega_{A^{\prime}}(xb_{1},b_{2},b_{3})+\omega_{A^{\prime}}(b_{1},xb_{2},b_{3})+\omega_{A^{\prime}}(b_{1},b_{2},xb_{3})$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\omega_{A^{\prime\prime}}(xb_{1},b_{2})+\omega_{A^{\prime\prime}}(b_{1},xb_{2})$
$\displaystyle=$ $\displaystyle 0$

for all $b_{1},b_{2},b_{3}\in B$.

The form $\omega_{A^{\prime}}$ agrees with the restriction of the form $\rho$
calculated by Macaulay 2 when restricted to $B^{\prime}\to B$, with the
inclusion given by multiplication by $x$: Indeed, the restriction of $\rho$
takes value 1 on each of:

$z\wedge x\wedge x^{2},\quad x\wedge zx\wedge x^{2},\quad 1\wedge x\wedge
zx^{2}$

and $-q$ on $z\wedge zx\wedge zx^{2}$. This exactly detects the coefficient of
$z$ when these wedges are written in terms of the $A^{\prime}$ basis $1\wedge
x\wedge x^{2}$ for $\wedge^{3}_{A^{\prime}}B^{\prime}$, which matches
$\omega_{A^{\prime}}$ since the generator of $\mathrm{Hom}_{A}(A^{\prime},A)$
as an $A^{\prime\prime}$ module detects the coefficient of $A^{\prime}$.

We now build a 3-form on $B$ out of the 3-form $\omega_{A^{\prime}}$ and
2-form $\omega_{A^{\prime\prime}}$ on $B^{\prime}$. Since $B=A[x]/(xf_{0})$,
$B^{\prime}=A[x]/(f_{0})$ we have exact sequences of free $A$-modules

$0\to A\to B\to B^{\prime}\to 0\mbox{ and }0\to B^{\prime}\to B\to A\to 0$

where the map $A\to B$ in the first sequence is given by $1\mapsto f_{0}$ and
the map $B^{\prime}\to B$ in the second sequence is given by $1\mapsto x$. It
follows an exact sequences

$0\to A\oplus B^{\prime}\to B\to Q\to 0\mbox{ and }0\to B\to A\oplus
B^{\prime}\to Q\to 0$

where $Q=A/(q)=B^{\prime}/(x)$. It follows an exact sequence

$0\to\wedge_{A}^{3}B^{*}\to\wedge^{3}_{A}(B^{\prime})^{*}\oplus\wedge^{2}_{A}(B^{\prime})^{*}\to\wedge^{2}(B^{\prime})^{*}/(q)\to
0$

where the map
$\wedge^{2}_{A}(B^{\prime})^{*}\to\wedge^{2}(B^{\prime})^{*}/(q)$ is the
reduction modulo $q$, and the map
$\wedge^{3}_{A}(B^{\prime})^{*}\to\wedge^{2}(B^{\prime})^{*}/(q)$ is obtained
by the composition

$\wedge^{3}_{A}(B^{\prime})^{*}\to\wedge^{3}_{A}B^{*}\to\wedge^{2}_{A}B^{*}\to\wedge^{2}_{A}(B^{\prime})^{*}\to\wedge^{2}_{A}(B^{\prime})^{*}/(q)$

where the first map is induced by the projection $B\to B^{\prime}$, the second
is given by contraction with $f_{0}$, the third map is induced by the
inclusion $B^{\prime}\to B$ sending $1\mapsto x$, and the final map is the
quotient map. Since
$q\wedge^{2}(B^{\prime})^{*}\simeq\wedge^{2}(B^{\prime})^{*}$ is a free, rank
1 module over the special component of $S^{2}_{A}(B^{\prime})$, there is a
unique generator as an $A^{\prime\prime}$ module. The 3-form
$\omega_{A^{\prime}}$ and the 2-form $\omega_{A^{\prime\prime}}$ do not have
the same image in $\wedge^{2}_{A}(B^{\prime})^{*}/(q)$; however, the form
$z\omega_{A^{\prime\prime}}$ is and it gives a generator for the
$A^{\prime\prime}$ submodule of 2-forms compatible with $\omega_{A^{\prime}}$.
The pair $(\omega_{A^{\prime}},z\omega_{A^{\prime\prime}})$ comes from an
element of $\wedge_{A}^{3}B^{*}$ which agrees with the 3-form calculated by
Macaulay2.

##  5 Lattice description of affine Springer fibers of classical groups

Let us recall Kazhdan-Lusztig’s definition [KL88] of affine Springer fibers.
Let $G$ be a split reductive group defined over a field $k$ and $\mathfrak{g}$
its Lie algebra. Let $F=k((\varpi))$ the field of Laurent formal series and
$\mathcal{O}=k[[\varpi]]$ its ring of integers. Let $\gamma\in\mathfrak{g}(F)$
be a regular semisimple element. The affine Springer fiber associated with
$\gamma$ is an ind-scheme defined over $k$ whose set of $k$-points is

$\mathcal{M}_{\gamma}(k)=\\{g\in
G(F)/G(\mathcal{O})|\mathrm{ad}(g)^{-1}\gamma\in\mathfrak{g}(\mathcal{O})\\}.$

We note that $\mathcal{M}_{\gamma}$ is non-empty only if the image
$a\in\mathfrak{c}(F)$ lies in $\mathfrak{c}(\mathcal{O})$ where
$\mathfrak{c}=\mathfrak{g}{/\\!\\!/}G$ is the invariant theoretic quotient of
$\mathfrak{g}$ by the adjoint action of $G$. As argued in [Ngo10], using the
Kostant section, we can define an affine Springer fiber $\mathcal{M}_{a}$
depending only on $a$ instead of $\gamma$, which is isomorphic to
$\mathcal{M}_{\gamma}$.

For $G=\mathrm{GL}_{n}$, the affine Springer fiber $\mathcal{M}_{a}$ has a
well-known lattice description. In this case, $\mathfrak{c}={\mathbb{A}}^{n}$.
If $a=(a_{1},\ldots,a_{n})\in\mathcal{O}_{n}$, we form the finite flat
$\mathcal{O}$-algebra

$B_{a}=\mathcal{O}[x]/(f_{a})$

where $f_{a}=x^{n}+a_{1}x^{n-1}+\cdots+a_{n}$ by the base change from the
universal spectral cover. As $\gamma\in\mathfrak{g}(F)$ is a regular
semisimple element, $B_{a}\otimes_{\mathcal{O}}F$ is finite and étale over
$F$. We have a well-known lattice description of the affine Springer fiber
$\mathcal{M}_{a}$ in this case.

######  Theorem 5.1.

For $G=\mathrm{GL}_{n}$ and
$a\in\mathfrak{c}^{\mathrm{r}s}(F)\cap\mathfrak{c}(\mathcal{O})$, the set
$\mathcal{M}_{a}(k)$ consists of lattices $\mathcal{V}$ in the $n$-dimensional
vector space $V=B_{a}\otimes F$ which are also $B_{a}$-modules.

See for example, Section 2 of [Yun16] for an exposition.

For computational purposes, it is desirable to have a lattice description of
affine Springer fibers similar to Theorem 5.1 for classical groups, which is
as simple as in the linear case. This is possible due to the construction of
the companion matrix, and in fact, this was our original motivation.

In the cases we have investigated in the paper, i.e., symplectic, special
orthogonal, and $G_{2}$, we have constructed a finite, flat spectral cover
$\mathfrak{s}=\mathrm{Spec}(B)$ of the invariant theoretic quotient
$\mathfrak{c}=\mathrm{Spec}(A)$ which is étale over the regular semisimple
locus of $\mathfrak{c}$. The degree $d=\deg(B/A)$ is the degree of the
standard representation which is $2n$ for $\mathrm{Sp}_{2n}$, $2n+1$ for
$\mathrm{SO}_{2n+1}$, $2n$ for $\mathrm{SO}_{2n}$ and $7$ for $G_{2}$. In the
case $\mathrm{SO}(2n)$, we must consider the normalization $\tilde{B}$ of $B$.
In each of these cases, we constructed a form $\omega$, which is

  * •

a non-degenerate symplectic form $\omega:B\times B\to A$ satisfying
$\omega(xb_{1},b_{2})+\omega(b_{1},xb_{2})=0$ for $\mathrm{Sp}_{2n}$

  * •

a non-degenerate symmetric form $\omega:B\times B\to A$ satisfying
$\omega(xb_{1},b_{2})+\omega(b_{1},xb_{2})=0$ for $\mathrm{SO}_{2n+1}$

  * •

a non-degenerate symmetric form $\omega:\tilde{B}\times\tilde{B}\to A$
satisfying $\omega(xb_{1},b_{2})+\omega(b_{1},xb_{2})=0$ for
$\mathrm{SO}_{2n}$

  * •

a non-degenerate alternating form $\omega:B\times B\times B\to A$ satisfying

$\omega(xb_{1},b_{2},b_{3})+\omega(b_{1},xb_{2},b_{3})+\omega(b_{1},b_{2},xb_{3})=0$

for $G_{2}$

We also constructed a trivialization of the determinant $\bigwedge^{d}_{A}B=A$
in all these cases.

For every $a\in\mathfrak{c}(\mathcal{O})\cap\mathfrak{c}^{rs}(F)$, we
construct a finite flat $\mathcal{O}$-algebra $B_{a}$ by base change from the
spectral cover $\mathfrak{s}\to\mathfrak{c}$. Because
$a\in\mathfrak{c}^{rs}(F)$, the generic fiber
$V_{a}=B_{a}\otimes_{\mathcal{O}}F$ is a finite étale $F$-algebra of degree
$d$. By pulling back $\omega$, we get a form $\omega_{a}$ which is a non-
degenerate alternating $F$-bilinear form on $V_{a}$ in the symplectic case, a
non-degenerate symmetric $F$-bilinear form on $V_{a}$ in the orthogonal case,
and a non-degenerate alternating $F$-trilinear form on $V_{a}$ in the $G_{2}$
case. Moreover, it extends to a non-degenerate form valued in $\mathcal{O}$ on
$B_{a}$ in $\mathrm{Sp}_{2n}$, $\mathrm{SO}_{2n+1}$ and $G_{2}$ cases and on
$\tilde{B}_{a}$ in the $\mathrm{SO}_{2n}$-case.

######  Theorem 5.2.

The set of $k$-points of the affine Springer fiber $\mathcal{M}_{a}$ is the
set of $\mathcal{O}$-lattices $\mathcal{V}$ of $V_{a}$, which are
$B_{a}$-modules, such that the restriction of $\omega_{a}$ has value in
$\mathcal{O}$ and such that $\deg(\mathcal{V}:B_{a})=0$ in
$\mathrm{Sp}_{2n},\mathrm{SO}_{2n+1},G_{2}$ cases and
$\deg(\mathcal{V}:\tilde{B}_{a})=0$ in the $\mathrm{SO}_{2n}$ case.

The proof of this result follows immediately from the proof of Theorem 5.1, as
lattices preserved by the nondegenerate form $\omega_{a}$ constructed above
are exactly those for which there is a reduction of structure to the classical
group $G$.

##  6 Application to the Hitchin fibration

Let $X$ be a smooth, projective curve over an algebraically closed field $k$
and let $G$ be a reductive group over $k$ with Lie algebra $\mathfrak{g}$. Fix
a line bundle $L$ on $X$ such that either $\deg(L)>2g-2$ or $L=K$ is the
canonical bundle. Denote by $\mathcal{M}$ the moduli stack of Higgs bundles on
$X$, whose $k$ points are given by the set of Higgs bundles

$\mathcal{M}(k)=\\{(E,\phi)\colon E\to X\text{ is a $G$ bundle,
}\phi\in\Gamma(X,{\rm ad}(E)\otimes L)\\}$

More succinctly, $\mathcal{M}$ is the mapping stack $\mathcal{M}={\rm
Maps}(X,[\mathfrak{g}_{L}/G])$ where
$\mathfrak{g}_{L}=\mathfrak{g}\wedge^{{\mathbb{G}}_{m}}L$ is the twisted
bundle of Lie algebras on $X$.

Recall that under mild hypotheses on the characteristic of $k$ (${\rm
char}(k)>2$ for $G=\mathrm{SO}_{n}$ and $\mathrm{Sp}_{2n}$ and ${\rm
char}(k)>3$ for $G=G_{2}$), the Chevalley isomorphism shows

$\mathfrak{g}{/\\!\\!/}G\simeq\mathfrak{t}{/\\!\\!/}W\simeq{\mathbb{A}}^{n}$

is an affine space with ${\mathbb{G}}_{m}$ action by weights
$d_{1},\dots,d_{n}$. Let

$\mathcal{A}={\rm
Maps}(X,\mathfrak{g}_{L}{/\\!\\!/}G)\simeq\otimes_{i=1}^{n}\Gamma(X,L^{\otimes
d_{i}})$

Hitchin, in [Hit87], studied the space $\mathcal{M}$, with appropriate
stability conditions imposed, through the fibration that now bears his name:

$h\colon\mathcal{M}\to\mathcal{A},\quad(E,\phi)\mapsto{\rm char}(\phi)$

where ${\rm char}(\phi)$ is given by composition with the quotient map
$[\mathfrak{g}/G]\to\mathfrak{g}{/\\!\\!/}G$. Let $\mathcal{M}_{a}$ denote the
fiber of the map $h$ over a point $a\in\mathcal{A}$. In the case that
$G=\mathrm{GL}_{n}$, $d_{i}=i$ and ${\rm char}(\phi)=\sum_{i}a_{i}x^{i}$ is
the characteristic polynomial of $\phi$, whose coefficients are then sections
$a_{i}\in\Gamma(X,L^{\otimes i})$.

The companion section $[x]:\mathfrak{g}{/\\!\\!/}G\to[\mathfrak{g}/G]$ can be
used to construct an explicit section to the Hitchin map after extracting a
square root of $L$. This section in many cases is almost the same as the
section constructed by Hitchin [Hit87] and [Hit07], but can be different from
the section constructed in [Ngo10] which is based on the Kostant section. In
every case, the Higgs bundle constructed from the companion section will be
built out of the structural sheaf of the spectral curve. Note that the
following assumes basic ${\mathbb{G}}_{m}$ equivariance properties of the
relevant forms. For example, in the case of $G=\mathrm{Sp}_{2n}$, we have
constructed a canonical alternating form $\omega\colon\wedge^{2}_{A}B\to A$
which satisfies $\omega(\lambda\xi)=\lambda^{1-2n}\omega(\xi)$ for any
$\lambda\in{\mathbb{G}}_{m}$ and $\xi\in\wedge^{2}_{A}B$.

In [Ngo10], it is shown that over a large open subset of $\mathcal{A}$, there
is a close connection, depending on a choice of section, between Hitchin
fibers and affine Springer fibers given by the Product Formula. More
precisely, let
$\mathfrak{D}=\bigcup_{\alpha}\mathfrak{t}^{s_{\alpha}}{/\\!\\!/}W$ be the
divisor consisting of the union of the image of each root hyperplane in
$\mathfrak{t}$; in particular, the complement of $\mathfrak{D}$ in
$\mathfrak{c}$ is the regular, semisimple locus $\mathfrak{c}^{rs}$. Fix
$a\in\mathcal{A}$ such that $a(X)\not\subset\mathfrak{D}$, and let $U\subset
X$ be the preimage of $\mathfrak{c}^{rs}$ in $X$. Given trivialization of the
line bundle $D$ on some neighborhood of each point $v\in X\setminus U$, we
have a map

$\prod_{v\in X\setminus U}\mathcal{M}_{x,a}\to\mathcal{M}_{a}.$

from the product of affine Springer fibers at the points $x\in X\setminus U$
to the Hitchin fiber, which consists of gluing with the companion section
restricted to $U$. It it induces a universal homeomorphism

$\prod_{\gamma\in X\setminus
U}\mathcal{M}_{\gamma,a}\wedge^{\prod_{\gamma}\mathcal{P}_{\gamma}(J_{a})}\mathcal{P}_{a}\to\mathcal{M}_{a}.$

The groups $\mathcal{P}_{\gamma}(J_{a})$ and $\mathcal{P}_{a}$ are discussed
in detail in [Ngo10]; we will not describe them here. This is proved in
[Ngo10] under the assumption that $\pi_{0}(\mathcal{P}_{a})$ is finite, and by
Bouthier and Cesnavicius in [BČ22] under the only assumption that
$a(X)\not\subset\mathfrak{D}$.

As Section 5 describes the affine springer fibers $\mathcal{M}_{\gamma,a}$,
the product formula above gives an explicit description of Hitchin fibers in
the case that $a(X)\not\subset\mathfrak{D}$. Namely, we have the following
descriptions for Hitchin fibers under this assumption.

  * •

for $G=\mathrm{GL}_{n}$, and $a\in\mathcal{A}$ we have a spectral cover
$p_{a}:Y_{a}\to X$ embedded in the total space $|L|$ of $L$. We then associate
with $a$ the Higgs bundle $E_{a}=p_{a*}\mathcal{O}_{Y_{a}}$ and the Higgs
fields $\phi:E_{a}\to E_{a}\otimes L$ given by the structure of
$\mathcal{O}_{Y_{a}}$ as an $\mathcal{O}_{|L|}$-module.

  * •

for $G=\mathrm{Sp}_{2n}$, and $a\in\mathcal{A}$, we have a spectral cover
$p_{a}:Y_{a}\to X$ embedded in the total space $|L|$ of $L$. If
$E_{a}=p_{a*}\mathcal{O}_{Y_{a}}$ then we have a canonical symplectic form
$\wedge^{2}E_{a}\to L^{\otimes(1-2n)}$. If $L^{\prime}$ is a square root of
$L$ then $E^{\prime}_{a}=E_{a}\otimes{L^{\prime}}^{\otimes 1-2n}$ will be
equipped with a canonical symplectic form with value in $\mathcal{O}_{X}$ and
also equipped with a Higgs fields derived from the the structure of
$\mathcal{O}_{Y_{a}}$ as a $\mathcal{O}_{|L|}$-module.

  * •

for $G=\mathrm{SO}_{2n+1}$, and $a\in\mathcal{A}$, we have a spectral cover
$p_{a}:Y_{a}\to X$ embedded in the total space $|L|$ of $L$. If
$E_{a}=p_{a*}\mathcal{O}_{Y_{a}}$ then we have a canonical non-degenerate
symmetric form $S^{2}E_{a}\to L^{\otimes(-2n)}$ so that the vector bundle
$E^{\prime}_{a}=E_{a}\otimes L^{\otimes n}$ affords a canonical no-degenerate
symmetric form with value in $\mathcal{O}_{X}$, and also equipped with a Higgs
fields derived from the the structure of $\mathcal{O}_{Y_{a}}$ as a
$\mathcal{O}_{|L|}$-module. It also affords a trivialization of the
determinant depending on the choice of a square root of $L$.

  * •

for $G=\mathrm{SO}_{2n}$, and $a\in\mathcal{A}$, we have a spectral cover
$p_{a}:Y_{a}\to X$ embedded in the total space $|L|$ of $L$. Using the
normalization of the universal spectral cover, we obtain a partial
normalization $\tilde{Y}_{a}$ of $Y_{a}$. If
$E_{a}=p_{a*}\mathcal{O}_{\tilde{Y}_{a}}$ then we have a canonical non-
degenerate symmetric form $S^{2}E_{a}\to L^{\otimes(2-2n)}$ so that the vector
bundle $E^{\prime}_{a}=E_{a}\otimes L^{\otimes 1-n}$ affords a canonical non-
degenerate symmetric form with values in $\mathcal{O}_{X}$, and also equipped
with a Higgs fields derived from the the structure of $\mathcal{O}_{Y_{a}}$ as
a $\mathcal{O}_{|L|}$-module. It also affords a canonical trivialization of
the determinant depending on the choice of a square root of $L$.

  * •

for $G=G_{2}$, and $a\in\mathcal{A}$, we have a spectral cover $p_{a}:Y_{a}\to
X$ embedded in the total space $|L|$ of $L$. If
$E_{a}=p_{a*}\mathcal{O}_{\tilde{Y}_{a}}$ then we have a canonical non-
degenerate 3-form $\wedge^{3}E_{a}\to L^{-9}$ so that the vector bundle
$E^{\prime}_{a}=E_{a}\otimes L^{\otimes 3}$ affords a canonical non-degenerate
3-form with value in $\mathcal{O}_{X}$, and also equipped with a Higgs fields
derived from the the structure of $\mathcal{O}_{Y_{a}}$ as a
$\mathcal{O}_{|L|}$-module. It also affords a canonical trivialization of the
determinant depending on the choice of a square root of $L$.

##  Appendix A Computer algebra code and $G_{2}$ computations

In this appendix, we give the computer code used to compute the 3-form $\rho$
in Section 3.5.

###  A.1 Construction of $\rho$

To construct $\rho$, we will use the connection between nondegenerate
alternating 3-forms and cross products. Let $V$ be a vector space with a
nondegenerate, symmetric bilinear form $\nu$.

######  Definition A.1.

A _cross product_ on $(V,\nu)$ is a bilinear map

$c\colon V\otimes V\to V$

satisfying the following three properties for all $v_{1},v_{2}\in V$:

  1. 1.

(Skew symmetry) $c(v_{1},v_{2})=-c(v_{2},v_{1})$;

  2. 2.

(Orthogonality) $\nu(c(v_{1},v_{2}),v_{1})=0$;

  3. 3.

(Normalization)
$\displaystyle\nu\big{(}c(v_{1},v_{2}),c(v_{1},v_{2})\big{)}=\det\begin{pmatrix}\nu(v_{1},v_{1})&\nu(v_{1},v_{2})\\\
\nu(v_{1},v_{2})&\nu(v_{2},v_{2})\end{pmatrix}$

The data of a cross product on $(V,\nu)$ is equivalent to the data of a
nondegenerate 3-form on $V$ whose associated symmetric bilinear form (see
equation (1)) is a scalar multiple of $\nu$. Indeed, to a cross product $c$,
one associates the 3-form

$\rho(v_{1},v_{2},v_{3})=\nu\big{(}c(v_{1},v_{2}),v_{3}\big{)}$ (8)

while for any non-degenerate 3-form $\rho$, there is a unique cross product
$c$ satisfying equation (8).

Now, consider the free, rank 7 $A$-module $B$ as in Section 3.5 equipped with
the symmetric, nondegenerate form $\omega$ defined by the formula

$\omega(g_{1},g_{2})={\rm
tr}_{B/A}\left(\frac{g_{1}\tau(g_{2})}{f^{\prime}}\right)$

as in the $\mathrm{SO}_{7}$ case. Here, $\tau(x)=-x$ is the natural involution
on $B$, and the trace is taken after inverting $f^{\prime}$ in $A$. To
construct a 3-form on $B$ which is nondegenerate over every $k$ point of $A$,
it suffices to construct a cross product

$c\colon B\otimes_{A}B\to B$

for $(B,\omega)$. Moreover, the equation

$\rho(xg_{1},g_{2},g_{3})+\rho(g_{1},xg_{2},g_{3})+\rho(g_{1},g_{2},xg_{3})=0$

is equivalent to the condition

$c(xg_{1},g_{2})+c(g_{1},xg_{2})=xc(g_{1},g_{2}).$ (9)

To simplify computations further, we note that any form $c\colon
B\otimes_{A}B\to B$ satisfying the conditions of Definition A.1 and equation
(9) can be recovered from its trace:

$tc\colon B\otimes_{A}B\to A,\quad(g_{1},g_{2})\mapsto{\rm
tr}_{B/A}\big{(}c(g_{1},g_{2})\big{)}$

Indeed, if we express

$c(x^{i},x^{j})=\sum_{l=0}^{6}c_{i,j}^{(l)}x^{l}$

then $c_{i,j}^{(6)}=tc(x^{i},x^{j})$ and

$tr_{B/A}\big{(}x^{l}c(x^{i},x^{j})\big{)}=\sum_{r=0}^{l}\binom{l}{r}tc(x^{i+r},x^{j+l-r})$

can be expressed in terms of $c_{i,j}^{(m)}$ for $6-l\leq m\leq 6$. This
allows us to recover the coefficients $c_{i,j}^{(l)}$ by downward induction on
$l$.

This idea is implemented in the following Macaulay2 code. There is a one-
dimensional solution space, which is specialized at a particular point to give
the form stated in equation (6). Note that it is immediate from the computer
calculation that the form $\rho$ is valued in $B$ and satisfies the conclusion
of Proposition 3.5.

    ΨS=QQ[e,q];
    ΨF=frac(S);
    ΨR=F[p_(0,0) .. p_(6,6)]; -- ring with p_(i,j)=tc(x^i,x^j),
    Ψ0\leq i,j\leq 6
    Ψ
    Ψ-- The following three commands define tc(x^i,x^j) for i or j between
    Ψ7 and 12 using the relation x^7-e*x^5+e^4/4*x^3+q*x=0.
    Ψfor l from 0 to 5 do [for k from 0 to 6 do p_(k,7+l)=e*p_(k,5+l)-
    Ψ(1/4)*e^2*p_(k,3+l)-q*p_(k,1+l)];
    Ψfor l from 0 to 5 do [for k from 0 to 6 do p_(7+l,k)=e*p_(5+l,k)-
    Ψ(1/4)*e^2*p_(3+l,k)-q*p_(1+l,k)];
    Ψfor l from 0 to 5 do [for k from 7 to 12 do p_(k,7+l)=e*p_(k,5+l)-
    Ψ(1/4)*e^2*p_(k,3+l)-q*p_(k,1+l)];
    Ψ
    Ψ-- I encodes orthogonality:
    ΨI = ideal(flatten for a from 0 to 6 list for k from 0 to 6 list
    Ψsum(0..k,j->binomial(k,j)*p_(k+j,a+k-j)));
    Ψ
    Ψ-- J encodes skew symmetry:
    ΨJ = ideal( flatten for a from 0 to 6 list for b from 0 to 6 list
    Ψp_(a,b)+p_(b,a) );
    Ψ
    Ψ-- The following encodes the normalization condition:
    ΨB=R[x]/(x^7-e*x^5+(1/4)*e^2*x^3+q*x);
    Ψ-- determinant of norms of x^i,x^j:
    Ψf = (i,j) -> coefficient(x^6,(-1)^i*x^(2*i))*coefficient(x^6,(-1)^j*
    Ψx^(2*j))-coefficient(x^6,(-1)^j*x^(i+j))*coefficient(x^6,(-1)^j*
    Ψx^(i+j));
    Ψ-- norm of c(x^i,x^j):
    Ψg = (i,j) -> coefficient(x^6, (p_(i,j)*(x^6-e*x^4+(1/4)*e^2*x^2+q)+
    Ψsum(0..1,l->binomial(1,l)*p_(i+l,j+1-l))*(x^5-e*x^3+(1/4)*e^2*x)+
    Ψsum(0..2,l->binomial(2,l)*p_(i+l,j+2-l))*(x^4-e*x^2+(1/4)*e^2)+
    Ψsum(0..3,l->binomial(3,l)*p_(i+l,j+3-l))*(x^3-e*x)+sum(0..4,l->
    Ψbinomial(4,l)*p_(i+l,j+4-l))*(x^2-e)+sum(0..5,l->binomial(5,l)*
    Ψp_(i+l,j+5-l))*(x)+sum(0..6,l->binomial(6,l)*p_(i+l,j+6-l)))
    Ψ*(p_(i,j)*((-x)^6-e*(-x)^4+(1/4)*e^2*(-x)^2+q)+sum(0..1,l->
    Ψbinomial(1,l)*p_(i+l,j+1-l))*((-x)^5-e*(-x)^3+(1/4)*e^2*(-x))+
    Ψsum(0..2,l->binomial(2,l)*p_(i+l,j+2-l))*((-x)^4-e*(-x)^2+(1/4)*
    Ψe^2)+sum(0..3,l->binomial(3,l)*p_(i+l,j+3-l))*((-x)^3-e*(-x))+
    Ψsum(0..4,l->binomial(4,l)*p_(i+l,j+4-l))*((-x)^2-e)+sum(0..5,l->
    Ψbinomial(5,l)*p_(i+l,j+5-l))*(-x)+sum(0..6,l->binomial(6,l)*
    Ψp_(i+l,j+6-l))) );
    Ψ-- K encodes the normalization condition:
    ΨK = ideal(flatten for i from 0 to 6 list for j from 0 to 6 list
    Ψf(i,j)-g(i,j));
    Ψ
    ΨQ=R/(I+J+K); -- imposing the relations on our ring of variables
    ΨQ2=Q/ideal(p_(6,3)-1,p_(6,4),p_(6,5)-5*e/2); -- specializes to our
    Ψparticular form rho
    Ψ
    Ψ-- Computation of c from tc:
    ΨP=Q2[x]/(x^7-e*x^5+e^2/4*x^3+q);
    ΨC=table(for k from 0 to 6 list k, for k from 0 to 6 list k, (i,j) ->
    Ψ(p_(i,j)*(x^6-e*x^4+(1/4)*e^2*x^2+q)+sum(0..1,l->binomial(1,l)*
    Ψp_(i+l,j+1-l))*(x^5-e*x^3+(1/4)*e^2*x)+sum(0..2,l->binomial(2,l)*
    Ψp_(i+l,j+2-l))*(x^4-e*x^2+(1/4)*e^2)+sum(0..3,l->binomial(3,l)*
    Ψp_(i+l,j+3-l))*(x^3-e*x)+sum(0..4,l->binomial(4,l)*p_(i+l,j+4-l))*
    Ψ(x^2-e)+sum(0..5,l->binomial(5,l)*p_(i+l,j+5-l))*(x)+sum(0..6,l->
    Ψbinomial(6,l)*p_(i+l,j+6-l))));
    Ψ-- This is the matrix for c with respect to the basis x^i, i=0,..,6
    ΨnetList C -- displays C

###  A.2 Nondegeneracy of $\rho$

Let $\rho$ be the form computed in the previous section, stated explicitly in
equation (6). Note that since we specialized to a particular form in the
previous section, it is not yet clear that this form is nondegenerate. For
this, we produce the following code in Macaulay2 to explicitly compute the
associated bilinear form is as in Proposition 3.4. The following uses some
basic operations on permutations from the package SpechtModule authored by
Jonathan Niño in Macaulay2.

    ΨT=permutations {0,1,2,3,4,5,6};
    Ψn = (v,w) -> sum(0..7!-1, k-> permutationSign(T_k)*coefficient(x^6,
    Ψv*(C_((T_k)_0))_((T_k)_1))*coefficient(x^6,w*(C_((T_k)_2))_((T_k)_3))
    Ψ*coefficient(x^6,(-x)^((T_k)_4)*(C_((T_k)_5))_((T_k)_6))  );
    ΨS=table(for k from 0 to 6 list k, for k from 0 to 6 list k, (i,j) ->
    Ψn((-x)^i,(-x)^j);
    ΨnetList S

## References

  * [AFV18] Jeffrey D Adler, Jessica Fintzen, and Sandeep Varma, _On kostant sections and topological nilpotence_ , Journal of the London Mathematical Society 97 (2018), no. 2, 325–351.
  * [BČ22] Alexis Bouthier and Kęstutis Česnavičius, _Torsors on loop groups and the hitchin fibration_ , Ann. Sci. École Norm. Sup 55 (2022), 791–864.
  * [Eng00] Friedrich Engel, _Ein neues, dem linearen komplexe analoges gebilde_ , 1900.
  * [Hit87] Nigel Hitchin, _Stable bundles and integrable systems_ , Duke mathematical journal 54 (1987), no. 1, 91–114.
  * [Hit00] , _The geometry of three-forms in six dimensions_ , Journal of Differential Geometry 55 (2000), no. 3, 547–576.
  * [Hit07] , _Langlands duality and $g_{2}$ spectral curves_, Quarterly Journal of Mathematics 58 (2007), no. 3, 319–344.
  * [KL88] David Kazhdan and George Lusztig, _Fixed point varieties on affine flag manifolds_ , Israel Journal of Mathematics 62 (1988), no. 2, 129–168.
  * [Kos63] Bertram Kostant, _Lie group representations on polynomial rings_ , American Journal of Mathematics 85 (1963), no. 3, 327–404.
  * [Ngo10] B. C. Ngo, _Le lemme fondamental pour les algebres de lie_ , Publications mathématiques de l’IHÉS 111 (2010), no. 1, 1–169.
  * [Ser13] Jean-Pierre Serre, _Local fields_ , vol. 67, Springer Science & Business Media, 2013.
  * [Yun16] Zhiwei Yun, _Lectures on springer theories and orbital integrals_ , arXiv preprint arXiv:1602.01451 (2016).