# Nilpotent varieties in symmetric spaces and twisted affine Schubert
varieties

Jiuzu Hong  Department of Mathematics, University of North Carolina at Chapel
Hill, Chapel Hill, NC 27599-3250, U.S.A<EMAIL_ADDRESS>and  Korkeat
Korkeathikhun  Department of Mathematics, University of North Carolina at
Chapel Hill, Chapel Hill, NC 27599-3250, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We relate the geometry of Schubert varieties in twisted affine Grassmannian
and the nilpotent varieties in symmetric spaces. This extends some results of
Achar-Henderson in the twisted setting. We also get some applications to the
geometry of the order 2 nilpotent varieties in certain classical symmetric
spaces.

##  1\. Introduction

Let $G$ be a reductive group over $\mathbb{C}$. Let $\mathcal{N}$ denote the
nilpotent cone of the Lie algebra $\mathfrak{g}$ of $G$. Let
$\operatorname{Gr}_{G}$ be the affine Grassmannian of $G$. Each spherical
Schubert cell $\operatorname{Gr}_{\lambda}$ is parametrized by a dominant
coweight $\lambda$. When $G=\mathrm{GL}_{n}$, Lusztig [Lu] defined an
embedding from $\mathcal{N}$ to $\operatorname{Gr}_{G}$, and showed that each
nilpotent variety in $\mathfrak{gl}_{n}$ can be openly embedded into certain
affine Schubert variety $\overline{\operatorname{Gr}}_{\lambda}$. This
embedding identifies the geometry of nilpotent varieties and certain affine
Schubert varieties in type $A$. However, there is no direct generalization for
general reductive groups.

In [AH], Achar-Henderson took a different idea for a general algebraic simple
group $G$. Let $\operatorname{Gr}_{0}^{-}$ be the opposite open Schubert cell
in $\operatorname{Gr}_{G}$. One can naturally define a map
$\pi:\operatorname{Gr}_{0}^{-}\to\mathfrak{g}$. Achar-Henderson showed that
$\pi(\operatorname{Gr}_{0}^{-}\cap\operatorname{Gr}_{\lambda})$ is contained
in $\mathcal{N}$ if and only if $\lambda$ is small in the sense of Broer [Br]
and Reeder [Re], i.e. $\lambda\nsucceq 2\gamma_{0}$, where $\gamma_{0}$ is the
highest short coroot of $G$. They also proved that $\pi:\operatorname{Gr}_{\rm
sm}\cap\operatorname{Gr}_{0}^{-}\to\pi(\operatorname{Gr}_{\rm
sm}\cap\operatorname{Gr}_{0}^{-})$ is a finite map whose fibers admits
transitive $\mathbb{Z}/2\mathbb{Z}$-actions, where $\operatorname{Gr}_{\rm
sm}$ is the union of all $\operatorname{Gr}_{\lambda}$ such that $\lambda$ is
small. Moreover, with respect to $\pi$, Achar-Henderson [AH, AH2] related the
geometric Satake correspondence and Springer correspondence.

In this paper, we consider a twisted analogue, and we will extend some results
of Achar-Henderson in [AH]. Let $\sigma$ be a diagram automorphism of order
$2$, and let $\sigma$ act on the field $\mathcal{K}=\mathbb{C}((t))$ via
$\sigma(t)=-t$ and $\sigma|_{\mathbb{C}}={\rm Id}_{\mathbb{C}}$. Then, we may
define a twisted affine Grassmannian
$\mathcal{G}r:=G(\mathcal{K})^{\sigma}/G(\mathcal{O})^{\sigma}$, where
$\mathcal{O}=\mathbb{C}[[t]]$. Each twisted Schubert cell
$\mathcal{G}r_{\bar{\lambda}}$, i.e. a $G(\mathcal{O})^{\sigma}$-orbit, is
parametrized by the image $\bar{\lambda}$ of a dominant coweight $\lambda$ in
the coinvariant lattice $X_{*}(T)_{\sigma}$ with respect to the induced action
of $\sigma$, where $X_{*}(T)$ is the coweight lattice of $G$. In fact,
$X_{*}(T)_{\sigma}$ can be regarded as the weight lattice of a reductive group
$H:=(\check{G})^{\sigma}$, where $\check{G}$ is the Langlands dual group of
$G$.

Let $\mathcal{G}r_{0}^{-}$ be the opposite open Schubert cell in
$\mathcal{G}r$. We may naturally define a map
$\pi:\mathcal{G}r_{0}^{-}\to\mathfrak{p}$, where $\mathfrak{p}$ is the
(-1)-eigenspace of $\sigma$ in $\mathfrak{g}$. Let
$\mathcal{M}_{\bar{\lambda}}$ denote the intersection
$\mathcal{G}r_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}$, which is a nonempty
open subset of $\mathcal{G}r_{\bar{\lambda}}$. The following theorem is the
main result of this paper, and it can follow from Proposition 2.3 in Section
2.1 and Theorem 4.2 in Section 4, based on case-by-case analysis.

######  Theorem 1.1.

Assume that $G$ is of type $A_{\ell}$ or $D_{\ell+1}$. The image
$\pi(\mathcal{M}_{\lambda})$ is contained in the nilpotent cone
$\mathcal{N}_{\mathfrak{p}}$ of $\mathfrak{p}$, if and only if $\bar{\lambda}$
is a small dominant weight with respect to $H$.

Actually, in Theorem 4.2 we describe precisely
$\pi(\mathcal{M}_{\bar{\lambda}})$ as a union of nilpotent orbits in
$\mathfrak{p}$ for each small $\bar{\lambda}$. In Theorem 4.5, Theorem 4.6,
and Theorem 4.14, we also determine all small $\bar{\lambda}$ such that
$\pi(\mathcal{M}_{\bar{\lambda}})$ is a nilpotent orbit and
$\pi:\mathcal{M}_{\bar{\lambda}}\to\pi(\mathcal{M}_{\bar{\lambda}})$ is an
isomorphism. Furthermore, We describe all fibers of
$\pi:\mathcal{M}\to\pi(\mathcal{M})$ in Proposition 4.11 and Proposition 4.15,
where $\mathcal{M}$ is the union of all small twisted Schubert cells. The
fibers are closely related to anti-commuting nilpotent varieties for symmetric
spaces. When $G$ is of type $A_{2\ell-1}$ (resp. $D_{\ell+1}$), the fiber
$\pi^{-1}(0)$ is actually the minimal (resp. maximal) order 2 nilpotent
variety in $\mathfrak{sp}_{2\ell}$ (resp. $\mathfrak{so}_{2\ell+1}$). This is
a very different phenomenon with the untwisted setting in the work of Achar-
Henderson [AH], and it actually makes the twisted setting more challenging.

For general simple Lie algebra $\mathfrak{g}$ and general diagram automorphism
$\sigma$, it was proved in [HLR, Appendix C] by Haines-Lourenço-Richarz that,
when $\bar{\lambda}$ is quasi-miniscule and $\overline{\mathcal{O}}$ is the
minimal nilpotent variety in $\mathfrak{p}$, the map
$\pi:\overline{\mathcal{G}r}_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}\to\overline{\mathcal{O}}$
is an isomorphism. In fact, we have also obtained this result independently,
cf. [Ko]. Also, under the same assumption as in Theorem 1.1, this isomorphism
is a special case of our Theorem 4.5, Theorem 4.6, and Theorem 4.14.

The geometric Satake correspondence for $\mathcal{G}r$ was proved by Zhu [Zh],
and it exactly recovers the Tannakian group $H$. On the other hand, the
Springer correspondence for symmetric spaces is more sophisticated than the
usual Lie algebra setting, see a survey on this subject [Sh]. It would be
interesting to relate these two pictures as was done in [AH, AH2]. Y. Li [Li]
defined the symmetric space analogue called $\sigma$-quiver variety in the
setting of Nakajima quiver variety, and he showed that certain $\sigma$-quiver
variety can be identified with null-cone of symmetric spaces. It is an
interesting question to investigate a connection between $\sigma$-quiver
variety and twisted affine Grassmannian in the spirit of the work of Mirković-
Vybornov [MV].

From Theorem 1.1, we can deduce some applications for the order 2 nilpotent
varieties in classical symmetric spaces. Let $\langle,\rangle$ be a symmeric
or symplectic non-degenerate bilinear form on a vector space $V$. Let
$\mathcal{A}$ be the space of self-adjoint linear maps with respect to
$\langle,\rangle$. We consider ${\rm Sp}_{2n}$-action on $\mathcal{A}$ when
$\langle,\rangle$ is symplectic and $\dim V=2n$, and ${\rm SO}_{n}$-action
when $\langle,\rangle$ is symmetric and $\dim V=n$. In Section 5, we obtain
the following results.

######  Theorem 1.2.

  1. (1)

If $\langle,\rangle$ is symmetric and $\dim V$ is odd, then any order 2
nilpotent variety in $\mathcal{A}$ is normal.

  2. (2)

If $\langle,\rangle$ is symplectic, then there is a bijection of order 2
nilpotent varieties in $\mathfrak{so}_{2n+1}$ and in $\mathcal{A}$, such that
they have the same cohomology of stalks of IC-sheaves.

  3. (3)

If $\langle,\rangle$ is symplectic, the smooth locus of any order 2 nilpotent
variety in $\mathcal{A}$ is the open nilpotent orbit.

It is known that when $\langle,\rangle$ is symplectic, any nilpotent variety
in $\mathcal{A}$ is normal, but it is not always true when $\langle,\rangle$
is symmetric, cf. [Oh]. Using our methods, we can also prove that there is a
bijection of order 2 nilpotent varieties in $\mathfrak{sp}_{2n}$ and in the
space of symmetric $(2n+1)\times(2n+1)$ matrices, such that they have the same
cohomlogy of stalks of IC-sheaves. This was already proved earlier by Chen-
Vilonen-Xue [CVX] using different methods. Also, Part 3) of Theorem 1.2 is not
true when $\langle,\rangle$ is symmetric and $\dim V$ is odd, see more
detailed discussions in Section 5.

Acknowledgments: This project grew out from a conversation with Yiqiang Li in
March 2019. We would like to thank him for inspiring discussions. J. Hong is
partially supported by NSF grant DMS-2001365.

##  2\. Notation and Preliminaries

###  2.1. Root datum

Let $G$ be a simply-connected simple algebraic group over $\mathbb{C}$, and
let $\mathfrak{g}$ be its Lie algebra. Let $\sigma$ be a diagram automorphism
of $G$ of order $r$, preserving a maximal torus $T$ and a Borel subgroup $B$
containing $T$ in $G$. Then $G$ has a root datum
$(X_{*}(T),X^{*}(T),\langle\cdot,\cdot\rangle,\check{\alpha}_{i},\alpha_{i},i\in
I)$ with the action of $\sigma$, where

  * •

$X_{*}(T)$ (resp. $X^{*}(T)$ ) is the coweight (resp. weight) lattice;

  * •

$I$ is the set of vertices of the Dynkin diagram of $G$;

  * •

$\alpha_{i}$ (resp. $\check{\alpha}_{i}$) is the simple root (resp. coroot)
for each $i\in I$;

  * •

$\langle\cdot,\cdot\rangle:X_{*}(T)\times X^{*}(T)\to\mathbb{Z}$ is the
perfect pairing.

The automorphism $\sigma$ of this root datum satisfies

  * •

$\sigma(\alpha_{i})=\alpha_{\sigma(i)}$ and
$\sigma(\check{\alpha}_{i})=\check{\alpha}_{\sigma(i)}$;

  * •

$\langle\sigma(\check{\lambda}),\sigma(\mu)\rangle=\langle\check{\lambda},\mu\rangle$
for any $\check{\lambda}\in X_{*}(T)$ and $\mu\in X^{*}(T)$.

As a diagram autormophism on $G$, $\sigma$ also preserves a pinning with
respect to $B$ and $T$, i.e. there exists root subgroups $x_{i},y_{i}$
associated to $\alpha_{i},-\alpha_{i}$ for each $i\in$, such that

$\sigma(x_{i}(a))=x_{\sigma(i)}(a),\quad\sigma(y_{i}(a))=y_{\sigma(i)}(a),\quad\text{
for any }a\in\mathbb{C}.$

Let $I_{\sigma}$ be the set of $\sigma$-orbits in $I$. Denote
$X^{*}(T)^{\sigma}=\\{\lambda\in X^{*}(T)\mid\sigma\lambda=\lambda\\}$ and
$X_{*}(T)_{\sigma}=X_{*}(T)/(\mathrm{Id}-\sigma)X_{*}(T)$. For each $\imath\in
I_{\sigma}$, define $\gamma_{\imath}=\bar{\check{\alpha}}_{i}\in
X_{*}(T)_{\sigma}$ for any $i\in\imath$, and define
$\check{\gamma}_{\imath}\in X^{*}(T)^{\sigma}$ by

$\check{\gamma}_{\imath}=\begin{cases}\sum_{i\in\imath}\alpha_{i}&\quad\text{if
no pairs in }\imath\text{ is adjacent, }\\\
2\sum_{i\in\imath}\alpha_{i}&\quad\text{if }\imath=\\{i,\sigma(i)\\}\text{ and
}i\text{ and }\sigma(i)\text{ are adjacent,}\\\ \alpha_{i}&\quad\text{if
}\imath=\\{i\\}.\end{cases}$

Let $\check{G}$ denote the Langlands dual group of $G$, and we still denote
the induced diagram automorphism on $\check{G}$ by $\sigma$. Denoted by
$H=(\check{G})^{\sigma}$ the $\sigma$-fixed subgroup of $\check{G}$. Then, $H$
has the root datum
$(X^{*}(T)^{\sigma},X_{*}(T)_{\sigma},\check{\gamma_{\imath}},\gamma_{\imath},\imath\in
I_{\sigma})$, cf. [HS, Section 2.2]. For $\bar{\lambda},\bar{\mu}\in
X_{*}(T)_{\sigma}$, define the partial order $\bar{\mu}\preceq\bar{\lambda}$
if $\bar{\lambda}-\bar{\mu}$ is a sum of positive roots of $H$. Let
$X_{*}(T)_{\sigma}^{+}$ be the set of dominant weight of $H$. In fact,
$X_{*}(T)_{\sigma}^{+}$ is the image of the quotient map $X_{*}(T)^{+}\to
X_{*}(T)_{\sigma}$, where $X_{*}(T)^{+}$ is the set of dominant weights of
$G$.

###  2.2. Twisted affine Grassmannian

Let $\sigma$ be a diagram automorphism of $G$ of order $r$. Let $\mathcal{O}$
denote the set of formal power series in $t$ with coefficients in $\mathbb{C}$
and denote $\mathcal{K}$ the set of Laurent series in $t$ with coefficients in
$\mathbb{C}$. Denote the automorphism $\sigma$ of order $r$ on $\mathcal{K}$
and $\mathcal{O}$ given by $\sigma$ acts trivially on $\mathbb{C}$ and maps
$t\to\epsilon t$ where we fix the primitive $r$-root of unity $\epsilon$. We
consider the following twisted affine Grassmannian  attached to $G$ and
$\sigma$,

$\mathcal{G}r_{G}=G(\mathcal{K})^{\sigma}/G(\mathcal{O})^{\sigma}.$

This space has been studied intensively in [BH, HR, PR, Ri]. The ramified
geometric Satake correspondence [Zh] asserts that there is an equivalence
between the category of spherical perverse sheaves on $\mathcal{G}r_{G}$ and
the category of representations of the algebraic group
$H=(\check{G})^{\sigma}$. If there is no confusion, we write $\mathcal{G}r$
for convenience.

Let $e_{0}$ be the based point in $\mathcal{G}r$. For any $\lambda\in
X_{*}(T)$, we attach an element $t^{\lambda}\in T(\mathcal{K})$ naturally and
define the norm $n^{\lambda}\in T(\mathcal{K})^{\sigma}$ of $t^{\lambda}$ by

(1)
$n^{\lambda}:=\prod_{i=0}^{r-1}\sigma^{i}(t^{\lambda})=\epsilon^{\sum_{i=1}^{r-1}i\sigma^{i}(\lambda)}t^{\sum\sigma^{i}\lambda}.$

Let $\bar{\lambda}$ be the image of $\lambda$ in $X_{*}(T)_{\sigma}$. Set
$e_{\bar{\lambda}}=n^{\lambda}\cdot e_{0}\in\mathcal{G}r$. Then
$e_{\bar{\lambda}}$ only depends on $\bar{\lambda}$. Following [BH, Zh],
$\mathcal{G}r$ admits the following Cartan decomposition

(2) $\mathcal{G}r=\bigsqcup_{\bar{\lambda}\in
X_{*}(T)_{\sigma}^{+}}\mathcal{G}r_{\bar{\lambda}}$

where $\mathcal{G}r_{\bar{\lambda}}=G(\mathcal{O})^{\sigma}\cdot
e_{\bar{\lambda}}$ is a Schubert cell. Let
$\overline{\mathcal{G}r}_{\bar{\lambda}}$ be the closure of
$\mathcal{G}r_{\bar{\lambda}}$. Then

$\overline{\mathcal{G}r}_{\bar{\lambda}}=\bigsqcup_{\bar{\mu}\preceq\bar{\lambda}}\mathcal{G}r_{\bar{\mu}},$

and $\dim\overline{\mathcal{G}r}_{\bar{\lambda}}=\langle
2\rho,\bar{\lambda}\rangle,$ where $\rho$ is the half sum of all positive
coroots of $H$.

By abuse of notation, we still use $\sigma$ to denote the induced automorphism
on $\mathfrak{g}$ of order $r$. Then there is a grading on $\mathfrak{g}$,

$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{r-1}$

where $\mathfrak{g}_{i}$ is the $\epsilon^{i}$-eigenspace. Set

$\mathfrak{p}=\mathfrak{g}_{1}.$

Set $\mathcal{O}^{-}=\mathbb{C}[t^{-1}]$. Consider the evaluation map ${\rm
ev}_{\infty}:G(\mathcal{O}^{-})\to G$. Let $G(\mathcal{O}^{-})_{0}$ denote its
kernel. The map ${\rm ev}_{\infty}$ factors through
$G(\mathbb{C}[t^{-1}]/(t^{-2}))\to G$. Note that the kernel of
$G(\mathbb{C}[t^{-1}]/(t^{-2}))\to G$ is canonical identified with the vector
space $\mathfrak{g}\otimes t$ with respect to the adjoint action of $G$ and
$\sigma$. It induces a $G\rtimes\langle\sigma\rangle$-equivariant map

$G(\mathcal{O}^{-})_{0}\to\mathfrak{g}\otimes t.$

Taking $\sigma$-invariance, we get a $K$-equivariant map

(3) $G(\mathcal{O}^{-})_{0}^{\sigma}\to\mathfrak{p},$

where $K:=G^{\sigma}$. Note that $K$ is a connected simply-connected simple
algebraic group, as $G$ is simply-connected.

Set $\mathcal{G}r_{0}^{-}:=G(\mathcal{O}^{-})^{\sigma}\cdot e_{0}\simeq
G(\mathcal{O}^{-})_{0}^{\sigma}$. Then $\mathcal{G}r_{0}^{-}$ is the open
opposite Schubert cell in $\mathcal{G}r$. From (3), we have the following
$G^{\sigma}$-equivariant map

(4) $\pi:\mathcal{G}r_{0}^{-}\to\mathfrak{p}.$

######  Lemma 2.1.

$\mathcal{G}r_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}$ is nonempty for any
$\lambda\in X_{*}(T)_{\sigma}^{+}$.

###### Proof.

It suffices to show that
$G(\mathcal{O})^{\sigma}n^{\lambda}G(\mathcal{O})^{\sigma}\cap
G(\mathcal{O}^{-})^{\sigma}$. Let $I$ be the Iwahori subgroup contained in
$G(\mathcal{O})^{\sigma}$ and let $I^{-}$ be the opposite Iwahori subgroup
contained in $G(\mathcal{O}^{-})^{\sigma}$. It suffices to show that
$In^{\lambda}I\cap I^{-}\not=\emptyset$.

Let $\mathcal{G}$ be the Kac-Moody group associated to the twisted loop group
$G(\mathcal{K})^{\sigma}$, see a construction in [BH, p.14]. There is a
projection map $p:\mathcal{G}\to G(\mathcal{K})^{\sigma}$. Let $\mathcal{I}$
(resp. $\mathcal{I}^{-}$) be the preimage of $I$ (resp. $I^{-}$) via the
projection $p$. Then we are reduced to show that
$\mathcal{I}w\mathcal{I}\cap\mathcal{I}^{-}\not=\emptyset$, where $w$ is an
element of the Weyl group $\mathcal{W}$ of $\mathcal{G}$. This is true, since
$\mathcal{I}s\mathcal{I}\cap\mathcal{I}^{-}\not=\emptyset$ for any simple
reflection $s\in\mathcal{W}$, and for any $y\in\mathcal{W}$ and simple
reflection $s$, $\mathcal{I}y\mathcal{I}s\mathcal{I}=\mathcal{I}ys\mathcal{I}$
if $\ell(ys)=\ell(y)+1$, cf. [Ku, 5.1.3 (d)]. ∎

Following [Br, Re, AH], an element $\bar{\lambda}$ of $X_{*}(T)_{\sigma}^{+}$
is called small, if $\bar{\lambda}\nsucceq 2\gamma_{0}$, where $\gamma_{0}$ is
the highest short root of $H$. The set of all small dominant weights is a
lower order ideal of $X_{*}(T)_{\sigma}^{+}$, i.e., if
$\bar{\mu}\preceq\bar{\lambda}$ and $\bar{\lambda}$ is small, then $\bar{\mu}$
is also small. Let $\mathcal{G}r_{\mathrm{sm}}$ be the union of
$\mathcal{G}r_{\bar{\lambda}}$ for small dominant weights $\bar{\lambda}$. Set

$\mathcal{M}=\mathcal{G}r_{\mathrm{sm}}\cap\mathcal{G}r_{0}^{-}.$

For each small dominant weight $\bar{\lambda}$, set

$\mathcal{M}_{\bar{\lambda}}=\mathcal{G}r_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}.$

Let $\mathcal{N}_{\mathfrak{p}}$ denote the nilpotent cone of $\mathfrak{p}$.
We shall prove in Section 4 that $\pi(\mathcal{M})$ is contained
$\mathcal{N}_{\mathfrak{p}}$, when $G$ is of type $A_{n}$ and $D_{n}$ and
$\sigma$ is of order $2$.

Recall that $\gamma_{0}$ is the highest short root of $H$. The following lemma
is a twisted analogue of [AH, Lemma 3.3].

######  Lemma 2.2.

If $\sigma$ is a diagram automorphism of order $r$, then
$\pi(\mathcal{G}r_{2\gamma_{0}}\cap\mathcal{G}r_{0}^{-})\nsubseteq\mathcal{N}_{\mathfrak{g}_{1}}$.

###### Proof.

Let $X_{N}$ be the Dynkin diagram of $G$. Following [Ka, p.128-129], we choose
the following root of $G$,

${\theta}_{0}=\begin{cases}{\alpha}_{1}+\cdots+{\alpha}_{2\ell-2},&\quad(X_{N},r)=(A_{2\ell-1},2);\\\
{\alpha}_{1}+\cdots+{\alpha}_{2\ell},&\quad(X_{N},r)=(A_{2\ell},2);\\\
{\alpha}_{1}+\cdots+{\alpha}_{\ell},&\quad(X_{N},r)=(D_{\ell+1},2);\\\
{\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3},&\quad(X_{N},r)=(D_{4},3);\\\
{\alpha}_{1}+2{\alpha}_{2}+2{\alpha}_{3}+{\alpha}_{4}+{\alpha}_{5}+{\alpha}_{6},&\quad(X_{N},r)=(E_{6},2).\end{cases}$

where the label of simple roots ${\alpha}_{i}$ follows from [Ka, TABLE Fin,
p.53]. Recall from the section 2.1 that for each $\iota\in I_{\sigma}$, we
define simple roots of $H$, $\gamma_{\imath}=\bar{\check{\alpha}}_{i}\in
X_{*}(T)_{\sigma}$.

Let $\check{\theta}_{0}$ be the coroot of ${\theta}_{0}$. Then,

$\bar{\check{\theta}}_{0}=\begin{cases}\gamma_{0}&\quad\text{if
}(X_{N},r)\not=(A_{2\ell},2)\\\ 2\gamma_{0}&\quad\text{if
}(X_{N},r)=(A_{2\ell},2)\end{cases}.$

Suppose $(X_{N},r)\not=(A_{2\ell},2)$. Note that $\theta_{0}\in X^{*}(T)$ and
$\check{\theta}_{0}:\mathbb{C}^{\times}\to T$. Each $a\in\mathbb{C}^{\times}$
can be identified with $\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}\in\mathrm{SL}_{2}.$ For each $i=0,...,r-1$, define a
homomorphism $\phi_{\sigma^{i}(\theta_{0})}:\mathrm{SL}_{2}\to G$ given by

$\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}\mapsto
x_{\sigma^{i}(\theta_{0})}(a),\hskip 14.22636pt\begin{pmatrix}1&0\\\
a&1\end{pmatrix}\mapsto y_{\sigma^{i}(\theta_{0})}(a),\hskip
14.22636pt\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}\mapsto\sigma^{i}(\check{\theta}_{0})(a).$

Let $\mathcal{S}$ be the product of $r$ copies of $\mathrm{SL}_{2}$. Then
$\check{\theta}_{0}$ can be extended to $\phi:\mathcal{S}\to G$ given by

$\phi(g_{0},...,g_{r-1})=\prod_{i=0}^{r-1}\phi_{\sigma^{i}(\theta_{0})}(g_{i}).$

This $\phi$ can extend scalar to $\mathcal{K}$. Abusing notation, define
$\sigma:\prod_{i=1}^{r}(\operatorname{SL}_{2}(\mathbb{\mathcal{K}}))_{i}\to\prod_{i=1}^{r}(\operatorname{SL}_{2}(\mathbb{\mathcal{K}}))_{i}$
by

$\sigma(g_{1}(t),g_{2}(t),...,g_{r}(t))=(g_{r}(\epsilon t),g_{1}(\epsilon
t),...,g_{r-1}(\epsilon t)).$

There exists an isomorphism

$\varphi:\mathrm{SL}_{2}(\mathcal{K})\to(\prod_{i=1}^{r}(\operatorname{SL}_{2}(\mathbb{\mathcal{K}}))_{i})^{\sigma}=\\{(g(t),g(\epsilon
t),...,g(\epsilon^{r-1}t))\mid g(t)\in\operatorname{SL}_{2}(\mathcal{K})\\}.$

Hence

$\phi\circ\varphi:\begin{pmatrix}t&0\\\
0&t^{-1}\end{pmatrix}\mapsto\left(\begin{pmatrix}\epsilon^{i}t&0\\\
0&(\epsilon^{i}t)^{-1}\end{pmatrix}\right)_{i=0,...,r-1}\mapsto\prod_{i=0}^{r-1}(\epsilon^{i}t)^{\sigma^{i}\check{\theta}_{0}}=n^{\check{\theta}_{0}}.$

Let $\mathfrak{s}$ be the product of $r$ copies of $\mathfrak{sl}_{2}$. Define
$\sigma:\mathfrak{s}\to\mathfrak{s}$ by

$\sigma(x_{1},,...,x_{r-1},x_{r})=(\epsilon x_{r},\epsilon x_{1},...,\epsilon
x_{r-1}).$

Since $\sigma$ has order $r$, we have
$\mathfrak{s}=\oplus_{i=0}^{r-1}\mathfrak{s}_{i}$ where $\mathfrak{s}_{i}$ is
the eigenspace of eigenvalue $\epsilon^{i}$. Then
$\mathfrak{s}_{1}=\\{(x,\epsilon x,...,\epsilon^{r-1}x)\mid
x\in\mathfrak{sl}_{2}\\}\cong\mathfrak{sl}_{2}$. The derivative of $\phi$ is
$\mathrm{d}\phi:\mathfrak{s}\to\mathfrak{g}$ which induces
$\mathfrak{s}_{1}\to\mathfrak{g}_{1}$. Hence we have the map
$\Psi:\mathfrak{sl}_{2}\to\mathfrak{g}_{1}$.

Consider the matrix $g(t)\in\operatorname{SL}_{2}(\mathcal{O}^{-})$,

$g(t)=\begin{pmatrix}1+t^{-1}&t^{-2}\\\
t^{-1}&1-t^{-1}+t^{-2}\end{pmatrix}=\begin{pmatrix}0&1\\\
-1&t^{2}-t+1\end{pmatrix}\begin{pmatrix}t^{2}&0\\\
0&t^{-2}\end{pmatrix}\begin{pmatrix}1&0\\\ t^{2}+t&1\end{pmatrix}.$

Then $(\phi\circ\varphi)(g(t))\in
G(\mathcal{O})^{\sigma}n^{2\check{\theta}_{0}}G(\mathcal{O})^{\sigma}$. Since
$G$ is not type $A_{2l}$, $\bar{\check{\theta}}_{0}=\gamma_{0}$ and then
$(\phi\circ\varphi)(g(t))\cdot
e_{0}\in\mathcal{G}r_{2\gamma_{0}}\cap\mathcal{G}r_{G,0}^{-}$. We have the
commutative diagram

${{\operatorname{Gr}_{\mathrm{SL}_{2},0}^{-}}}$${{\mathcal{G}r_{\mathcal{S},0}^{-}}}$${{\mathcal{G}r_{G,0}^{-}}}$${{\mathfrak{sl}_{2}}}$${{\mathfrak{s}_{1}}}$${{\mathfrak{g}_{1}}}$$\scriptstyle{g(t)\cdot
L_{0}\mapsto\varphi(g(t))\cdot
e_{0}}$$\scriptstyle{(g_{i}(t))_{i=0}^{r-1}\cdot
e_{0}\mapsto\phi(g_{i}(t))_{i=0}^{r-1})\cdot
e_{0}}$$\scriptstyle{\pi_{\mathrm{SL}_{2}}}$$\scriptstyle{\pi}$$\scriptstyle{x\mapsto(x,\epsilon
x,...,\epsilon^{r-1}x)}$

where
$\operatorname{Gr}_{\mathrm{SL}_{2},0}^{-}:=\mathrm{SL}_{2}(\mathcal{O}^{-})_{0}\cdot
e_{0}\subset\operatorname{Gr}_{\mathrm{SL}_{2}}$, and
${\mathcal{G}r_{\mathcal{S},0}^{-}}$ is defined similarly. The commutativity
follows from

$\pi((\phi\circ\varphi)(g(t))\cdot
e_{0})=\Psi(\pi_{\operatorname{SL}_{2}}(g(t)\cdot
e_{0}))=\Psi\begin{pmatrix}1&0\\\ 1&-1\end{pmatrix},$

where the latter is not nilpotent. It follows that,
$\pi(\mathcal{G}r_{2\gamma_{0}})\not\subseteq\mathcal{N}_{\mathfrak{p}}$.

Suppose that $(X_{N},r)=(A_{2n},2)$. In this case,
$\bar{\check{\theta}}_{0}=2\gamma_{0}$ and
$\sigma(\check{\theta}_{0})={\check{\theta}_{0}}$. Then $\check{\theta}_{0}$
can be extended to $\phi:\mathrm{SL}_{2}\to G$ defined by

$\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}\mapsto x_{\theta_{0}}(a),\hskip
14.22636pt\begin{pmatrix}1&0\\\ a&1\end{pmatrix}\mapsto
y_{\theta_{0}}(a),\hskip 14.22636pt\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}\mapsto\check{\theta}_{0}(a).$

$\phi$ can extend the scalar to $\mathcal{K}$. Define a group homomorphism
$\sigma:\mathrm{SL}_{2}(\mathcal{K})\to\mathrm{SL}_{2}(\mathcal{K})$ by

$\begin{pmatrix}a(t)&b(t)\\\
c(t)&d(t)\end{pmatrix}\mapsto\begin{pmatrix}a(-t)&-b(-t)\\\
-c(-t)&d(-t)\end{pmatrix}$

where $a(t)\in\mathcal{K}$. Then $\phi:\mathrm{SL}_{2}(\mathcal{K})\to
G(\mathcal{K})$ is $\sigma$-equivariant. The induced homomorphism
$\sigma:\mathfrak{sl}_{2}\to\mathfrak{sl}_{2}$ is given by

$\begin{pmatrix}a&b\\\ c&-a\end{pmatrix}\mapsto\begin{pmatrix}a&-b\\\
-c&-a\end{pmatrix}.$

The derivative $\mathrm{d}\phi:\mathfrak{sl}_{2}\to\mathfrak{g}$ induces the
map $\Psi:(\mathfrak{sl}_{2})_{1}\to\mathfrak{g}_{1}$. Similarly to the above
arguement, we have the commutative diagram

${{\mathcal{G}r_{\mathrm{SL}_{2},0}^{-}}}$${{\mathcal{G}r_{\mathrm{G},0}^{-}}}$${{(\mathfrak{sl}_{2})_{1}}}$${{\mathfrak{g}_{1}}}$$\scriptstyle{\pi_{\mathrm{SL}_{2}}}$$\scriptstyle{\Psi}$$\scriptstyle{\pi}$

where ($\mathfrak{sl}_{2})_{1}$ is the eigenspace of eigenvalue $-1$ under
$\sigma$. Now consider $g(t)\in\mathrm{SL}_{2}(\mathcal{O}^{-})^{\sigma}$

$g(t)=\begin{pmatrix}1&t^{-1}\\\
t^{-1}&1+t^{-2}\end{pmatrix}=\begin{pmatrix}0&t\\\
-t^{-1}&1+t^{2}\end{pmatrix}\begin{pmatrix}t^{2}&0\\\
0&t^{-2}\end{pmatrix}\begin{pmatrix}1&0\\\ t&1\end{pmatrix}.$

Then $\phi(g(t))\in
G(\mathcal{O})^{\sigma}n^{\check{\theta}_{0}}G(\mathcal{O})^{\sigma}$ and
$\phi(g(t))\cdot e_{0}\in\mathcal{G}r_{2\gamma_{0}}\cap\mathcal{G}r_{0}^{-}$.
The result follows from

$\pi(\phi(g(t))\cdot e_{0})=\Psi(\pi_{\operatorname{SL}_{2}}(g(t)\cdot
e_{0}))=\Psi\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$

where the latter is not nilpotent. It also follows that,
$\pi(\mathcal{G}r_{2\gamma_{0}})\not\subseteq\mathcal{N}_{\mathfrak{p}}$. ∎

######  Proposition 2.3.

For $\bar{\lambda}\in X_{*}(T)_{\sigma}^{+}$, if
$\pi(\mathcal{G}r_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-})\subset\mathcal{N}_{\mathfrak{p}}$,
then $\bar{\lambda}$ is small.

###### Proof.

Since $\mathcal{G}r_{0}^{-}$ is an open subset of $\mathcal{G}r$,
$\pi(\overline{\mathcal{G}r}_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-})\subset\mathcal{N}_{\mathfrak{p}}$.
By Lemma 2.2,
$\mathcal{G}r_{2\gamma_{0}}\nsubseteq\overline{\mathcal{G}r}_{\bar{\lambda}}$
which means $\bar{\lambda}\nsucceq 2\gamma_{0}$. ∎

Define the following anti-involution

$\iota:G(\mathcal{K})\to G(\mathcal{K}),\hskip 14.22636ptg(t)\mapsto
g(-t)^{-1}.$

It can be checked that $\iota$ commutes with $\sigma$, and $\iota$ preserves
$G(\mathcal{K})^{\sigma},G(\mathcal{O})^{\sigma}$ and $K^{-}$. This induces
the map

$\iota:\mathcal{G}r_{0}^{-}\to\mathcal{G}r_{0}^{-},\hskip 14.22636ptg(t)\cdot
e_{0}\mapsto g(-t)^{-1}\cdot e_{0}.$

The following lemma will be used in Section 4.

######  Lemma 2.4.

For $\bar{\lambda}\in X_{*}(T)_{\sigma}^{+}$,
$\iota(\mathcal{M}_{\bar{\lambda}})\subset\mathcal{M}_{\bar{\lambda}}$.

###### Proof.

It suffices to prove $\iota(n^{\lambda})\in\mathcal{M}_{\bar{\lambda}}$ for
each $\lambda\in X_{*}(T)^{+}$.

$\displaystyle\iota(n^{\lambda})$
$\displaystyle=\iota(\epsilon^{\sigma\lambda+2\sigma^{2}\lambda+...+(r-1)\sigma^{r-1}\lambda}t^{\sum_{i=0}^{r-1}\sigma^{i}\lambda})$
$\displaystyle=\epsilon^{-(\sigma\lambda+2\sigma^{2}\lambda+...+(r-1)\sigma^{r-1}\lambda)}(-1)^{\sum_{i=0}^{r-1}\sigma^{i}\lambda}t^{-\sum_{i=0}^{r-1}\sigma^{i}\lambda}$
$\displaystyle=(-1)^{\sum_{i=0}^{r-1}\sigma^{i}\lambda}n^{-\lambda}.$

Since $(-1)^{\sum_{i=0}^{r-1}\sigma^{i}\lambda}$ is fixed by $\sigma$,
$\iota(n^{\lambda})\in
G(\mathcal{O})^{\sigma}n^{-\lambda}G(\mathcal{O})^{\sigma}$. Let $W$ be the
Weyl group of $G$ with respect to the maximal torus $T$ and $\omega_{0}$ the
longest element of $W$. We can choose a representative $\dot{\omega}_{0}\in G$
of $\omega_{0}$ such that $\sigma(\dot{\omega}_{0})=\dot{\omega}_{0}$, cf.
[HS, Section 2.3].

When $G$ is of type $D_{2\ell}$ with $\ell\geq 2$, $w_{0}=-1$; otherwise,
$w_{0}=-\sigma$ and $\sigma$ is of order $2$, cf. [Hu2, Ex 5, p.71]. If
$w_{0}=-1$, it is easy to see that $n^{-\lambda}=w_{0}n^{\lambda}w_{0}^{-1}$.
If $w_{0}=-\sigma$ and $\sigma$ has order 2,

$n^{-\lambda}=(-1)^{-\sigma\lambda}t^{-(\lambda+\sigma\lambda)}=(-1)^{w_{0}\lambda}t^{w_{0}(\lambda+\sigma\lambda)}=w_{0}(-1)^{\lambda}t^{\lambda+\sigma\lambda}w_{0}^{-1}=w_{0}(-1)^{\lambda+\sigma\lambda}n^{\lambda}w_{0}^{-1}.$

In any case, $\iota(n^{\lambda})\in
G(\mathcal{O})^{\sigma}n^{\lambda}G(\mathcal{O})^{\sigma}$. ∎

##  3\. Nilpotent orbits in the space of self-adjoint maps

In this section, we will review some facts on the nilpotent orbits in certain
symmetric spaces. These results are known, cf. [Se]. We provide proofs here,
as the proofs in [Se] are omitted.

Let $B=\langle\cdot,\cdot\rangle$ be a nondegenerate symmetric or skew-
symmetric bilinear form on a vector space $V=\mathbb{C}^{m}$ and $\mathcal{A}$
the set of self-adjoint linear maps under the bilinear form. In this section,
we describe the classification of nilpotent orbits in the space $\mathcal{A}$
in Theorem 3.3, Theorem 3.4 and Theorem 3.6.

The isometry group of the form $B$ is

$I_{B}=\\{g\in\mathrm{GL}(V)\mid\langle gu,gv\rangle=\langle u,v\rangle\text{
for all }u,v\in V\\},$

whose Lie alegbra is

(5) $\mathfrak{g}_{B}:=\\{X\in\mathfrak{sl}(V)\mid\langle Xu,v\rangle+\langle
u,Xv\rangle=0\text{ for all }u,v\in V\\}.$

When $B$ is symplectic, $\dim V$ is even, $I_{B}\cong\mathrm{Sp}_{2n}$ and
$\mathfrak{g}_{B}\simeq\mathfrak{sp}_{2n}$ where $m=2n$. When $B$ is
symmetric, $I_{B}\cong\mathrm{O}_{m}$ and
$\mathfrak{g}_{B}\cong\mathfrak{so}_{m}$.

The group $I_{B}$ acts on the space of self-adjoint linear maps

(6) $\mathcal{A}=\\{X\in\operatorname{End}(V)\mid\langle Xu,v\rangle=\langle
u,Xv\rangle\text{ for all }u,v\in V\\}$

by conjugation. The orbit is called nilpotent if it is the orbit of a
nilpotent element of $\mathcal{A}$.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Suppose that
$\mathfrak{g}$ has $\mathbb{Z}_{m}$-grading

$\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}_{m}}\mathfrak{g}_{i}$

so that $[\mathfrak{g}_{k},\mathfrak{g}_{\ell}]\subset\mathfrak{g}_{k+\ell}$.
We have the following graded version of Jacobson-Morozov theorem.

######  Lemma 3.1.

Let $X$ be a nonzero nilpotent element in $\mathfrak{g}_{i}$. There exists an
$\mathfrak{sl}_{2}$-triple $H,X,Y$ such that $H\in\mathfrak{g}_{0}$ and
$Y\in\mathfrak{g}_{-i}$.

###### Proof.

This Lemma follows from the usual Jacobson-Morozov Theorem, and the proof is
similar to [EK, Lemma 1.1]. ∎

For each $A\in\mathfrak{sl}(V)$, as a linear map, we denote its adjoint by
$A^{*}$ under the form $B$. We define an involution $\sigma$ on
$\mathfrak{g}=\mathfrak{sl}(V)$ by

(7) $\sigma(A)=-A^{*}.$

Then $\mathfrak{g}$ is the direct sum of eigenspaces,
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$. Thus
$\mathfrak{g}_{0}=\mathfrak{g}_{B}$ and $\mathfrak{g}_{1}=\mathcal{A}$. Fix a
nonzero nilpotent element $X\in\mathcal{A}$. By Lemma 3.1, there exists
$Y\in\mathcal{A}$ and $H\in\mathfrak{g}_{B}$, such that $X,Y,H$ is an
$\mathfrak{sl}_{2}$-triple. This induces a representation of
$\mathfrak{sl}_{2}$ on $V$ and hence we have a decomposition

(8) $V=\bigoplus_{r\geq 0}M(r)$

where $M(r)$ is a finite direct sum of irreducible representation of
$\mathfrak{sl}_{2}$ of highest weight $r$. For $r\geq 0$, let $H(r)$ be the
highest weight space in $M(r)$. Define a new bilinear form $(\cdot,\cdot)$ on
$H(r)$ by

$(u,v)_{r}=\langle u,Y^{r}v\rangle.$

######  Lemma 3.2.

For any $r\geq 0$, $(\cdot,\cdot)_{r}$ is symplectic (resp. symmetric) if $B$
is symplectic (resp. symmetric).

###### Proof.

We assume $B$ is symplectic. The proof is similar when $B$ is symmetric. It is
easy to see that $(\cdot,\cdot)_{r}$ is skew-symmetric. It remains to show
that $(\cdot,\cdot)_{r}$ is nondegerate. Let $V_{r}$ be an $r$-weight space in
$\mathbb{C}^{2n}$. For any $u\in V_{r}$, $v\in V_{s}$ with $s\neq-r$,

$(r+s)\langle u,v\rangle=\langle ru,v\rangle+\langle u,sv\rangle=\langle
Hu,v\rangle+\langle u,Hv\rangle=0$

This implies that $V_{r}$ and $V_{s}$ are
$\langle\cdot,\cdot\rangle$-orthogonal. Let

$W=\text{Span}\\{u\in V_{r}\mid u=Yv\text{ for some }v\in\mathbb{C}^{2n}\\}.$

It can be seen that $V_{r}=H(r)\oplus W$. For $u\in H(r)$ and $v\in W$, write
$v=Yv^{\prime}$,

$(u,v)_{r}=\langle u,Y^{r}v\rangle=\langle u,Y^{r+1}v^{\prime}\rangle=\langle
Y^{r+1}u,v^{\prime}\rangle=0.$

Hence $H(r)$ is $(\cdot,\cdot)_{r}$-orthogonal to $W$.

We claim that $\langle\cdot,\cdot\rangle:(Y^{r}\cdot H(r))\times
H(r)\to\mathbb{C}$ is nondegenerate. Let $u=Y^{r}u^{\prime}\in Y^{r}\cdot
H(r)$ be such that $\langle u,v\rangle=0$ for all $v\in H(r)$. For each
$w\in\mathbb{C}^{2n}$, write $w=\sum_{s}w_{s}$ where each $w_{s}$ belongs to
$V_{s}$. Since $u\in V_{-r}$, $\langle u,w_{s}\rangle=0$ for $s\neq r$. Write
$w_{r}=w_{1}+w_{2}$ where $w_{1}\in H(r)$ and $w_{2}=Yw^{\prime}_{2}\in W$. By
the assumption $\langle u,w_{1}\rangle=0$ and hence

$\langle u,w_{r}\rangle=\langle u,w_{2}\rangle=\langle
Y^{r}u^{\prime},Yw^{\prime}_{2}\rangle=\\-\langle
Y^{r+1}u^{\prime},w^{\prime}_{2}\rangle=0.$

We obtain $\langle u,w\rangle=0$ for any $w$ and hence $u=0$. This claim
implies that $(\cdot,\cdot)_{r}$ is nondegenerate. ∎

A partition of a positive integer is denoted by a tuple
$[d_{1},d_{2},...,d_{k}]$ of positive integers. We use the exponent notation

$[a_{1}^{i_{1}},...,a_{r}^{i_{r}}]$

to denote a partition where $a_{j}^{i_{j}}$ means there are $i_{j}$ copies of
$a_{j}$. For example, $[3^{2},1^{4}]=[3,3,1,1,1,1]$ is a partition of 10. Put
$r_{i}=|\\{j\mid d_{j}=i\\}|$ and $s_{i}=|\\{j\mid d_{j}\geq i\\}|$. In fact,
each partition can be illustrated by Young diagram and then $s_{i}$ is the
$i$-th part of the dual diagram. The following Theorem gives the
parametrization of nilpotent $I_{B}$-orbits in $\mathcal{A}$.

######  Theorem 3.3.

There exists one-to-one correspondences

$\\{\text{nilpotent }\mathrm{Sp}_{2n}\text{-orbits in
}\mathcal{A}\\}\leftrightarrow\left\\{\begin{array}[]{c}\text{partitions of
}2n\text{ such that}\\\ \text{every part occurs with even
multiplicity}\end{array}\right\\}.$

and

$\\{\text{nilpotent }\mathrm{O}_{m}\text{-orbits in
}\mathcal{A}\\}\leftrightarrow\\{\text{partitions of }m\\}.$

###### Proof.

The proof is similar to [CM, Lemma 5.1.17]. For the case that $B$ is
symplectic, it suffices to show that any nilpotent element in $\mathcal{A}$
gives arise the partition of $2n$ such that every part occurs with even
multiplicity. Given nilpoent $X\in\mathcal{A}$, a number of Jordan blocks of
size $r+1$ equals to the multiplicity of $M(r)$ in $\mathbb{C}^{2n}$ which is
exactly $\dim H(r)$. By Lemma 3.2, $\dim H(r)$ is even for every $r$.

If $B$ is symmetric, there are no constraints on $\dim H(r)$ which means there
are no conditions on partitions of $m$. ∎

######  Theorem 3.4.

There exists one-to-one correspondence

$\\{\text{nilpotent }\mathrm{SO}_{2n+1}\text{-orbits in
}\mathcal{A}\\}\leftrightarrow\left\\{\text{partitions of }2n+1\right\\}.$

###### Proof.

Since $\mathrm{O}_{2n+1}=\mathrm{SO}_{2n+1}\times\\{\pm I_{2n+1}\\}$, the
orbits under $\mathrm{O}_{2n+1}$ and $\mathrm{SO}_{2n+1}$ coincide. The
results immediately follows from Theorem 3.3 ∎

Consider the case that $B$ is symmetric and $m=2n$. Given nilpotent elements
$X,X^{\prime}\in\mathcal{A}$ whose partitions are the same and have at least
one odd part. Say that they are conjugated by an element
$g\in\mathrm{O}_{2n}$. If $\det g=1$, we conclude that $X,X^{\prime}$ are in
the same $\mathrm{SO}_{2n}$-orbits. Suppose that $\det g=-1$. We modify this
$g$ so that it has determinant $1$. Note that either $X$ or $X^{\prime}$ gives
the same decomposition (8). An odd part in the partition corresponds to an odd
dimensional irreducible representation $S$ of $\mathfrak{sl}_{2}$ in
$\mathbb{C}^{2n}$. We put $h=g$ except that $h(v)=-g(v)$ for $v\in S$.
Therefore, $\det h=1$, and $X$ and $X^{\prime}$ are conjugated by $h$. If
there is no odd parts, we need the following Lemma.

######  Lemma 3.5.

Let $X$ be a nilpotent element in $\mathcal{A}$ whose partition contains only
even parts, and $k\in\mathrm{O}_{2n}$ such that $k\cdot X=kXk^{-1}=X$. Then
$\det k=1$.

###### Proof.

Let $\mathrm{O}_{2n}^{X}$ be the stabilizer group of $O_{2n}$ at $X$. Then
$k\in\mathrm{O}_{2n}^{X}$. By multiplicative Jordan decomposition, cf. [Bo,
Theorem 4.4, p.83], let $k_{s}\in O_{2n}^{X}$ be the semisimple part of $k$.
Then $\det k_{s}=\det k$. Hence we may assume that $k$ is semisimple. Let
$\sigma$ be an automorphism on $\mathfrak{g}=\mathfrak{sl}(V)$ defined by (7).
Then $\sigma$ commutes with ${\rm Ad}k$ on $\mathfrak{g}$, as
$k\in\mathrm{O}_{2n}$. Then
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$, and we have the
decomposition of $k$-stabilizers
$\mathfrak{g}^{k}=\mathfrak{g}_{0}^{k}\oplus\mathfrak{g}_{1}^{k}$ where
$\mathfrak{g}_{i}^{k}=\mathfrak{g}_{i}\cap\mathfrak{g}^{k}$. Since
$\mathfrak{g}^{k}$ is reductive and $X\in\mathfrak{g}_{1}^{k}$, by Lemma 3.1,
there exists an $\mathfrak{sl}_{2}$-triple $H,X,Y$ such that
$X,Y\in\mathfrak{g}_{1}^{k}$ and $H\in\mathfrak{g}_{0}^{k}$ and hence we have
the decomposition (8). It is easy to see that $k(M(r))\subset M(r)$, and also,
$k$ stabilizes each weight space of $M(r)$.

Recall that $\langle\cdot,\cdot\rangle$ is a nondegenerate symmetric form on
$V=\mathbb{C}^{2n}$ and a form on $H(r)$ given by $(u,v)_{r}=\langle
u,Y^{r}v\rangle$ is also symmetric for any $r\geq 0$. For any $u,v\in H(r)$,

$(ku,kv)_{r}=\langle ku,Y^{r}kv\rangle=\langle ku,kY^{r}v\rangle=\langle
u,Y^{r}v\rangle=(u,v)_{r}.$

Hence $k\big{|}_{H(r)}\in\mathrm{O}(H(r))$ for any $r$. In particular
$\det\left(k\big{|}_{H(r)}\right)=\pm 1$.

Let $M(r)_{\ell}$ be an $\ell$-weight space, and $L(r)$ the lowest weight
space in $M(r)$. Observe that $X\big{|}_{M(r)_{\ell}}$ is an isomorphism from
$M(r)_{\ell}$ to $M(r)_{\ell+2}$ and the diagram

${{L(r)}}$${{M(r)_{\ell}}}$${{M(r)_{\ell+2}}}$${{H(r)}}$${{L(r)}}$${{M(r)_{\ell}}}$${{M(r)_{\ell+2}}}$${{H(r)}}$$\scriptstyle{X\big{|}_{M(r)_{\ell}}}$$\scriptstyle{k\big{|}_{M(r)_{\ell}}}$$\scriptstyle{k\big{|}_{M(r)_{\ell+2}}}$$\scriptstyle{X\big{|}_{M(r)_{\ell}}}$$\scriptstyle{k\big{|}_{L(r)}}$$\scriptstyle{k\big{|}_{H(r)}}$

commutes. Then $k\big{|}_{M(r)_{\ell}}$ has the same determinant for all
$\ell$. Since $X$ has only even parts, a number of weight spaces in $M(r)$ is
even for each $r$. Then

$\det
k=\prod_{r}\det\left(k\big{|}_{M(r)}\right)=\prod_{r}\prod_{\ell}\det\left(k\big{|}_{M(r)_{\ell}}\right)=\prod_{r}\left(\det\left(k\big{|}_{H(r)}\right)\right)^{\ell}=1$

as desired. ∎

Now suppose that $X,X^{\prime}\in\mathcal{A}$ have the same partition and
contain only even parts. If $\det g=1$, they are in the same
$\mathrm{SO}_{2n}$-orbits. Suppose that $\det g=-1$ and they are conjugated by
another element $h\in\mathrm{SO}_{2n}$. Say $g\cdot X=X^{\prime}=h\cdot X$ and
let $k=g^{-1}h$. Then $\det k=(\det g^{-1})(\det h)=-1$ but this contradicts
to Lemma 3.5. In this case, it means $X,X^{\prime}$ are conjugated by an
element in $\mathrm{O}_{2n}$ of determinant -1 only. We have the following
theorem:

######  Theorem 3.6.

Nilpotent $\mathrm{SO}_{2n}$-orbits in $\mathcal{A}$ are parametrized by
partitions of $2n$ except that the partitions with only even parts correspond
to two orbits.

For each nilpotent element $X$ having the partition $[d_{1},d_{2},...,d_{k}]$,
we denote the $I_{B}$-orbits of $X$ by
$\mathcal{O}_{X}$,$\mathcal{O}_{[d_{1},d_{2},...,d_{k}]}$, or simply
$[d_{1},d_{2},...,d_{k}]$. We are now ready to compute the dimension of
nilpotent $I_{B}$-orbits.

######  Theorem 3.7.

Let $X$ be a nilpotent element in $\mathcal{A}$. Then the dimension of
$I_{B}$-orbit of $X$ is

$\dim\mathcal{O}_{X}=\frac{1}{2}\left(m^{2}-\sum_{i}s_{i}^{2}\right).$

###### Proof.

Suppose that $B$ is symplectic on $\mathbb{C}^{m}$, $m=2n$. Recall that we
have the decomposition (8). For each $Z\in\mathfrak{g}_{B}^{X}$, we consider
how $Z$ sends $M(d)$. Suppose that $Z$ sends $M(d)$ to $M(e)$ for $d\neq e$.
Since $Z$ and $X$ commute, by theory of representations of
$\mathfrak{sl}_{2}$, $Z$ is uniquely determined by a linear map $L(d)\to M(e)$
where $L(d)$ is the lowest weight space in $M(d)$. In this case, $Z$ sends
$L(e)$ to $M(d)$. Thus we can assume $d<e$. For $v\in L(d)$, $X^{d+1}v=0$ and
then $ZX^{d+1}v=X^{d+1}Zv=0$ is the constraint of $Zv=0$. Note that
$r_{d+1}=\dim L(d)$. Therefore the set of all linear maps from $L(d)$ to
$M(e)$ forms a vector space of dimension $(d+1)r_{d+1}r_{e+1}$.

Now consider the case $Z$ sends $M(d)$ to itself. We consider where $Z$ sends
$L(d)$ to. Suppose that $Z$ sends $L(d)$ to $H(d)$. We define a new bilinear
form $(\cdot,\cdot)_{d}$ on $L(d)$ given by $(u,v)_{d}=\langle u,Zv\rangle$.
It can be checked $(\cdot,\cdot)_{d}$ is symmetric and completely determine
the map $Z$. The set of all such $(\cdot,\cdot)_{d}$ forms a vector space of
dimension $\frac{1}{2}r_{d+1}(r_{d+1}+1)$. If $Z$ sends $L(d)$ to
$(d-2)$-weight space in $M(d)$, we define the new form by $(u,v)_{d-2}=\langle
u,XZv\rangle$. Again, this form is symmetric and completely determine the map
$Z$. Continue this process up to the case $Z$ sends $L(d)$ to itself. We
obtain

$\displaystyle\dim\mathfrak{k}^{X}$ $\displaystyle=\sum_{d\geq
0}\left[(d+1)\left(\sum_{e>d}r_{d+1}r_{e+1}\right)+\dfrac{d+1}{2}r_{d+1}(r_{d+1}+1)\right]$
$\displaystyle=\left[r_{1}(r_{2}+r_{3}+\cdots)+\dfrac{1}{2}r_{1}(r_{1}+1)\right]+\left[2r_{2}(r_{3}+r_{4}+\cdots)+\dfrac{2}{2}r_{2}(r_{2}+1)\right]$
$\displaystyle\hskip
56.9055pt+\left[3r_{3}(r_{4}+r_{5}+\cdots)+\dfrac{3}{2}r_{3}(r_{3}+1)\right]+\cdots$
$\displaystyle=\left[\dfrac{1}{2}r_{1}(r_{1}+2r_{2}+2r_{3}+...)+\dfrac{1}{2}r_{1}\right]+\left[\dfrac{2}{2}r_{2}(r_{2}+2r_{3}+2r_{4}+...)+\dfrac{2}{2}r_{2}\right]$
$\displaystyle\hskip
56.9055pt+\left[\dfrac{3}{2}r_{3}(r_{3}+2r_{4}+2r_{5}+...)+\dfrac{3}{2}r_{3}\right]+\cdots$
$\displaystyle=\left[\dfrac{1}{2}(s_{1}-s_{2})(s_{1}+s_{2})+\dfrac{1}{2}r_{1}\right]+\left[\dfrac{2}{2}(s_{2}-s_{3})(s_{2}+s_{3})+\dfrac{2}{2}r_{2}\right]$
$\displaystyle\hskip
56.9055pt+\left[\dfrac{3}{2}(s_{3}-s_{4})(s_{3}+s_{4})+\dfrac{3}{2}r_{3}\right]+\cdots$
$\displaystyle=\dfrac{1}{2}\sum_{i}s_{i}^{2}+\dfrac{1}{2}(r_{1}+2r_{2}+3r_{3}+\cdots)+\cdots$
$\displaystyle=\dfrac{1}{2}\sum_{i}s_{i}^{2}+\dfrac{1}{2}\sum_{i}s_{i}.$
$\displaystyle=n+\dfrac{1}{2}\sum_{i}s_{i}^{2}$

and hence

$\dim\mathcal{O}_{X}=\dim\mathfrak{g}_{B}-\dim\mathfrak{g}_{B}^{X}=(2n^{2}+n)-\left(n+\dfrac{1}{2}\sum_{i}s_{i}^{2}\right)=\dfrac{m^{2}}{2}-\frac{1}{2}\sum_{i}s_{i}^{2}.$

If $B$ is symmetric, the arguement is similar except that the form $(u,v)_{d}$
is symplectic and hence the vector space consisting of such forms
$(\cdot,\cdot)_{d}$ has dimension $\frac{1}{2}r_{d+1}(r_{d+1}-1)$. ∎

######  Remark 3.8.

The dimension of $I_{B}$-orbits can also be obtained from [Se, 3.1.c, 3.2.b],
where the formulae are not uniform and the proofs are also omitted.

The closure relation on the set of nilpotent orbits in $\mathcal{A}$ is given
by

$\mathcal{O}_{Y}\preceq\mathcal{O}_{X}\text{ if
}\mathcal{O}_{Y}\subset\overline{\mathcal{O}}_{X}$

for nilpotent elements $X,Y\in\mathcal{A}$. Given two partitions
$\bar{d}=[d_{1},...,d_{N}],\bar{f}=[f_{1},...,f_{N}]$ of $N$ (put some
$d_{i},f_{i}=0$ if needed). We say that $\bar{d}$ dominates $\bar{f}$, denoted
by $\bar{d}\succeq\bar{f}$ if

$d_{1}\geq f_{1}$

$d_{1}+d_{2}\geq f_{1}+f_{2}$

$\vdots$

$d_{1}+...+d_{N}\geq f_{1}+...+f_{N}.$

######  Theorem 3.9.

Let $X,Y$ be nilpotent elements in $\mathcal{A}$ having partition
$\bar{d},\bar{f}$, respectively. Then
$\mathcal{O}_{\bar{d}}\succeq\mathcal{O}_{\bar{f}}$ if and only if $\bar{d}$
dominates $\bar{f}$.

###### Proof.

See [Oh, Theorem 1]. ∎

For example, all nilpotent $\mathrm{Sp}_{10}$-orbits in
$\mathcal{A}\subset\mathfrak{sl}_{10}$ are

$\mathcal{O}_{[5^{2}]}\succeq\mathcal{O}_{[4^{2},1^{2}]}\succeq\mathcal{O}_{[3^{2},2^{2}]}\succeq\mathcal{O}_{[3^{2},1^{4}]}\succeq\mathcal{O}_{[2^{4},1^{2}]}\succeq\mathcal{O}_{[2^{2},1^{6}]}\succeq\mathcal{O}_{[1^{10}]}.$

The dimensions are 40, 36, 32, 28, 24, 16, 0, respectively.

##  4\. The connection between Schubert cells and nilpotent $K$-orbits

The goal of this section is to show that for any small $\bar{\lambda}$,
$\mathcal{M}_{\bar{\lambda}}$ is sent to the nilpotent cone
$\mathcal{N}_{\mathfrak{p}}$ by the map $\pi$, and show that how each
$\mathcal{M}_{\bar{\lambda}}$ is sent to nilpotent $K$-orbits in the
$\mathcal{N}_{\mathfrak{p}}$. Theorem 4.2 describes the image
$\pi(\mathcal{M}_{\bar{\lambda}})$ where the proofs are provided by case-by-
case consideration in this section.

Let $X_{N}$ be the type of Dynkin diagram of $G$ and $\sigma$ the diagram
automorphism on $G$ of order $r$, denoted by the pair $(X_{N},r)$. We consider
the cases $(X_{N},r)=(A_{2\ell},2),(A_{2\ell-1},2)$, and $(D_{\ell+1}2)$. Then
$H=(\check{G})^{\sigma}$ is either of type $B_{\ell}$ or $C_{\ell}$. We make
the following labelling for simple roots $\gamma_{i}$:

(9)
$\begin{cases}\gamma_{1}=\gamma_{\\{1,2\ell-1\\}},\dots,\gamma_{\ell-1}=\gamma_{\\{\ell-1,n+1\\}},\gamma_{n}=\gamma_{\\{\ell\\}}&\quad(X_{N},r)=(A_{2\ell-1},2);\\\
\gamma_{1}=\gamma_{\\{1,2\ell\\}},\dots,\gamma_{\ell-1}=\gamma_{\\{\ell-1,\ell+2\\}},\gamma_{\ell}=\gamma_{\\{\ell,\ell+1\\}}&\quad(X_{N},r)=(A_{2\ell},2);\\\
\gamma_{1}=\gamma_{\\{1\\}},\dots,\gamma_{\ell-1}=\gamma_{\\{\ell-1\\}},\gamma_{\ell}=\gamma_{\\{\ell,\ell+1\\}}&\quad(X_{N},r)=(D_{\ell+1},2).\\\
\end{cases}$

This labelling of vertices of type $B_{\ell}$ and $C_{\ell}$ agrees with the
labelling in [Ka, TABLE Fin, p.53]. Then the highest short root $\gamma_{0}$
of $H$ can be described in the following table.

$(X_{N},r)$ | $G$ | $H$ | Simple roots of $H$ | Highest short root $\gamma_{0}$ of $H$
---|---|---|---|---
$(A_{2\ell},2)$ | $\mathrm{SL}_{2\ell+1}$ | $\mathrm{PSO}_{2\ell+1}$ |  $\gamma_{i}=\bar{\check{\alpha}}_{i}=\bar{\check{\alpha}}_{2\ell-i+1}$, $i=1,...,\ell$ | $\gamma_{1}+\gamma_{2}+...+\gamma_{\ell}.$
$(A_{2\ell-1},2)$ | $\mathrm{SL}_{2\ell}$ | $\mathrm{PSp}_{2\ell}$ |  $\gamma_{i}=\bar{\check{\alpha}}_{i}=\bar{\check{\alpha}}_{2\ell-i}$, $i=1,...,\ell$ | $\gamma_{1}+2\gamma_{2}+...+2\gamma_{\ell-1}+\gamma_{\ell}.$
$(D_{\ell+1},2)$ | $\mathrm{Spin}_{2\ell+2}$ | $\mathrm{PSO}_{2\ell+1}$ |  $\gamma_{i}=\bar{\check{\alpha}}_{i},i=1,...,\ell-1$ $\gamma_{\ell}=\bar{\check{\alpha}}_{\ell}=\bar{\check{\alpha}}_{\ell+1}$ | $\gamma_{1}+\gamma_{2}+...+\gamma_{\ell}.$

From Section 2.1, we can identify the weight lattice of $H$ with
$X_{*}(T)_{\sigma}$, and the set of dominant weights of $H$ with
$X_{*}(T)^{+}_{\sigma}$. Then, from the construction of root system of
classical Lie algebras given in [Hu2, §12], we can make the following
identifications:

$X_{*}(T)_{\sigma}\cong\begin{cases}\mathbb{Z}^{\ell}&\text{if }H=B_{\ell};\\\
\\{(a_{1},...,a_{\ell})\in\mathbb{Z}^{\ell}\mid a_{1}+\cdots+a_{\ell}\in
2\mathbb{Z}\\}&\text{if }H=C_{\ell}\end{cases}$

and

$X_{*}(T)_{\sigma}^{+}\cong\\{(a_{1},...,a_{\ell})\in X_{*}(T)_{\sigma}\mid
a_{1}\geq\cdots\geq a_{\ell}\geq 0\\}$

for any cases of $H$. This identification preserves the relation on
$X_{*}(T)_{\sigma}^{+}$ and the dominance relation on
$\\{(a_{1},...,a_{\ell})\in X_{*}(T)_{\sigma}\mid a_{1}\geq\cdots\geq
a_{\ell}\geq 0\\}$.

In the Table 1, we can further make the following identifications for simple
roots and fundamental weights of $H$. Those fundamental weights follows from
[Hu2, Table 1., p.69].

$H$ | Simple roots | $\gamma_{0}$ | Fundamental weights
---|---|---|---
$B_{\ell}$ |  $\gamma_{1}=(1,-1,0,0,...,0)$ $\gamma_{2}=(0,1,-1,0,...,0)$ $\vdots$ $\gamma_{\ell-1}=(0,0,0,...,1,-1)$ $\gamma_{\ell}=(0,0,0,...,0,1)$ | $(1,0,0,...,0,0)$ |  $\omega_{j}=(1^{j}0^{\ell-j})$, $j=0,...,\ell-1$ $\omega_{\ell}=(\dfrac{1}{2},\dfrac{1}{2},...,\dfrac{1}{2})$
$C_{\ell}$ |  $\gamma_{1}=(1,-1,0,0,...,0)$ $\gamma_{2}=(0,1,-1,0,...,0)$ $\vdots$ $\gamma_{\ell-1}=(0,0,0,...,1,-1)$ $\gamma_{\ell}=(0,0,0,...,0,2).$ | $(1,1,0,...,0,0)$ | $\omega_{j}=(1^{j}0^{\ell-j}),j=0,...,\ell$
Table 1. Simple roots, highest short root, and fundamental weights of $H$ in
term of tuples

The following lemma is well-known, cf.[AH]. We give a self-contained proof
here.

######  Lemma 4.1.

All small dominant weights of $H$ are

  1. (1)

$\omega_{j}=(1^{j}0^{\ell-j})$, $j=0,...,\ell-1$,
$2\omega_{\ell}=(1,1,...,1)$, if $H$ has the type $B_{\ell}$.

  2. (2)

$\omega_{1}+\omega_{2j+1}=(21^{2j}0^{\ell-2j-1})$,
$j=0,...,\lfloor\frac{\ell-1}{2}\rfloor$
$\omega_{2j}=(1^{2j}0^{\ell-2j})$, $j=0,...,\lfloor\frac{\ell}{2}\rfloor$, if
$H$ has the type $C_{\ell}$.

###### Proof.

Suppose that $H$ has the type $B_{\ell}$. The highest short root is
$\gamma_{0}=(1,0,...,0)$. By definition, a dominant weight
$(a_{1},...,a_{\ell})\in X_{*}(T)_{\sigma}^{+}$ is small if and only if
$(a_{1},...,a_{\ell})\nsucceq(2,0,...,0)$ which is equivalent to $a_{1}\leq
1$. This proves the first part.

Now assume that $H$ has the type $C_{\ell}$. The highest short root is
$\gamma_{0}=(1,1,...,0)$. Let $(a_{1},...,a_{\ell})\in X_{*}(T)_{\sigma}^{+}$
be a small dominant weight. Then $(a_{1},...,a_{\ell})\nsucceq(2,2,...,0)$ and
so $a_{1}\leq 2$. If $a_{1}=1$, then
$(a_{1},...,a_{\ell})=(1^{2j}0^{\ell-2j})$. If $a_{1}=2$, then $a_{2}<2$ and
hence $(a_{1},...,a_{\ell})=(21^{2j}0^{\ell-2j-1})$. ∎

Let $\bar{\mu}$ be the maximal element among all small dominant weights of
$H$, then

$\mathcal{G}r_{\mathrm{sm}}=\coprod_{{\bar{\lambda}}\preceq\bar{\mu},\\\
\bar{\lambda}\text{
small}}\mathcal{G}r_{\bar{\lambda}}=\overline{\mathcal{G}r}_{\bar{\mu}}.$

Since $\mathcal{G}r_{\mathrm{sm}}$ is irreducible and $\mathcal{M}$ is an open
subset of $\mathcal{G}r_{sm}$, $\mathcal{M}$ is irreducible.

The following theorem is the main result of this section.

######  Theorem 4.2.

If $\bar{\lambda}$ is small, then $\pi(\mathcal{M}_{\bar{\lambda}})$ is
contained in $\mathcal{N}_{\mathfrak{p}}$. Moreover, the image
$\pi(\mathcal{M}_{\bar{\lambda}})$ can be described as the union of nilpotent
orbits as the following table:

$(X_{N},r)$ |  Small dominant weight $\overline{\lambda}$ of $H$ |  Orbits in $\pi(\mathcal{M}_{\bar{\lambda}})$
---|---|---
$(A_{2\ell},2)$ | $(1^{j}0^{\ell-j}),j=0,1,\dots,\ell$ | $[2^{j}1^{2\ell-2j+1}]$
$(A_{2\ell-1},2)$ | $(1^{2j}0^{\ell-2j}),j=0,1,\dots,\lfloor\frac{\ell}{2}\rfloor$ | $[2^{2j}1^{2\ell-4j}]$
$(20^{\ell-1})$ | $0,[2^{2}1^{2\ell-4}]$
$(21^{2}0^{\ell-3})$ | $[2^{2}1^{2\ell-4}],[2^{4}1^{2\ell-8}],[3^{2}1^{2\ell-6}]$
$(21^{2j}0^{\ell-2j-1}),j=2,\dots,\lfloor\frac{\ell-3}{2}\rfloor$ |  $[2^{2j}1^{2\ell-4j}],[2^{2j+2}1^{2\ell-4j-4}],$ $[3^{2}2^{2j-2}1^{2\ell-4j-2}],[3^{2}2^{2j-4}1^{2\ell-4j+2}]$
$(21^{2\lfloor\frac{\ell-1}{2}\rfloor}0^{\ell-2\lfloor\frac{\ell-1}{2}\rfloor-1})$ |  $[2^{\ell-2}1^{4}],[2^{\ell}],$ $[3^{2}2^{\ell-4}1^{2}],[3^{2}2^{\ell-6}1^{4}]$ , if $\ell$ is even
|  $[2^{\ell-1}1^{2}]$, $[3^{2}2^{\ell-3}],[3^{2}2^{\ell-5}1^{4}]$ , if $\ell$
is odd
$(D_{\ell+1},2)$ | $(1^{j}0^{\ell-j}),j=0,1,\dots,\ell$ |  $0$, if $j=0$
$0,[31^{2\ell-1}]$, if $j$ is even, $j\geq 2$
|  |  $[31^{2\ell-1}]$, if $j$ is odd

where the nilpotent orbit $[a_{1}^{i_{1}},...,a_{r}^{i_{r}}]$ in the above
table means empty set if the associated partition is invalid for some small
$\ell$.

This theorem follows from Theorem 4.5, Theorem 4.6, Theorem 4.10, and Theorem
4.14, which will be proved separately case by case.

The partial orders of small dominant weights of $H$ are shown in the below
picture, where the partial order is compatible with the height.

${{2\omega_{\ell}}}$${{\omega_{1}+\omega_{2\lfloor\frac{\ell-1}{2}\rfloor+1}}}$${{\omega_{\ell-1}}}$${{\omega_{1}+\omega_{2\lfloor\frac{\ell-1}{2}\rfloor-1}}}$${{\omega_{\ell-2}}}$${{\omega_{1}+\omega_{2\lfloor\frac{\ell-1}{2}\rfloor-3}}}$${{\omega_{2\lfloor\frac{\ell}{2}\rfloor}}}$${{\omega_{2\lfloor\frac{\ell}{2}\rfloor-2}}}$${{\omega_{1}+\omega_{3}}}$${{\omega_{2}}}$${{2\omega_{1}}}$${{\omega_{4}}}$${{\omega_{1}}}$${{\omega_{2}}}$${{0}}$${{0}}$${\text{Type
}B_{\ell}}$${\text{Type }C_{\ell}}$

We first recall a crucial lemma from [AH, Lemma 4.3].

######  Lemma 4.3.

Let $g=\sum_{i=N}^{\infty}x_{i}t^{i}\in\mathrm{SL}_{n}(\mathcal{K})$, where
$x_{N}\neq 0$. Let ${\lambda}=(a_{1},a_{2},...,a_{n})$ be a tuple of integers
such that $a_{1}\geq a_{2}\cdots\geq a_{n}$ and $\sum a_{i}=0$, and
$g(t)\in\mathrm{SL}_{n}(\mathcal{O})t^{{\lambda}}\mathrm{SL}_{n}(\mathcal{O})$.
Then

  1. (1)

$N=a_{n}$.

  2. (2)

The rank of $x_{N}$ equals to the number of $j$ such that $a_{j}=a_{n}$.

  3. (3)

For any $s\geq 1$,

$\operatorname{rk}\begin{pmatrix}x_{N}&x_{N+1}&\cdots&x_{N+s-2}&x_{N+s-1}\\\
0&x_{N}&\cdots&x_{N+s-3}&x_{N+s-2}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&x_{N}&x_{N+1}\\\
0&0&\cdots&0&x_{N}\end{pmatrix}=\sum_{j=1}^{n}\max\\{s-(a_{j}-a_{n}),0\\}.$

We have the following lemma for the twisted version.

######  Lemma 4.4.

Let $g(t)=\sum_{i=N}^{\infty}x_{i}t^{i}\in G(\mathcal{K})^{\sigma}$ where
$x_{i}\in\operatorname{Mat}_{m\times m}$, $x_{N}\neq 0$. Let
$\bar{\lambda}=(a_{1},...,a_{\ell})\in X_{*}(T)_{\sigma}^{+}$ be such that
$g(t)\in G(\mathcal{O})^{\sigma}n^{\lambda}G(\mathcal{O})^{\sigma}$. Then

  1. (1)

$N=\begin{cases}-a_{1}&\text{if }(X_{N},r)=(A_{2\ell},2),(A_{2\ell-1},2);\\\
-2a_{1}&\text{if }(X_{N},r)=(D_{\ell+1},2).\end{cases}$

  2. (2)

The rank of $x_{N}$ is equal to the number of $j$ such that $a_{j}=a_{1}$.

###### Proof.

We can write

$\bar{\lambda}=(a_{1},...,a_{\ell})=\sum_{i=1}^{\ell}\left(\sum_{j=1}^{i}a_{j}\right)\gamma_{i}$

where $\gamma_{i}$ are simple roots of $H$ as labelled by (9). We choose a
representative $\lambda\in X_{*}(T)$ of $\bar{\lambda}$ by

$\lambda=\begin{cases}\sum_{i=1}^{\ell}\left(\sum_{j=1}^{i}a_{j}\right)\check{\alpha}_{i}&\text{if
}(X_{N},r)=(A_{2\ell},2),(D_{\ell+1},2);\\\
\sum_{i=1}^{\ell-1}\left(\sum_{j=1}^{i}a_{j}\right)\check{\alpha}_{i}+\dfrac{1}{2}\left(\sum_{j=1}^{\ell}a_{j}\right)\check{\alpha}_{\ell}&\text{if
}(X_{N},r)=(A_{2\ell-1},2)\\\ \end{cases}$

so that

$\lambda+\sigma\lambda=\begin{cases}\sum_{i=1}^{\ell}\left(\sum_{j=1}^{i}a_{j}\right)\check{\alpha}_{i}+\sum_{i=\ell+1}^{2\ell}\left(\sum_{j=1}^{2\ell-i+1}a_{j}\right)\check{\alpha}_{i}&\text{if
}(X_{N},r)=(A_{2\ell},2);\\\
\sum_{i=1}^{\ell}\left(\sum_{j=1}^{i}a_{j}\right)\check{\alpha}_{i}+\sum_{i=\ell+1}^{2\ell-1}\left(\sum_{j=1}^{2\ell-i}a_{j}\right)\check{\alpha}_{i}&\text{if
}(X_{N},r)=(A_{2\ell-1},2);\\\
\sum_{i=1}^{\ell-1}\left(\sum_{j=1}^{i}2a_{j}\right)\check{\alpha}_{i}+\left(\sum_{j=1}^{\ell}a_{j}\right)(\check{\alpha}_{\ell}+\check{\alpha}_{\ell+1})&\text{if
}(X_{N},r)=(D_{\ell+1},2).\\\ \end{cases}$

The simple coroots of ${G}$ are identified with tuples of integers through the
construction of root system given from [Hu2, §12].

Let $\rho:G\to{\rm GL}(V)$ be the standard representation of $G$. We will
determine the double ${\rm SL(V_{\mathcal{O}})}$-coset in ${\rm
SL}(V_{\mathcal{K}})$ that $\rho(g(t))$ belongs to.

If ${G}$ is of the type $A_{m}$, then $\check{\alpha}_{i}$, $i=1,...,m$, are
identified with the following $(m+1)$-tuples

$\check{\alpha}_{1}=(1,-1,0,0,...,0)$

$\check{\alpha}_{2}=(0,1,-1,0,...,0)$

$\vdots$

$\check{\alpha}_{m}=(0,0,0,...,1,-1)$

and hence, as the coweight of $\mathrm{SL}_{m}$, $\lambda+\sigma\lambda$
corresponds to the following tuples

$\lambda+\sigma\lambda=\begin{cases}(a_{1},a_{2},...,a_{\ell},0,-a_{\ell},...,-a_{2},-a_{1})&\text{if
}(X_{N},r)=(A_{2\ell},2);\\\
(a_{1},a_{2},...,a_{\ell},-a_{\ell},...,-a_{2},-a_{1})&\text{if
}(X_{N},r)=(A_{2\ell-1},2).\end{cases}$

Assume that ${G}$ has the type $D_{\ell+1}$. Then $\check{\alpha}_{i}$,
$i=1,...,\ell+1$, are identified with following $(\ell+1)$-tuples

$\check{\alpha}_{1}=(1,-1,0,0,...,0)$

$\check{\alpha}_{2}=(0,1,-1,0,...,0)$

$\vdots$

$\check{\alpha}_{\ell}=(0,0,0,...,1,-1)$

$\check{\alpha}_{\ell+1}=(0,0,0,...,1,1).$

Then $\lambda+\sigma\lambda=(2a_{1},2a_{2},...,2a_{\ell},0)$ as the coweight
of $G=\mathrm{Spin}_{2\ell+2}$. Choose an appropriate maximal torus and a
positive root system in $G$. Composing with $\rho:G\to{\rm SL}_{2\ell+2}$, as
the coweight of $\mathrm{SL}_{2\ell+2}$, $\lambda+\sigma\lambda$ corresponds
to the following tuple

$(2a_{1},2a_{2},...,2a_{\ell},0,0,-2a_{\ell},...,-2a_{2},-2a_{1}).$

We write $g(t)=A(t)n^{\lambda}B(t)$, where $n^{\lambda}$ is a norm of
$t^{\lambda}$ defined by (1) and $A(t),B(t)\in G(\mathcal{O})^{\sigma}$. Hence
$\rho(g(t))\in\mathrm{SL}_{m}(\mathcal{O})\rho(t^{\lambda+\sigma\lambda})\mathrm{SL}_{m}(\mathcal{O})$.
By above descriptions of $\rho(t^{\lambda+\sigma\lambda})$, this lemma follows
from Lemma 4.3. ∎

###  4.1. Case $(X_{N},r)=(A_{2\ell},2)$

Let $\langle\cdot,\cdot\rangle$ be a nondegenerate symmetric bilinear form on
$V=\mathbb{C}^{2\ell+1}$ whose matrix is

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

The diagram automorphism $\sigma$ on $\mathfrak{g}$ given by (7) becomes
$\sigma(A)=-JA^{T}J^{-1}$ and the diagram automorphism $\sigma$ on $G$ is
given by

(10) $\sigma(A)=JA^{-T}J^{-1}.$

This $\sigma$ gives the decomposition
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ to 1 and -1 eigenspaces
$\mathfrak{k}$ and $\mathfrak{p}$, respectively. Let
$K:=(\mathrm{SL}_{2\ell+1})^{\sigma}=\mathrm{SO}_{2\ell+1}$. The
classification of nilpotent $K$-orbits in $\mathfrak{p}$ and their dimensions
follow from the Theorem 3.4 and 3.7.

Set

(11)
$\mathcal{N}_{\mathfrak{p},2}=\\{x\in\mathcal{N}_{\mathfrak{p}}\,|\,x^{2}=0\\}.$

######  Theorem 4.5.

$\pi$ maps $\mathcal{M}$ isomorphically onto $\mathcal{N}_{\mathfrak{p},2}$.
Moreover, $\pi$ maps $\mathcal{M}_{(1^{j}0^{\ell-j})}$ isomorphically to
$[2^{j}1^{2\ell-2j+1}]$.

###### Proof.

Let $g(t)\cdot e_{0}\in\mathcal{M}$. Then $g(t)\cdot
e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$ for some $j$. By Lemma 4.4,
$g(t)=I+xt^{-1}$ for some $x\in\operatorname{Mat}_{2\ell+1\times 2\ell+1}$. By
Lemma 2.4, $\iota(g(t)\cdot e_{0})\in\mathcal{M}_{(1^{j}0^{\ell-j})}$. Hence
$\iota(g(t))=(I-xt^{-1})^{-1}=I+zt^{-1}$ for some
$z\in\operatorname{Mat}_{2\ell+1\times 2\ell+1}$, and so $x^{2}=0$.
Conversely, let $x\in\mathcal{N}_{\mathfrak{p}}$ be such that $x^{2}=0$. Then
$I+xt^{-1}\in G(\mathcal{K})^{\sigma}$. By (2), $(I+xt^{-1})\cdot
e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ for some
$\bar{\lambda}=(a_{1},...,a_{\ell})\in X_{*}(T)_{\sigma}^{+}$. By lemma 4.4,
$\bar{\lambda}=(1^{j}0^{\ell-j})$. We have proved that
$\mathcal{M}=\\{(I+xt^{-1})\cdot e_{0}\mid x\in\mathfrak{p},x^{2}=0\\}$.

Let $(I+xt^{-1})\cdot e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-i})}$. Then $x^{2}=0$
and $x$ has the Jordan blocks of size at most 2. By Lemma 4.4,
$\operatorname{rk}x=j$ and then $x$ has the partition $[2^{j}1^{2\ell-2j+1}]$.
It is obvious that $\pi$ is injective. To prove surjectivity, let
$x\in\mathcal{N}_{\mathfrak{p},2}$ having the partition
$[2^{j}1^{2\ell-2j+1}]$. Then, $(I+xt^{-1})\cdot e_{0}\in\mathcal{M}$ and
hence $(I+xt^{-1})\cdot e_{0}\in\mathcal{M}_{(1^{k}0^{\ell-k})}$ for some $k$.
In fact, $k=j$ since $\pi((I+xt^{-1})\cdot e_{0})=x\in[2^{k}1^{2\ell-2k+1}].$
∎

###  4.2. Case $(X_{N},r)=(A_{2\ell-1},2)$

Let $\langle\cdot,\cdot\rangle$ be a symplectic bilinear form on
$V=\mathbb{C}^{2\ell}$ whose matrix is

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

The diagram automorphism $\sigma$ on $\mathfrak{g}$ given by (7) becomes
$\sigma(A)=-JA^{T}J^{-1}$ and the action on $G$ is given by

(12) $\sigma(A)=JA^{-T}J^{-1}.$

This $\sigma$ gives the decomposition
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ to 1 and -1 eigenspaces
$\mathfrak{k}$ and $\mathfrak{p}$, respectively. Let
$K:=(\mathrm{SL}_{2\ell})^{\sigma}=\mathrm{Sp}_{2\ell}$. The classification of
nilpotent $K$-orbits in $\mathfrak{p}$ and their dimensions follow from the
Theorem 3.3 and 3.7.

Define constructible sets

$\mathcal{M}^{\prime}:=\bigcup_{j=0}^{\lfloor\frac{\ell}{2}\rfloor}\mathcal{M}_{(1^{2j}0^{\ell-2j})},\hskip
14.22636pt\mathcal{M}^{\prime\prime}:=\bigcup_{j=0}^{\lfloor\frac{\ell-1}{2}\rfloor}\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}.$

By Lemma 4.4, the element of $\mathcal{M}^{\prime}$ is of the form
$(I+xt^{-1})\cdot e_{0}$ and the element of $\mathcal{M}^{\prime\prime}$ is of
the form $(I+xt^{-1}+yt^{-2})\cdot e_{0}$. Let $\mathcal{N}_{\mathfrak{p},2}$
be the order 2 nilpotent cone defined as in (11).

######  Theorem 4.6.

$\pi$ maps $\mathcal{M^{\prime}}$ isomorphically onto
$\mathcal{N}_{\mathfrak{p},2}$. Moreover, $\pi$ maps
$\mathcal{M}_{(1^{2j}0^{\ell-2j})}$ isomorphically to $[2^{2j}1^{2\ell-4j}]$.

###### Proof.

The proof is the same as the proof of Theorem 4.5, where in this case we use
Lemma 4.4 for $(A_{2\ell-1},2)$. ∎

Before we describe elements of $M^{\prime\prime}$, we need the following
lemma.

######  Lemma 4.7.

Let $g(t)\cdot e_{0}=(I+xt^{-1}+yt^{-2})\cdot
e_{0}\in\mathcal{M}^{\prime\prime}$. Then

  1. (1)

$\iota(g(t))\neq g(t)$

  2. (2)

If $g(t)\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}$, then
$\operatorname{rk}\begin{pmatrix}y&x\\\ 0&y\end{pmatrix}=2j+2.$

###### Proof.

Note that $\iota(g(t))=g(t)$ if and only if

$(I+xt^{-1}+yt^{-2})(I-xt^{-1}+yt^{-2})=I$

which is equivalent to $y=\dfrac{1}{2}x^{2}$ and $x^{4}=0$. Suppose that
$\iota(g(t))=g(t)$. Observe that
$\operatorname{rk}x^{3}\leq\operatorname{rk}x^{2}=\operatorname{rk}y=1$. If
$\operatorname{rk}x^{3}=1$, then
$\operatorname{rk}x^{4}=\operatorname{rk}x^{3}=1$ which is impossible. Hence
$x^{3}=0$. Since $\operatorname{rk}x^{2}=1$, $x\in\mathfrak{p}$ is nilpotent
having the partition $[32^{j}1^{2l-2j-3}]$ but this contradicts to Theorem
3.3. This proves the first part.

Assume that $g(t)\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}$. We write

$g(t)=At^{(2,1^{2j},0^{2\ell-4j-2},(-1)^{2j},-2)}B$

where $A=\sum_{i=0}A_{i}t^{i},B=\sum_{i=0}B_{i}t^{i}\in
G(\mathcal{O})^{\sigma}$. In particular,
$g(t)\in\mathrm{SL}_{2\ell}(\mathcal{O})t^{\check{\lambda}}\mathrm{SL}_{2\ell}(\mathcal{O})$
where $\check{\lambda}=(2,1^{2j},0^{2\ell-4j-2},(-1)^{2j},-2)$. By Lemma 4.3,

$\operatorname{rk}\begin{pmatrix}y&x\\\
0&y\end{pmatrix}=\sum_{j=1}^{2\ell}\max\\{-a_{j},0\\}=2j+2$

as desired. ∎

Let $g(t)=I+xt^{-1}+yt^{-2}$. By Lemma 2.4,
$\iota(g(t))=I+x^{\prime}t^{-1}+y^{\prime}t^{-2}$ for some matrices
$x^{\prime},y^{\prime}$. Hence

$(I-xt^{-1}+yt^{-2})(I+x^{\prime}t^{-1}+y^{\prime}t^{-2})=I=(I+x^{\prime}t^{-1}+y^{\prime}t^{-2})((I-xt^{-1}+yt^{-2})$

which implies

(13) $x^{\prime}=x,\hskip 7.11317ptx^{2}=y+y^{\prime},\hskip
7.11317ptxy=y^{\prime}x,\hskip 7.11317ptxy^{\prime}=yx,\hskip
7.11317ptyy^{\prime}=y^{\prime}y=0.$

By Lemma 4.4 and Lemma 4.7, $\operatorname{rk}y=\operatorname{rk}y^{\prime}=1$
and $y^{\prime}\neq y$. Since $\sigma(g(t))=g(t)$, $y^{\prime}=Jy^{T}J^{-1}$
which means that $y$ and $y^{\prime}$ are adjoint each other. We set

$\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}:=\\{(I+xt^{-1}+yt^{-2})\cdot
e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}\mid L=L^{\prime}\\},$
$\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}}:=\\{(I+xt^{-1}+yt^{-2})\cdot
e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}\mid L\neq L^{\prime}\\}.$

The following lemma will be used in the proofs of Lemma 4.9 and Theorem 4.14.

######  Lemma 4.8.

Let $(\cdot,\cdot)$ be a nondegenerate bilinear form on a vector space $V$
over a field $\mathbb{C}$ and let $T$ a linear map on $V$. Denote the adjoint
of $T$ by $T^{*}$. Assume that $\operatorname{Im}T=\operatorname{Im}T^{*}$ and
$\operatorname{rk}T=1$. Then $T$ is self-adjoint or skew-adjoint.

###### Proof.

It is easy to see that $\ker T=(\operatorname{Im}T^{*})^{\perp}$ and $\ker
T^{*}=(\operatorname{Im}T)^{\perp}$. Say that $\operatorname{Im}T=\mathbb{C}v$
and $\operatorname{Im}T^{*}=\mathbb{C}v^{\prime}$ for some $v,v^{\prime}\in
V$. Then $Tw=v$ for some $w\in V$. Since
$\operatorname{Im}T=\operatorname{Im}T^{*}$, we have $T^{*}w=\lambda v$ for
some $\lambda\in\mathbb{C}$. Let $u\in V$. Then $Tu=kv$ for some
$k\in\mathbb{C}$. Since $T(u-kw)=0$, $u-kw\in\ker
T=(\operatorname{Im}T^{*})^{\perp}=(\operatorname{Im}T)^{\perp}=\ker T^{*}$.
Hence $T^{*}u=T^{*}(kw)=k\lambda v=\lambda Tu$. Since $u$ is arbitrary,
$T^{*}=\lambda T$. Consider $T+T^{*}=(1+\lambda)T$. Then

$(1+\lambda)T=T+T^{*}=(T+T^{*})^{*}=(1+\lambda)T^{*}.$

Therefore $1+\lambda=0$ or $T=T^{*}$ which means that $T$ is skew-adjoint or
self-adjoint. ∎

######  Lemma 4.9.

If $(I+xt^{-1}+yt^{-2})\cdot
e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}$, then
$y^{\prime}=-y$.

###### Proof.

We know that $y\neq y^{\prime}$ are adjoint each other, they have the same
images, and $\operatorname{rk}y=1$. By Lemma 4.8, $y$ is skew-adjoint, i.e.,
$y^{\prime}=-y$. ∎

######  Theorem 4.10.

  1. (1)

If $\ell$ is even, then

$\pi(\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}})=[2^{2j}1^{2\ell-4j}]\cup[2^{2j+2}1^{2\ell-4j-4}]$

for $j=0,1,...,\frac{\ell-2}{2}$. If $\ell$ is odd, then

$\pi(\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}})=\begin{cases}[2^{2j}1^{2\ell-4j}]\cup[2^{2j+2}1^{2\ell-4j-4}]&\hskip
1.42271pt\text{if }\ell\geq 3,\,0\leq j\leq\frac{\ell-3}{2};\\\
[2^{\ell-1}1^{2}]&\hskip 1.42271pt\text{if }j=\frac{\ell-1}{2}.\end{cases}$

  2. (2)

When $\ell\geq 3$, for $j=1,...,\lfloor\frac{\ell-1}{2}\rfloor$, we have

$\pi(\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}})=\begin{cases}[3^{2}1^{2\ell-6}]&\hskip
1.42271pt\text{if }j=1;\\\
[3^{2}2^{2j-2}1^{2\ell-4j-2}]\cup[3^{2}2^{2j-4}1^{2\ell-4j+2}]&\hskip
1.42271pt\text{if }\ell\geq 4,\,2\leq
j\leq\lfloor\frac{\ell-1}{2}\rfloor.\end{cases}$

Moreover, for any $\ell\geq 1$, $\mathcal{M}_{(20^{\ell-1})}^{\mathrm{II}}$ is
empty.

###### Proof.

Let $g(t)\cdot e_{0}=(I+xt^{-1}+yt^{-2})\cdot
e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}$. By Lemma 4.7,

$\operatorname{rk}x\leq\operatorname{rk}\begin{pmatrix}y&x\\\
0&y\end{pmatrix}=2j+2\leq
2\operatorname{rk}y+\operatorname{rk}x=2+\operatorname{rk}x.$

Since $x^{2}=y+y^{\prime}=0$, $x$ is nilpotent whose partition is
$[2^{2k}1^{2\ell-4k}]$ so that $\operatorname{rk}x=2k$. Hence $k=j$ or $j+1$.
If $\ell$ is odd and $j=\frac{\ell-1}{2}$, then $k=j$.

Let $E_{ij}\in\operatorname{Mat}_{2\ell\times 2\ell}$ be the matrix which has
1 at the entry $i,j$ and 0 elsewhere. For each
$j=0,1,\dots,\lfloor\frac{\ell-1}{2}\rfloor$, let

$x_{j}=\operatorname{diag}(0,J_{2},..,J_{2},0_{2\ell-4j-2},-J_{2},..,-J_{2},0)$

where there are $j$ blocks of $J_{2}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$
and $j$ blocks of $-J_{2}$, and $0_{2\ell-4j-2}$ is the square zero matrix of
size $2\ell-4j-2$. Then $x_{j}\in\mathfrak{p}$ is nilpotent and has the
partition $[2^{2j}1^{2\ell-4j}]$. It is easy to check that
$g(t)=I+x_{j}t^{-1}+E_{1,2\ell}t^{-2}\in G(\mathcal{K})^{\sigma}$ and
$\iota(g(t))=I+x_{j}t^{-1}-E_{1,2\ell}t^{-2}$. By (2), $g(t)\cdot
e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ for some
$\bar{\lambda}=(a_{1},...,a_{\ell})\in X_{*}(T)_{\sigma}^{+}$ with $a_{1}\geq
a_{2}...\geq a_{\ell}\geq 0$ and $\sum a_{i}$ is even. By Lemma 4.4, since
$\operatorname{rk}E_{1,2\ell}=1$, $\bar{\lambda}=(21^{2k}0^{\ell-2k-1})$ for
some $k$. By Lemma 4.7,

$2k+2=\operatorname{rk}\begin{pmatrix}E_{1,2\ell}&x_{j}\\\
0&E_{1,2\ell}\end{pmatrix}=2j+2.$

Then $(I+x_{j}t^{-1}+E_{1,2\ell}t^{-2})\cdot
e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}$. For
$j=0,1,\dots,\lfloor\frac{\ell-2}{2}\rfloor$, let

$x^{\prime}_{j}=x_{j}+E_{1,2j+2}-E_{2\ell-2j-1,2\ell}.$

Then $x^{\prime}_{j}\in\mathfrak{p}$ is nilpotent and has the partition
$[2^{2j+2}1^{2\ell-4j-4}]$. Similarly, one can show that
$(I+x^{\prime}_{j}t^{-1}+E_{1,2\ell}t^{-2})\cdot
e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}$. Since $\pi$ is
$K$-invariant, this proves the first part.

Now, we prove the second part. Let $g(t)\cdot e_{0}=(I+xt^{-1}+yt^{-2})\cdot
e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}}$. Set
$U=L+L^{\prime}$. Since $y\neq y^{\prime}$, $\dim U=2$ and
$U=\operatorname{Im}x^{2}$. Assume that
$L=\mathbb{C}v,L^{\prime}=\mathbb{C}v^{\prime}$. By (13), we have
$xy=y^{\prime}x$ and $xy^{\prime}=yx$. Hence $xv=bv^{\prime}$ and
$xv^{\prime}=av$ for some $a,b\in\mathbb{C},v,v^{\prime}\in\mathbb{C}^{2l}$.
Then

$x\big{|}_{U}=\begin{pmatrix}0&a\\\ b&0\end{pmatrix},\hskip
14.22636ptx^{2}\big{|}_{U}=\begin{pmatrix}ab&0\\\ 0&ab\end{pmatrix}$

Suppose that $ab\neq 0$. To show that
$\langle\cdot,\cdot\rangle\big{|}_{U\times U}$ is nondegenerate, let
$u\in\mathbb{C}^{2\ell}$ be such that $\langle x^{2}u,x^{2}v\rangle=0$ for all
$v\in\mathbb{C}^{2\ell}$. Since $x^{2}$ is self-adjoint, $\langle
x^{4}u,v\rangle=0$ for all $v$. Therefore, $x^{4}u=0$ and so $x^{2}u\in\ker
x^{2}\big{|}_{U}=0$. Since $y,y^{\prime}$ are adjoint each other and
$yy^{\prime}=0=y^{\prime}y$, $L^{\prime}\subset\ker y=(L^{\prime})^{\perp}$
and $L\subset\ker y^{\prime}=L^{\perp}$. This implies $\langle
v,v\rangle=\langle v^{\prime},v^{\prime}\rangle=0$ and $\langle
v,v^{\prime}\rangle\neq 0$. Observe that

$ab\langle v,v^{\prime}\rangle=\langle v,x^{2}v^{\prime}\rangle=\langle
xv,xv^{\prime}\rangle=\langle bv^{\prime},av\rangle=-ab\langle
v,v^{\prime}\rangle$

which implies $ab=0$, a contradiction. This proves $x^{4}=0$. Since
$x\in\mathfrak{p}$ and $\operatorname{rk}x^{2}=2$, by Theorem 3.3, $x$ is
nilpotent having the partition $[3^{2}2^{2k}1^{2\ell-4k-6}]$. By Lemma 4.7,

$\operatorname{rk}x\leq\operatorname{rk}\begin{pmatrix}y&x\\\
0&y\end{pmatrix}=2j+2\leq
2\operatorname{rk}y+\operatorname{rk}x=2+\operatorname{rk}x.$

Since $\operatorname{rk}x=2k+4$, $k=j-1$ or $j-2$. Here we see that $j\neq 0$
and hence $\mathcal{M}_{(20^{\ell-1})}^{\mathrm{II}}$ is empty. When $j=1$, we
see that $k=0$. Let

$J_{3}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 0&0&0\end{pmatrix},\hskip
5.69046ptJ_{2}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}.$

For each $j=1,...,\lfloor\frac{\ell-1}{2}\rfloor$, let

$x_{j-1}=\operatorname{diag}(J_{3},J_{2},...,J_{2},0_{2\ell-4j-2},-J_{2},...,-J_{2},-J_{3})$

where there are $j-1$ blocks of $J_{2}$, and $j-1$ blocks of $-J_{2}$. Then
$x_{j-1}\in\mathfrak{p}$ is nilpotent having the partition
$[3^{2}2^{2j-2}1^{2\ell-4j-2}]$. Note that
$g(t):=1+x_{j-1}t^{-1}+E_{13}t^{-2}\in G(\mathcal{K})^{\sigma}$ and
$\iota(g(t))=1+x_{j-1}t^{-1}+E_{2\ell-2,2\ell}t^{-2}$. By (2), $g(t)\cdot
e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ for some
$\bar{\lambda}=(a_{1},...,a_{\ell})\in X_{*}(T)_{\sigma}^{+}$ with $a_{1}\geq
a_{2}...\geq a_{\ell}\geq 0$ and $\sum a_{i}$ is even. By Lemma 4.4, since
$\operatorname{rk}E_{13}=1$, $\bar{\lambda}=(21^{2k}0^{\ell-2k-1})$ for some
$k$. By Lemma 4.7,

$2k+2=\operatorname{rk}\begin{pmatrix}E_{13}&x_{j-1}\\\
0&E_{13}\end{pmatrix}=2j+2.$

Then $(I+x_{j}t^{-1}+E_{13}t^{-2})\cdot
e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}}$. For
$j=2,...,\lfloor\frac{\ell-1}{2}\rfloor$, let

$x^{\prime}_{j-2}=\operatorname{diag}(0,J_{3},J_{2},...,J_{2},0_{2\ell-4j},-J_{2},...,-J_{2},-J_{3},0)$

where there are $j-2$ blocks of $J_{2}$, and $j-2$ blocks of $-J_{2}$. Then
$x^{\prime}_{j-2}\in\mathfrak{p}$ is nilpotent having the partition
$[3^{2}2^{2j-4}1^{2\ell-4j+2}]$. One can check that
$h(t):=1+x^{\prime}_{j-2}t^{-1}+(E_{24}+E_{1,2\ell})t^{-2}\in
G(\mathcal{K})^{\sigma}$ and
$\iota(h(t))=1+x^{\prime}_{j-2}t^{-1}+(E_{2\ell-3,2\ell-1}-E_{1,2\ell})t^{-2}$.
Similarly, $h(t)\cdot e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ where
$\bar{\lambda}=(21^{2k}0^{\ell-2k-1})$ for some $k$. By Lemma 4.7,

$2k+2=\operatorname{rk}\begin{pmatrix}E_{24}+E_{1,2\ell}&x^{\prime}_{j-2}\\\
0&E_{24}+E_{1,2\ell}\end{pmatrix}=2j+2.$

Then $(I+x^{\prime}_{j-2}t^{-1}+(E_{24}+E_{1,2\ell})t^{-2})\cdot
e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}}$. ∎

In the following proposition, we describe the fibers of
$\pi:\mathcal{M}\to\pi(\mathcal{M})$.

######  Proposition 4.11.

For any $x\in\pi(\mathcal{M})$,

$\pi^{-1}(x)\cong\\{z\in\mathfrak{sp}_{2\ell}\mid
xz+zx=0,z^{2}=0,\operatorname{rk}(z+\frac{1}{2}x^{2})\leq 1\\}.$

In particular, $\pi^{-1}(0)$ is isomorphic to the closure of nilpotent orbit
$[21^{2\ell-2}]$ in $\mathfrak{sp}_{2\ell}$ and $\dim\pi^{-1}(0)=2\ell+1$.

###### Proof.

Note that $(1+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}$ if and only if
$\det(1+xt^{-1}+yt^{-2})=1,\operatorname{rk}y\leq 1$ and

(14) $x^{T}J-Jx=0,\quad-x^{T}Jx+y^{T}J+Jy=0,\hskip
14.22636ptx^{T}Jy-y^{T}Jx=0,\hskip 14.22636pty^{T}Jy=0.$

Set $z=y-\frac{1}{2}x^{2}$, (14) is equivalent to

(15) $x\in\mathfrak{p},\quad z\in\mathfrak{k},\quad xz+zx=0,\quad z^{2}=0.$

If $xz+zx=0$, then $\det(1+xt^{-1}+(z+\frac{1}{2}x^{2})t^{-2})=1$. Since
$\pi^{-1}(0)=\\{(1+yt^{-2})\cdot e_{0}\mid
y\in\mathfrak{sp}_{2\ell},y^{2}=0,\operatorname{rk}y\leq 1\\}$, the dimension,
cf. [CM, Corollary 6.1.4], is

$\dim\pi^{-1}(0)=\dim[21^{2\ell-2}]=(2\ell^{2}+\ell)-\frac{1}{2}((2\ell-1)^{2}+1^{2})-\frac{1}{2}(2\ell-2)=2\ell+1.$

∎

${{\mathcal{M}_{(21^{4})}^{\mathrm{I}}}}$${{\mathcal{M}_{(21^{4})}^{\mathrm{II}}}}$${{[3^{2}2^{2}]}}$${{\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{I}}}}$${{[3^{2}1^{4}]}}$${{\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{II}}}}$${{\mathcal{M}_{(1^{4}0)}}}$${{[2^{4}1^{2}]}}$${{\mathcal{M}_{(20^{4})}^{\mathrm{I}}}}$${{\mathcal{M}_{(20^{4})}^{\mathrm{II}}=\emptyset}}$${{[2^{2}1^{6}]}}$${{\mathcal{M}_{(1^{2}0^{3})}}}$${{0}}$${{0}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

Figure 1. $\mathcal{M}_{\bar{\lambda}}$ for small dominant weight
$\bar{\lambda}$ and their image under the map $\pi$ in type
$(X_{N},r)=(A_{9},2)$

In [AH, Theorem 1.2], they proved that there are finitely many $G$-orbits in
$\mathrm{Gr}_{\lambda}\cap\mathrm{Gr}_{0}^{-}$ for small dominant coweight
$\lambda$. In the case of $(A_{2\ell},2)$, it is easy to see that $K$ acts
transitively on $\mathcal{M}_{(1^{j}0^{\ell-j})}$ and hence there are finitely
many $K$-orbits in $\mathcal{M}$. For the case $(A_{2\ell-1},2)$, it is not
obvious to determine if there are finitely many $K$-orbits in
$\mathcal{M}_{(21^{j}0^{\ell-j-1})}$. If
$g(t)=1+xt^{-1}+(z+\frac{1}{2}x^{2})t^{-2}\in\mathcal{M}_{(21^{j}0^{\ell-j-1})}$,
then $g(t)$ satisfies (15). If the action of $K$ on the following anti-
commuting nilpotent variety

$\\{(x,z)\in\mathcal{N}_{\mathfrak{p}}\times\mathcal{N}_{\mathfrak{k}}\mid
xz+zx=0\\}$

by diagonal cojugation has finitely many orbits, then there are finitely many
$K$-orbits in $\mathcal{M}_{(21^{j}0^{\ell-j-1})}$.

######  Example 4.12.

Consider the case $(X_{N},r)=(A_{9},2)$ . In this case, $G=\mathrm{SL}_{10}$
and $\mathfrak{g}=\mathfrak{sl}_{10}=\mathfrak{k}\oplus\mathfrak{p}$. The
diagram as shown in Figure 1 describes the image of
$\mathcal{M}_{\bar{\lambda}}$ for each small dominant weight $\bar{\lambda}$.
For instance, $\mathcal{M}_{(21^{2}0^{2})}$ consists of two parts,
$\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{I}}$ and
$\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{II}}$. By Theorem 4.10,
$\pi(\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{I}})$ is precisely the union of two
nilpotent orbits $[2^{4}1^{2}]$ and $[2^{2}1^{6}]$ in $\mathfrak{p}$ while
$\pi(\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{II}})$ is the single nilpotent orbit
$[3^{2}1^{4}]$ in $\mathfrak{p}$. Since $(21^{2}0^{2})\succeq(1^{4}0)$,
$\mathcal{M}_{(21^{2}0^{2})}\succeq\mathcal{M}_{(1^{4}0)}$. Similarly,
$\mathcal{M}_{(21^{2}0^{2})}\succeq\mathcal{M}_{(20^{4})}$. According to the
table in Theorem 4.2, the image of certain $\mathcal{M}_{\bar{\lambda}}$ is a
union of 4 nilpotent orbits. It does not happen in this case since $\ell=5$ is
not large enough. In the case of $(X_{N},r)=(A_{13},2)$,
$\pi(\mathcal{M}_{(21^{4}0^{2})})$ is a union of nilpotent orbits
$[2^{4}1^{6}],[2^{6}1^{2}],[3^{2}2^{2}1^{4}],[3^{2}1^{8}]$ in
$\mathfrak{p}\subset\mathfrak{sl}_{14}$.

###  4.3. Case $(X_{N},r)=(D_{\ell+1},2)$

In this case, it is more convenient to work with $G=\mathrm{SO}_{2\ell+2}$ and
$\sigma$ is a diagram automorphism on $G$. It is known that
$G^{\sigma}\simeq\mathrm{SO}_{2\ell+1}\times\\{\pm I\\}$. Let
$G(\mathcal{O})^{\sigma,\circ}$ denote the identity component of the group
$G(\mathcal{O})^{\sigma}$. Then, the action of ${\rm
Spin}_{2\ell+2}(\mathcal{O})^{\sigma}$ on the twisted affine Grassmanian
$\mathcal{G}r$ of ${{\rm Spin}_{2n+2}}$ factors through
$G(\mathcal{O})^{\sigma,\circ}$. Let $G(\mathcal{O}^{-})_{0}^{\sigma}$ be the
kernel of the evaluation map $G(\mathcal{O}^{-})^{\sigma}\to G^{\sigma}$. The
action of ${\rm Spin}_{2\ell+2}(\mathcal{O})_{0}^{-}$ on $\mathcal{G}r$
factors through $G(\mathcal{O}^{-})_{0}^{\sigma}$. Hence, the opposite open
Schubert cell $\mathcal{G}r_{0}^{-}$ is a
$G(\mathcal{O}^{-})_{0}^{\sigma}$-orbit. In fact, $\mathcal{G}r$ is naturally
the neutral component of the twisted affine Grassmannian associated to
$(G,\sigma)$, whose definition is a bit more sophisticated.

We can realize the group $G$ as $\\{g\in{\rm SL}_{2\ell+2}|gJg^{T}=J\\}$, and
the Lie algebra of $G$ as
$\mathfrak{g}=\mathfrak{so}_{2\ell+2}(J)=\\{x\in\mathfrak{gl}_{2n}\mid
Jx+x^{T}J=0\\}$ where

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

The diagram automorphism $\sigma$ of order 2 on $\mathfrak{g}$ can be given by
$\sigma(x)=wxw$ where

$w=\operatorname{diag}\left(I_{\ell},\begin{pmatrix}0&1\\\
1&0\end{pmatrix},I_{\ell}\right).$

The diagram automorphism $\sigma$ on $G$ is also defined the same. We also
have the decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$. Let $K$
be the identity component of $G^{\sigma}$. $K$ has Lie algebra $\mathfrak{k}$
and acts on $\mathfrak{p}$ by conjugation. It can be checked that $J=A^{T}A$
where

$A=\begin{pmatrix}\frac{1}{\sqrt{2}}&0&\cdots&\cdots&0&\frac{1}{\sqrt{2}}\\\
0&\ddots&\cdots&\cdots&\reflectbox{$\ddots$}&0\\\
\vdots&\vdots&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\vdots&\vdots\\\
\vdots&\vdots&\frac{i}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&\vdots&\vdots\\\
0&\reflectbox{$\ddots$}&\cdots&\cdots&\ddots&0\\\
\frac{i}{\sqrt{2}}&0&\cdots&\cdots&0&-\frac{i}{\sqrt{2}}\end{pmatrix}.$

Another realization of $\mathfrak{so}_{2\ell+2}$ is
$\mathfrak{so}_{2\ell+2}(I)=\\{x\in\mathfrak{gl}_{2\ell}\mid x+x^{T}=0\\}$.
There exists an isomorphism from $\mathfrak{so}_{2\ell+2}(J)$ to
$\mathfrak{so}_{2\ell+2}(I)$ given by $x\mapsto AxA^{-1}$. Under
$\mathfrak{so}_{2\ell+2}(I)$, the diagram automorphism $\sigma_{0}$ is defined
by $\sigma_{0}(x)=w_{0}xw_{0}$ where
$w_{0}=(PA)w(PA)^{-1}=\operatorname{diag}(-1,1,1,...,1)$ and $P$ is some
matrix of change of basis.

######  Proposition 4.13.

  1. (1)

If $x$ is a nonzero nilpotent element in $\mathfrak{p}$, then $x$ has the
partition $[31^{2\ell-1}]$.

  2. (2)

There are exactly 2 nilpotent $K$-orbits in $\mathfrak{p}$: $\\{0\\}$ and
$\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}.$

###### Proof.

Since $w_{0}xw_{0}=-x$, $x$ has the form

$x=\begin{pmatrix}0&-u^{t}\\\ u&0\end{pmatrix}$

where $u\in\mathbb{C}^{2l+1}$ is a nonzero column vector. Then

$x^{2}=\begin{pmatrix}-u^{t}u&0\\\ 0&-uu^{t}\end{pmatrix}.$

If $x^{2}=0$, then $uu^{t}=0$ which implies $u=0$, a contradiction. Since
$\operatorname{rk}x=2$ and $x^{2}\neq 0$, $x$ has the partition $[31^{2n-1}]$.
This proves the first part.

The element of $K$ has the form

$k=\begin{pmatrix}1&0\\\ 0&g\end{pmatrix}.$

where $g\in\mathrm{SO}_{2\ell+1}$ and $k$ acts on $x\in\mathfrak{p}$ by

$k\cdot x=kxk^{-1}=\begin{pmatrix}1&-(gu)^{t}\\\ gu&0\end{pmatrix}.$

Hence the action of $K$ on $\mathfrak{p}$ is the same as the action of
$\mathrm{SO}_{2\ell+1}$ on $\mathbb{C}^{2\ell+1}$. Note that for every $k\geq
3$, $x^{k}$ has the scalar $u^{T}u$ on every nonzero entry. Since $x$ is
nilpotent, $u^{T}u=0$. The result immediately follows since the action of
$\mathrm{SO}_{2\ell+1}$ on $\\{z\in\mathbb{C}^{2\ell+1}\mid z^{T}z=0\\}$ has
two orbits. ∎

######  Theorem 4.14.

For $j=0,1,...,\ell$, we have

$\pi(\mathcal{M}_{(1^{j}0^{\ell-j})})=\begin{cases}\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}&\quad\text{
if }j\text{ is odd};\\\ \\{0\\}&\quad\text{ if }j=0;\\\
\mathcal{N}_{\mathfrak{p}}&\quad\text{ if }j\text{ is even and }j\geq
2.\end{cases}.$

Moreover,
$\mathcal{M}_{(10^{\ell-1})}=\\{(1+xt^{-1}+\frac{1}{2}x^{2}t^{-2})\cdot
e_{0}\mid x\in\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}\\}$. Consequently,
$\pi$ maps $\mathcal{M}_{(10^{\ell-1})}$ isomorphically onto
$\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}$.

###### Proof.

By Lemma 4.4, let $g(t)\cdot e_{0}=(1+xt^{-1}+yt^{-2})\cdot
e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}.$ We work under
$\mathfrak{so}_{2\ell+2}(I)$ and $\mathrm{SO}_{2\ell+2}(I)$. Since $g(t)$ is
fixed by $\sigma_{0}$, $w_{0}xw_{0}=-x$ and $w_{0}yw_{0}=y$. Similarly to the
proof of Proposition 4.13, $x,y$ are in the form

(16) $x=\begin{pmatrix}0&-u^{T}\\\ u&0\end{pmatrix},\hskip
14.22636pty=\begin{pmatrix}y_{0}&0\\\ 0&D\end{pmatrix}$

where $u\in\mathbb{C}^{2\ell+1}$ is a column vector, $y_{0}\in\mathbb{C}$, and
$D\in\operatorname{Mat}_{(\ell-1)\times(\ell-1)}$. Since $g^{T}g=I$, the
following equations hold:

(17) $x^{T}+x=0,\hskip 14.22636ptx^{T}x+y^{T}+y=0,\hskip
14.22636ptx^{T}y+y^{T}x=0,\hskip 14.22636pty^{T}y=0.$

Then $y_{0}=0$ and $u^{T}u=0$ which implies $x^{3}=0$. Hence
$x\in\mathcal{N}_{\mathfrak{p}}$.

Suppose that $j$ is odd and $x=0$. We have $y+y^{T}=0$ and $y^{T}y=0$. Then
$y_{0}=0$ and $D$ is a nilpotent element of $\mathfrak{so}_{2\ell+1}$ with
$D^{2}=0$. Since $\operatorname{rk}D=\operatorname{rk}y=j$, $D$ has the
partition $[2^{j}1^{2\ell-2j+2}]$. This contradicts to the classification of
nilpotent orbits of type B, [CM, Theorem 5.1.2]. Hence $x\neq 0$. Now,
consider the matrix $x_{0}\in\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}$
defined by

$x_{0}=\begin{pmatrix}0&\cdots&0&-1&-i\\\ \vdots&&&&\\\ 0&&&&\\\ 1&&&&\\\
i&&&&\end{pmatrix}.$

Let $N$ be a nilpotent element in $\mathfrak{so}_{2\ell-1}(I)$ having the
partition $[2^{j-1}1^{2\ell-j+1}].$ Such a matrix $N$ exists in view of [CM,
Theorem 5.1.2]. Then the matrices $x_{0}$ and

$y_{0}:=\operatorname{diag}(0,...,0,N,0,0)+\dfrac{1}{2}x_{0}^{2}$

satisfy the relations in (17). Hence $g(t):=1+x_{0}t^{-1}+y_{0}t^{-2}\in
G(\mathcal{O})^{\sigma}$. By (2), $g(t)\cdot
e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ for some
$\bar{\lambda}=(a_{1},a_{2},...,a_{\ell})\in X_{*}(T)_{\sigma}^{+}$ with
$a_{1}\geq...\geq a_{\ell}$. Since $\operatorname{rk}y_{0}=j$, by Lemma 4.4,
$\bar{\lambda}=(1^{j}0^{\ell-j})$ and hence $g(t)\cdot
e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$. Since $\pi$ is $K$-equivariant, the
first part is done.

Suppose that $j$ is even. For each
$j=0,2,4,...,2\lfloor\frac{\ell}{2}\rfloor$, consider the $\ell\times\ell$
matrix

$\begin{pmatrix}&&1&&&&&\\\ &\reflectbox{$\ddots$}&&&&&&\\\ 1&&&&&&&\\\
&&&&&&&\\\ &&&&&&&\\\ &&&&&&&-1\\\ &&&&&&\reflectbox{$\ddots$}&\\\
&&&&&-1&&\end{pmatrix}$

where there are $\frac{j}{2}$ copies of each 1 and -1. Denote $z_{j}$ the
square zero matrix of size $2\ell+2$ whose $\ell\times\ell$ submatrix on the
right top is replaced by the above matrix. Now we work under
$\mathfrak{so}_{2\ell+2}(J)$ and $\mathrm{SO}_{2\ell+2}(J)$. Since
$wz_{j}w=z_{j}$ and $\operatorname{rk}z_{j}=j$, we have $(1+z_{j}t^{-2})\cdot
e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$ and then $\pi((1+z_{j}t^{-2})\cdot
e_{0})=0$. Let $x_{0}$ be the square zero matrix of size $2\ell+2$ whose
$4\times 4$ submatrix at the center is replaced by

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

Then $x_{0}\in\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}$, Set
$y_{0}=\frac{1}{2}x_{0}^{2}+z_{j}$. Then $\operatorname{rk}y_{0}=j$. It can be
checked that $x_{0},y_{0}$ satisfy $wx_{0}w=-x_{0},wy_{0}w=y_{0}$, and

(18) $\displaystyle x_{0}^{T}J+Jx_{0}=0,\hskip
14.22636ptx_{0}^{T}Jx_{0}+y_{0}^{T}J+Jy_{0}=0,$ $\displaystyle
x_{0}^{T}Jy_{0}+y_{0}^{T}Jx_{0}=0,\hskip 14.22636pty_{0}^{T}Jy_{0}=0.$

Hence $h(t):=1+x_{0}t^{-1}+y_{0}t^{-2}\in G(\mathcal{O})^{\sigma}$. Similarly,
one can show that $h(t)\cdot e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$. This
proves the second part.

To prove the last part, let $x$ be a nonzero nilpotent element in
$\mathfrak{p}$. Since $x$ has the partition $[31^{2\ell}]$,
$\operatorname{rk}x^{2}=1$. It is easy to check that
$(1+xt^{-1}+\frac{1}{2}x^{2}t^{-2})\cdot e_{0}\in\mathcal{M}_{(10^{\ell-1})}$.
Conversely, let $g(t)\cdot e_{0}=(1+xt^{-1}+yt^{-2})\cdot
e_{0}\in\mathcal{M}_{(10^{\ell-1})}$. Let
$\iota(g(t))=1+xt^{-1}+y^{\prime}t^{-2}$. Since
$g(t)=g(t)^{-T}=(\iota(g(-t)))^{T}$, $y=(y^{\prime})^{T}$. Then $y$ and
$y^{\prime}$ are adjoint each other under the symmetric form whose matrix is
$I$. Note that $\operatorname{rk}y=\operatorname{rk}y^{\prime}=1$. If
$\operatorname{Im}y\neq\operatorname{Im}y^{\prime}$, then
$\operatorname{rk}x^{2}=\operatorname{rk}y+\operatorname{rk}y^{\prime}=2$, a
contradiction. Hence $\operatorname{Im}y=\operatorname{Im}y^{\prime}$. By
Lemma 4.8, $y^{\prime}=y$ or $y^{\prime}=-y$. By (13), $x^{2}=y+y^{\prime}$
and hence $y^{\prime}=y$. By (17), $x^{T}+x=0$ and $x^{T}x+y^{T}+y=0$. Then
$y+y^{\prime}=x^{2}=y+y^{T}$, so $y=y^{\prime}=y^{T}$. Therefore,
$g(t)=1+xt^{-1}+yt^{-2}=1+xt^{-1}+\frac{1}{2}x^{2}t^{-2}.$ ∎

######  Proposition 4.15.

For $x\in\mathcal{N}_{\mathfrak{p}}$, write $x$ as in (16). Then

$\pi^{-1}(x)\cong\\{D\in\mathfrak{so}_{2\ell+1}\mid Du=0,D^{2}=0\\}.$

In particular, $\pi^{-1}(0)$ is isomorphic to the maximal order 2 nilpotent
variety in $\mathfrak{so}_{2\ell+1}$, and

$\dim\pi^{-1}(0)=\begin{cases}\ell^{2}&\quad\text{if }\ell\text{ is even;}\\\
\ell^{2}-1&\quad\text{if }\ell\text{ is odd.}\\\ \end{cases}$

###### Proof.

Under the realization $\mathfrak{so}_{2\ell+2}(I)$ and
$\mathrm{SO}_{2\ell+2}(I)$, and the diagram automorphism $\sigma_{0}$, we have
that $1+xt^{-1}+yt^{-2}\in\mathcal{M}$ if and only if $w_{0}yw_{0}=y$ and the
conditions (17) hold. Set $z=y-\frac{1}{2}x^{2}$, these conditions are
equivalent to

(19) $z=\begin{pmatrix}0&\\\ &D\end{pmatrix},\quad
D\in\mathfrak{so}_{2\ell+1},\quad D^{2}=0,\quad Du=0.$

Hence $\pi^{-1}(0)\cong\\{D\in\mathfrak{so}_{2\ell+1}\mid D^{2}=0\\}$ which is
$\overline{\mathcal{O}}_{[2^{k}1^{2\ell-2k+1}]}$ in $\mathfrak{so}_{2\ell+1}$
where $k$ is the maximal even integer. By the dimension formula, cf. [CM,
Corollary 6.1.4],

$\dim\pi^{-1}(0)=\begin{cases}\dim\mathcal{O}_{[2^{\ell}1]}=\ell^{2}&\quad\text{if
}\ell\text{ is even;}\\\
\dim\mathcal{O}_{[2^{\ell-1}1^{3}]}=\ell^{2}-1&\quad\text{if }\ell\text{ is
odd}\\\ \end{cases}$

as desired. ∎

Similar to the case $(A_{2\ell-1},2)$, it is not obvious to see if there are
finitely many $K$-orbits in $\mathcal{M}_{(1^{j}0^{\ell-j})}$. If
$g(t)=1+xt^{-1}+(z+\frac{1}{2}x^{2})t^{-2}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$,
then $g(t)$ satisfies (19). If the action of $K$ on the following anti-
commuting nilpotent variety

$\\{(x,z)\in\mathfrak{so}_{2\ell+2}(I)\times\mathfrak{so}_{2\ell+2}(I)\mid
xz+zx=0,x,z\text{ nilpotent}\\}$

by diagonal cojugation has finitely many orbits, then there are finitely many
$K$-orbits in $\mathcal{M}_{(1^{j}0^{\ell-j})}$.

##  5\. Applications

In this section, we describe some applications to the geometry of order 2
nilpotent varieties in the certain classical symmetric spaces.

Let $\langle,\rangle$ be a symmetric or symplectic non-degenerate bilinear
form on a vector space $V$. Recall that $\mathcal{A}$ is the space of all
self-adjoint linear maps with respect to $\langle,\rangle$. Set
$\mathcal{N}_{\mathcal{A},2}$ denote the space of all nilpotent operators $x$
in $\mathcal{A}$ such that $x^{2}=0$. If $\langle,\rangle$ is symmetric and
$\dim V=2n+1$, then ${\rm SO}_{2n+1}$-orbits in $\mathcal{N}_{\mathcal{A},2}$
are classified by the partitions $[2^{j}1^{2n+1-2j}]$ with $0\leq j\leq n$; if
$\langle,\rangle$ is symplectic and $\dim V=2n$, then ${\rm Sp}_{2n}$-orbits
in $\mathcal{N}_{\mathcal{A},2}$ are classified by the partitions
$[2^{2j}1^{2n-2j}]$ with $0\leq j\leq\lfloor\frac{n}{2}\rfloor$.

######  Theorem 5.1.

Assume that $\langle,\rangle$ is symplectic or symmetric and $\dim V$ is odd.
Then any order 2 nilpotent variety in $\mathcal{A}$ is normal.

###### Proof.

By Theorem 4.5 and Theorem 4.6, for any order 2 nilpotent variety
$\overline{\mathcal{O}}$ in $\mathcal{A}$, $\overline{\mathcal{O}}$ is
isomorphic to
$\overline{\mathcal{M}}_{\bar{\lambda}}:=\overline{\mathcal{G}r}_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}$
for a small dominant weight $\bar{\lambda}$ of $H$. Note that
$\overline{\mathcal{M}}_{\bar{\lambda}}$ is an open subset of the twisted
Schubert variety $\overline{\mathcal{G}r}_{\bar{\lambda}}$ and
$\overline{\mathcal{G}r}_{\bar{\lambda}}$ is a normal variety. It follows that
$\overline{\mathcal{O}}$ is also normal. ∎

In fact, when $\langle,\rangle$ is symplectic, any nilpotent variety in
$\mathcal{A}$ is normal, see [Oh]. In loc.cit., Ohta also showed that not all
nilpotent varieties are $\mathcal{N}_{\mathfrak{p}}$ is normal, when
$\langle,\rangle$ is symmetric. When $\langle,\rangle$ is symmetric and $\dim
V$ is odd, this theorem seems to be new.

######  Remark 5.2.

Theorem 5.1 is true for any field $\mathrm{k}$ of characteristic $p>2$, as one
can easily see that the classification theorem in Section 3 still holds for
order 2 nilpotent orbits, and the arguments in Theorem 4.5, Theorem 4.6
applies as well. The same remark applies to the following Theorem 5.3 and
Theorem 5.4

For any variety $X$, let ${\rm IC}_{X}$ denote the intersection cohomology
sheaf on $X$. The perverse sheaf ${\rm IC}_{X}$ captures the singularity of
the variety $X$. For any $x\in X$, we denote by $\mathscr{H}^{k}_{x}({\rm
IC}_{X})$ the k-th cohomology of the stalk of ${\rm IC}_{X}$ at $x$.

######  Theorem 5.3.

  1. (1)

When $\langle,\rangle$ is symmetric and $\dim V=2n+1$, for any $0\leq j\leq
n$, let $\mathcal{O}_{j}$ denote the nilpotent orbit in $\mathcal{A}$
associated to the partition $[2^{j}1^{2n+1-2j}]$ and let
$\mathcal{O}^{\prime}_{j}$ denote the nilpotent orbit in $\mathfrak{sp}_{2n}$
associated to the partition $[2^{j}1^{2n-2j}]$, we have

$\dim\mathcal{O}_{j}=\dim\mathcal{O}^{\prime}_{j}=j(2n+1-j).$

Moreover, for any $x\in\mathcal{O}_{[2^{i}1^{2n+1-2i}]}$ and
$x^{\prime}\in\mathcal{O}^{\prime}_{[2^{i}1^{2n-2i}]}$, and for any
$k\in\mathbb{Z}$,

$\dim\mathscr{H}_{x}^{k}({\rm
IC}_{\overline{\mathcal{O}}_{j}})=\dim\mathscr{H}_{x}^{k}({\rm
IC}_{\overline{\mathcal{O}^{\prime}}_{j}}).$

  2. (2)

When $\langle,\rangle$ is symplectic and $\dim V=2n$, for any $0\leq
j\leq\lfloor\frac{n}{2}\rfloor$, let $\mathcal{O}_{2j}$ denote the nilpotent
orbit in $\mathcal{A}$ associated to the partition $[2^{2j}1^{2n-4j}]$ and let
$\mathcal{O}^{\prime}_{2j}$ denote the nilpotent orbit in
$\mathfrak{so}_{2n+1}$ associated to the partition $[2^{2j}1^{2n+1-4j}]$, we
have

$\dim\mathcal{O}_{2j}=\dim\mathcal{O}^{\prime}_{2j}=4j(n-j),$

Moreover, for any integer $0\leq i\leq j$, $x\in\mathcal{O}_{2i}$,
$x^{\prime}\in\mathcal{O}^{\prime}_{2i}$, and for any $k\in\mathbb{Z}$, we
have

$\dim\mathscr{H}_{x}^{k}({\rm
IC}_{\overline{\mathcal{O}}_{2j}})=\dim\mathscr{H}_{x}^{k}({\rm
IC}_{\overline{\mathcal{O}^{\prime}}_{2j}}).$

###### Proof.

We first prove part 1). By Theorem 3.7 and [CM, Corollary 6.1.4], it is easy
to verify $\dim\mathcal{O}_{j}=\dim\mathcal{O}^{\prime}_{j}=j(2n+1-j)$. By
Theorem 4.5, $\overline{\mathcal{O}}_{j}$ can be embedded into an open subset
in the twisted affine Schubert variety $\overline{\mathcal{G}r}_{\omega_{j}}$
associated to $(\mathrm{SL}_{2n+1},\sigma)$. On the other hand, in view of
[AH], $\overline{\mathcal{O}^{\prime}}_{j}$ can be embedded into the untwisted
affine Schubert variety
$\overline{\operatorname{Gr}}^{\omega_{j}}_{\mathrm{Sp}_{2n}}$ in the affine
Grassmannian $\overline{\operatorname{Gr}}_{\mathrm{Sp}_{2n}}$ of
$\mathrm{Sp}_{2n}$. Set

$\mathcal{F}={\rm
IC}_{\overline{\mathcal{O}}_{j}}[-\dim\overline{\mathcal{O}}_{j}],\quad\text{
and }\mathcal{F}^{\prime}={\rm
IC}_{\overline{\mathcal{O}^{\prime}}_{j}}[-\dim\overline{\mathcal{O}^{\prime}}_{j}].$

By purity vanishing property of intersection cohomology sheaf of Schubert
varieties (cf. [KL]),
$\mathscr{H}_{x}^{k}(\mathcal{F})=\mathscr{H}_{x^{\prime}}^{k}(\mathcal{F}^{\prime})=0$
when $k$ is odd. Equivalently,

$\mathscr{H}_{x}^{k}({\rm
IC}_{\overline{\mathcal{O}}_{j}})=\mathscr{H}_{x}^{k}({\rm
IC}_{\overline{\mathcal{O}^{\prime}}_{j}})=0$

for any odd integer $k$, as
$\dim\overline{\mathcal{O}}_{j}=\dim\overline{\mathcal{O}^{\prime}}_{j}$ is
even.

Note that the affine Grassmannian $\operatorname{Gr}_{\mathrm{Sp}_{2n}}$ and
the twisted affine Grassmannian $\mathcal{G}r_{{\rm SL}_{2n+1}}$ have the same
underlying affine Weyl group. Applying the results in [KL], the polynomials
$\sum\dim\mathscr{H}_{x}^{2k}(\mathcal{F})q^{k}$ and
$\sum\mathscr{H}_{x}^{2k}(\mathcal{F}^{\prime})q^{k}$ are both equal to the
same Kazhdan-Lusztig polynomial $P_{\omega_{i},\omega_{j}}(q)$ for the affine
Weyl group of $\mathfrak{so}_{2n+1}$. It follows that

$\dim\mathscr{H}_{x}^{k}({\rm
IC}_{\overline{\mathcal{O}}_{j}})=\dim\mathscr{H}_{x}^{k}({\rm
IC}_{\overline{\mathcal{O}^{\prime}}_{j}})$

for all even integer $k$. Alternatively, one can see these two polynomials are
equal, as they both coincide with the jump polynomial of the Brylinsky-Kostant
filtration on the irreducible representation $V_{\omega_{j}}$ of $H$, see
[Bry, Zh].

For the second part of the theorem, the proof is almost the same, except that
by Theorem 4.6, $\overline{\mathcal{O}}_{2j}$ can be openly embedded into the
twisted affine Schubert variety $\overline{\mathcal{G}r}_{\omega_{2j}}$
associated to $(\mathrm{SL}_{2n},\sigma)$, and
$\overline{\mathcal{O}}^{\prime}_{2j}$ can be openly embedded into the affine
Schubert variety
$\overline{\operatorname{Gr}}^{\omega_{2j}}_{\mathrm{Spin}_{2n+1}}$. ∎

Part 1) of this theorem was due to Chen-Xue-Vilonen [CVX] by different
methods. This theorem shows that there is a natural bijection between order 2
nilpotent varieties in $\mathcal{A}$ and order 2 nilpotent varieties in its
dual classical Lie algebras, such that they share similar geometry and
singularities. However, under such bijection the associated two nilpotent
varieties are not necessarily isomorphic. For example, when $\langle,\rangle$
is symmetric and $\dim V=2n+1$, $\overline{\mathcal{O}}_{[21^{2n-1}]}$ is the
minimal nilpotent variety in $\mathcal{A}$. In fact
$\overline{\mathcal{O}}_{[21^{2n-1}]}$ is smooth, since the quasi-miniscule
twisted affine Schubert variety $\overline{\mathcal{G}r}_{\omega_{1}}$ is
smooth, cf. [HR, Section 5.1]. On the other hand, the minimal nilpotent
variety $\overline{\mathcal{O}^{\prime}}_{[21^{2n-2}]}$ in
$\mathfrak{sp}_{2n}$ is not smooth, as the smooth locus of the quasi-miniscule
affine Schubert variety $\overline{\operatorname{Gr}}_{\omega_{1}}$ is the
open cell $\operatorname{Gr}_{\omega_{1}}$.

We now describe another application.

######  Theorem 5.4.

If $\langle,\rangle$ is symplectic, then the smooth locus of any order 2
nilpotent variety in $\mathcal{A}$ is the open nilpotent orbit.

###### Proof.

Let $\overline{\mathcal{O}}$ be any order 2 nilpotent variety in
$\mathcal{A}$. By Theorem 4.6, $\overline{\mathcal{O}}$ can be openly embedded
into a twisted Schubert variety $\overline{\mathcal{G}r}_{\bar{\lambda}}$ with
$\bar{\lambda}$ small, in the twisted affine Grassmannian
$\mathcal{G}r_{\mathrm{SL}_{2n}}$. Then this theorem follows from [BH, Theorem
1.2]. ∎

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