Let $J_G|_\fc = J_G\times_{\fc_G}\fc$. Composing (<ref>) with the closed embedding $J\hookrightarrow J_G|_\fc$ gives a locally closed embedding

\[
J\hookrightarrow \mathrm{Res}^{\canft}_{\fc_G}(\canT\times \canft)^W|_\fc = \mathrm{Res}^{\canft\times_{\fc_G}\fc}_{\fc_G}\big(\canT\times (\canft\times_{\fc_G}\fc)\big)^W
\]

We describe the image of this embedding in the following theorem. First, we introduce some notation. Recall that we have an almost direct sum decomposition $\canT  = C_1\canA_1$. For each $\nu\in W/W_X$, we also have an abelian subgroup scheme $C_\nu = \nu(C_1)$. As $C_1$ is $W_X$ stable, this is a well-defined subgroup of $\canT$. We have the following description of regular centralizers.

The closure of the image of the locally closed embedding

\[
J\to J_G|_\fc\to \mathrm{Res}^{\fc\times_{\fc_G}\canft}_{\fc}\big(\canT\times (\fc\times_{\fc_G}\canft)\big)^{W}
\]

is the closure of the subgroup scheme

\[
\left(\coprod_{\nu\in W/W_\fa} \mathrm{Res}^{\canfa_\nu^\reg}_{\fc^\reg}(C_\nu\times \canfa_\nu^{\reg})\right)^{W}
\]

in $\mathrm{Res}^{\fc\times_{\fc_G}\canft}_{\fc}\big(\canT\times (\fc\times_{\fc_G}\canft)\big)^{W}$, which we denote by $\cC$. Moreover, the open embedding $J\to \cC$ is an isomorphism when $G$ has derived group simply connected.

It is a direct computation that the image of $J$ in $\mathrm{Res}^{\fc\times_{\fc_G}\canft}_{\fc}\big(\canT\times (\fc\times_{\fc_G}\canft)\big)^{W}$ over the regular semisimple locus is exactly

\[
\left(\coprod_{\nu\in W/W_\fa} \mathrm{Res}^{\canfa_\nu^\reg}_{\fc^\reg}(C_\nu\times \canfa_\nu^{\reg})\right)^{W}.
\]

and hence the first statement follows. Moreover, we have a Cartesian diagram

\[
\xymatrix{
J\ar@{^{(}->}[d]\ar@{^{(}->}[r] & \cC\ar@{^{(}->}[d] \\
J_G|_{\fc}\ar@{^{(}->}[r] & \mathrm{Res}^{\fc\times_{\fc_G}\canft}_{\fc}\big(\canT\times (\fc\times_{\fc_G}\canft)\big)^{W}
\]

When $G$ has derived group simply connected, the lower morphism is an isomorphism, and so therefore is the map $J\to \cC$.

Note that although

\[
\left(\coprod_{\nu\in W/W_\fa} \mathrm{Res}^{\canfa_\nu^\reg}_{\fc^\reg}(C_\nu\times \canfa_\nu^{\reg})\right)^{W} \simeq  \mathrm{Res}^{\canfa^\reg}_{\fc^\reg}(C_1\times \canfa^{\reg})^{W_X},
\]

it is not true that $\cC$ is isomorphic to $\mathrm{Res}^{\canfa}_{\fc}(C_1\times \canfa)^{W_X}$.

The error in the argument of <cit.> was to assume that the map

\[
\mathrm{Res}^{\fc\times_{\fc_G}\canft}_\fc(\canT\times \canft)^W\to \mathrm{Res}^{\fa}_\fc(\canT\times \canft)^{W_\fa}
\]

is a closed embedding. Indeed, in example <ref> the source has unipotent part over zero which lies in the kernel of the morphism to the latter, which has only a multiplicative group over zero.

§.§ Application to the Cokernel Group Scheme $J_A$

We continue the notation of the previous section, with $\canfa = \mathrm{Lie}(\canA)$ the Lie algebra of the canonical torus. Recall that for quasisplit symmetric spaces, the natural map $\fc\to \fc_G$ is an embedding, and we have an inclusion $J\to J_G|_\fc$. Let us denote the cokernel of this inclusion by $J_A$. The goal of this section, is to give a Galois description for $J_A$ using the finite flat cover $\canfa\to \fc$.

Indeed, this idea is not new to this paper:  While Knop did not use the construction of regular centralizers in <cit.>, he did consider the Weil restriction $J_A^1:=\mathrm{Res}^\canfa_\fc(\canA\times \canfa)^{W_\fa}$. The key result of loc. cit. was providing an action of the fiberwise neutral component of $J_A^1$, which we denote by $J_A^0$, on the cotangent bundle $T^*X$ over $\fc$ for $X$ a $G$-variety. He used this action to characterize the group of $G$-equivariant automorphisms of $X$.

In fact, we will show that Knop's groups $J_A^0$ and $J_A^1$ are closely related to the cokernel $J_A$ in the case where $X$ is a symmetric space. We prove the following comparison.

There exist open embeddings

\[
J_A^0\hookrightarrow J_A\hookrightarrow J_A^1.
\]

When $G$ has derived group simply connected, $J_A\simeq J_A^1$.

We first construct a morphism $\mathrm{Res}^{\fc\times_{\fc_G}\canft}_\fc\big(\canT\times(\fc\times_{\fc_G}\canft)\big)^W\to J_A^1$. Note that there is one such morphism by using the projection $\canT\to \canA$, but $J$ does not lie in the kernel of this map. Instead, we consider a map defined as follows:  Let $\nu\in W/W_X$ and let $\ol{\nu}\in W$ be a lift of $\nu$. On the component $\canfa_\nu\subset \canft\times_{\fc_G}\fc$, we define a map

\[
\mathrm{Res}^{\canfa_\nu}_{\fc}(\canT\times \canfa_\nu)\to \mathrm{Res}^{\canfa_\nu}_{\fc}(\canA\times \canfa_\nu)
\]

induced by the composition $\canT\xrightarrow{\ol{\nu}}\canT\to \canA$.
On $W_X$-invariant parts, this is independant of the choice of lift $\ol{\nu}$, and so we obtain a well-defined arrow

\begin{equation}
\label{eqn: map of weil restrictions on components}
\mathrm{Res}^{\canfa_\nu}_{\fc}(\canT\times \canfa_\nu)^{W_X}\to \mathrm{Res}^{\canfa_\nu}_{\fc}(\canA\times \canfa_\nu)^{W_X}
\end{equation}

depending only on $\nu\in W/W_X$. Now, for any point

\[
\alpha\colon S\times_{\fc_G}\canft\to \canT
\]

we have well-defined $W_X$-equivariant maps $\alpha_\nu\colon S\times_{\fc}\canfa_\nu\to \canA$ defined by (<ref>). The maps $\alpha_\nu$ differ only by the action of $W$, and hence, identifying $\canfa_\nu\simeq \canfa$, we obtain a well-defined $W_X$- equivariant morphism

\[
S\times_\fc\canfa\to \canA.
\]

Hence we have constructed a map

\[
\xi\colon \mathrm{Res}^{\fc\times_{\fc_G}\canft}_\fc\big(\canT\times(\fc\times_{\fc_G}\canft)\big)\to J_A^1
\]

This map is surjective by the surjectivity of the map $\canT\to \canA$. We claim that this map has kernel $\cC$, where the notation comes from Theorem <ref>. Indeed, an $S$ point

\[
S\times_\fc (\fc\times_{\fc_G}\ft)\to \canT
\]

of the Weyl restriction $\mathrm{Res}^{\fc\times_{\fc_G}\canft}_\fc(\canT\times (\fc\times_{\fc_G}\canft))^W$ is in this kernel if and only if its restriction to each component $\fa_\nu\subset \fc\times_{\fc_G}\ft$ for $\nu\in W/W_\fa$ lies in constant group scheme $C_\nu\times \canfa_\nu$. This is exactly the condition for this $S$ point to lie in the subgroup scheme $\cC$.

Now, we see that the composition

\[
J_G|_{\fc}\to \mathrm{Res}^{\fc\times_{\fc_G}\canft}_\fc(\canT\times (\fc\times_{\fc_G}\canft))^W\to J_A^1
\]

has kernel given by the preimage of $\cC$ in $J_G|_{\fc}$, and the image is an open embedding. It is easy to see that this preimage is exactly $J$ over the regular semisimple locus. As $J$ is a closed subgroup of $J_G|_\fc$ which is conatined in this kernel, and as $J$ is equal to the kernal on a dense open subset of $\fc$, we conclude that $J$ is equal to the kernel of the above map.

Therefore, we have produced an embedding $J_A\to J_A^1$. It is clear that this map has open image. For the statement on $J_A^0$, we note that by <cit.>, $J_A^0$ is the unique minimal open subgroup scheme of $J_A^1$, and so there is a natural open embedding $J_A^0\to J_A$.

When the map $J_G|_\fc\to \mathrm{Res}^{\fc\times_{\fc_G}\canft}_\fc(\canT\times (\fc\times_{\fc_G}\canft))^W$ is an isomorphism, the map $J\to J_A^1$ is surjective and hence is an isomorphism.

One can ask exactly which subgroup $J_A$ is inside of $J_A^1$.
We state and prove such a description below.

For any restricted root $\alpha\in \Phi_r$, define the restricted root hyperplane $\fh_\alpha$ to be the fixed points of the reflection $s_\alpha$ on $\canfa$. Define the open subgroup $J_A'\subset J_A^1$ to be the subscheme of $J_A^1$ consisting of $S$-points $x$ for which the composition

\[
S\times_\fc \fh_\alpha \hookrightarrow S\times_{\fc}\canfa\xrightarrow{x} \canA\xrightarrow{\alpha} \bG_m
\]

avoids $-1$. Then, the embedding $J_A\hookrightarrow J_A^1$ of Theorem <ref> identifies $J_A$ with $J_A'$.

By Lemma <ref>, the image of $J_G|_\fc$ in $J_A^1$ consists of $S$-points $x$ which have a lift $\tilde{x}$ satisfying

\[
S\times_{\fc_G}\fh_{G,\alpha}\to S\times_{\fc_G} \canft\to \canT\xrightarrow{\alpha}\bG_m
\]

avoids $-1$ for all roots $\alpha\in \Phi$. A preimage $\wt{x}$ of $x$ has this property if and only if, on each component $\canfa_\nu$, the restriction

\[
S\times_\fc \fh_{\alpha,\nu}\to S\times_\fc\canfa_\nu\to \canA\xrightarrow{\alpha}\bG_m
\]

avoids $-1$, where $\fh_{\alpha,nu}\subset \canfa_\nu$ is the fixed points of $s_\alpha$ acting on $\canfa_\nu$. It is clear that a lift exists if and only if $x$ satisfies the same property on $\canfa$.

In <cit.>, it was shown that the group scheme appearing in Lemma <ref> is the largest subgroup of $J_A^1$ acting on the cortangent bundle $T^*X$ over $\fc$.

One may note that the dual group $G_X^\vee$ is constructed with maximal torus the dual of the canonical torus of $X$. Hence, Lemma <ref> applied to the group $G_X^\vee$ also sandwiches the regular centralizers for the dual group between Weil restrictions along the finite flat map $\canfa\to \fc$. In fact, one will find that the group scheme constructed in this way is dual in some sense to $J_A$. This duality will be made precise and explored in greater detail in our forthcoming work on relative duality in Hitchin systems.

§.§ Constrution of the Regular Section

In this section, we use our description of regular centralizers to construct a section of the map $\fp^\reg/H\to \fp^\reg\myfatslash H$. To begin, using Theorem <ref>, we choose sections $\sigma_j\colon \fc\xrightarrow{\sim} \mathcal{S}_j\subset \fp^\reg$ of the Chevalley quotient map, with exactly one section $\mathcal{S}_j$ intersecting each regular, nilpotent $H$-orbit in $\cN_{\fp}^\reg$. Together, these sections define a morphism

\begin{equation}
\label{eqn: coprod section}
\sigma = \coprod \sigma_j\colon\coprod_{\cN_\fp^\reg/H}\fc\to \fp^\reg\to \fp^\reg/H
\end{equation}

where the final map is by projection.

The morphism $\sigma$ of equation (<ref>) factors through the gluing map

\[
\coprod_{\cN_\fp^\reg/H}\fc\to \fp^\reg\myfatslash H
\]

of Theorem <ref>.

Suppose that the sheets corresponding to $\sigma_i$ and $\sigma_j$ are glued over the open subset $U\subset \fc$. We must show that the images of these two sections are equal in the stack $\fp^\reg/H$. We can assume without loss of generality that the derived subgroup of $G$ is simply connected. Let $\cF$ denote the sheaf on $U$ whose sections correspond to elements $g\in H$ which conjugate $\sigma_1$ to $\sigma_2$. Then, $\cF$ is a torsor under the action of the group scheme $J|_U$. We claim that $\cF$ is trivial and so admits a global section.

In fact, we prove a stronger statement:  Namely, that $J|_U$ admits no nontrivial torsors. Recall that torsors over $J|_U$ are classified by $H^1(U,J)$. From the short exact sequence

\[
1\to J\to J_G|_\fc\to J_A\to 1,
\]

we obtain an exact sequence on cohomology groups

\[
H^0(U,J_G|_\fc)\to H^0(U,J_A)\to H^1(U,J)\to H^1(U,J_G|_\fc).
\]

Since $G$ has derived group simply connected, we have

\[
H^1(U,J_G|_\fc) = H^1(\canft\times_{\fc_G}U,\canT)^W
\]

which is zero by Hilbert's Theorem 90. On the other hand, the subset $U$ is associated to a subroot system of $\Phi_r$ generating a subgroup $W_X'\subset W_X$, with preimage in $\Phi$ generating $W'\subset W$. We have $W_X' = W'\cap W_X$. Then the map $H^0(U,J_G|_\fc)\to H^0(U,J_A)$ is identified with the map

\[
\canT^{W'}\to \canA^{W_X'}.
\]

By Lemma <ref>, a choice of identification $T\simeq \canT$ and maximal split torus $A\subset T$ induces an isogeny of tori $A\to \canA$ intertwining the actions of the respective Weyl groups $W_\fa\simeq W_X$. In particular, for $W_\fa'\subset W_\fa$ the subgroup of $W_\fa$ correspopnding to $W_X'$, we have that

\[
A^{W_\fa'}\subset T^{W'}\simeq \canT^{W'}\to \canA^{W_X'}
\]

is surjective. We deduce that $H^1(U,J) = 0$. In particular, the images of $\sigma_1$ and $\sigma_2$ coincide on $U$, and so $\sigma$ induces the desired section.

§.§ The Parabolic Cover

We now state our results on the parabolic cover. This is a fundamentally different type of cover, modeling on parabolics of $H$ rather than Borels of $G$ satisfying some $\theta$ compatibility condition, but it shares many of the features of the cover used in section <ref>. In particular, it will be reducible.

§.§.§ Relations Between Weyl groups

We begin by discussing some various Weyl groups that will emerge in later sections and the relationships between them. Fix a fixed maximal $\theta$-split torus $T$ containing a maximally $\theta$-split torus $A$. We define $C = C_H(A)$. The connected component $C^\circ$ is a (not necessarily maximal) torus of $H$. Let $W_C = N_H(C^\circ)/Z_H(C^\circ)$ be the Weyl group of $C^\circ$ in $H$.

* There are inclusions $N_H(A)\subset N_H(C)$ and $C_H(A)\subset C_H(C)$.

* There is a canonical map $\xi\colon W_A\to W_C$.

Since the form is quasisplit, Proposition <ref> implies that $C^\circ \subset C_H(A)\subset T$. In particular, $C_H(C^\circ)$ is abelian and since $C^\circ\subset C_H(A)$, we have

\[ C_H(A)\subset C_H(C_H(A))\subset C_H(C). \]

Moreover, it is elementary to see that $N_H(A)\subset N_H(C_H(A)^\circ)$. Since conjugation preserves the identity component of $C_H(A)$, we conclude that $N_H(A)\subset N_H(C)$.

The above implies that there is a well-defined map $\xi\colon W_A\to W_C$.

We will denote the image of the morphism $\xi$ above by $W_{im}\subset W_C$. In all classical cases, $W_{im} = W_C$; that is, $\xi$ is surjective. The author conjectures that this is true in general, though it has not been checked for exceptional types.

Let $W_{\ker}\subset W_A$ be the kernel of the map $\xi\colon W_A\to W_C$ constructed above.

The projection map $\pi\colon \fa\to \fc$ as well as its subcover $\fa\git W_{\ker}\to \fc$ are finite and flat.

By the assumption on the characteristic, the characteristic $p$ of the field does not divide $|W_{\ker}|$ and so $\fa\git W_{\ker}$ is Cohen-Macaulay, see 6.4.6 of <cit.>. Since $\fc$ is regular, both claims follow by Miracle Flatness.

§.§.§ Definition and Geometry of Parabolic Covers

We begin with a preliminary lemma relating borels/parabolics of $H$ to those of $G$.

Any Borel $B$ of $H$ is the intersection $B_G\cap H$ for some $\theta$-stable Borel $B_G$ of $G$. Moreover, suppose we have a Levi $M\subset H$ of $H$ and suppose that there exists a Levi $M_G\subset G$ such that $M_G\cap H = M$. Then, there exists parabolics $Q\subset H$ with Levi $M$ and $Q_G\subset G$ with Levi $M_G$ such that for any parabolic $P$ of $H$ conjugate to $Q$, there exists a parabolic $P_G$ of $G$ conjugate to $Q_G$ such that $P = P_G\cap H$.

We begin with the statement on Borels, first showing that the intersection any $\theta$-stable Borel of $G$ with $H$ is a Borel of $H$. Let $B_G$ be a $\theta$-stable Borel of $G$ and put $B = B_G\cap H$. It is clear that $B$ is solvable; we must show that $H/B$ is projective, or equivalently that $H/B\simeq H\cdot B_G\subset G/B_G$ is a closed embedding. We claim that, in fact, $H\cdot B_G =  (G/B_G)^\theta$. Let $B_G'$ be another $\theta$-stable Borel of $G$, and let $g\in G$ be such that $g\cdot B_G = B_G'$. Applying $\theta$ gives $\theta(g)\cdot B_G = B_G'$, so that $g^{-1}\theta(g)\in N_G(B_G) = B_G$ and further

\[
g^{-1}\theta(g)\in \{b\in B_G\colon \theta(b) = b^{-1}\}.
\]

Recall that the image of the morphism $G\to G$ sending $g\mapsto g^{-1}\theta(g)$ is precisely those $g$ in $G$ such that $\theta(g) = g^{-1}$. As $B_G$ is $\theta$-stable, the image of the morphism $B_G\to B_G$ is therefore the intersection

\[
\{g\in G\colon \theta(g) = g^{-1}\}\cap B_G = \{b\in B_G\colon \theta(b) = b^{-1}\},
\]

and so there exists $b\in B_G$ so that $g^{-1}\theta(g) = b^{-1}\theta(b)$. We have $gb^{-1}\cdot B_G = B_G'$ while also

\[
(gb^{-1})^{-1}\theta(gb^{-1}) = b(g^{-1}\theta(g))\theta(b)^{-1} = b(b^{-1}\theta(b))\theta(b)^{-1} = 1.
\]

We conclude that $B_G$ and $B_G'$ are $G^\theta$ conjugate. Hence, $G^\theta\cdot B_G = (G/B_G)^\theta$ and since the flag variety is invariant under central isogeny, also $H\cdot B_G = (G/B_G)^\theta$. Now, since the fixed locus of an algebraic involution is closed, we conclude that $B_G\cap H$ is Borel.

Since all Borels of $H$ are conjugate under $K$, and since $K$ conjugacy preserves $\theta$-stability, we conclude that every Borel of $H$ is the intersection of a Borel of $G$ with $H$.

Now, suppose we have a Levi $M\subset H$ of $H$ and suppose that there exists a Levi $M_G\subset G$ such that $M_G\cap H = M$. Choose any parabolic $Q_G$ of $G$ with Levi factor $M_G$, and put $Q = Q_G\cap H$. We must show first that $Q$ is a parabolic with Levi $M$. Let $B_G\subset Q_G$ be a Borel in $Q_G$ and $B = B_G\cap H$ the corresponding Borel of $H$. Since we have a commutative diagram

\[
\xymatrix{
H/B\ar[r]\ar[d] & H/Q\ar[d] \\
G/B_G\ar[r] & G/Q_G
\]

with the horizontal arrows being surjective and the left vertical arrow being a closed immersion, it follows that $H/Q$ is a closed subvariety of $G/Q_G$. In particular, $H/Q$ is projective, and $Q$ is a parabolic in $H$. That the Levi factor of $Q$ is $M$ follows from intersecting the decomposition $Q_G = M_G\cdot U_G$ where $U_G$ is the unipotent radical of $Q_G$.

Now, for any parabolic $P$ of $H$ conjugate to $Q$ by $h\in H$, we may take the corresponding conjugate parabolic $h\cdot Q_G$ of $G$. The result follows.

We now introduce the central object of study; the parabolic cameral cover.

To make sense of the definition, we need the following.

For any $x\in \fa^\reg$, $C^\circ = C_H(x)^\circ$ is a (not necessarily maximal) torus of $H$, and $C_\fh(X) = {\mathrm Lie}(C^\circ)$.

By Proposition <ref>, $C_H(x)^\circ\subset C_H(A)$ is a connected abelian subgroup of a maximal torus, so is a torus in $H$. The statement on Lie algebras is Lemma 4.2 of <cit.>.

Define the Levi subgroup $M:=C_H(C^\circ)^\circ$ of $H$ and $M_G:=Z_G(C^\circ)^\circ$ denote the analoguous Levi of $G$.

Let $Q$ and $Q_G$ be parabolics of $H$ and $G$, respectively, satisfying the conditions of Lemma <ref> for the Levis $M\subset M_G$.

Note that $M$, $M_G$ are well defined up to conjugation by $H$.  The parabolics $Q$ and $Q_G$ may involve further choice as parabolics with fixed Levi type are not necessarily conjugate.

Fix the data of Definition <ref>. Then, let

\[
\wt{\fp}^{\reg} = \left\{(X,P)\colon \begin{array}{c} X\in \fp^{\reg},\; {\mathrm Lie}(P)\supset C_\fk(C_\fk(X))\text{ and }P\text{ is a}\\ \text{parabolic of $H$ which is $H$ conjugate to $Q$}\end{array}\right\}
\]

and denote by $\pi_\fp\colon \wt{\fp}^{\reg}\to \fp^{\reg}$ the projection map. We denote by $\tilde{\fp}^{\mathrm rs} = \tilde{\fp}^\reg|_{\fp^{\mathrm rs}}$.

Let $W_{H,M} = N_H(M)/M$ denote the relative Weyl group of $M$. There is a map $\wt{\fp}^{\mathrm rs}\to \fa\git W_{\ker}\times_{W_{\fa}/W_{\ker}}W_{H,M} = \fa\times_{W_\fa}W_{H,M}$ fitting into a Cartesian diagram

\[
\begin{tikzcd}
\tilde{\fp}^{\mathrm rs} \arrow{rr} \arrow{d} & & \fa\times_{W_{\fa}}W_{H,M}\arrow{d}\\
\fp^{\mathrm rs} \arrow{r} & \fp\git H \arrow{r}{\cong} & \fa\git W_{a}
\end{tikzcd}
\]

Let $\wt{\fa}^{\reg} = \pi_\fp^{-1}(\fa^\reg)$ be the closed subscheme of $\wt{\fp}^{\reg}$ over the subscheme $\fa^{\reg}\subset \fp^{\mathrm rs}$. Then, we have a projection map $\wt{\fa}^{\reg}\to \fa^{\reg}$.

The relative Weyl group $W_{H,M}$ acts on the set of all parabolics of $H$ with Levi $M$, and the set of such parabolics conjugate to $Q$ is a $W_{H,M}$ torsor under this action.

The map $\fa^{\reg}\times W_{H,M}\to \wt{\fa}^{\reg}$ sending $(X,w)\mapsto (X,w\cdot Q)$ is an isomorphism, and the resulting diagram

\[
\begin{tikzcd}
\tilde{\fa}^{\reg} \arrow{r}{\cong} \arrow{d} & \fa^\reg\times W_{H,M}\arrow{r} & \fa\times_{W_{\fa}}W_{H,M}\arrow{d}\\
\fa^{\reg} \arrow{r} & \fp\git H \arrow{r}{\cong} & \fa\git W_{a}
\end{tikzcd}
\]

with the top right arrow given by quotienting by the diagonal action of $W_\fa$, is commutative and Cartesian.

Recall from Lemma <ref> that $H\cdot \fa^\reg = \fp^{\mathrm rs}$. The $H$-orbit $H\cdot(\wt{\fa}^{reg})\to \fp^{\mathrm rs}$ is therefore a surjective $W_{H,M}$ cover. Since there is an inclusion of $W_{H,M}$ covers

\[ H\cdot(\wt{\fa}^{reg})\hookrightarrow \wt{\fp}^{\mathrm rss} \]

we conclude that this map is an isomorphism. In particular, since the map $\wt{\fa}^\reg\to \fa\times_{W_\fa}W_{H,M}$ is $Z_H(\fa^\reg)$ equivariant, it extends to a map

\[ p^{\mathrm rs}\colon \wt{\fp}^{\mathrm rs}\to \fa\times_{W_{\fa}}W_{H,M} \]

by taking

\[ p^{\mathrm rs}(h\cdot \gamma) = p^{\mathrm rs}(\gamma). \]

This map is $H$ invariant, and hence the diagram

\[
\begin{tikzcd}
\tilde{\fp}^{\mathrm rs} \arrow{rr} \arrow{d} & & \fa\times_{W_{\fa}}W_{H,M}\arrow{d}\\
\fp^{\mathrm rs} \arrow{r} & \fp\git H \arrow{r}{\cong} & \fa\git W_{a}
\end{tikzcd}
\]

is a Cartesian square.

We now seek to extend Proposition <ref> over the regular locus. For this, we will need Lemma <ref>, which demonstrates the structure of the parabolic cover over the regular nilpotent locus. We begin with some preliminary lemmas.

First, let us set some notation. Fix $e\in \cN_\fp^\reg$. We will be interested in the fiber of $\pi_\fp$ over $e$. Let $B_G$ be the unique Borel in $G$ such that $e\in {\mathrm Lie}(B_G)$, and let $T_G\subset B_G$ be the unique maximal torus of $B_G$. Note that $B_G$ is necessarily $\theta$ stable, and hence $T_G$ contains a maximal torus $T_H$ of $H$. Choose $\fa$ so that $C = C_H(\fa)\subset T_H$, and let $S_H$ denote the set of simple roots of $H$ with respect to $T_H$ and the Borel $B_G\cap H$ (which is Borel by the proof of Lemma <ref>). Denote by $V\subset S_H$ the set of simple roots of $H$ which are trivial on $C$.

We furthermore fix pinnings $\{e_\alpha^\fg\}_{\alpha\in \Phi}$ and $\{e_\alpha^\fk\}$ of the groups $G$ and $H$ with respect to the choice of tori $T_G\supset T_H$, respectively.

For all simple quasisplit forms except possibly the quasisplit form on $E_6$, we have:

* The nilpotent

\[
\wt{e} = \sum_{\alpha\in S_G} e_{\alpha}^{\fg}
\]

lies in $\fp$.

* Consider the nilpotent element

\[
e' := \sum_{\alpha\in S_H\setminus V}e_\alpha^{\fk}\in \fh
\]

The nilpotents $\wt{e}\in \fp$ and $e'\in \fh$ commute.

We proceed case-by-case through the classification of quasisplit simple symmetric pairs and some explicit computations. As the definitions of $e,e'$ do not depend on isogeny class or center, we further assume that all pairs are of the form $(G,\theta,K)$ for $G$ simple semisimple.

In the case of any split pair $(G,\theta,K)$, we have that $C = Z(G)\cap K$, and so $V = S_K$. Hence, $e' = 0$ and the result is trivial.

In the case of $(G,\theta,K) = (\SL_{2n},\theta,S(\GL_n\times \GL_n))$ from example <ref>, we have $T_K$ the set of diagonal matrices, and

\[
\wt{e} = \begin{pmatrix}
0 & I_n \\ N_n & 0
\end{pmatrix}\quad \text{where }N_n = \resizebox{1.5in}{.6in}{$\begin{pmatrix}
0 & & & &    \\
1 & 0 & & & \\
& 1 & 0 & &  \\
& & \ddots & \ddots & \\
& & &  1 & 0
\end{pmatrix}$}\text{ is $n\times n$}
\]

On the other hand, $V = \emptyset$ and

\[
e'= \begin{pmatrix}
N_n & \\ & N_n
\end{pmatrix}
\]

It is easy to check $[\wt{e},e'] = 0$.

In the case of $(G,\theta, K) = (\SL_{2n+1},\theta,S(\GL_n\times \GL_{n+1}))$, we have $T_K$ is again the diagonal matrices, and

\[
\wt{e} = \begin{pmatrix}
& \vec{0} & I_n \\
I_n & & \\
\vec{0}^t & & \\
\end{pmatrix}\quad \text{where $\vec{0}$ is the $n\times 1$ zero vector.}
\]

Then, again $V=\emptyset$ and

\[
e' = \begin{pmatrix}
N_n & \\ & N_{n+1}
\end{pmatrix}
\]

It is again an easy check that $[\wt{e},e'] = 0$.

Now consider the case $(G,\theta,K) = (\SO_{2n+2},\theta,SO_n\times\SO_{n+2})$ of example <ref>, with $\theta$ given by

\[
\theta\begin{pmatrix}
A & B \\ -B^t & D
\end{pmatrix} = \begin{pmatrix}
A & -B \\ B^t & D
\end{pmatrix}
\]

In this case, we take

\[
T_G = \left\{ \resizebox{2.4in}{.6in}{ $\begin{pmatrix}
& a_1 &  & & & & & \\
-a_1 & & & & & & & \\
& & & a_2 & & & & \\
& & -a_2 & & & & & \\
& & & &  \ddots & & \\
%        & & & & \ddots & & & \\
& & & & &   & a_{n+1} \\
& & & & &  -a_{n+1} & \\
\end{pmatrix}$ } \colon a_j\in k^\times \right\}
\]

When $n$ is even, this gives $T_K = T_G$ while when $n$ is odd, $T_K$ is the $n-1$ dimensional subtorus given by $a_{(n+1)/2} = 0$. We divide into cases based on the parity of $n$:  If $n$ is even, we get:

\[
\wt{e} = \begin{pmatrix}
0 & B \\ -B^t & 0
\end{pmatrix} \quad \text{where }B = \resizebox{2.5in}{.7in}{$\begin{pmatrix}
i & 1 & i & -1 & & & &&& \\
-1 & i & 1 & i & & & &&& \\
& & i & 1 & i & -1 & &&& \\
& & -1 & i & 1 & i & && &  \\
& &  & \ddots & & \ddots& & \\
& && &  i & 1 & i & -1 & & \\
& & & & -1 & i & 1 & i & & \\
& & && &  & i & 1 & 2i & 0 \\
& & && &  & -1 & i & 2 & 0
\end{pmatrix}$} \text{ is $n\times (n+1)$}
\]

Moreover, $V$ consists of all but two roots of $\SO_{n+1}$, i.e.

\[
e' = \resizebox{1.8in}{.6in}{$\begin{pmatrix}
\mathbf{0}_n & & \\
& \mathbf{0}_{n-2} & && & \\
& & & &0 &  2 \\
& & & &0 &  2i \\
& &0 &0 & &   \\
& &-2 &-2i & &
\end{pmatrix}$ }
\]

So that $[\wt{e},e'] = 0$.

In the case of $n$ odd, we instead get

\[
\wt{e} = \begin{pmatrix}
0 & B \\ -B^t & 0
\end{pmatrix}\quad \text{where }B = \resizebox{2.3in}{.8in}{$\begin{pmatrix}
0&i & 1 & i & -1 & & & &\\
0&-1 & i & 1 & i & & & & \\
&& & i & 1 & i & -1 & & \\
& & & -1 & i & 1 & i & &  \\
&& &  & \ddots & & \ddots\\
&& && &  i & 1 & i & -1  \\
& && & & -1 & i & 1 & i \\
& & &&& &  & 2i & 2
\end{pmatrix}$} \text{is }n\times (n+1)
\]

Again, $V$ consists of all but two roots of $\SO_{n+1}$, and we have

\[
e' = \resizebox{2in}{.65in}{$\begin{pmatrix}
\mathbf{0}_n & &&  \\
& \mathbf{0}_{n-2} & && & & \\
& & & &0 &  2 & 1 \\
& & & &0 &  2i & i \\
& &0 &0 & & &   \\
& &-2 &-2i & & & \\
& & -1 &-i & & &
\end{pmatrix}$}
\]

We can again verify the commuting property.

The only remaining simple, semisimple quasisplit involution is the quasisplit form on $E_6$.

Let $e' = \sum_{\alpha\in S_H\setminus V}e_\alpha^{\fk}\in \cN_\fp$ for $S_H$ a set of simple roots of $H$ and $V\subset S_H$. Then one can choose an associated cocharacter $\lambda'$ (See Definition <ref> and Lemma <ref>.) such that $C_H(\lambda') = C_H(h')$ for

\[
h' = \sum_{\alpha\in S_H\setminus V} \alpha\in \fh
\]

In each $H$ conjugacy class of parabolics $P$ of $H$, there is a unique parabolic $P$ such that $\mathrm{Lie}(P)\supset C_\fh(e')$ and $P$ has Levi factor $C_H(\lambda')$.

First, we note that $h'$ satisfies the $\fsl_2$ relation

\[
[h',e'] = 2e'
\]

Similarly, it is elementary to check that the cocharacter $\lambda' = \prod_{\alpha\in S_H\setminus V}\alpha$ is associated to $e'$, and that $C_H(\lambda') = C_H(h')$.

A parabolic $P$ in each conjugacy class of parabolics of $H$ with Levi factor $L=C_H(\lambda')$ is determined by a choice of polarization of the root system generated by $S_H\setminus V$. In particular, we note that for the choice of polarization agreeing with the polarization determining $S_H\subset \Phi_H$, we have $\mathrm{Lie}(P)\supset C_\fh(e')$ and this $P$ is unique with this property.

We keep notation as in Definition <ref>. Let $G$ satisfy condition $(*)$. For any regular, nilpotent element $e\in \cN_\fp^\reg$, there exists a parabolic $P$ of $H$ so that $P$ is conjugate to $Q$ and $\mathrm{Lie}(P)\supset C_\fh(C_\fh(e))$. Moreover, there are finitely many parabolics $P$ satisfying these properties.

We may assume that $G$ is simple of type not $E_6$. Moreover, as the fibers of $\pi_\fp$ are invariant with respect to the $H$ action, we may reduce to the case $e = \wt{e}$ in the notation of Lemma <ref>. For such elements, we have $e'\in C_\fh(\wt{e})$, and so $C_\fh(e')\supset C_\fh(C_\fh(\wt{e}))$.
Now, by Lemma <ref> applied for $H$, there is a unique parabolic $P$ of $H$ containing $C_\fh(e')$ with the Levi factor $C_\fh(\lambda')$. But we can compute

\[
C_\fh(\lambda') = C_\fh\left( \sum_{\alpha\in S_H\setminus V}\alpha \right) = C_\fh(C^\circ) = M.
\]

So the existence is proved.

For the finiteness condition, we note that we can also apply the above argument to show that there exist parabolics $P_G$ of $G$ which are $H$ conjugate to $Q_G$ and which contain $C_\fg(C_\fh(e))$. By Lemma <ref>, each parabolic $P$ of $H$ is obtained by intersecting one of these $P_G$ with $H$. In particular, the number of parabolics $P$ of $H$ satisfying the conditions of the Lemma is bounded by the number of parabolics $P_G$ of $G$ satisfying the conditions of the Lemma. Since $e\in \mathrm{Lie}(P_G)$ for any such $P_G$, it follows that each $P_G$ contains the unique Borel $B_G$ whose Lie algebra contains $e$. As there are finitely many parabolics of fixed type extending a Borel, finiteness follows.

The map $\wt{\fp}^\reg\to \fp^\reg$ is quasifinite, and there is an action of $H$ on $\wt{\fp}^\reg$ such that the GIT quotient

\[
\wt{\fc}:=\wt{\fp}^\reg\git H
\]

fits into a Cartesian diagram

\[
\xymatrix{
\wt{\fp}^\reg\ar[d]\ar[r] & \wt{\fc}\ar[d]\\
\fp^\reg \ar[r] & \fc
\]

and the map $\wt{\fc}\to \fc$ is finite with the components of $\wt{\fc}$ indexed by $W_{H,M}/W_C$, each isomorphic to the finite flat cover $\fa\git W_{\ker}\to \fc$.

We first prove the map $\wt{\fp}^\reg\to \fp^\reg$ is quasifinite. Let $x\in \fp^\reg$ and consider the Jordan decomposition $x = s+n$ as in Lemma <ref>. Let $L$ be the distinguished Levi $G_s^\circ$. Then, there exists a minimal parabolic $P_L\subset L$ such that $P_L$ is conjugate to a fixed parabolic $Q_L\supset C_{H_L}(C)^\circ$ by Corollary <ref>. In particular, the Levi factor for $P_L$ has

\[
\mathrm{Lie}(M_L) = C_{\fh_L}(C_{\fh_L}(n)) = C_{\fh_L}\big( C_\fh(s)\cap C_\fh(n) \big) = C_{\fh_L}(C_\fh(x)) \subset C_\fh(C_\fh(x)) = \mathrm{Lie}(M)
\]

and since there exist finitely many parabolics $P$ conjugate to $Q$ extending the parabolic $P_L$ we conclude the fiber over $x$ is finite.

Observe also that $\wt{\fp}^\reg\to \fp^\reg$ factors through

\[
\wt{\fp}^\reg\subset \fp^\reg\times H/Q\to \fp^\reg
\]

and so is projective. In particular, we also deduce that the map $\wt{\fp}^\reg\to \fp^\reg$ is finite. In particular, we deduce a unique extension of the morphism

\[
(\fa^\reg\times_{W_\fa}W_{H,M})\times_{\fa\git W_\fa}\fp^{rs}\to \wt{\fp}^{rs}
\]

constructed in Proposition <ref> to a surjective map

\[
\xi\colon (\fa\times_{W_\fa}W_{H,M})\times_{\fa\git W_\fa}\fp^{\reg}\to \wt{\fp}^{\reg}
\]

By Zariski's Main Theorem, this map induces isomorphisms on each irreducible component

\[
(\fa\times_{W_\fa}W_C)\times_{\fa\git W_\fa}\fp^\reg = (\fa\git W_{\ker})\times_{\fa\git W_\fa} \fp^\reg\simeq \wt{\fp}^\reg_{[w]}
\]

In particular, the map $\xi$ constructed above is equivariant with respect the action of the group $H$, and hence, the GIT quotient $\wt{\fp}^\reg\git H$ is a cover of $\fc$. We will denote $\wt{\fc}:=\wt{\fp}^\reg\git H$. The resulting properties of the map

\[
\wt{\fc}\to \fc
\]

are now immediate.

§.§.§ A Proposed Galois Description of Regular Centralizers.

For the purposes of this section, we restrict to the setting of quasisplit symmetric pairs with connected regular centralizers. Due to the reliance on Lemma <ref>, we exclude the quasisplit form on $E_6$, though we expect the results of this section should extend to this case.

Let $\Par\to \wt{\fp}^\reg$ denote the universal parabolic over $\wt{\fp}^\reg$ whose fiber over a $k$-point $(x,P)$ of $\wt{\fp}^\reg$ is the parabolic $P$. For each parabolic $P$, we have a projection $P\to M_P:=P/U_P$ to its Levi quotient. Let $\LeviM\to \wt{\fp}^\reg$ be the universal Levi and

\[
\pi\colon \Par\to \LeviM
\]

the projection map.

Let $Z(\LeviM)$ denote the subgroup scheme of $\LeviM$ whose fiber over $(x,P)\in \wt{\fp}^\reg$ consists of the center of the corresponding Levi quotient $Z(M_P)$. Then, $Z(\LeviM)\simeq Z(M)\times \wt{\fp}^\reg$ is a constant group scheme over $\wt{\fp}^\reg$.

For any two parabolics $P,P'$ which are $H$ conjugate to $Q$, and any element $h\in H$ conjugating $P' = h \cdot P$, there is an induced map on the quotients

\[
h\colon M_P\to M_{P'}
\]

and hence also on the corresponding centers. The choice of such an $h\in H$ conjugating $P$ to $P'$ is unique only up to right multiplication by elements of $P$. However, for any $g\in P$, the induced map

\[
g\colon M_P\to M_P
\]

is multiplication by an element of $M_P$ and hence the induced map on $Z(M_P)$ is the identity. We conclude that the isomorphism $h\colon Z(M_P)\to Z(M_{P'})$ is canonical, and the lemma follows.

For every $[w]\in W_{H,M}/W_C$, let $C_{[w]}$ denote the embedding $w\cdot C\subset Z(M)$, where $w\cdot C$ is the conjugation of $C$ by any lift $\dot{w}\in N_H(M)$. Recall the decomposition of $\wt{\fp}^\reg$ into components

\[
\wt{\fp}^\reg = \bigcup_{[w]\in W_{H,M}/W_C} \wt{\fp}^\reg_{[w]}.
\]

There is a canonical $W_{H,M}$ equivariant map

\[
\iota\colon \pi_\fp^*I^\reg\to Z(M)\times \wt{\fp}^\reg
\]

which restricts to a map

\[
\iota_{[w]}\colon \pi_\fp^*I^\reg|_{\wt{\fp}^\reg_{[w]}}\to C_{[w]}\times \wt{\fp}^\reg_{[w]}
\]

for every $[w]\in W_{H,M}/W_C$.

We describe the map $\iota$ first. For this, we need the following claim.

Claim. For every $(x,P)\in \wt{\fp}^\reg$, there is an inclusion $\pi_\fp^*I_x^\reg\subset P$.

This follows from the definition of $\wt{\fp}^\reg$ and the assumption that the form be quasi-split.

For $h\in I_x^\reg$ and $(x,P)\in \wt{\fp}^\reg$, we consider its image

\[
h \mod [\mathrm{Lie}(P),\mathrm{Lie}(P)]\in M_P=\LeviM_{(x,P)} .
\]

This defines a map $\iota\colon \pi_\fp^*I^\reg\to \LeviL$. Note that when $h$ is regular semisimple, its image lies in the center $Z(M_P)$ and moreover $h\in Z_{M_P}(h)$, where the latter is identified with $C_{[w]}\subset Z(M)$. Hence, the image of $\iota$ lands in $Z(\LeviM)\simeq Z(M)\times \wt{\fp}^\reg$, while the image of its restriction $\iota_{[w]}$ lies in $C_{[w]}\times \wt{\fp}^\reg_{[w]}$. The $W_{H,M}$ equivariance follows from the construction of $\iota$.

The below is probably wrong!! Restate the map $\iota_\fc$ and state the rest as conjecture.
Let $w,w'\in W_{H,M}/W_C$. We can lift identification $\wt{\fp}^\reg_{[w]}\simeq \wt{\fp}^\reg_{[w']}$ to an isomorphism of the constant group schemes $C_{[w]}\times \wt{\fp}^\reg_{[w]}\simeq C_{[w']}\times \wt{\fp}^\reg_{[w']}$ which is well defined up to action by $W_{C}$. Therefore, we can descend $\iota$ to a morphism

\begin{equation}
\label{eqn: galois map}
\iota_\fc\colon J\to \mathrm{Res}^{\wt{\fc}}_\fc(C\times\wt{\fc}^\reg)^{W_{H,M}}
\end{equation}

We conjecture that $\iota_\fc$ is an open embedding. As it is an isomorphism over the regular, semisimple locus, this is equivalent to checking that $\iota_\fc$ is injective.

To prove that $\iota_\fc$ is an embedding, we will introduce a notion of Levi induction for the parabolic cover. For $x\in \fa$, let $L = G_x^\circ$ be the distinguished Levi introduced in section <ref>. We use a subscript $L$ to denote corresponding objects for the symmetric pair $(L,\theta,H_L)$, e.g. $W_{\fa,L},W_{C,L},$ etc. Note that since $C = C_H(\fa)^\circ\subset L$, we have $C = C_L$ and so it is immediate that $W_{C,L}\subset W_C$. Moreover, since $W_{\fa,L}\subset W_\fa$, we get an inclusion of exact sequences

\[
\xymatrix{
1\ar[r] & W_{\ker,L}\ar[r]\ar@{^{(}->}[d] & W_{\fa,L}\ar[r]\ar@{^{(}->}[d] & W_{C,L}\ar@{^{(}->}[d] \\
1\ar[r] & W_{\ker}\ar[r] & W_{\fa}\ar[r] & W_{C} \\
\]

Moreover, we have an inclusion of relative Weyl groups given as follows.

For $L$ the distinguished Levi associated to $x\in \fa$ as above, there is a canonical inclusion $W_{H_L,M_L}\subset W_{H,M}$.

Let $\mathfrak{z} = \mathfrak{z}(M)$, respectively $\mathfrak{z}_L$ be the Lie algebra of the center of $M$. We have identities

\[
W_{H,M} = N_H(Z(M))/C_H(Z(M)) = N_{H}(\mathfrak{z})/C_H(\mathfrak{z})
\]

and likewise for $W_{H_L,M_L} = N_{H_L}(\mathfrak{z}_L)/C_{H_L}(\mathfrak{z}_L)$. Consider the action of $\mathrm{Ad}(x)$ on $\mathfrak{z}$. There is an eigen-decomposition

\[
\mathfrak{z} = \bigoplus_{\lambda} \mathfrak{z}_\lambda
\]

where $\mathfrak{z}_\lambda$ is the $\lambda$ weight space for the action of $\mathrm{Ad}(x)$ and $\mathfrak{z}_0 = \mathfrak{z}_L$. Suppose that $h\in H_L = H\cap G_x^\circ$ normalizes $\mathfrak{z}_L$. Then, for any $\lambda$ and $y\in \mathfrak{z}_\lambda$,

\[
[x,\mathrm{Ad}(h)\cdot y] = [\mathrm{Ad}(h)\cdot x,\mathrm{Ad}(h)\cdot y] = \mathrm{Ad}(h)\cdot[x,y] = \lambda h\cdot y
\]

where we used in the first equality that $\mathrm{Ad}(h)\cdot x = x$. Therefore, $\mathrm{Ad}(h)$ preserves $\mathfrak{z}$, and we conclude that $N_{H_L}(\mathfrak{z}_L)\subset N_H(\mathfrak{z})$. The desired inclusion now holds.

Let $\wh{\fc} = \wt{\fc}\times_{\fa\git W_{\ker}}\fa$. The induced map $\wh{\fc}\to \fc$ is a reducible $W_{\wh{\fc}}:=W_\fa\rtimes W_{H,M}$ cover. Let $N(\wh{\fc})$ be the normalization of $\wh{\fc}$. $N(\wh{\fc})$ is a disjoint union

\[
N(\wh{\fc}) \simeq \coprod_{[w]\in W_{H,M}/W_C} \fa
\]

Then, for any choice of $[w]\in W_{H,M}/W_C$, we have a natural embedding $N(\wh{\fc}_L)\hookrightarrow N(\wh{\fc})$ from Lemma <ref>. This map descends to embeddings

\[
\wh{\xi}_{[w]}\colon \wh{\fc}_L\to \wh{\fc} \quad \text{ and }\quad \wt{\xi}_{[w]}\colon \wt{\fc}_L\to \wt{\fc}.
\]

Recall the notation $\varphi_L\colon \fc_L\to \fc$. We have a commutative diagram

\[
\xymatrix{
N(\wh{\fc}_L)\ar@{^{(}->}[d]\ar[r] & \wh{\fc}_L\ar@{^{(}->}[d]\ar[r] & \wt{\fc}_L\ar[r]\ar[d] & \fc_L\ar[d] \\
N(\wh{\fc})\ar[r] & \wh{\fc}\ar[r] & \wt{\fc} \ar[r] & \fc
\]

Let $\fd_L\subset \fc_L$ and $\fd\subset \fc$ denote the ramification divisor of $\varphi_L \colon \fc_L\to \fc$ and its preimage. (Recall that an explicit description of these divisors pulled back to $\fa$ was given in Section <ref>.) Let $U_L\subset \fc_L$ and $U\subset \fc$ be the complement of $\fd_L$ and $\fd$, respectively, and let $\wh{U}_L$ and $\wh{U}$ denote the preimages of $U_L$ and $U$ in $\wt{\fp}^\reg_L$ and $\wt{\fp}^\reg$, respectively.

The map $\iota_\fc$ of Proposition <ref> is an inclusion. Moreover, it induces an isomorphism when $G$ is simply connected.

Let $L$ be a distinguished Levi of $G$ associated to $x\in \fa$. Over $U_L\subset \fc_L$, the map $\wh{\xi}_{[w]}$ induces a commutative diagram

\[
\xymatrix{
\varphi_L^*\mathrm{Res}^{\wh{\fc}}_{\fc}(C\times \wh{\fc})^{W_{\wh{\fc}}} \ar@{^{(}->}[r] & \mathrm{Res}^{\wh{\fc}_L}_{\fc_L}(C\times \wh{\fc}_L)^{W_{\wh{\fc}_L}} \\
\varphi_L^*J\ar[u]^-{\iota}\ar[r]^{\simeq} & J_L\ar[u]^-{\iota_L}
\]

with the top arrow being an injection as the map $\wh{\fc}_L\to \wh{\fc}$ is a $W_{\wh{\fc}_L}$ equivariant embedding which is an isomorphism onto a union oc components of $\wh{\fc}$ and the bottom arrow being an isomorphism by Lemma <ref>. Hence, $\iota$ is an injection on $U\subset \fc$ if and only if $\iota_L$ is an injection on $U_L$.

For the first claim, it suffices to check away from a codimension 2 locus. Hence, it suffices to check for Levis $L$ such that the associated symmetric pair $(L,\theta,H_L)$ is of rank 1 in the sense that

\[
\dim(\fa) - \dim({\mathrm Lie}(Z_G)^{\theta = -1}) = 1.
\]

Such forms can be classified easily: in particular, the restricted roots systems must be rank 1, and there are only 3 isogeny classes of semisimple symmetric pairs. We may then do a simple computation to verify $\iota_L$ is an injection.

Now, note that if $\iota_L$ is an isomorphism on $U_L$, then $\iota$ must also be an isomorphism on $U$. Therefore, to check the second statement, it suffices to check for simply connected symmetric pairs of rank 1. This follows from the computations above.

It is important that the Galois comparison group scheme $J^1 = \mathrm{Res}^{\wh{\fc}}_{\fc}(C\times \wh{\fc})^{W_{\wh{\fc}}}$ is sufficiently well behaved. For this, we prove the following.

The map $\wh{\fc}\to \fc$ is Cohen-Macaulay over an open set of $\fc$ whose complement has codimension at least 2.

In particular, the space $J^1$ is smooth on an open set whose complement has codimension at least 2.

By Levi induction, it suffices to check for rank one Levis. If $\wt{\fc}\to \fc$ is irreducible, then it agrees with the map $\fa\git W_{\ker}\to \fc$, which is flat by Lemma <ref>. That leaves only the case $(\SL_3,S(\bG_m\times \GL_2))$. For this case, let $\mathcal{S}\subset \fp^\reg$ denote the Kostant-Rallis section

\[
\mathcal{S} = \left\{ x_a = \begin{pmatrix}
& a & 1 \\
1 & & \\
a & &
\end{pmatrix}\colon a\in k \right\}
\]

Then, after identifying $\mathcal{S}\simeq \bA^1$ and $H/B\simeq \bP^1$, we may write $\wt{\fp}^\reg\to \fp^\reg$ as

\[
\wt{\fp}^\reg\simeq \{ (a,\pm a)\in \bA^1\times \bP^1\}\to \bA^1
\]

In particular, it is a complete intersection and so Cohen-Macaulay. We conclude that $\wt{\fp}^\reg\to \fp$ is flat in this case by Miracle Flatness.

\[
\coprod_{W_{H,M}/W_{H_L,M_L}}\xi_{[w]}\colon \coprod_{W_{H,M}/W_{H_L,M_L}}\wh{U}_L\to \wh{U}
\]

is an isomorphism. In particular, there is an isomorphism

\[
\mathrm{Res}^{\wh{\fc}_L}_{\fc_L}(C\times \wh{\fc}_L)^{W_{\wh{\fc}_L}}|_{U_L}\xrightarrow{\simeq}  \chi_L^*\mathrm{Res}^{\wh{\fc}}_{\fc}(C\times \wh{\fc})^{W_{\wh{\fc}}}|_{U_L}
\]

over $U_L$

Using the commutative diagram

\[
\xymatrix{
\wh{\fc}_L\ar[r]\ar[d] & \wh{\fc}\ar[d] \\
\fc_L\ar[r] & \fc
\]

we have a

The map $\iota$ is an embedding.

It suffices to check away from codimension 2. We apply Lemma <ref> to reduce to the case of semisimple rank 1 quasisplit forms, which can be checked explicitly.

§ GENERAL STRUCTURE OF THE HITCHIN FIBRATION

In this section, we introduce the Hitchin morphism and prove the basic structure theorems <ref> and <ref>. Our discussion here will be very limited. Describing the geometry in a more comprehensive way, with the goal of proving support theorems similar to <cit.> and <cit.>, is the subject of ongoing work.

Fix a smooth projective curve $C$ of genus at least 2 and a line bundle $D$ on $C$ of degree $\deg(D)\geq 2g$. Taking $D = K_C$ the canonical bundle will also suffice. Also fix a symmetric pair $(G,\theta,H)$, and let $\cM = \mathrm{Maps}(C,\fp_D/H)$ denote the stack of maps from the curve $C$ to the twisted stack quotient $\fp_D/H$ where $\fp_D = \fp\otimes D$. On $k$-points, $\cM$ classifies pairs

\[
\cM(k) = \{(\cT_H,\sigma)\colon \cT_H\text{ is a $H$ torsor and }\sigma\in \Gamma(C,\cT_H\wedge^{\mathrm{Ad}} \fp_D)\}
\]

We have a Hitchin base $\cA = \mathrm{Maps}(C,\fc_D)$ classifying maps from $C$ to the twisted GIT quotient $\fc_D = (\fp\git H)_D$. In particular, by Theorem <ref>, there is a $\bG_m$ equivariant isomorphism $\fc\simeq \bA^r$ where the $\bG_m$ action on $\bA^r$ is given by exponents $(e_1,\dots, e_r)$. This induces an identification of affine spaces

\[
\cA \simeq \oplus_{i=1}^r H^0(C,D^{\otimes e_i})
\]

There is a natural Hitchin morphism

\[
h\colon \cM\to \cA
\]

induced by the Chevalley map $\fp/H\to \fc$. We restrict our attention to the regular locus in $\cM$; namely, we let $\cM^\reg = \mathrm{Maps}(C,\fp^\reg_D/H)$ be the substack of $\cM$ classifying maps $C\to \fp_D/H$ which factor through the open substack $\fp^\reg_D/H\subset \fp_D/H$. We abuse notation to denote $h\colon \cM^\reg\to \cA$.

To study the geometry of $h$ over the regular locus, we introduce the space
$$\cA^{\reg} = \mathrm{Maps}(C,(\fp^\reg\myfatslash H)_D).$$
The factorization $\fp^\reg/H\to \fp^\reg\myfatslash H\to \fc$ induces a factorization

\[
\cM^\reg\xrightarrow{h^{\reg}} \cA^{\reg}\xrightarrow{\phi} \cA.
\]

Let $\mathfrak{D}\subset \fc$ be the divisor defined by the function $\prod_{\alpha\in \Phi_r} d\alpha\in k[\fa]$. Following the notation of <cit.>, we will restrict to the following dense open subset of $\cA$.

Let $\cA^{\diamondsuit}\subset \cA$ denote the locus of maps $C\to \fc_D$ whose image intersects transversely with $\mathfrak{D}$.

We will denote by $\cA^{\myfatslash,\diamondsuit}$ denote the base change $\cA^{\reg}\times_{\cA}\cA^\diamondsuit$. We prove the first part of the structure theorem now.

The map $\phi\colon \cA^{\reg}\to \cA$ is étale when restricted to $\cA^{\diamondsuit}$.

It suffices to prove the étale lifting property. Fix a henselian scheme $\bD$. Then, the lifting property is equivalent to proving that, for any diagram

\[
\xymatrix{
C\ar[d]\ar[r] & (\fp^{\reg}\myfatslash H)_D\ar[d] \\
C\times \bD\ar[r] & \fc_D
\]

there is exists a unique lift $C\times \bD\to (\fp^\reg\myfatslash H)_D$. This lift can be constructed locally on $C$:  By the assumption that we work over $\cA^{\reg}$, the intersection of the preimage of $\mathfrak{D}$ under $C\times \bD\to \fc_D$ is a disjoint collection of maps $\bD\to \fc_D$. For each such map, there is a unique lift lying in the sheet of $(\fp^\reg\myfatslash H)_D$ dictated by the map $C\to (\fp^{\reg}\myfatslash H)_D$. We conclude that the map is étale.

In addition, we have a general result on the structure of the map $h^{\reg}$. Namely, for $a\colon S\times C\to (\fp^\reg\myfatslash H)_D$, we put $J_a :=a^*J$ for $J$ the band on $\fp^\reg\myfatslash H$ of Lemma <ref>. We may define a scheme over $\cA^{\reg}$ whose values over a $\cA^{\reg}$ scheme $S\times X\to(\fp^\reg\myfatslash H)_D$ are given by the space of $J_a$ torsors on $S\times C$,

\[
\cP(S) = (\text{$J_a$ torsors on $S\times C$}).
\]

Note that although $J_a$ is in general a band and not a group scheme, the notion of a $J_a$ torsor is well defined as the space of torsors is invariant under inner automorphisms. When $(G,\theta,H)$ is quasisplit, $\cP$ is a commutative group scheme over $\cA$.

Given a section $[\epsilon]\colon \cA^{\reg}\to \cM^{\reg}$ of $h^{\reg}$, there is an identification

\[
\cM^\reg|_{\cA^{\myfatslash,\diamondsuit}}\simeq \cP|_{\cA^{\myfatslash,\diamondsuit}}
\]

The morphism $\fp^\reg/H\to \fp^\reg\myfatslash H$ is a gerbe banded by $J$ which is trivialized by the section $[\epsilon]$. The result is immediate.

A section $[\epsilon]\colon \cA^{\reg}\to \cM^{\reg}$ is naturally produced by a $\bG_m$-equivariant section of the map

\[
\fp^{\reg}/H\to \fp^\reg\myfatslash H.
\]

Such a section was constructed in Lemma <ref>.

§ THE SYMMETRIC PAIR (GL2N,GLN X GLN)

In this section, we provide a spectral description of $J$ and the gerbe $\fp^{reg}/K\rightarrow \fp^{reg}\myfatslash K$ for the example of the symmetric pair $(\GL_{2n},\GL_n\times\GL_n)$.  This is very closely related to the description of Hitchin fiber in <cit.>.

§.§ The Spectral Cover

The map $\fa\git W_\fa\to \ft\git W$ (here $\ft$ and $W$ are for the group $GL(2n)$) is an embedding by Lemma <ref>. Over $\ft\git W$, there is a natural spectral cover $\ol{\fc}_{\GL(2n)}\to \ft\git W$ given by $\ol{\fc}_{\GL(2n)} = \ft\git S_{2n-1}$ for $S_{2n-1}\subset S_{2n}=W$ the index $2n$ subgroup which is the stabilizer of a fixed element in $\{1,2,...,2n\}$. Explicitly, we have $k[\ft\git W] = k[a_1,\dots, a_{2n}]$ where the $a_{i}$ are the degree $i$ elementary symmetric polynomials in $k[\ft]$, and

\[
k[\ol{\fc}_{\GL(2n)}] = k[\ft\git W][x]/(x^{2n}+a_1x^{2n-1}+\cdots+a_{2n}).
\]

The map $\fa\git W_{a}\rightarrow \ft\git W$ corresponds to the map $k[a_{1},...,a_{2n}]\rightarrow k[a_{2},...,a_{2n}]$ which sends $a_{2i+1}\mapsto 0$ for each $0\leq i <n$.

Let $\ol{\fc} = \ol{\fc}_{\GL(2n)}\times_{\ft\git W}\fa\git W_\fa$ be the restriction of this spectral cover to $\fa\git W_\fa$. Explicitly, we have

\[
k[\ol{\fc}] = k[\fa\git W_\fa][x]/(x^{2n}+a_{2}x^{2n-2}+\cdots +a_{2n-2}x^2+a_{2n})
\]

The map $ k[\fa\git W_\fa][x]\rightarrow k[\fa\git W_\fa][x]/(x^{2n}+a_{2}x^{2n-2}+\cdots +a_{2n-2}x^2+a_{2n})$ gives a $\bbG_{m}$-equivariant embedding $\overline{\fc}\hookrightarrow (\fa\git W_{a})\times \bbA^{1}$, where $\bbG_{m}$ acts with weight one on $\bbA^{1}$.

The involution $\theta$ on $GL(2n)$ acts on $\ft$ and also on the quotient $\ft\git S_{n-1}$. Explicitly, this action takes $x\mapsto -x$ and $a_i\mapsto (-1)^ia_i$. The spectral cover $\ol{\fc}\subset \ft\git S_{2n-1}$ is preserved by this action, and so we have an involution $i\colon \ol{\fc}\to \ol{\fc}$ defined over $\fa\git W_\fa$ taking $x\mapsto -x$. We denote by

\[ p\colon \ol{\fc}/i\to \fa\git W_\fa \]

the quotient of the cover $\ol{\fc}\to \fa\git W_\fa$. We will refer to the map $p$ as the (generic) spectral cover of $\fa\git W_\fa$. We note that $p$ corresponds to the inclusion

\[k[\fa\git W_{a}]\hookrightarrow  k[\fa\git W_{a}][y]/(y^{n}+a_{2}y^{n-1}+...+a_{n}).\]

The map

\[ k[\fa\git W_{a}][y]\rightarrow  k[\fa\git W_{a}][y]/(y^{n}+a_{2}y^{n-1}+...+a_{n})\]

gives a $\bbG_{m}$-equivariant embedding

\[\overline{\fc}/i\hookrightarrow \fc\times \bbA^{1},\]

where $\bbG_{m}$ acts on $\bbA^{1}$ with weight two.

Finally we want to note that there is a particularly nice description of the set $U$ of Proposition <ref> via the spectral cover.

The set $U$ of proposition <ref> is $k[\fa\git W_{a}][a_{2n}^{-1}]$, that is to say it is the complement of the vanishing locus of $a_{2n}$.

The interpretation of this in terms of the spectral cover is that the vanishing locus of $a_{2n}$ is precisely the image in $\fa\git W_{a}$ of the intersection $((\fa\git W_{a})\times \{0\})\times_{(\fa\git W_{a})\times \bbA^{1}}\overline{\fc}/i$

This is immediate from the definition of $U$ in proposition <ref>.

Recall that the regular centralizer group scheme $I_{K}^\reg\rightarrow \fp^{\reg}$ descends to a smooth group scheme $J$ on the GIT quotient $\fp^{\reg}\git (\GL_n\times \GL_n)\simeq \fa\git W_\fa$ since the form is quasi-split. We give a description of $J$ using the spectral cover above.

There is a natural map $J\to {\mathrm Res}_{\fa\git W_\fa}^{\ol{\fc}/i}(\bG_m)$ where ${\mathrm Res}_{\fa\git W_\fa}^{\ol{\fc}/i}(\bbG_{m})$ denotes the Weil restriction of $\bG_m$ along the map $p\colon \ol{\fc}/i\to \fc$. This map is an isomorphism.

Note that one has the description of regular centralizers of the adjoint action of $G$ on $\fg$ as the Weil restriction of $\bG_m$ along the spectral cover $\ol{\fc}_{\GL(2n)}\to \ft\git W$. In particular, restricting to $\fa\git W_\fa\subset \ft\git W$, it follows that there is an isomorphism

\[
J\xrightarrow{\sim} {\mathrm Res}^{\ol{\fc}}_{\fa\git W_\fa}(\bG_m)^i
\]

of $J$ with the $i$-invariant locus in ${\mathrm Res}^{\ol{\fc}}_{\fa\git W_\fa}(\bG_m)^i$. Since

\[
{\mathrm Res}^{\ol{\fc}}_{\fa\git W_\fa}(\bG_m)^i = {\mathrm Res}^{\ol{\fc}/i}_{\fa\git W_\fa}\left({\mathrm Res}^{\ol{\fc}}_{\ol{\fc}/i}(\bG_m)\right)^i
\]

it therefore suffices to show that

\[
{\mathrm Res}^{\ol{\fc}}_{\ol{\fc}/i}(\bG_m)^i \simeq \bG_m.
\]

Note that we have a map

\[
\xi\colon \bG_m\to {\mathrm Res}^{\ol{\fc}}_{\ol{\fc}/i}(\bG_m)^i.
\]

Namely, an $S$-point $S\to \bG_m\times \ol{\fc}/i$ has image givn by the base change

\[
\xymatrix{
S\ar[r] & \bG_m\times \ol{\fc}/i \\
S\times_{\ol{\fc}/i}\ol{\fc}\ar[u]\ar[r] & \bG_m\times \ol{\fc}\ar[u]
\]

We claim this map is an isomorphism. Let $\cD\subset \ol{\fc}/i$ be the ramification locus of the map $\ol{\fc}\to \ol{\fc}/i$, i.e. the image of the fixed point locus of $i$ in $\ol{\fc}/i$. For $x\in (\ol{\fc}/i)\setminus \cD$, we have the stalk

\[
{\mathrm Res}^{\ol{\fc}}_{\ol{\fc}/i}(\bG_m)_x = \bG_m\times \bG_m
\]

with $i$ acting by swapping the two factors. As the preimage of such an $x$ is two points, it is easy to see that the map $\xi_x\colon \bG_m\to \bG_m\times \bG_m$ is the diagonal map. Hence, $\xi$ is an isomorphism away from $\cD$. For $x\in \cD$, we have the fiber

\[
{\mathrm Res}^{\ol{\fc}}_{\ol{\fc}/i}(\bG_m)_x = \bG_m\times \bG_a
\]

with $i$ acting by $(y,z)\mapsto (y,-z)$. The map $\xi_x\colon \bG_m\to \bG_m\times \bG_a$ is the inclusion into the first factor, and we conclude that the map $\xi$ is an isomorphism.

§.§ Applications to the Hitchin Fibration for (GL2n,GLn x GLn)

The regular quotient for the case of $(\GL_{2n},\GL_n\times \GL_n)$ was computed in Example <ref>. Now that we have constructed spectral covers, we give an alternate description. This recovers the work of Schapostnik on spectral covers <cit.>.

We will make use of the notation from Section <ref>; in particular, we fix a smooth projective curve $C$ of genus at least 2 and a line bundle $D$ on $C$ of degree at lest $2g$. For any $S$ point $S\times C\to \fc_D$, we define the spectral cover at $a$ to be the base change

\[
\xymatrix{
\ol{C}_a\ar[d]\ar[r] & (\ol{\fc}/i)_D\ar[d] \\
S\times C\ar[r] & \fc_D
\]

In particular, we will set $\ol{C}$ to be the base change along the evaluation map $\cA\times C\to \fc_D$. We can realize $\ol{C}_a$ as a subvariety of the total space $\mathrm{Tot}(D)$ by considering the vanishing locus of the characteristic polynomial equation.

It is immediate from the Weil restriciton description of Proposition <ref> that we have the following description of fibers.

A section $[\epsilon]\colon \cA^{\reg}\to \cM^\reg$ induces an isomorphism of schemes over $\cA^{\myfatslash,\diamondsuit}$

\[
\cM^\reg|_{\cA^{\myfatslash,\diamondsuit}}\simeq \Pic(\phi^*\ol{C}/\cA^{\myfatslash,\diamondsuit}).
\]

We note that such a section $[\epsilon]$ is induced by the Kostant-Rallis sections of Section <ref>.

We denote by $\mathfrak{D}^{ns}\subset \fc$ the locus over which the map

\[
\fp^\reg\myfatslash H \to \fc
\]

has 2 preimages. In general, we have shown that $\mathfrak{D}^{ns}$ is a Zariski closed subset, but for the example $(\GL_{2n},\GL_n\times\GL_n)$, $\mathfrak{D}^{ns}$ is a divisor.

For any point $a\in \cA(S)$, the image of the map $a\colon S\times C\to \fc_D$ meets the nonseparated divisor $\mathfrak{D}^{ns}$ exactly at the image of the zero section of $\ol{C}_a$ in $\mathrm{Tot}(D)$.

As the nonseparated locus was found to be given exactly by the coordinate axes $\delta^* = 0$ in $\fa$, the claim is immediate.

We now derive the basic geometry of the map $\phi\colon \cA^{\myfatslash,\diamondsuit}\to \cA^{\diamondsuit}$. We do so in families, generalizing the earlier work that focused on studying fibers.

Let $d = \deg(D)$. The map  $\phi\colon \cA^{\myfatslash,\diamondsuit}\to \cA^{\diamondsuit}$ is an étale map of degree $2^d$.

This follows from Theorem <ref> and the fact that $d$ is the self intersection number of the zero divisor in $\mathrm{Tot}(D)$, see Proposition 9.16 of <cit.>.

Let $\Delta\subset C^d$ be the pairwise diagonal subscheme. We have an evaluation map

\[
\cA^{\diamondsuit}\to (C^d\setminus \Delta)/S_d
\]

sending a point $a$ of $\cA^{\diamondsuit}$ to the preimage under $a$ of the non-separated divisor $\mathfrak{D}^{ns}$.

Let $Z_1,Z_2$ be the two distinct sheets of $\fp^\reg\myfatslash H$ over $\mathfrak{D}^{ns}$. There is a decomposition

\[
\cA^{\myfatslash,\diamondsuit} = \coprod_{i=0}^d \cA^{\myfatslash,\diamondsuit}_i
\]

where the closed points of $\cA^{\myfatslash,\diamondsuit}_i$ classify maps $a\colon C\to (\fp^\reg\myfatslash H)_D$ for which $Z_1$ has $i$ preimages. Then, we can define

\[
\cA^{\myfatslash,\diamondsuit}\to (C^d\setminus \Delta)/(S_i\times S_{d-i})
\]

by sending a point $a$ of $\cA^{\myfatslash,\diamondsuit}$ to the preimage under $a$ of $Z_1$ and $Z_2$, respectively.

For every $0\leq i\leq d$, there is a Cartesian diagram

\[
\xymatrix{
\cA^{\myfatslash,\diamondsuit}_i\ar[d]\ar[r] & (C^d\setminus \Delta)/(S_i\times S_{d-i})\ar[d] \\
\cA^{\diamondsuit}\ar[r] & (C^d\setminus \Delta)/S_d
\]

where both vertical arrows are étale $\binom{d}{i}$ covers.

It is immediate that the map $\cA^{\myfatslash,\diamondsuit}\to \cA^{\diamondsuit}$ is an étale $\binom{d}{i}$ cover and the diagram above commutes. As the right vertical arrow is also étale of degree $\binom{d}{i}$, this implies that the diagram is Cartesian.

§ SECTIONS FROM SL2 TRIPLES

In this appendix, we review the construction of <cit.> and note an extension of those results to positive characteristic when $p$ is greater than the Coxeter number of $G$ based on the results of <cit.>, <cit.>, <cit.>, and <cit.>. In particular, we review the theory of normal $\fsl_2$ triples, and derive the Kostant-Rallis section from the construction of the Kostant section. We compare this with the results of <cit.>, reviewed in Section <ref>. It is likely that all results in this section are well known to experts.

We say an $\fsl_2$ triple $(e,h,f)$ is normal if $e,f\in \fp$ and $h\in \fk$. We say that an $\fsl_2$-triple is principal if $e$ is regular as an element of $\fp$.

Note that a principal, normal $\fsl_2$ triple in the sense of Definition <ref> is a principal $\fsl_2$ triple of $\fg$ in the usual sense only in the case of a quasi-split involution.

In the characteristic $p$ case, we will need to pass to associated characters.

Fix a nilpotent $e\in \cN$. An associated character of $e$ is a character $\lambda\colon \bG_m\to G$ such that $e\in \fg(2;\lambda)$ (where $\fg(k;\lambda)$ is the $k$-th graded piece of $\fg$ under the grading induced by $\lambda$) and there is a Levi subgroup $L\subset G$ such that $\lambda(\bG_m)\subset L^{\mathrm der}$ and $e$ is distinguished in ${\mathrm Lie}(L)$, i.e. $Z_{L^{\mathrm der}}(e)^\circ$ is unipotent.

(<cit.>, Prop. 4) Given an associated character $\lambda$ to a nilpotent $e$, one can extend $e$ to a unique $\fsl_2$ triple $(e,h,f)$ with $h\in {\mathrm Lie}({\mathrm image}(\lambda))$ and $f\in \fg(-2;\lambda)$.

We recall the following facts about associated characters and $\fsl_2$ of $G$ up to conjugation.

(<cit.>, Prop 18 and <cit.>, Theorem 1.1) Consider the projection

\[
\{(e,\lambda)\colon e\in \cN\text{ and $\lambda$ is an associated character for $e$}\}/G\to \cN/G
\]

where $G$ acts by conjugation on each set.

* This map is a bijection in good characteristic.

* The bijection above factors through

\[
\{(e,\lambda)\colon e\in \cN\text{ and $\lambda$ is an associated character for $e$}\}/G\to \{\fsl_2\text{-triples}\}/G\to\cN/G
\]

where the first map comes from Lemma <ref>. The map from $G$-orbits of $\fsl_2$-triples to $\cN/G$ is a bijection if and only if the characteristic of the field is greater than the Coxeter number.

Part (1) follows from Prop. 18, part 2, of <cit.>. Part 2 follows from Thoerem 1.1 of <cit.>.

We will demand, in addition, that associated characters be compatible with the involution on $G$ in the following sense.

We say that a character $\lambda$ is a normal associated character with respect to a nilpotent $e\in \cN_\fp$ of $\fp$ if it is an associated character or $e$ and ${\mathrm Image}(\lambda)\subset K$.

(<cit.>, Cor. 5.4) For any $e\in \cN_\fp$, there exists a normal associated character $\lambda$ for $e$. Moreover, such a character is unique up to conjugation by the connected component of the centralizer $Z_K(e)^\circ$.

We now deduce the results on $\fsl_2$ triples relevant to our paper.

The map

\[
\{H\text{-orbits of normal $\fsl_2$ triples}\}\to \{H\text{-orbits of nilpotents in $\fp$}\}
\]

is surjective, i.e. for any $e\in \cN_\fp$, there exists a normal $\fsl_2$ triple $(e,h,f)$ extending $e$. Assuming that the characteristic of the field is greater than the Coxeter number, this map is a bijection.

It suffices to prove this Lemma for $H=K$. In characteristic zero, this follows from <cit.>, Proposition 4.

In characteristic $p>0$, surjectivity follows from Lemma <ref> and Lemma <ref>. Now assume the characteristic is greater than the Coxeter number. Then, we have a sequence of maps

\[
\xymatrix{
%\{G\text{-orbits of pairs }(e,\lambda)\text{ for $\lambda$ associated to $e\in \fp$}\}\ar[r]^-{\simeq} & \{G\text{-orbits of normal $\fsl_2$-triples}\}\ar[r]^-{\simeq} & \{G\text{-orbits of nilpotents in $\fp$}\} \\
\left\{\substack{K\text{-orbits of pairs }(e,\lambda)\text{ for} \\ \text{$\lambda$ associated to $e$, valued in<EMAIL_ADDRESS>& \left\{\substack{ K\text{-orbits of normal} \\ \text{$\fsl_2$-triples}}\right\}\ar@{->>}[r] & \{K\text{-orbits of nilpotents in $\fp$}\}
\]

Since the composite map is an isomorphism, the map $\phi$ is injective. We claim that it is also surjective. Suppose that a normal $\fsl_2$ triple $(e,h,f)$ is not in the image of $\phi$. Then, by Lemma <ref>, there is a character $\lambda$ valued in $K$ associated to $e$. Moreover, by Lemma <ref>, any two associated characters of $e$ are conjugate by an element of $Z_G(e)^\circ$, and there is a unique character $\lambda'$ associated to $e$ for which $(e,h,f)$ is the corresponding $\fsl_2$ triple. Let $g\in Z_G(e)^\circ$ conjugate $\lambda'$ and $\lambda$, so that $g$ also conjugates $(e,h,f)$ to a normal $\fsl_2$ triple $(e,h',f')$. Since this $g$ preserves normality of the $\fsl_2$ triple, ${\mathrm Lie}({\mathrm image}\; g\cdot \lambda)\subset \fk$. Since $g\cdot \lambda$ is a one-parameter subgroup, it is connected and hence has image in $K$. We conclude that $g\cdot \lambda$ is an associated character to $e$ valued in $K$ whose associated $\fsl_2$ triple is $(e,h,f)$.

Now let $e\in \cN_\fp^{\reg}$ be a regular nilpotent. From a principal, normal $\fsl_2$ triple $(e,h,f)$, one produces a Kostant-Rallis section by considering the slice $e+\fc_\fp(f)$.

The map $e+\fc_\fp(f)\to \fa\git W_\fa$ is an isomorphism. We will call its inverse a Kostant-Rallis section associated to $e$.

Moreover, for a given regular nilpotent $e$ in $\fp$, this section is unique up to conjugation by $Z_K(e)^\circ$. In particular, this gives a bijection

\[
\{K\text{-orbits of Kostant-Rallis sections}\} \to \{K\text{-orbits of regular nilpotents in $\fp$}\}.
\]

In characteristic zero, this is the content of <cit.>, Theorem 11.

In characteristic $p>0$, by <cit.>, Lemma 6.29, it suffices to check that $e+\fc_\fp(f)$ is an ${\mathrm Ad}(\lambda)$-graded complement of $[\fk,e]$, where $\lambda\colon k^\times\to K$ is an associated character to $e$. Certainly the slice is ${\mathrm Ad}(\lambda)$ graded as $e$ and $f$ are homogeneous with respect to the grading. To show that the slice gives a complement, it suffices to show

\[
\fp = \big( e+\fc_\fp(f) \big)\oplus [\fk,e]
\]

By the proof of Lemma 3.1.3 of <cit.>, we have that

\[
\fg = \big(e+\fc_\fg(f)\big)\oplus [e,\fg].
\]

Intersecting this with $\fp$ and using the fact that $e\in \fp$ gives this result.

Let $\fs = e+\fc_{\fp}(f)$ be the Kostant-Rallis slice in $\fp$. Then, $\fp = \fs+[e,\fp]$. In particular, if

\[ a\colon H\times \fs\to \fp^\reg\]

is the action map. Then the differential of $a$ at $0\in \fs$ is surjective.

The first claim follows from the proof of Theorem <ref>. For the second, we note that the differential is identified with the map

\[ \fh\times \fs\to \fh,\quad (x,s)\mapsto [x,e]+s. \]

By the first claim together with the observation that $\fh = \fk$, this is surjective.

§ TABLE OF SIMPLE SYMMETRIC PAIRS

\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
G & K & \text{Type of $\Phi_r$} &\text{Holomorphic Involution} & \text{Antiholomorphic Involution} & \text{Real Form} & \text{Split/Quasi-split} & \text{Dual (Nadler) Group}\\
\hline
\SL_n & S(\GL_m\times \GL_{n-m}), \, m\leq n/2 & C_{m} & g\mapsto I_{m,n-m}gI_{m,n-m} & g\mapsto -I_{m,n-m}\ol{g}^tI_{m,n-m} & \SU(m,n-m) & \substack{\text{quasi-split when}\\ \text{ $n=2m$ or $n=2m+1$}} & \Sp_{2m}\\
\SL_{2n} & \Sp_{2n} & A_{n-1} & g\mapsto \Omega_n(g^t)^{-1}\Omega_n^t & g\mapsto \Omega_n\ol{g}\Omega_n  & \SU^*(2n) & & \SL_n \\
\SL_n & \SO_n & A_{n-1} & g\mapsto (g^t)^{-1} & g\mapsto -\ol{g}^t & \SL_n(\R) & \text{split} & \SL_n \\
\hline
\Sp_{2n} & \Sp_{2m}\times \Sp_{2(n-m)}, \, 2m\leq n & BC_m\text{ if $2m<n$ and $C_m$ if $2m=n$} & g\mapsto K_{m,n-m}gK_{m,n-m} & g\mapsto -K_{m,n-m}\ol{g}^tK_{m,n-m} & \Sp(2m,2(n-m)) &  & \Sp_{2m} \\
\Sp_{2n} & \GL_n & C_n & g\mapsto I_{n,n}gI_{n,n} & g\mapsto \ol{g} & \Sp_n(\R) & \text{split} & \SO_{2n+1} \\
\hline
\SO_{2n} & \SO_n\times \SO_{n} & D_n & g\mapsto I_{n,n}gI_{n,n} & g\mapsto I_{n,n}\ol{g}I_{n,n} & \SO(n,n)&  \text{split} &  \SO_{2n} \\
\SO_n & \SO_m\times \SO_{n-m},\, m<n/2 & B_m & g\mapsto I_{m,n-m}gI_{m,n-m} & g\mapsto I_{m,n-m}\ol{g}I_{m,n-m} & \SO(m,n-m) &  \substack{\text{quasi-split when $n=2m+2$;} \\ \text{split when $n=2m+1$}} &  \SO_{2m+1} \\
\SO_{2n} & \GL_n & C_n\text{ if $n$ is even and $BC_n$ if $n$ is odd} & g\mapsto \Omega_ng\Omega_n^t & g\mapsto \Omega_n\ol{g}\Omega_n & \SO^*(2n) &  &  \Sp_{2\lfloor n/2 \rfloor} \\
\hline
E_6 & A_5\times A_1 & F_4 &&& E_{6(2)} & \text{quasi-split} & F_4 \\
E_6 & D_5\times k^\times & BC_2 & & & E_{6(-14)} & & \SO_5 \\
E_6 & C_4 & E_6 & && E_{6(6)} & \text{split} & E_6 \\
E_6 & F_4 & A_2 &&& E_{6(-26)} & & \SL_3 \\
\hline
E_7 & A_7 & E_7 & && E_{7(7)} &  \text{split} & E_7 \\
E_7 & D_6\times A_1 & F_4 & && E_{7(-5)} & &F_4 \\
E_7 & E_6\times k^\times & C_3 &&& E_{7(-25)} & & \Sp_6\\
\hline
E_8 & D_8 & E_8 &&& E_{8(8)} & \text{split} & E_8 \\
E_8 & E_7 \times A_1 & F_4 & & & E_{8(-24)} & &F_4 \\
\hline
F_4 & B_4 & BC_1 &&& F_{4(-20)} && BC_1\\
F_4 & C_3\times A_1 & F_4 &&& F_{4(4)} & \text{split} & F_4\\
\hline
G_2 & A_1\times A_1 & G_2 & && G_{2(2)} & \text{split} & G_2 \\
\hline
G\times G & G\overset{\Delta}{\subset} G\times G & \text{Type of }G & (g_1,g_2)\mapsto (g_2,g_1) & \substack{(g_1,g_2)\mapsto (\tau(g_1),-\tau(g_2)) \\ \text{ $\tau$ the compact form on $G_1$}} & (G_1)_\R\subset (G_1)_\C & \text{quasi-split} & \substack{\check{G}\subset \check{G}\times \check{G} \\ g\mapsto (g,g^{-1})}\\
\hline
\end{array}$

Table 1:  A list of all simple symmetric pairs $(\fp,K)$ and the corresponding real forms. This is a compilation of table 26.3 of <cit.>, table 1 of <cit.>, and tables 2.1 and 2.2 of <cit.>. Notation for real forms of exceptional groups follows notation of Cartan. We only list forms up to isogeny. The group $\SO_n$ is defined above relative to the standard symmetric inner product $(x,y) = x^ty$. The group $\Sp_{2n}$ is defined relative to the skew symmetric form

\[
\langle x,y\rangle = x^tJy \quad\text{where }J = \begin{pmatrix}
0 & I_n \\ -I_n & 0
\end{pmatrix}.
\]

The following matrices are used in the table:

\[
I_{p,q} = \begin{pmatrix}
I_p & \\
& -I_q
\end{pmatrix}\quad \Omega_n = \begin{pmatrix}
& I_n \\ -I_n &
\end{pmatrix} \quad K_{p,q} = \begin{pmatrix}
I_{p,q} & \\ & I_{p,q}
\end{pmatrix}
\]

In the exceptional cases, only the isogeny class of $K$ is listed.