#  $B_{n-1}$-bundles on the flag variety, I

Mark Colarusso  Department of Math and Stats, University of South Alabama,
Mobile, Al, 36608<EMAIL_ADDRESS>and  Sam Evens  Department of
Mathematics, University of Notre Dame, Notre Dame, IN, 46556<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider the orbits of a Borel subgroup $B_{n-1}$ of
$G_{n-1}=GL(n-1)$ (respectively $SO(n-1)$) acting on the flag variety
$\mathcal{B}_{n}$ of $G=GL(n)$ (resp. $SO(n)$). The group $B_{n-1}$ acts on
$\mathcal{B}_{n}$ with finitely many orbits, and we use the known description
of $G_{n-1}$-orbits on $\mathcal{B}$ to study these orbits. In particular, we
show that a $B_{n-1}$-orbit is a fibre bundle over a $B_{n-1}$-orbit in a
generalized flag variety of $G_{n-1}$ with fibre a $B_{m-1}$-orbit on
$\mathcal{B}_{m}$ for some $m<n$. We further use this fibre bundle structure
to study the orbits inductively and describe the monoid action on the fibre
bundle. In a sequel to this paper, we use these results to give a complete
combinatorial model for these orbits and show how to understand the closure
relations on these orbits in terms of the monoid action.

###### Key words and phrases:

$K$-orbits on flag variety, algebraic group actions

###### 2020 Mathematics Subject Classification:

14M15, 14L30, 20G20

##  1\. Introduction

In this paper, we consider certain solvable subgroups acting on the flag
variety with finitely many orbits. The subgroups we consider arise naturally
in the complex Gelfand-Zeitlin integrable system and in the theory of Gelfand-
Zeitlin modules. The goal of this paper, and its sequel, is to provide a
combinatorial description of the orbits and in particular to understand
monoidal actions on the orbits. We then hope to use these results to
understand representations of the Lie algebra realized via the Beilinson-
Bernstein correspondence using local systems on these orbits.

In more detail, let $G_{n}=GL(n)$ or $SO(n)$, and embed $G_{n-1}$ as a
symmetric subgroup of $G_{n}$, up to center in the $GL(n)$-case. We note that
the pairs $(G_{n},G_{n-1})$ are (up to center and isogeny) the multiplicity
free symmetric pairs. We let $B_{n-1}$ denote a Borel subgroup of $G_{n-1}$
and recall that $B_{n-1}$ acts on the flag variety of Borel subgroups
$\mathcal{B}_{n}$ of $G_{n}$ with finitely many orbits. Our main result is
that $B_{n-1}$-orbits on $\mathcal{B}_{n}$ fibre over $B_{n-1}$-orbits on a
generalized flag variety of $G_{n-1}$ with fibre given by $B_{m-1}$-orbits on
$\mathcal{B}_{m}$ for some $m<n$. More precisely, we construct a class of
parabolic subgroups of $G_{n}$, which we call special parabolic subgroups.

######  Theorem 1.1.

For each $B_{n-1}$-orbit $Q$ on $\mathcal{B}_{n}$, there is a special
parabolic $R$ of $G_{n}$ so that $R\cap G_{n-1}$ is a parabolic subgroup of
$G_{n-1}$, and

  1. (1)

there is a fibre bundle $Q\to Q_{1}$, where $Q_{1}$ is a $B_{n-1}$-orbit on
$G_{n-1}/R\cap G_{n-1}$;

  2. (2)

the fibre of $Q\to Q_{1}$ may be identified with a $B_{m-1}$-orbit $Q_{2}$ on
$\mathcal{B}_{m}$.

If we denote the above $B_{n-1}$-orbit by $Q=\mathcal{O}(Q_{1},Q_{2})$, then
$\mathcal{O}(Q_{1},Q_{2})=\mathcal{O}(Q_{1}^{\prime},Q_{2}^{\prime})$ if and
only if $Q_{i}=Q_{i}^{\prime}$ for $i=1,2.$

Our theorem allows for an inductive description of $B_{n-1}$-orbits on the
flag variety $\mathcal{B}_{n}$ in terms of special parabolics, Weyl group
data, and data derived inductively from the same problem for a smaller index.
The special parabolic subalgebras are in one-to-one correspondence with
$G_{n-1}$-orbits on $\mathcal{B}_{n}$. For the closed $G_{n-1}$-orbits on
$\mathcal{B}_{n}$, the corresponding special parabolics are the
representatives of these orbits given in Part (2) of Propositions 2.8, 2.9,
and 2.10. The special parabolics corresponding to $G_{n-1}$-orbits that are
not closed are given in the orthogonal cases by taking parabolics containing
the upper triangular matrices and corresponding to the choice of simple roots
$\\{\alpha_{i},\dots,\alpha_{l}\\}$ where $i=1,\dots,l$ for type B and
$i=1,\dots,l-1$ for type D using the usual ordering of simple roots (see (3.4)
and (3.7)). In the general linear case the special parabolics are given by
choosing a consecutive subset of simple roots and a corresponding element of
the Weyl group (see (3.3)).

In a sequel to this paper, we use our results to give a combinatorial
description of the $B_{n-1}$-orbits on $\mathcal{B}_{n}$. Further, we use the
action of a monoid using both simple roots of $G_{n-1}$ and of $G_{n}$ to show
that every $B_{n-1}$-orbit arises from a zero dimensional $B_{n-1}$-orbit
using the monoid action. Richardson and Springer [RS90] have pointed out the
seminal role of verifying this result in the description of the orbit closure
relation. As a consequence of work of Richardson and Springer [RS90], we use
this result to determine the closure relation between $B_{n-1}$-orbits in
terms of the monoid action. An essential step in this process is to describe
instances when this monoid action may be described efficiently using the fibre
bundle description of the orbits of Theorem 1.1 (see Theorems 4.7, 4.11, and
4.12).

In the $GL(n)$-case, Hashimoto [Has04] describes the $B_{n-1}$-orbits on
$\mathcal{B}_{n}$, and discusses part of the monoid action that we consider in
this paper. Our work advances on [Has04] in that our approach also describes
orbits both in general linear and orthogonal cases in an essentially uniform
manner, elucidates the bundle structure, and gives a complete description of
the closure relation. While Hashimoto’s approach is based on analyzing orbits
of a Borel subgroup, our approach is based more on studying the orbits of
$G_{n-1}$ on the flag manifold. It would also be interesting to relate our
results to the work of Gandini and Pezzini [GP18], who study the general case
of orbits of solvable subgroups of $G$ with finitely many orbits on the flag
variety of $G$. In particular, it would be of interest to understand some of
the invariants they consider in their more general context in our situation.

In Section 2 of this paper, we introduce notation and recall standard facts
about orbits and embeddings of $G_{n-1}$ in $G_{n}$ for later use. In Section
3, we establish the fibre bundle structure for $B_{n-1}$-orbits and use it to
give an inductive method to describe $B_{n-1}$-orbits on $\mathcal{B}_{n}$. In
Section 4, we discuss the monoid action and compute it in cases where the role
of the bundle structure is salient. The remaining cases are treated in the
sequel to this paper using a more explicit combinatorial description of the
orbits.

We would like to thank Friedrich Knop, Jacopo Gandini, and Guido Pezzini for
useful discussions which aided in the development of our results. During the
preparation of this paper, the first author was supported in part by the
National Security Agency grant H98230-16-1-0002 and the second author was
supported in part by the Simons Foundation grant 359424.

##  2\. Preliminaries

We discuss some notation and standard results that will be used throughout the
paper.

###  2.1. Notation

By convention, all algebraic groups and Lie algebras discussed in this paper
have complex coefficients. If an algebraic group $M$ is defined, we denote by
the corresponding Gothic letter ${\mathfrak{m}}$ its Lie algebra, and if a Lie
subalgebra ${\mathfrak{m}}$ of the Lie algebra of a complex Lie group $G$ is
given, we let $M$ denote the connected subgroup of $G$ with Lie algebra
${\mathfrak{m}}$. For an algebraic group $M$ and $g\in M$, we denote by ${\rm
Ad}(g)$ both the adjoint action of $M$ on its Lie algebra, and the conjugation
action of $M$ on itself. As a convention, we let
$\mathcal{B}_{\mbox{${\mathfrak{m}}$}}$ denote the flag variety of Borel
subalgebras of the Lie algebra ${\mathfrak{m}}$, and let
$\mathcal{B}_{n}=\mathcal{B}_{{\mbox{${\mathfrak{g}}$}}_{n}}$ for
${\mbox{${\mathfrak{g}}$}}_{n}=\mathfrak{gl}(n)$ or $\mathfrak{so}(n)$. For
$(V,\beta)$ a vector space with symmetric bilinear form $\beta$ and subspace
$U$, we let $U^{\perp}:=\\{v\in V:\beta(u,v)=0\ \forall u\in U\\}.$ Throughout
the paper, we let $\imath$ denote a fixed choice of $\sqrt{-1}$.

###  2.2. Realizations

We use the conventions for realizing $\mathfrak{so}(n)$ from Chapters 1 and 2
of [GW98] (see also [CE19]). More explicitly, we let $\mathfrak{so}(n)$ be the
symmetry Lie algebra of the non-degenerate, symmetric bilinear form $\beta$ on
$\mbox{${\mathbb{C}}$}^{n}$ given by $\beta(x,y)=x^{T}Jy$, where $J$ is the
automorphism of $\mbox{${\mathbb{C}}$}^{n}$ such that $J(e_{i})=e_{n+1-i}$ for
the standard basis $e_{1},\dots,e_{n}$ of $\mbox{${\mathbb{C}}$}^{n}$ and
$i=1,\dots,n$. Note that $\beta(e_{j},e_{n+1-j})=1$ for $j=1,\dots,n$. We let
$SO(n)$ be the connected algebraic group with Lie algebra $\mathfrak{so}(n).$
Then the subalgebra $\mbox{${\mathfrak{h}}$}_{n}$ consisting of diagonal
matrices in $\mathfrak{so}(n)$ is a Cartan subalgebra, and the subalgebra
$\mbox{${\mathfrak{b}}$}_{+}$ consisting of upper triangular matrices in
$\mathfrak{so}(n)$ is a Borel subalgebra, and we refer to these as the
standard Cartan and Borel subalgebras. We refer also to
$\mbox{${\mathfrak{h}}$}_{n}$ and $\mbox{${\mathfrak{b}}$}_{+}$ to denote the
diagonal and upper triangular matrices in $\mathfrak{gl}(n).$ For
$\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n)$ or $\mathfrak{so}(n)$, any
parabolic subalgebra
$\mbox{${\mathfrak{p}}$}\supset\mbox{${\mathfrak{b}}$}_{+}$ we refer to as a
standard parabolic subalgebra. We also let
$\Phi=\Phi(\mbox{${\mathfrak{g}}$},\mbox{${\mathfrak{h}}$}_{n})$ denote the
roots of ${\mathfrak{g}}$ with respect to the Cartan subalgbra
$\mbox{${\mathfrak{h}}$}_{n}$. If $\alpha\in\Phi$, we let
$\mbox{${\mathfrak{g}}$}_{\alpha}$ denote the root space corresponding to the
root $\alpha$.

It will be convenient to relabel the standard basis $\\{e_{1},\dots,e_{n}\\}$
of $\mbox{${\mathbb{C}}$}^{n}$ depending on whether $n$ is even or odd. If
$n=2l$ is even, then we let $e_{-i}:=e_{2l+1-i}$ for $i=1,\dots,l$. Note that
for $j,\,k\in\\{-l,\dots,-1,1,\dots,l\\}$, we have
$\beta(e_{j},e_{k})=\delta_{j,-k}$. If $n=2l+1$, then we let $e_{0}:=e_{l+1}$
and $e_{-i}:=e_{2l+2-i}$ for $i=1,\dots,l$. For
$j,\,k\in\\{-l,\dots,-1,0,1,\dots,l\\}$, we have
$\beta(e_{j},e_{k})=\delta_{j,-k}$.

We recall the real rank one subalgebras for $\mathfrak{gl}(n)$ and
$\mathfrak{so}(n)$. For $\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n)$, let
$t=\mbox{diag}[1,\dots,1,-1]\in GL(n)$, and let $\theta(x)=txt^{-1}$ for
$x\in\mbox{${\mathfrak{g}}$}$. Then the fixed set
$\mbox{${\mathfrak{g}}$}^{\theta}=\mathfrak{gl}(n-1)\oplus\mathfrak{gl}(1)$,
regarded as block matrices, and
$\mbox{${\mathfrak{g}}$}_{n-1}=\mathfrak{gl}(n-1)\oplus 0$.

For $\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l+1)$, let $t$ be an element of
the Cartan subgroup with Lie algebra $\mbox{${\mathfrak{h}}$}_{2l+1}$ with the
property that ${\rm Ad}(t)|_{\mbox{${\mathfrak{g}}$}_{\alpha_{i}}}=\mbox{${\rm
id}$}$ for $i=1,\dots,l-1$ and ${\rm
Ad}(t)|_{\mbox{${\mathfrak{g}}$}_{\alpha_{l}}}=-\mbox{${\rm id}$}$. Consider
the involution $\theta_{2l+1}:={\rm Ad}(t)$. Then
$\mbox{${\mathfrak{k}}$}:=\mbox{${\mathfrak{g}}$}^{\theta_{2l+1}}\cong\mathfrak{so}(2l)$,
realized as block matrices with the $l+1$st rows and columns equal to $0$ (see
[Kna02], p. 700). Note that
$\mbox{${\mathfrak{h}}$}_{2l+1}=\mbox{${\mathfrak{h}}$}_{2l}\subset\mbox{${\mathfrak{k}}$}$.
For later use, let $\sigma_{2l+1}$ be the diagonal matrix with respect to the
basis $\\{e_{1},\dots,e_{l},e_{0},e_{-l},\dots,e_{-1}\\}$ such that
$\sigma_{2l+1}\cdot e_{j}=e_{j}$ for $j\not=0$ and $\sigma_{2l+1}\cdot
e_{0}=-e_{0}$ and note that $\theta=\theta_{2l+1}={\rm Ad}(\sigma_{2l+1})$.

In the case $\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l)$, consider the
involutive linear map
$\sigma_{2l}:\mbox{${\mathbb{C}}$}^{2l}\to\mbox{${\mathbb{C}}$}^{2l}$ such
that $\sigma_{2l}(e_{l})=e_{-l}$ and $\sigma_{2l}(e_{i})=e_{i}$ for $i\not=l$
or $-l.$ Define an involution $\theta$ of ${\mathfrak{g}}$ by $\theta={\rm
Ad}(\sigma_{2l})$. Then
$\mbox{${\mathfrak{k}}$}:=\mbox{${\mathfrak{g}}$}^{\theta_{2l}}\cong\mathfrak{so}(2l-1)$,
and $\theta_{2l}$ is the involution induced by the diagram automorphism
interchanging the simple roots $\alpha_{l-1}$ and $\alpha_{l}$ (see [Kna02],
p. 703). If we realize $\mbox{${\mathfrak{h}}$}_{2l}$ as diagonal matrices of
the form $x=\mbox{diag}[x_{1},\dots,x_{l},-x_{l},\dots,-x_{1}]$ and define for
$i=1,\dots,l$, $\epsilon_{i}\in\mbox{${\mathfrak{h}}$}_{2l}^{*}$ by
$\epsilon_{i}(x)=x_{i}$, then the induced action of $\theta_{2l}$ is via
$\theta_{2l}(\epsilon_{l})=-\epsilon_{l}$ and
$\theta_{2l}(\epsilon_{i})=\epsilon_{i}$ for $i=1,\dots,l-1$. We will omit the
subscripts $2l+1$ and $2l$ from $\theta$ when ${\mathfrak{g}}$ is understood.

We also denote the involution of the corresponding classical group $G$ by
$\theta$. For $G=GL(n)$, $G^{\theta}\cong GL(n-1)\times GL(1)$ is connected,
while for $G=SO(n)$, the group $G^{\theta}\cong S(O(n-1)\times O(1))$ is
disconnected. We let $K:=(G^{\theta})^{0}$ be the identity component of
$G^{\theta}$. Then $K\cong SO(n-1)$, and $K$ has Lie algebra
$\mbox{${\mathfrak{k}}$}=\mbox{${\mathfrak{g}}$}^{\theta}$.

######  Remark 2.1.

For $SO(n)$, note that we can realize the subgroups $SO(n-1)$ and $SO(n-2)$ as
follows. For $n$ either odd or even, note that the $-1$ eigenspace
$(\mbox{${\mathbb{C}}$}^{n})^{-\sigma_{n}}$ is one dimensional and is
nondegenerate for the restriction of $\beta$. If
$(\mbox{${\mathbb{C}}$}^{n})^{-\sigma_{n}}=\mbox{${\mathbb{C}}$}v_{n}$, then
since the perpindicular of an $SO(n-1)$-stable subspace is $SO(n-1)$-stable,
it follows that

(2.1) $SO(n-1)=\\{g\in SO(n):g\cdot v_{n}=v_{n}\\}.$

In the case $n=2l,$ let
$V_{2}=\mbox{${\mathbb{C}}$}e_{l}+\mbox{${\mathbb{C}}$}e_{-l}.$ Then

(2.2) $SO(2l-2)=\\{g\in SO(2l):g\cdot v=v\ \forall v\in V_{2}\\}.$

Here $SO(2l-2)$ is realized as determinant one orthogonal transformations of
the orthogonal complement $V_{2}^{\perp}$ to $V_{2}$. In the case $n=2l+1$,
one similarly observes that

(2.3) $SO(2l-1)=\\{g\in SO(2l+1):g\cdot e_{0}=e_{0},\
g\cdot(e_{l}-e_{-l})=e_{l}-e_{-l}.\\}$

For later use, we also note that if $\alpha_{1},\dots,\alpha_{l}$ are the
simple roots for $SO(2l+1)$, then the simple roots for $SO(2l)$ are given by
$\alpha_{1},\dots,\alpha_{l-1},\alpha_{l-1}+2\alpha_{l}$. We may take as root
vectors for the last two simple roots of the Lie algebra $\mathfrak{so}(2l)$
the vectors $e_{\alpha_{l-1}}=E_{l-1,l}-E_{-l,-(l-1)}$ and
$e_{\alpha_{l}}=E_{l-1,-l}-E_{l,-(l-1)}$ and then their sum
$e_{\alpha_{l-1}}+e_{\alpha_{l}}$ is the root vector for for
$\mathfrak{so}(2l-1)$ for the simple root $\alpha_{l-1}$.

###  2.3. Description of $K$-orbits on $\mathcal{B}_{n}$ in the multiplicity
free setting

Note that the cases
$(\mbox{${\mathfrak{g}}$},\mbox{${\mathfrak{g}}$}_{n-1})=(\mathfrak{gl}(n),\mathfrak{gl}(n-1))$
or $(\mathfrak{so}(n),\mathfrak{so}(n-1))$ are up to center essentially the
only situations where each irreducible representation of ${\mathfrak{g}}$ has
multiplicity free branching law when regarded as a representation of a
symmetric subalgebra ${\mathfrak{k}}$. By abuse of notation, we use
${\mathfrak{k}}$ to denote $\mbox{${\mathfrak{g}}$}_{n-1}$ and $K$ to denote
$G_{n-1}$. This property is well-known to be equivalent to the property that a
Borel subgroup $B_{K}$ of $K$ has finitely many orbits on
$\mathcal{B}_{\mbox{${\mathfrak{g}}$}}$, the variety of Borel subalgebras of
${\mathfrak{g}}$ (see for example, Proposition 3.1.3 of [CE19]).

######  Notation 2.2.

For a group $A$ acting on a variety $X$, we let $A\backslash X$ denote the
collection of $A$-orbits on $X$

We will need to use an explicit description of
$K\backslash\mathcal{B}_{\mbox{${\mathfrak{g}}$}}$. We present this mostly
without proof, and note that it follows from explicit study of the Richardson-
Springer discussion of orbits in [RS90], and is worked out explicitly for
$GL(n)$ in Section 4.4 of [CE14] (see especially Example 4.16, and Examples
4.30 to 4.32) and for $SO(n)$ in section 2.7 of [CE].

We first recall the monoidal action. Suppose $M$ is a subgroup of $G$ acting
with finitely many orbits on the flag variety
$\mathcal{B}_{\mbox{${\mathfrak{g}}$}}$ of Borel subalgebras of
${\mathfrak{g}}$, the Lie algebra of connected reductive group $G$. For a
simple root $\alpha$ for $G$, consider the
${\mbox{${\mathbb{P}}$}}^{1}$-bundle
$p:\mathcal{B}_{\mbox{${\mathfrak{g}}$}}\to{{\mathcal{P}}}_{\alpha,\mbox{${\mathfrak{g}}$}}$,
where ${{\mathcal{P}}}_{\alpha,\mbox{${\mathfrak{g}}$}}$ consists of the
parabolic subalgebras of ${\mathfrak{g}}$ with Lie algebra $G$-conjugate to
$\mbox{${\mathfrak{b}}$}_{0}+\mbox{${\mathfrak{g}}$}_{-\alpha}$, where
$\mbox{${\mathfrak{b}}$}_{0}$ is a Borel subalgebra for which $\alpha$ is a
simple root. We call the parabolic subalgebras in
${{\mathcal{P}}}_{\alpha,\mbox{${\mathfrak{g}}$}}$ simple parabolics of type
$\alpha$. Then if $Q=M\cdot\mbox{${\mathfrak{b}}$}$ is a $M$-orbit in
$\mathcal{B}_{\mbox{${\mathfrak{g}}$}}$, $p^{-1}p(Q)$ contains a unique open
$M$-orbit, and we call this orbit $m(s_{\alpha})*Q$ and note that if
$m(s_{\alpha})*Q\not=Q$, then $\dim(m(s_{\alpha})*Q)=\dim(Q)+1.$ We let
$\mbox{${\mathfrak{p}}$}_{\alpha}=p(\mbox{${\mathfrak{b}}$})$ and let
$P_{\alpha}$ denote the corresponding parabolic subgroup of $G$ with unipotent
radical $U_{\alpha}$ and if $T$ is a maximal torus of the group $B$ with Lie
algebra ${\mathfrak{b}}$, let $L_{\alpha}$ be the Levi subgroup of
$P_{\alpha}$ containing $T$ and let $Z_{\alpha}$ be the center of
$L_{\alpha}$. We denote by $M_{\alpha}$ the image of $M\cap P_{\alpha}$ in the
group $S_{\alpha}:=P_{\alpha}/Z_{\alpha}U_{\alpha}$ locally isomorphic to
$SL(2)$. Then one of the following occurs.

  1. (1)

Complex Case: $M_{\alpha}$ contains a maximal unipotent subgroup of
$S_{\alpha}$ and is contained in a Borel subgroup of $S_{\alpha}$. In this
case, $m(s_{\alpha})*Q$ meets $p^{-1}p(\mbox{${\mathfrak{b}}$})$ in a
$M_{\alpha}$-orbit isomorphic to ${\mathbb{C}}$ and the complement is a point;

  2. (2)

Noncompact/Real Case: $M_{\alpha}$ is locally isomorphic to a maximal torus of
$S_{\alpha}$. In this case, $m(s_{\alpha})*Q$ meets
$p^{-1}p(\mbox{${\mathfrak{b}}$})$ in a $M_{\alpha}$-orbit isomorphic to
$\mbox{${\mathbb{C}}$}^{\times}$, and if $M_{\alpha}$ is a torus, the
complement is a union of two $M_{\alpha}$-orbits, and otherwise, the
complement is a single $M_{\alpha}$-orbit;

  3. (3)

Compact Case: $M_{\alpha}=S_{\alpha}.$ In this case, $m(s_{\alpha})*Q=Q$ and
$p^{-1}p(\mbox{${\mathfrak{b}}$})\cong\mbox{${\mathbb{P}}$}^{1}$ is a single
$M_{\alpha}$-orbit.

######  Definition 2.3.

  1. (1)

In the complex case above, we say that the simple root $\alpha$ is complex
stable for $Q$ if $m(s_{\alpha})*Q\not=Q$ and complex unstable for $Q$ if
$m(s_{\alpha})*Q=Q;$

  2. (2)

In the noncompact/real case above, we say that the simple root $\alpha$ is
noncompact for $Q$ if $m(s_{\alpha})*Q\not=Q$ and real for $Q$ if
$m(s_{\alpha})*Q=Q;$

  3. (3)

In the compact case above, we say the simple root $\alpha$ is compact for $Q$,
and note that in this case $m(s_{\alpha})*Q=Q.$

The type of a simple root $\alpha$ for $Q$ depends only on $Q$ and not on the
choice of ${\mathfrak{b}}$. We introduce some notation needed to describe
$K$-orbits on $\mathcal{B}_{\mbox{${\mathfrak{g}}$}}.$

######  Remark 2.4.

When $M$ is the fixed point set of an involution $G$, then our description of
roots by type coincides with the usual definition from real groups [RS90].

######  Notation 2.5.

For each simple root $\alpha$, choose root vectors
$e_{\alpha}\in\mbox{${\mathfrak{g}}$}_{\alpha}$,
$f_{\alpha}\in\mbox{${\mathfrak{g}}$}_{-\alpha}$, and
$h_{\alpha}=[e_{\alpha},f_{\alpha}]$ so that
$e_{\alpha},f_{\alpha},h_{\alpha}$ spans a subalgebra of ${\mathfrak{g}}$
isomorphic to $\mbox{${\mbox{${\mathfrak{s}}$}\mbox{${\mathfrak{l}}$}}$}(2)$.
Consider the unique Lie algebra homomorphism
$\phi_{\alpha}:\mbox{${\mbox{${\mathfrak{s}}$}\mbox{${\mathfrak{l}}$}}$}(2)\to\mbox{${\mathfrak{g}}$}$
such that:

(2.4) $\phi_{\alpha}:\left[\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right]\to
e_{\alpha},\;\phi_{\alpha}:\left[\begin{array}[]{cc}0&0\\\
1&0\end{array}\right]\to
f_{\alpha},\;\phi_{\alpha}:\left[\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right]\to h_{\alpha}\;$

Let $\phi_{\alpha}:SL(2)\to G$ be the induced Lie group homomorphism, and let

(2.5)
$u_{\alpha}=\phi_{\alpha}\left(\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}1&\imath\\\
\imath&1\end{array}\right]\right).$

In the remainder of this section, we consider the cases where $K=G_{n-1}$ as
above, and consider the monoid action relative to the $K$-action on
$\mathcal{B}_{n}$. We now introduce some notation which will be used
throughout the paper.

######  Notation 2.6.

Let

$\mathcal{F}=(V_{0}=\\{0\\}\subset V_{1}\subset\dots\subset
V_{i}\subset\dots\subset V_{n}=\mbox{${\mathbb{C}}$}^{n})$

be a full flag in $\mbox{${\mathbb{C}}$}^{n}$, with $\dim V_{i}=i$ and
$V_{i}=\mbox{span}\\{v_{1},\dots,v_{i}\\}$, with each
$v_{j}\in\mbox{${\mathbb{C}}$}^{n}$. We will denote this flag $\mathcal{F}$ by

$\mathcal{F}=(v_{1}\subset v_{2}\subset\dots\subset v_{i}\subset
v_{i+1}\subset\dots\subset v_{n}).$

We will also make use of the following notation for partial flags in
$\mbox{${\mathbb{C}}$}^{n}$. Let

$\mathcal{P}=(V_{0}=\\{0\\}\subset V_{1}\subset\dots\subset
V_{i}\subset\dots\subset V_{k}=\mbox{${\mathbb{C}}$}^{n})$

denote a $k$-step partial flag with $\dim V_{j}=i_{j}$ and
$V_{j}=\mbox{span}\\{v_{1},\dots,v_{i_{j}}\\}$ for $j=1,\dots,k$. Then we
denote $\mathcal{P}$ as

$\mathcal{P}=(\\{v_{1},\dots,v_{i_{1}}\\}\subset\\{v_{i_{1}+1},\dots,v_{i_{2}}\\}\subset\dots\subset\\{v_{i_{k-1}+1},\dots,v_{i_{k}}\\}).$

######  Notation 2.7.

Let $T$ be the maximal torus of $G$ with Lie algebra ${\mathfrak{t}}$, and let
$W$ be the Weyl group with respect to $T$. For an element $w\in W$, let
$\dot{w}\in N_{G}(T)$ be a representative of $w$. If
$\mbox{${\mathfrak{m}}$}\subset\mbox{${\mathfrak{g}}$}$ is a Lie subalgebra
which is normalized by $T$, then ${\rm Ad}(\dot{w})\mbox{${\mathfrak{m}}$}$ is
independent of the choice of representative of $w$, and we denote it by
$w(\mbox{${\mathfrak{m}}$})$. Similarly, we denote by $w(M)$ the corresponding
group.

######  Proposition 2.8.

Let $\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n)$ and
$\mbox{${\mathfrak{k}}$}=\mathfrak{gl}(n-1)$.

  1. (1)

The number of $K$-orbits on $\mathcal{B}$ is ${n+1\choose 2}$.

  2. (2)

For $i=1,\dots,n$, let $w_{i}$ be the cycle $(n,n-1,\dots i)$ in the symmetric
group $\mathcal{S}_{n}$, and let
$\mbox{${\mathfrak{b}}$}_{i}:=w_{i}(\mbox{${\mathfrak{b}}$}_{+})$. The
distinct closed $K$-orbits on $\mathcal{B}$ are the $K$-orbits
$Q_{i}=K\cdot\mbox{${\mathfrak{b}}$}_{i}$ so there are exactly $n$ closed
$K$-orbits. Further, the Borel subalgebra $\mbox{${\mathfrak{b}}$}_{i}$ is the
stabilizer of the flag:

(2.6) $\mathcal{F}_{i}:=(e_{1}\subset\dots\subset
e_{i-1}\subset\underbrace{e_{n}}_{i}\subset e_{i}\subset\dots\subset
e_{n-1}).$

  3. (3)

The non-closed $K$-orbits are of the form:

(2.7) $Q_{i,j}=m(s_{\alpha_{j-1}})\dots m(s_{\alpha_{i}})*Q_{i}$

with $1\leq i<j\leq n$. The codimension of $Q_{i,j}$ is $n-1-(j-i)$. Further,
the Borel subalgebra:

(2.8) $\mbox{${\mathfrak{b}}$}_{i,j}:={\rm
Ad}(\hat{v})(\mbox{${\mathfrak{b}}$}_{+})\in Q_{i,j},\ \
\hat{v}:={\dot{w}}_{i}\cdot
u_{\alpha_{i}}\cdot\dot{s}_{\alpha_{i+1}}\dots\dot{s}_{\alpha_{j-1}},$

and $\mbox{${\mathfrak{b}}$}_{i,j}$ is the stabilizer of the flag:

(2.9) $\mathcal{F}_{i,j}:=(e_{1}\subset\dots\subset
e_{i-1}\subset\underbrace{e_{i}+e_{n}}_{i}\subset e_{i+1}\subset\dots\subset
e_{j-1}\subset\underbrace{e_{n}}_{j}\subset e_{j}\subset\dots\subset
e_{n-1}).$

In particular, the unique open orbit is $Q_{1,n}$, and it contains the Borel
subalgebra

$\mbox{${\mathfrak{b}}$}_{1,n}={\rm Ad}(w_{1}){\rm
Ad}(u_{\alpha_{1}})s_{\alpha_{2}}\dots
s_{\alpha_{n-1}}(\mbox{${\mathfrak{b}}$}_{+})$

which stabilizes the flag:

(2.10) $\mathcal{F}_{1,n}:=(e_{1}+e_{n}\subset e_{2}\subset\dots\subset
e_{n-1}\subset e_{n}).$

The following diagram indicates the $K$-orbits on $\mathcal{B}_{4}$, together
with the order relations given by closure. For general $n$, the diagram has
the same shape, with $n$ closed orbits, $n-1$ orbits one line below, and so
forth, until we have one orbit on the last line.

$Q_{1}$$Q_{2}$$Q_{3}$$Q_{4}$$Q_{1,2}$$Q_{2,3}$$Q_{3,4}$$Q_{1,3}$$Q_{2,4}$$Q_{1,4}$

We now classify the orbits of $K=SO(n-1)$ on the flag variety for
$\mbox{${\mathfrak{g}}$}=\mathfrak{so}(n)$. When $n=2l+1$ is odd, there is a
bijection between $\mathcal{B}_{\mathfrak{so}(n)}$ and the set of all maximal
isotropic flags in $\mbox{${\mathbb{C}}$}^{n}$, i.e., partial flags of the
form

$\mathcal{I}\mathcal{F}=(V_{1}\subset\dots\subset V_{i}\subset\dots\subset
V_{l}),$

with $\dim(V_{i})=i$ and $B(u,w)=0$ for all $u,w\in V_{i}.$ When $n=2l$ is
even, the space of maximal isotropic flags in $\mbox{${\mathbb{C}}$}^{n}$
consists of two $SO(2n)$-orbits. To distinguish them, we choose a maximal
isotropic flag $U_{1}\subset\dots\subset U_{l}$, where $U_{i}$ is the span of
the standard basis $e_{1},\dots,e_{i}$ from Section 2.2. Then we identify
$\mathcal{B}_{\mathfrak{so}(n)}$ with the maximal isotropic flags
$V_{1}\subset\dots\subset V_{l}$ such that $\dim(V_{l}\cap U_{l})\equiv
l\pmod{2}.$ As before, if $V_{i}$ is the span of vectors
$\\{v_{1},\dots,v_{i}\\},$ we write the above maximal isotropic flag as

$\mathcal{I}\mathcal{F}=(v_{1}\subset v_{2}\subset\dots\subset
v_{[\frac{n}{2}]}).$

We consider the type $B$ and $D$ cases separately, and use the enumeration of
simple roots in [GW98].

######  Proposition 2.9.

Let $\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l+1)$ and
$\mbox{${\mathfrak{k}}$}=\mathfrak{so}(2l)$.

  1. (1)

There are exactly $l+2$ $K$-orbits on the flag variety $\mathcal{B}$ of
${\mathfrak{g}}$.

  2. (2)

We let
$\mbox{${\mathfrak{b}}$}_{-}:=s_{\alpha_{l}}(\mbox{${\mathfrak{b}}$}_{+})$.
Exactly two $K$-orbits on $\mathcal{B}$ are closed, and they are
$Q_{+}:=K\cdot\mbox{${\mathfrak{b}}$}_{+}$ and
$Q_{-}:=K\cdot\mbox{${\mathfrak{b}}$}_{-}.$ Further,
$m(s_{\alpha_{l}})*Q_{+}=m(s_{\alpha_{l}})*Q_{-}=K\cdot{\rm
Ad}(u_{\alpha_{l}})\mbox{${\mathfrak{b}}$}_{+}.$

  3. (3)

The non-closed orbits are of the form

$Q_{i}:=m(s_{\alpha_{i+1}})*m(s_{\alpha_{i+2}})\cdots
m(s_{\alpha_{l-1}})*m(s_{\alpha_{l}})*Q_{+}$

for $i=0,\dots,l-1$. Moreover, the codimension of $Q_{i}$ in $\mathcal{B}$ is
$i$. Further,

(2.11) $\mbox{${\mathfrak{b}}$}_{i}={\rm
Ad}(\hat{v})(\mbox{${\mathfrak{b}}$}_{+})\in Q_{i},\
\hat{v}:=u_{\alpha_{l}}\dot{s}_{\alpha_{l-1}}\dot{s}_{\alpha_{l-2}}\dots\dot{s}_{\alpha_{i+1}}.$

In particular, the unique open $K$-orbit contains the Borel subalgebra

(2.12) $\mbox{${\mathfrak{b}}$}_{0}={\rm
Ad}(u_{\alpha_{l}})s_{\alpha_{l-1}}s_{\alpha_{l-2}}\dots
s_{\alpha_{1}}(\mbox{${\mathfrak{b}}$}_{+}).$

######  Proposition 2.10.

Let $\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l)$ and
$\mbox{${\mathfrak{k}}$}=\mathfrak{so}(2l-1)$.

  1. (1)

There are exactly $l$ $K$-orbits in the flag variety $\mathcal{B}$ of
${\mathfrak{g}}$.

  2. (2)

The orbit $Q_{+}:=K\cdot\mbox{${\mathfrak{b}}$}_{+}$ is the only closed
$K$-orbit.

  3. (3)

Let

$Q_{i}:=m(s_{\alpha_{i}})\dots m(s_{\alpha_{l-1}})*Q_{+}$

and let

(2.13) $\mbox{${\mathfrak{b}}$}_{i}={\rm
Ad}(\hat{v})(\mbox{${\mathfrak{b}}$}_{+}),\
\hat{v}:={\dot{s}}_{\alpha_{l-1}}{\dot{s}}_{\alpha_{l-2}}\dots\dot{s}_{\alpha_{i}}(\mbox{${\mathfrak{b}}$}_{+})\mbox{
for }i=1,\dots,l-1.$

Then $Q_{i}=K\cdot\mbox{${\mathfrak{b}}$}_{i}$ has codimension $i-1$ in
$\mathcal{B}$. The distinct $K$-orbits are $Q_{+},Q_{l-1},\dots,Q_{1}$. In
particular, the unique open orbit is $Q_{1}$ and contains the Borel subalgebra

(2.14) $\mbox{${\mathfrak{b}}$}_{1}=s_{\alpha_{l-1}}s_{\alpha_{l-2}}\dots
s_{\alpha_{1}}(\mbox{${\mathfrak{b}}$}_{+}).$

######  Remark 2.11.

In case $\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l)$ or $\mathfrak{so}(2l+1),$
the Borel subalgebra $\mbox{${\mathfrak{b}}$}_{+}$ corresponds to the maximal
isotropic flag $(e_{1}\subset e_{2}\subset\dots\subset e_{l})$. By Proposition
2.10, the Borel subalgebra $\mbox{${\mathfrak{b}}$}_{1}$ in the open $K$-orbit
in $\mathcal{B}_{\mathfrak{so}(2l)}$ corresponds to the maximal isotropic flag
$(e_{l}\subset e_{1}\subset\dots\subset e_{l-1}).$ Similarly, by Proposition
2.9, it follows that the Borel subalgebra $\mbox{${\mathfrak{b}}$}_{0}$ in the
open $K$-orbit in $\mathcal{B}_{\mathfrak{so}(2l+1)}$ corresponds to the
maximal isotropic flag $(u_{\alpha_{l}}(e_{l})\subset e_{1}\subset
e_{2}\subset\dots\subset e_{l-1}).$ We may choose our root vectors so that
$\mbox{${\mathbb{C}}$}\cdot
u_{\alpha_{l}}(e_{l})=\mbox{${\mathbb{C}}$}\cdot(e_{l}+\imath\sqrt{2}e_{0}+e_{-l}).$
We consider the $\alpha_{l}$ root vector $e=E_{l,0}-E_{0,-l}$ and take $f$ to
be the transpose of $e$, and let $x=[e,f]$, and note that $\\{e,x,f\\}$
generates the three dimensional subalgebra
${\mbox{${\mathfrak{s}}$}}_{\alpha_{l}}:=\phi_{\alpha_{l}}(\mbox{${\mbox{${\mathfrak{s}}$}\mbox{${\mathfrak{l}}$}}$}(2)).$
Let $V_{3}$ be the span of $e_{l},e_{0},$ and $e_{-l}$, and note that $V_{3}$
is a representation of ${\mbox{${\mathfrak{s}}$}}_{\alpha_{l}}$ and is
isomorphic to the adjoint representation via an isomorphism mapping $e_{-l}\to
f$, $e_{0}\to-x$, and $e_{l}\to e.$ A brief computation using the composed
isomorphism
$\mbox{${\mbox{${\mathfrak{s}}$}\mbox{${\mathfrak{l}}$}}$}(2)\to\mathfrak{so}(3)\to
V_{3}$ above then gives the above formula for $u_{\alpha_{l}}(e_{l})$.

##  3\. Structure of $B_{n-1}$-orbits

Recall that $G=GL(n)$ or $SO(n)$ and $K=G_{n-1}$ from Section 2.2. We show
that $B_{n-1}$-orbits on the flag variety $\mathcal{B}$ of $G$ have a useful
bundle structure and describe the data required to classify orbits.

###  3.1. Fibre Bundle Structure

Let $Q_{K}=K\cdot\mbox{${\mathfrak{b}}$}\subset\mathcal{B}$ be a $K$-orbit.
Let $R$ be a parabolic subgroup of $G$ containing $B$ and consider the
projection $\pi:\mathcal{B}\to\mathcal{R}:=G/R$. Then the morphism

(3.1) $\phi:K\times_{K\cap R}R/B\to\pi^{-1}\pi(Q_{K}),\ \
\phi(k,\mbox{${\mathfrak{b}}$}_{1})={\rm
Ad}(k)\cdot\mbox{${\mathfrak{b}}$}_{1},$

is an isomorphism, and the preimage $\phi^{-1}(Q_{K})$ is $K\times_{K\cap
R}(K\cap R)\cdot\mbox{${\mathfrak{b}}$}.$ This realizes both
$\pi^{-1}\pi(Q_{K})$ and $Q_{K}$ as bundles over $K/K\cap R.$ Let $R=L\cdot U$
be a Levi decomposition of the parabolic $R$. Recall that the morphism
$R\cdot\mbox{${\mathfrak{b}}$}\to\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$ given
by
$\mbox{${\mathfrak{b}}$}_{1}\mapsto\mbox{${\mathfrak{b}}$}_{1}\cap\mbox{${\mathfrak{l}}$}$
is an isomorphism, where $\mbox{${\mathfrak{b}}$}_{1}\in
R\cdot\mbox{${\mathfrak{b}}$}\cong\mathcal{B}_{R}.$

We now assume that $R$ is $\theta$-stable. Then by Theorem 2 of [BH00], there
is a $\theta$-stable Levi subgroup $L$ of $R$, $K\cap R$ is a parabolic
subgroup of $K$, and $K\cap R=(K\cap L)\cdot(K\cap U)$ is a Levi decomposition
of $K\cap R$. The group $K\cap R$ then acts on
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$ through its quotient $K\cap L$, and it
follows that we can identify $Q_{K}$ with $K\times_{K\cap R}(K\cap
L)\cdot(\mbox{${\mathfrak{b}}$}\cap\mbox{${\mathfrak{l}}$}).$ Since $K\cap L$
is a symmetric subgroup of $L$, $K\cap L$ acts with finitely many orbits on
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}.$

We now prove our basic result about $K$-orbits on $\mathcal{B}$ for the
multiplicity free pairs $(G,K)$.

######  Theorem 3.1.

Let $Q_{K}=K\cdot\mbox{${\mathfrak{b}}$}$ be a $K$-orbit on $\mathcal{B}$.
Then there exists a $\theta$-stable parabolic subgroup $R\supset B$ with
$\theta$-stable Levi decomposition $R=L\cdot U$ such that the pair
$(\mbox{${\mathfrak{l}}$},\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$})$
is of the same type as $(\mbox{${\mathfrak{g}}$},\mbox{${\mathfrak{k}}$})$ up
to abelian factors in the center of ${\mathfrak{l}}$ and the $K\cap L$-orbit
$(K\cap L)\cdot(\mbox{${\mathfrak{b}}$}\cap\mbox{${\mathfrak{l}}$})$ is open
in the flag variety $\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$. Thus,

(3.2) $Q_{K}\cong K\times_{K\cap R}(K\cap
L)\cdot(\mbox{${\mathfrak{b}}$}\cap\mbox{${\mathfrak{l}}$}).$

is a $K$-homogenous fibre bundle over the partial flag variety $K/K\cap R$
with fibre the open $K\cap L$-orbit on
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$.

###### Proof.

If $Q_{K}$ is closed, then $B$ is $\theta$-stable by Corollary 6.6 of [Spr85],
and then ${\mathfrak{l}}$ is abelian so that
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}=\\{\mbox{${\mathfrak{b}}$}\cap\mbox{${\mathfrak{l}}$}\\}$
is a point, and the assertion follows.

When $Q_{K}$ is not closed, we consider the different types $A,B,$ and $D$
separately. In type A, by Proposition 2.8, the orbit
$Q_{K}=Q_{i,j}=K\cdot\mbox{${\mathfrak{b}}$}_{i,j}$ for some $i,j$ with $1\leq
i<j\leq n$, and $\mbox{${\mathfrak{b}}$}_{i,j}$ is the Borel subalgebra in
Equation (2.8). Let $P_{i,j}$ be the standard parabolic subgroup of $GL(n)$
which is the stabilizer of the flag

${{\mathcal{P}}}_{i,j}:=(e_{1}\subset\dots\subset
e_{i-1}\subset\\{e_{i},\dots,e_{j}\\}\subset e_{j+1}\subset\dots\subset
e_{n})$

and corresponds to the choice of simple roots
$\\{\alpha_{i},\dots,\alpha_{j-1}\\}.$ Recall the cycle $w_{i}=(n,n-1,\dots
i)$ from Proposition 2.8(2) and let
$\mbox{${\mathfrak{r}}$}=\mbox{${\mathfrak{r}}$}_{i,j}:={w}_{i}(\mbox{${\mathfrak{p}}$}_{i,j})$.
Then $\mbox{${\mathfrak{r}}$}_{i,j}$ is the Lie algebra stabilizer of the
partial flag $w_{i}({{\mathcal{P}}}_{i,j})$ which is

(3.3) $\mathcal{R}_{i,j}:=(e_{1}\subset\dots\subset
e_{i-1}\subset\\{e_{i},e_{i+1},\dots,e_{j-1},e_{n}\\}\subset e_{j}\subset
e_{j+1}\subset\dots\subset e_{n-1}.)$

By Proposition 2.8(3), $\mbox{${\mathfrak{b}}$}_{i,j}$ is the stabilizer of
the full flag in Equation (2.9). It follows that
$\mbox{${\mathfrak{b}}$}_{i,j}$ also stabilizes the partial flag
$\mathcal{R}_{i,j}$ in (3.3), so that $\mbox{${\mathfrak{b}}$}_{i,j}$ is in
the parabolic $\mbox{${\mathfrak{r}}$}_{i,j}.$ Since the Cartan subalgebra
$\mbox{${\mathfrak{h}}$}_{n}$ is contained in $\mbox{${\mathfrak{r}}$}_{i,j}$,
it follows that $\mbox{${\mathfrak{r}}$}_{i,j}$ is $\theta$-stable. Further,
$\mbox{${\mathfrak{r}}$}_{i,j}$ has Levi decomposition
$\mbox{${\mathfrak{r}}$}_{i,j}=\mbox{${\mathfrak{l}}$}\oplus\mbox{${\mathfrak{u}}$}$,
where
$\mbox{${\mathfrak{l}}$}\cong\mathfrak{gl}(1)^{i-1}\oplus\mathfrak{gl}(j-i+1)\oplus\mathfrak{gl}(1)^{n-j}.$
It follows from (3.3) that
$\mbox{${\mathfrak{r}}$}_{i,j}\cap\mbox{${\mathfrak{k}}$}$ is a standard
parabolic subalgebra of ${\mathfrak{k}}$ with Levi factor
$\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$}\cong\mathfrak{gl}(1)^{i-1}\oplus\mathfrak{gl}(j-i)\oplus\mathfrak{gl}(1)^{n-j}.$

To prove the Theorem when $\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n)$, it
remains to show that $(K\cap
L)\cdot(\mbox{${\mathfrak{b}}$}_{i,j}\cap\mbox{${\mathfrak{l}}$})$ is open in
the flag variety of ${\mathfrak{l}}$. In the Levi decomposition of
$\mbox{${\mathfrak{r}}$}_{i,j}$ above, $\mathfrak{gl}(j-i+1)$ is identified
with endomorphisms of the span of $\\{e_{i},\dots,e_{j-1},e_{n}\\}.$ Observe
that $\mbox{${\mathfrak{b}}$}_{i,j}\cap\mathfrak{gl}(j-i+1)$ is the stabilizer
of the flag

$(e_{i}+e_{n}\subset e_{i+1}\subset\dots\subset e_{j-1}\subset e_{n})$

in $\mathfrak{gl}(j-i+1).$ By part (3) of Proposition 2.8, the point
$\mbox{${\mathfrak{b}}$}_{i,j}\cap\mathfrak{gl}(j-i+1)$ is in the open $K\cap
GL(j-i+1)$-orbit on $\mathcal{B}_{\mathfrak{gl}(j-i+1)}$. Hence,
$\mbox{${\mathfrak{b}}$}_{i,j}\cap\mbox{${\mathfrak{l}}$}$ is in the open
$K\cap L$-orbit on the flag variety $\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$ of
${\mathfrak{l}}$.

For the type B case, the relevant symmetric pair is
$(\mathfrak{so}(2l+1),\mathfrak{so}(2l))$. Recall from Proposition 2.9 that
since $Q_{K}$ is not closed, $Q_{K}=Q_{i}$ with $i<l$ where
$Q_{i}=K\cdot\mbox{${\mathfrak{b}}$}_{i}={\rm
Ad}(u_{\alpha_{l}})s_{\alpha_{l-1}}\dots
s_{\alpha_{i+1}}(\mbox{${\mathfrak{b}}$}_{+})$. Let
$\mbox{${\mathfrak{r}}$}_{i}\subset\mbox{${\mathfrak{g}}$}$ be the standard
parabolic subalgebra generated by $\mbox{${\mathfrak{b}}$}_{+}$ and the
negative simple root spaces
$\mbox{${\mathfrak{g}}$}_{-\alpha_{l}},\mbox{${\mathfrak{g}}$}_{-\alpha_{l-1}},\dots,\mbox{${\mathfrak{g}}$}_{-\alpha_{i+1}}.$
Thus,

(3.4)
$\mbox{${\mathfrak{r}}$}_{i}=\mbox{${\mathfrak{l}}$}\oplus\mbox{${\mathfrak{u}}$}\mbox{
with
}\mbox{${\mathfrak{l}}$}\cong\mathfrak{so}(2(l-i)+1)\oplus\mathfrak{gl}(1)^{i}.$

Note that $\mbox{${\mathfrak{r}}$}_{i}$ is $\theta$-stable and that
$\theta|_{[\mbox{${\mathfrak{l}}$},\mbox{${\mathfrak{l}}$}]}=\theta_{2(l-i)+1}$.
Thus, $\mbox{${\mathfrak{r}}$}_{i}\cap\mbox{${\mathfrak{k}}$}$ is a parabolic
subalgebra of ${\mathfrak{k}}$ with Levi factor:

(3.5)
$\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$}\cong\mathfrak{so}(2(l-i))\oplus\mathfrak{gl}(1)^{i},$

so that $K\cap L\cong SO(2(l-1))\times GL(1)^{i}$. To see that
$\mbox{${\mathfrak{b}}$}_{i}\subset\mbox{${\mathfrak{r}}$}_{i}$, note that we
can choose the representative $\dot{s}_{\alpha_{j}}$ of $s_{\alpha_{j}}$ so
that $\dot{s}_{\alpha_{j}}\in L$ for $j=i+1,\dots,l$, and $u_{\alpha_{l}}\in
L$ by Equation (2.5). Thus, the element

(3.6) $v:=u_{\alpha_{l}}\dot{s}_{\alpha_{l-1}}\dots\dot{s}_{\alpha_{i+1}}\in
L\subset R.$

Hence, $\mbox{${\mathfrak{b}}$}_{i}={\rm
Ad}(v)\mbox{${\mathfrak{b}}$}_{+}\subset{\rm
Ad}(v)\mbox{${\mathfrak{r}}$}_{i}=\mbox{${\mathfrak{r}}$}_{i}$.

It remains to show that $(K\cap
L)\cdot(\mbox{${\mathfrak{b}}$}\cap\mbox{${\mathfrak{l}}$})$ can be identified
with the open $K\cap L$-orbit in the flag variety
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$. Note that
$\mbox{${\mathfrak{b}}$}_{+}\cap\mbox{${\mathfrak{l}}$}$ can be identified
with the standard Borel subalgebra
$\mbox{${\mathfrak{b}}$}_{+,\mbox{${\mathfrak{l}}$}}$ of upper triangular
matrices in ${\mathfrak{l}}$. Since the element $v$ in Equation (3.6) is in
$L$, we have:

$\mbox{${\mathfrak{b}}$}\cap\mbox{${\mathfrak{l}}$}=({\rm
Ad}(v)\mbox{${\mathfrak{b}}$}_{+})\cap\mbox{${\mathfrak{l}}$}={\rm
Ad}(v)(\mbox{${\mathfrak{b}}$}_{+}\cap\mbox{${\mathfrak{l}}$})={\rm
Ad}(v)\mbox{${\mathfrak{b}}$}_{+,\mbox{${\mathfrak{l}}$}}.$

It follows from Equations (2.12) and (3.6) that ${\rm
Ad}(v)\mbox{${\mathfrak{b}}$}_{+,\mbox{${\mathfrak{l}}$}}\subset\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$
is a representative of the open $K\cap L$-orbit on
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$.

The type D case is similar to type B. In this setting,
$\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l)$ and
$\mbox{${\mathfrak{k}}$}=\mathfrak{so}(2l-1)$, and since $Q_{K}$ is a
$K$-orbit that is not closed, then by Proposition 2.10, $Q_{K}=Q_{i}$ has
codimension $i-1<l-1$. By part (3) of Proposition 2.10, we can take
$\mbox{${\mathfrak{b}}$}=\mbox{${\mathfrak{b}}$}_{i}=s_{\alpha_{l-1}}s_{\alpha_{l-2}}\dots
s_{\alpha_{i}}(\mbox{${\mathfrak{b}}$}_{+})$. Let
$\mbox{${\mathfrak{r}}$}_{i}\subset\mbox{${\mathfrak{g}}$}$ be the standard
parabolic subalgebra generated by $\mbox{${\mathfrak{b}}$}_{+}$ and the
negative simple root spaces
$\mbox{${\mathfrak{g}}$}_{-\alpha_{l}},\mbox{${\mathfrak{g}}$}_{-\alpha_{l-1}},\dots,\mbox{${\mathfrak{g}}$}_{-\alpha_{i}}.$
Then $\mbox{${\mathfrak{r}}$}_{i}$ has the Levi decomposition:

(3.7)
$\mbox{${\mathfrak{r}}$}_{i}=\mbox{${\mathfrak{l}}$}\oplus\mbox{${\mathfrak{u}}$}\mbox{
with
}\mbox{${\mathfrak{l}}$}\cong\mathfrak{so}(2(l-i)+2)\oplus\mathfrak{gl}(1)^{i-1}.$

We claim that $\mbox{${\mathfrak{r}}$}_{i}$ is $\theta$-stable. Indeed, we saw
in Section 2.2 that $\theta(\epsilon_{l})=-\epsilon_{l}$ and
$\theta(\epsilon_{k})=\epsilon_{k}$ for $k\neq l$. It follows that
$\theta(\alpha_{i})=\alpha_{i}$ for $i=1,\dots,l-2$ and that $\alpha_{l-1}$
and $\alpha_{l}$ are complex $\theta$-stable with
$\theta(\alpha_{l-1})=\alpha_{l}$. It then follows easily that
$\mbox{${\mathfrak{r}}$}_{i}$ has Levi decomposition (3.7) and that
$\theta|_{\mbox{${\mathfrak{l}}$}_{ss}}=\theta_{2(l-i)+2}$, whence
$\mbox{${\mathfrak{l}}$}^{\theta}=\mbox{${\mathfrak{k}}$}\cap\mbox{${\mathfrak{l}}$}\cong\mathfrak{so}(2(l-i)+1)\oplus\mathfrak{gl}(1)^{i-1}$,
and $(L^{\theta})^{0}=K\cap L\cong SO(2(l-i)+1)\times GL(1)^{i-1}$. The
remainder of the proof proceeds as in the previous case.

Q.E.D.

######  Remark 3.2.

In Proposition 12 of the paper [BH00], Brion and Helminck prove a closely
related result which in the multiplicity free case, describes $K$-orbits on
$\mathcal{B}$ as bundles over a closed $K$-orbit with fibre an open orbit in a
smaller flag variety. In the orthogonal cases, our result coincides with the
result of [BH00], but in the general linear case, our bundle is in many cases
different from theirs.

######  Remark 3.3.

We call a parabolic of ${\mathfrak{g}}$ a _special parabolic subalgebra_ if it
is one of the representatives of the closed $K$-orbits given in Part (2) of
Propositions 2.8, 2.9, and 2.10 or is one of the parabolic subalgebras
$\mbox{${\mathfrak{r}}$}_{i,j}\supset\mbox{${\mathfrak{b}}$}_{i,j}$ and
$\mbox{${\mathfrak{r}}$}_{i}\supset\mbox{${\mathfrak{b}}$}_{i}$ constructed in
the proof of Theorem 3.1. We also call the corresponding parabolic subgroups
the special parabolic subgroups of $G$. Note that for ${\mathfrak{r}}$ a
special parabolic subalgebra of ${\mathfrak{g}}$, the standard Borel
subalgebra $\mbox{${\mathfrak{b}}$}_{n-1}$ of ${\mathfrak{k}}$ is in
${\mathfrak{r}}$. Hence, $\mbox{${\mathfrak{r}}$}\cap\mbox{${\mathfrak{k}}$}$
is a standard parabolic subalgebra of ${\mathfrak{k}}$.

######  Remark 3.4.

Theorem 3.1 fails for other real rank one symmetric pairs. For example, if
$\mbox{${\mathfrak{g}}$}=\mathfrak{sp}(6)$, and
$\mbox{${\mathfrak{k}}$}=\mathfrak{sp}(4)\oplus\mathfrak{sp}(2)$, then there
exists a $K$-orbit $Q_{K}=K\cdot\mbox{${\mathfrak{b}}$}$ where the minimal
$\theta$-stable parabolic subalgebra containing ${\mathfrak{b}}$ is all of
${\mathfrak{g}}$. However, $Q_{K}$ is not open.

For $\theta$-stable parabolics $R$, $K\cap R$ is a parabolic of $K$ and we let
$W_{K}^{K\cap R}\subset W_{K}$ be the representatives of shortest length for
the cosets in $W_{K}/W_{K\cap R}$. Now we use the fibre bundle structure of a
$K$-orbit $Q_{K}$ given in (3.2) to describe the $B_{n-1}$-orbits $Q$
contained in $Q_{K}$ as fibre bundles.

######  Theorem 3.5.

Let $Q_{K}=K\cdot\mbox{${\mathfrak{b}}$}$ be a $K$-orbit in $\mathcal{B}$. Let
$\mbox{${\mathfrak{r}}$}\supset\mbox{${\mathfrak{b}}$}$ be a special parabolic
subalgebra, and let $R\subset G$ be the corresponding parabolic subgroup with
$\theta$-stable Levi decomposition $R=LU$. Suppose $Q$ is a $B_{n-1}$-orbit
contained in $Q_{K}$. Then $Q$ fibres over a $B_{n-1}$-orbit in the partial
flag variety $K/K\cap R$ of $R$ with fibre isomorphic to a $B_{n-1}\cap
L$-orbit contained in the open $K\cap L$ -orbit of the flag variety
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$ of ${\mathfrak{l}}$. More precisely,

(3.8) $Q\cong B_{n-1}\times_{B_{n-1}\cap w(R)}(B_{n-1}\cap w(L))\cdot{\rm
Ad}(\dot{w})\mbox{${\mathfrak{b}}$},$

where $w\in W_{K}^{K\cap R}$,

(3.9) $B_{n-1}\cap w(L)=w(B_{n-1}\cap L)$

is a Borel subgroup of $K\cap w(L)$, and ${\rm
Ad}(\dot{w})\mbox{${\mathfrak{b}}$}$ is contained in the open $K\cap
w(L)$-orbit in the flag variety $\mathcal{B}_{w(R)}\cong\mathcal{B}_{w(L)}$ of
$R$.

Further, we have a one-to-one correspondence:

(3.10)
$B_{n-1}\backslash\mathcal{B}\leftrightarrow\displaystyle\coprod_{R\mbox{
special}}\left(B_{n-1}\backslash K/K\cap R\times B_{n-1}\cap
L\backslash\tilde{Q}_{K\cap L}\right),$

where $\tilde{Q}_{K\cap L}$ is the open $K\cap L$-orbit on
$\mathcal{B}_{\mbox{${\mathfrak{k}}$}\cap\mbox{${\mathfrak{l}}$}}$ (see
Notation 2.2).

###### Proof.

Consider the fibre bundle $Q_{K}=K\times_{K\cap R}(K\cap
L)\cdot\mbox{${\mathfrak{b}}$}$ in Equation (3.2). Let $\pi|_{Q_{K}}:Q_{K}\to
K/K\cap R=Q_{K,\mbox{${\mathfrak{r}}$}}$ be the canonical projection. Let
$Q=B_{n-1}\cdot\mbox{${\mathfrak{b}}$}^{\prime}$ be a $B_{n-1}$-orbit in
$Q_{K}$. Its image under $\pi$ is a $B_{n-1}$-orbit in the partial flag
variety $K/K\cap R$. Since $W_{K}^{K\cap R}$ indexes $B_{n-1}$-orbits in
$K/K\cap R$, then $\pi(Q)=B_{n-1}\cdot w(\mbox{${\mathfrak{r}}$})$ for some
$w\in W_{K}^{K\cap R},$ and we may assume that
$\mbox{${\mathfrak{b}}$}^{\prime}=w(\mbox{${\mathfrak{b}}$})$ by replacing
${\mathfrak{b}}$ by a $K\cap L$-conjugate. Then the fibre bundle structure
from Equation (3.1) identifies $Q$ as

(3.11) $Q\cong B_{n-1}\times_{B_{n-1}\cap w(R)}(B_{n-1}\cap w(R))\cdot{\rm
Ad}(\dot{w})(\mbox{${\mathfrak{b}}$}).$

Now the group $B_{n-1}\cap w(R)$ acts on
$w(R)/w(B)\cong\mathcal{B}_{w(\mbox{${\mathfrak{l}}$})}$ through its image in
the quotient $w(R)/w(U)$. Denote this image by $\hat{B}_{w(L)}$. We claim that

(3.12) $\hat{B}_{w(L)}=B_{n-1}\cap w(L)=w(L\cap B_{n-1}),$

so that $\hat{B}_{w(L)}$ is a Borel subgroup in $K\cap w(L)$. To verify the
claim, first consider
$w(\mbox{${\mathfrak{l}}$})\cap\mbox{${\mathfrak{b}}$}_{n-1}$. By Remark 3.3,
the special parabolic ${\mathfrak{r}}$ contains
$\mbox{${\mathfrak{b}}$}_{n-1}$, so $\mbox{${\mathfrak{b}}$}_{n-1}$ is a Borel
subalgebra of $\mbox{${\mathfrak{r}}$}\cap\mbox{${\mathfrak{k}}$}$. Let
$\Phi^{+}_{\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$}}$ be the roots
appearing in the Borel subalgebra
$\mbox{${\mathfrak{b}}$}_{n-1}\cap\mbox{${\mathfrak{l}}$}$. Since $w\in
W_{K}^{K\cap R}$, standard results imply that
$w(\Phi^{+}_{\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$}})\subset\Phi^{+}_{\mbox{${\mathfrak{b}}$}_{n-1}}$.
Thus $w(\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{b}}$}_{n-1})\subset
w(\mbox{${\mathfrak{l}}$})\cap\mbox{${\mathfrak{b}}$}_{n-1}$. It follows that
$w(\mbox{${\mathfrak{l}}$})\cap\mbox{${\mathfrak{b}}$}_{n-1}=w(\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{b}}$}_{n-1})$,
so that $w(\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{b}}$}_{n-1})$ is a
Borel subalgebra of $\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$}$ with
corresponding Borel subgroup $w(L\cap B_{n-1})=w(L)\cap B_{n-1},$ yielding the
second equality in Equation (3.12). To get the first equality of Equation
(3.12), we consider the Lie algebra
$\hat{\mbox{${\mathfrak{b}}$}}_{w(L)}=\mbox{Lie}(\hat{B}_{w(L)})$. Since
$\mbox{${\mathfrak{h}}$}_{n}\cap\mbox{${\mathfrak{k}}$}\subset
w(\mbox{${\mathfrak{r}}$})\cap\mbox{${\mathfrak{k}}$}$, we have the
decomposition

$w(\mbox{${\mathfrak{r}}$})\cap\mbox{${\mathfrak{b}}$}_{n-1}=w(\mbox{${\mathfrak{l}}$})\cap\mbox{${\mathfrak{b}}$}_{n-1}\oplus
w(\mbox{${\mathfrak{u}}$})\cap\mbox{${\mathfrak{b}}$}_{n-1}.$

Thus,
$\hat{\mbox{${\mathfrak{b}}$}}_{w(L)}=w(\mbox{${\mathfrak{l}}$})\cap\mbox{${\mathfrak{b}}$}_{n-1}$
and Equation (3.12) follows. Equation (3.8) now follows from (3.11). Note that
the orbit $(B_{n-1}\cap w(L))\cdot{\rm Ad}(\dot{w})\mbox{${\mathfrak{b}}$}$ in
the fibre of the bundle in (3.8) is isomorphic via translation by
$\dot{w}^{-1}$ to an orbit of the standard Borel subgroup $B_{n-1}\cap L$ of
$L\cap K$ contained in the open $K\cap L$-orbit of
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$.

The one-to-one correspondence in Equation (3.10) follows easily.

Q.E.D.

######  Notation 3.6.

We use the notation of Theorem 3.5. Given a $B_{n-1}-$orbit
$Q_{\mbox{${\mathfrak{r}}$}}$ in $K/K\cap R$ and a $B_{n-1}\cap L$-orbit
$Q_{\mbox{${\mathfrak{l}}$}}$ in the open $K\cap L$-orbit on
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$, we let
${\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})$ be
the corresponding $B_{n-1}$-orbit on $\mathcal{B}$ from this theorem. Theorem
3.5 implies that for any $K$-orbit $Q_{K}$ the $B_{n-1}$-orbits contained in
$Q_{K}$ are of the form
${\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})$ as
above.

######  Remark 3.7.

Let $\mbox{${\mathfrak{b}}$}_{\hat{v}}$ be one of the representatives for
$\mbox{${\mathfrak{b}}$}_{i,j},\mbox{${\mathfrak{b}}$}_{i},\mbox{${\mathfrak{b}}$}_{+}$,
or $\mbox{${\mathfrak{b}}$}_{-}$ for $K$-orbits on $\mathcal{B}$ discussed in
Propositions 2.8, 2.9, and 2.10. Let $Q$ be a $B_{n-1}$-orbit in one of the
above $K$-orbits $Q_{K}$ and let $\mbox{${\mathfrak{b}}$}_{\hat{v}}={\rm
Ad}(\hat{v})\mbox{${\mathfrak{b}}$}_{+}.$ Then

(3.13) $Q=B_{n-1}\cdot{\rm Ad}(\dot{w}){\rm
Ad}(\ell)\mbox{${\mathfrak{b}}$}_{\hat{v}}=B_{n-1}\cdot{\rm
Ad}(\dot{w}\ell\hat{v})\mbox{${\mathfrak{b}}$}_{+}$

for a representative $\dot{w}$ of unique $w\in W^{K\cap R}_{K}$ and for
$\ell\in K\cap L$ such that ${\rm
Ad}(\ell)(\mbox{${\mathfrak{b}}$}_{\hat{v}}\cap\mbox{${\mathfrak{l}}$})$ is a
representative of the $B_{n-1}\cap L$-orbit $Q_{\mbox{${\mathfrak{l}}$}}$
contained in the open $K\cap L$-orbit on the flag variety
$\mathcal{B}_{\mbox{${\mathfrak{l}}$}}$ of ${\mathfrak{l}}$. In the notation
given above in 3.6, the orbit $Q_{\mbox{${\mathfrak{r}}$}}=B_{n-1}\cdot
w(\mbox{${\mathfrak{r}}$})$, and $Q_{\mbox{${\mathfrak{l}}$}}=(B_{n-1}\cap
L)\cdot{\rm
Ad}(\ell)(\mbox{${\mathfrak{b}}$}_{\hat{v}}\cap\mbox{${\mathfrak{l}}$}).$

By Equation (3.10), the problem of describing the $B_{n-1}$-orbits on $G/B$
reduces to the description of $B_{n-1}$-orbits on partial flag varieties
$K/(K\cap R)$ via $W_{R\cap K}^{K}$ and the problem of determining
$B_{n-1}$-orbits in the open $K$-orbit in $\mathcal{B}$. We now address this
second issue.

###  3.2. $B_{n-1}$-orbits in the open $K$-orbit

Recall the pair $(G,K)$ with $G=G_{n}$ and $K=G_{n-1}$ from Section 2.2.

######  Theorem 3.8.

Let ${\tilde{Q}}_{K}$ be the open $K$-orbit on $\mathcal{B}_{n}$. There are
bijections

(3.14) $\\{B_{n-1}-\mbox{orbits on }{\widetilde{Q}}_{K}\\}\longleftrightarrow
B_{n-1}\backslash K/B_{n-2}\longleftrightarrow\\{B_{n-2}-\mbox{orbits on
}\mathcal{B}_{n-1}\\},$

where the last equivalence is induced by the self-map $g\mapsto g^{-1}$ of
$K$.

###### Proof.

To prove the first equivalence, it suffices to find a point
$\mbox{${\mathfrak{b}}$}_{y}$ in ${\tilde{Q}}_{K}$ with stabilizer $B_{n-2}$
in $K$. The second equivalence is then clear. We prove the first equivalence
by considering the three cases separately.

When $\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n)$, by Proposition 2.8, the open
$K$-orbit $Q_{1,n}$ on $\mathcal{B}_{n}$ is the orbit through the flag
$(e_{1}+e_{n}\subset e_{2}\subset\dots\subset e_{n-1}\subset e_{n})$. Thus,
$Q_{1,n}=GL(n-1)\cdot\mathcal{F}_{y}$, where
$\mathcal{F}_{y}:=(e_{n-1}+e_{n}\subset e_{1}\subset e_{2}\subset\dots\subset
e_{n-2}\subset e_{n}).$ Suppose that $g\in GL(n-1)$ fixes $\mathcal{F}_{y}.$
For $j=1,\dots,n-1$, let $U_{j}$ be the span of $e_{1},\dots,e_{j}$ and note
that $GL(n-1):=\\{g\in GL(n):g\cdot e_{n}=e_{n},\text{ and }\ g\cdot
U_{n-1}=U_{n-1}\\}.$ Since $g$ fixes the line
$\mbox{${\mathbb{C}}$}(e_{n-1}+e_{n})$ in the flag and $g\cdot e_{n}=e_{n},$
we see that $g\cdot e_{n-1}=e_{n-1}.$ Since $g\in GL(n-1)$, then $g\cdot
e_{j}\in U_{n-1}$ for $j=1,\dots,n-2$. But also $g\cdot
e_{j}\in\mbox{${\mathbb{C}}$}(e_{n-1}+e_{n})+U_{j}$ for $j=1,\dots,n-2$. Thus,
for $j=1,\dots,n-2$, the vector $g\cdot e_{j}\in
U_{n-1}\cap(\mbox{${\mathbb{C}}$}(e_{n-1}+e_{n})+U_{j})=U_{j}.$ Hence, $g$
stabilizes $U_{n-2}$, so $g\in GL(n-2)$, and also $g$ stabilizes the standard
flag $(e_{1}\subset e_{2}\subset\dots\subset e_{n-2})$ in $U_{n-2}$. Thus,
$g\in B_{n-2}$. Since $B_{n-2}$ fixes $\mathcal{F}_{y}$, this proves the
Theorem in case of $GL(n)$.

Next, consider the case $\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l)$ and recall
the basis $e_{1},\dots,e_{l},e_{-l},\dots,e_{-1}$. By Remark 2.11, the open
$SO(2l-1)$-orbit on $\mathcal{B}_{\mathfrak{so}(2l)}$ is identified with the
$SO(2l-1)$-orbit through the maximal isotropic flag
$\mathcal{I}\mathcal{F}_{y}:=(e_{l}\subset e_{1}\subset\dots\subset e_{l-1}).$
Let $g\in SO(2l-1)_{y}$, the stabilizer of $\mathcal{I}\mathcal{F}_{y}.$ Since
$g\in SO(2l-1)$, then $g$ commutes with the element $\sigma_{2l}$ on
$\mbox{${\mathbb{C}}$}^{2l}$ (see Section 2.2). Since $g$ stabilizes the flag
$\mathcal{I}\mathcal{F}_{y}$, we see that $g\cdot e_{l}=ae_{l}$ for some
$a\in\mbox{${\mathbb{C}}$}^{\times}$. Since $g$ commutes with $\sigma_{2l}$,
it follows that $g\cdot e_{-l}=ae_{-l}$, and since $\beta(e_{l},e_{-l})=1$,
the scalar $a=\pm 1$. But $g\in SO(2l-1)$, so by Equation (2.1), $g$ fixes the
vector $e_{l}-e_{-l}$. Thus, $a=1$ and by Equation (2.2), $g\in SO(2l-2),$ so
that $SO(2l-1)_{y}=SO(2l-2)_{y}.$ But $B_{2l-2}$ is clearly in $SO(2l-2)_{y}$
so since $SO(2l-2)_{y}$ is solvable and $B_{2l-2}$ is a Borel subgroup of
$SO(2l-2),$ it follows that $B_{2l-2}=SO(2l-2)_{y}=SO(2l-1)_{y}$, proving the
required assertion.

Now let $\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l+1)$ and recall the basis
$e_{1},\dots,e_{l},e_{0},e_{-l},\dots,e_{-1}$ from Section 2.2. By Remark
2.11, the open orbit is the $SO(2l)$-orbit through the point in
$\mathcal{B}_{\mathfrak{so}(2l+1)}$ corresponding to the maximal isotropic
flag $(e_{l}+\sqrt{2}\imath e_{0}+e_{-l}\subset e_{1}\subset
e_{2}\subset\dots\subset e_{l-1}).$ We can choose an element $s$ of $SO(2l)$
in the diagonal torus that $s\cdot(e_{l}+\sqrt{2}\imath e_{0}+e_{-l})=\imath
e_{l}+\sqrt{2}\imath e_{0}-\imath e_{-l}$, so that the open orbit is the
$SO(2l)$-orbit through the point $y$ in $\mathcal{B}_{\mathfrak{so}(2l+1)}$
corresponding to the maximal isotropic flag
$\mathcal{I}\mathcal{F}_{y}=(\imath e_{l}+\sqrt{2}\imath e_{0}-\imath
e_{-l}\subset e_{1}\subset e_{2}\subset\dots\subset e_{l-1})$. Note that by a
small calculation using the root vectors in $\mbox{${\mathfrak{b}}$}_{2l-1}$,
we see that $B_{2l-1}$ is contained in the stabilizer $SO(2l)_{y}$. Now let
$g\in SO(2l)_{y}$. Since $g\in SO(2l)$, then $g\cdot e_{0}=e_{0}$. Since $g$
stabilizes the line in the flag, we see that
$g\cdot(e_{l}-e_{-l})=e_{l}-e_{-l}$, so that $g\in SO(2l-1)$ by Equation
(2.3). As in the preceding paragraph, it follows that $B_{2l-1}=SO(2l-1)_{y}$.

Q.E.D.

######  Remark 3.9.

For later use, we make explicit the correspondence between $B_{n-1}$-orbits on
the open $K$-orbit ${\widetilde{Q}}_{K}$ in $\mathcal{B}_{n}$ and
$B_{n-2}$-orbits on $\mathcal{B}_{n-1}$ given in Equation (3.14). Let
$Q=B_{n-1}\cdot{\rm
Ad}(k)\tilde{\mbox{${\mathfrak{b}}$}}\subset{\widetilde{Q}}_{K}=K\cdot\tilde{\mbox{${\mathfrak{b}}$}}$,
where $\tilde{\mbox{${\mathfrak{b}}$}}$ is the stabilizer of the (isotropic)
flag $\mathcal{F}_{y}$ ($\mathcal{I}\mathcal{F}_{y}$) given in the proof of
Theorem 3.8. Denote the corresponding $B_{n-2}$-orbit in $\mathcal{B}_{n-1}$
by $Q^{op}$. Note that $Q^{op}=B_{n-2}\cdot{\rm
Ad}(k^{-1})\cdot\mbox{${\mathfrak{b}}$}_{n-1}$. We can realize this
correspondence geometrically as follows. Let $q:K\to{\widetilde{Q}}_{K}\cong
K/B_{n-2}$ and $\pi:K\to\mathcal{B}_{n-1}$ be the natural projections, and let
$\psi:K\to K$ be the inversion map, i.e., $\psi(k)=k^{-1}$. Then

(3.15) $Q^{op}=\pi(\psi(q^{-1}(Q))).$

More generally, if $\mathcal{Y}\subset{\widetilde{Q}}_{K}$ is any
$B_{n-1}$-stable subvariety, it corresponds to a $B_{n-2}$-stable subvariety
of $\mathcal{B}_{n-1}$:

(3.16) $\mathcal{Y}^{op}=\pi(\psi(q^{-1}(\mathcal{Y}))).$

Using (3.16), we note that

(3.17)
$\dim\mathcal{Y}^{op}=\dim\mathcal{Y}+\dim\mbox{${\mathfrak{b}}$}_{n-2}-\dim\mbox{${\mathfrak{b}}$}_{n-1}.$

Note the easily verified numerical identity

(3.18)
$\dim\mbox{${\mathfrak{b}}$}_{n-1}-\dim\mbox{${\mathfrak{b}}$}_{n-2}=\mbox{rank}(\mbox{${\mathfrak{k}}$}).$

Equation (3.17) then becomes

(3.19)
$\dim\mathcal{Y}^{op}=\dim\mathcal{Y}-\mbox{rank}(\mbox{${\mathfrak{k}}$}).$

##  4\. The monoid action for $B_{n-1}$-orbits on $\mathcal{B}_{n}$

In this section, we use Theorems 3.5 and Theorem 3.8 to describe several
monoid actions for the $B_{n-1}$-orbits on
$\mathcal{B}_{n}=\mathcal{B}_{\mbox{${\mathfrak{g}}$}_{n}}$. One of the major
points is to compute the monoid action on an $B_{n-1}$-orbit
${\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})$ in
$\mathcal{B}_{n}$ in terms of other monoidal actions. In later work, we apply
this action to the study the order relation on $B_{n-1}$-orbits on
$\mathcal{B}_{n}$.

###  4.1. Left and Right Monoid Actions

Recall Notation 2.2 and before let $K=G_{n-1}$. Let $K_{\Delta}\subset G\times
K$ be the diagonal copy of $K$ in the product $G\times K$. Note that there is
a canonical bijection

$\chi:B_{n-1}\backslash\mathcal{B}\to
K_{\Delta}\backslash\mathcal{B}\times\mathcal{B}_{K},$

given by $\chi(Q)=K_{\Delta}\cdot(Q,eB_{n-1})$ and we let $Q_{\Delta}$ denote
$\chi(Q)$. Further, $Q_{\Delta}\cong K\times_{B_{n-1}}Q$, and it follows that
the map $Q\mapsto Q_{\Delta}$ preserves topological properties like closure
relations and open sets, and
$\dim(Q_{\Delta})=\dim(Q)+\dim(\mathcal{B}_{n-1}).$

In particular, $K_{\Delta}$ has finitely many orbits on
$\mathcal{B}\times\mathcal{B}_{n-1}$, and hence the set
$K_{\Delta}\backslash\mathcal{B}\times\mathcal{B}_{n-1}$ has 2 different
monoid actions, coming from the monoid actions for the two different kinds of
simple roots of $\mbox{${\mathfrak{g}}$}\times\mbox{${\mathfrak{k}}$}$ via the
generalities from Section 2.3. We decompose the simple roots
$\Pi_{\mbox{${\mathfrak{g}}$}\times\mbox{${\mathfrak{k}}$}}=\Pi_{\mbox{${\mathfrak{k}}$}}\cup\Pi_{\mbox{${\mathfrak{g}}$}}.$

######  Notation 4.1.

For a simple root $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$, we call the monoid
action $Q_{\Delta}\to m(s_{\alpha})*Q_{\Delta}$ a left monoid action. If
$\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$, we call the monoid action
$Q_{\Delta}\to m(s_{\alpha})*Q_{\Delta}$ a right monoid action. We use the
equivalence between $B_{n-1}\backslash\mathcal{B}$ and
$K_{\Delta}\backslash\mathcal{B}\times\mathcal{B}_{n-1}$ to similarly define
left and right monoid actions $m(s_{\alpha})$ on
$B_{n-1}\backslash\mathcal{B}$.

In more detail, for $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$, the left action
uses the projection $p_{\alpha}:G/B_{n}\times K/B_{n-1}\to G/B_{n}\times
K/P_{\alpha}$ where $P_{\alpha}$ is the parabolic of $K$ with Lie algebra
$\mbox{${\mathfrak{b}}$}_{n-1}+\mbox{${\mathfrak{k}}$}_{-\alpha}.$ Thus, if
$Q_{\Delta}=K_{\Delta}\cdot(Q,eB_{n-1})$, then

$p_{\alpha}^{-1}p_{\alpha}(Q_{\Delta})=K_{\Delta}\cdot(Q,P_{\alpha}/B_{n-1})=K_{\Delta}\cdot(P_{\alpha}\cdot
Q,eB_{n-1}).$

It follows that for $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}},$ and
$Q=B_{n-1}xB_{n}/B_{n}$.

(4.1) $m(s_{\alpha})*Q\text{ \ is the unique open }B_{n-1}-\text{orbit in \
}P_{\alpha}\cdot Q=P_{\alpha}xB_{n}/B_{n}.$

Similar analysis shows that if $Q=B_{n-1}xB_{n}/B_{n}$ and
$\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$, then

(4.2) $m(s_{\alpha})*Q\mbox{ is the unique open }B_{n-1}-\mbox{orbit in
}B_{n-1}xP_{\alpha}/B_{n},$

where $P_{\alpha}$ here is the parabolic of $G$ whose Lie algebra is
$\mbox{${\mathfrak{b}}$}_{n}+\mbox{${\mathfrak{g}}$}_{-\alpha}.$

For a useful generalization of the right monoid action, we recall the notion
of strongly orthogonal roots.

######  Definition 4.2.

Two nonproportional roots
$\alpha,\,\beta\in\Phi(\mbox{${\mathfrak{g}}$},\mbox{${\mathfrak{h}}$})$ are
said to be _strongly orthogonal_ if $\alpha\pm\beta$ is not a root.

Recall that by elementary arguments, strongly orthogonal roots are necessarily
orthogonal and the corresponding
$\mbox{${\mbox{${\mathfrak{s}}$}\mbox{${\mathfrak{l}}$}}$}(2)$ subalgebras,
$\mbox{${\mathfrak{s}}$}_{\alpha}$ and $\mbox{${\mathfrak{s}}$}_{\beta}$,
commute. We also make use of the following notation.

######  Notation 4.3.

Let $S\subset\Pi_{\mbox{${\mathfrak{g}}$}}$ be a subset of the simple roots of
${\mathfrak{g}}$, and let
$\mbox{${\mathfrak{m}}$}_{S}\subset\mbox{${\mathfrak{g}}$}$ be the standard
Levi subalgebra of ${\mathfrak{g}}$ given by the subset $S$. We denote the
corresponding standard parabolic subalgebra by
$\mbox{${\mathfrak{p}}$}_{S}\supset\mbox{${\mathfrak{b}}$}_{+}$ and its Levi
decomposition by
$\mbox{${\mathfrak{p}}$}_{S}=\mbox{${\mathfrak{m}}$}_{S}\oplus\mbox{${\mathfrak{u}}$}_{S}$,
and if $S$ is understood, we let
$\mbox{${\mathfrak{m}}$}=\mbox{${\mathfrak{m}}$}_{S}$ and
$\mbox{${\mathfrak{u}}$}=\mbox{${\mathfrak{u}}$}_{S}.$ Let $P_{S},M_{S},U_{S}$
be the corresponding connected subgroups of $G.$

######  Remark 4.4.

Suppose $S\subset\Pi_{\mbox{${\mathfrak{g}}$}}$ is a subset of simple roots,
let $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$ be a simple root which is
strongly orthogonal to the roots in $S$, and let
$S^{\prime}=S\cup\\{\alpha\\}$. The projection $G/P_{S}\mapsto
G/P_{S^{\prime}}$ is a $\mathbb{P}^{1}$-bundle. Hence, if $H$ is a subgroup of
$G$ acting on $G/P_{S}$ with finitely many orbits, there is a right monoid
action $m(s_{\alpha})$ on $H\backslash G/P_{S}.$

######  Remark 4.5.

There are additional monoid actions we will use. Let
$\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$, and consider the finite set
$B_{n-1}\backslash K/P)$ where $P$ is a standard parabolic subgroup of $K$. As
above, there is a bijection

$\chi:B_{n-1}\backslash K/P\to K_{\Delta}\backslash K/B_{n-1}\times K/P$

given by $\chi(Q)=K_{\Delta}\cdot(eB_{n-1},Q)=:Q^{n-1}_{\Delta}.$ This
bijection satisfies similar properties as the above bijection also denoted
$\chi$.

(1) It follows that if $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$ is a simple
root of $\Pi_{\mbox{${\mathfrak{k}}$}\times\mbox{${\mathfrak{k}}$}}$ with root
space in the first factor, then there is another left monoid action
$m(s_{\alpha})$ on $K\backslash K/B_{n-1}\times K/P$ and hence by the
equivalence of orbits induced by this version of $\chi$, there is a left
monoid action $m(s_{\alpha})$ on $B_{n-1}\backslash K/P$. Similar analysis to
the previous left monoid action verifies the following assertion. For $Q\in
B_{n-1}\backslash K/P$,

(4.3) $m(s_{\alpha})*Q\text{ \ is the unique open orbit in \ }P_{\alpha}\cdot
Q.$

(2) There is also a right monoid action in the strongly orthogonal setting.
Again, we let $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$. The standard parabolic
$P\subset K$ is $P=P_{S}$ for $S\subset\Pi_{\mbox{${\mathfrak{k}}$}}$. Suppose
$\alpha$ is strongly orthogonal to the roots of $S$. Then if the root space of
$\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}\times\mbox{${\mathfrak{k}}$}}$ is in
the second factor, it follows from Remark 4.4 that there is a right monoid
action $m(s_{\alpha})$ on $K_{\Delta}\backslash K/B_{n-1}\times K/P$ and hence
on $B_{n-1}\backslash K/P$ via the bijection $\chi$. It can be shown that if
$Q=B_{n-1}\cdot xP_{S}/P_{S}$, then

(4.4) $m(s_{\alpha})*Q\text{ \ is the unique open orbit in \
}B_{n-1}xP_{S^{\prime}}/P_{S},$

where $P_{S^{\prime}}$ is the standard parabolic subgroup of $K$ corresponding
to the subset of simple roots $S^{\prime}=S\cup\\{\alpha\\}$.

######  Remark 4.6.

We will abuse notation by using the same symbol $m(s_{\alpha})$ to denote both
the left monoid action by a root $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$ on
$B_{n-1}\backslash\mathcal{B}$ and for the left monoid action on
$B_{n-1}\backslash K/P.$ This may require some parsing on the part of the
reader. For example, if we consider a $B_{n-1}$-orbit
${\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})$ on
$\mathcal{B}$, then we will prove that in certain circumstances,
$m(s_{\alpha})*{\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})={\mathcal{O}}(m(s_{\alpha})*Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}}).$
In this equality, on the left hand side of the equality, the monoid action is
the left monoid action on $B_{n-1}\backslash\mathcal{B}$ and on the right hand
side of the equality, the monoid action is the left monoid action on
$B_{n-1}\backslash K/P.$

###  4.2. Computation of Monoid Actions

It follows from definitions that if $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$
and $Q\subset Q_{K}$, then $m(s_{\alpha})*Q\subset Q_{K}$. We now examine the
effect of the left monoid action on the fibre bundle structure of
$Q={\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})\subset
Q_{K}$ after first recalling some notation from Remark 3.7. Suppose
$Q_{K}=K\cdot\mbox{${\mathfrak{b}}$}$, where
$\mbox{${\mathfrak{b}}$}=\mbox{${\mathfrak{b}}$}_{\hat{v}}={\rm
Ad}(\hat{v})\mbox{${\mathfrak{b}}$}_{+}\in\mathcal{B}$ is the representative
given in Propositions 2.8, 2.9, and 2.10 respectively. Let
$\mbox{${\mathfrak{r}}$}\supset\mbox{${\mathfrak{b}}$}$ be the special
parabolic subalgebra of ${\mathfrak{g}}$ associated to the $K$-orbit $Q_{K}$,
and let
$\mbox{${\mathfrak{r}}$}=\mbox{${\mathfrak{l}}$}\oplus\mbox{${\mathfrak{u}}$}$
be its Levi decomposition. Then $Q=B_{n-1}\cdot{\rm Ad}(\dot{w}){\rm
Ad}(\ell)\mbox{${\mathfrak{b}}$}$, where $w\in W_{K}^{K\cap R}$ and $\ell\in
K\cap L$ (see Equation (3.13)). Further
$Q_{\mbox{${\mathfrak{r}}$}}=B_{n-1}\cdot w(\mbox{${\mathfrak{r}}$})$ and
$Q_{\mbox{${\mathfrak{l}}$}}=(B_{n-1}\cap L)\cdot{\rm
Ad}(\ell)(\mbox{${\mathfrak{b}}$}_{\hat{v}}\cap\mbox{${\mathfrak{l}}$})$.

######  Theorem 4.7.

Let $\alpha$ be a root in $\Pi_{\mbox{${\mathfrak{k}}$}}$. Let
$\pi_{\mbox{${\mathfrak{r}}$}}:G/B_{+}\to G/R$ be the projection. Then

(4.5)
$\pi_{\mbox{${\mathfrak{r}}$}}(m(s_{\alpha})*{\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}}))=m(s_{\alpha})*Q_{\mbox{${\mathfrak{r}}$}},$

where the monoid action on $B_{n-1}\backslash K/K\cap R$ in the right hand
side is given in Remark 4.5.

Suppose also that $\alpha\in w(\Phi_{\mbox{${\mathfrak{l}}$}})$. Then $\alpha$
is a simple root of $\mbox{${\mathfrak{k}}$}\cap w(\mbox{${\mathfrak{l}}$})$
with respect to the Borel subalgebra $\mbox{${\mathfrak{b}}$}_{n-1}\cap
w(\mbox{${\mathfrak{l}}$})$ of $\mbox{${\mathfrak{k}}$}\cap
w(\mbox{${\mathfrak{l}}$})$, and

(4.6)
$m(s_{\alpha})*{\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})=\mathcal{O}(Q_{\mbox{${\mathfrak{r}}$}},m(s_{w^{-1}(\alpha)})*Q_{\mbox{${\mathfrak{l}}$}}).$

Suppose $\alpha\not\in w(\Phi_{\mbox{${\mathfrak{l}}$}})$. Then

(4.7)
$m(s_{\alpha})*{\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})=\mathcal{O}(m(s_{\alpha})*Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}}).$

###### Proof.

By Equation (4.1), $m(s_{\alpha})*Q$ is the open $B_{n-1}$-orbit in the
variety $\mathcal{Y}:=P_{\alpha}\dot{w}\ell B/B$. By equivariance of
$\pi_{\mbox{${\mathfrak{r}}$}}$, its restriction
$\pi_{\mbox{${\mathfrak{r}}$}}:\mathcal{Y}\to P_{\alpha}\dot{w}R/R$ is open
since quotient morphisms are open (Section 12.2 of [Hum75]). Thus,
$m(s_{\alpha})*Q$ projects to the open $B_{n-1}$-orbit in
$P_{\alpha}\dot{w}R/R$. Under the identification
$Q_{K,\mbox{${\mathfrak{r}}$}}\cong K/K\cap R$, the latter orbit is identified
with the open $B_{n-1}$-orbit in $P_{\alpha}\dot{w}(K\cap R)/(K\cap R)$. But
this orbit is exactly $m(s_{\alpha})*Q_{\mbox{${\mathfrak{r}}$}}$ by Equation
(4.3). Equation (4.5) follows.

Note that the $P_{\alpha}$-orbit $\mathcal{Y}$ fibres over
$P_{\alpha}/(P_{\alpha}\cap w(R))$ via the identification

(4.8) $\mathcal{Y}\cong P_{\alpha}\times_{P_{\alpha}\cap w(R)}(P_{\alpha}\cap
w(L)){\rm Ad}(\dot{w}\ell)\mbox{${\mathfrak{b}}$}.$

Indeed, the identification $\mathcal{Y}\cong P_{\alpha}\times_{P_{\alpha}\cap
w(R)}(P_{\alpha}\cap w(R)){\rm Ad}(\dot{w}\ell)\mbox{${\mathfrak{b}}$}$ is
formal, and since the diagonal torus of $K$ normalizes $P_{\alpha},w(L),$ and
$w(U)$, it follows that the action of $P_{\alpha}\cap w(R)$ on ${\rm
Ad}(\dot{w}\ell)\mbox{${\mathfrak{b}}$}$ is through $P_{\alpha}\cap w(L)$. Now
suppose that $\alpha$ is a root of
$w(\mbox{${\mathfrak{l}}$})\cap\mbox{${\mathfrak{k}}$}$. It follows that
$P_{\alpha}wR/R=B_{n-1}wR/R$, whence
$m(s_{\alpha})*Q_{\mbox{${\mathfrak{r}}$}}=Q_{\mbox{${\mathfrak{r}}$}}$.
Further the fibre bundle in Equation (4.8) may be written as

(4.9) $\mathcal{Y}\cong B_{n-1}\times_{B_{n-1}\cap w(R)}(P_{\alpha}\cap
w(L)){\rm
Ad}(\dot{w}\ell)(\mbox{${\mathfrak{b}}$}\cap\mbox{${\mathfrak{l}}$}).$

We observed in Equation (3.9) that $B_{n-1}\cap w(L)$ is a Borel subgroup of
the Levi subgroup $K\cap w(L)$ of $K$. Since $P_{\alpha}\cap w(L)\supset
B_{n-1}\cap w(L)$ and $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$, it immediately
follows that $P_{\alpha}\cap w(L)$ is a parabolic subgroup of $K\cap w(L)$ and
$\alpha$ is a simple root with respect to the Borel subalgebra
$\mbox{${\mathfrak{b}}$}_{n-1}\cap w(\mbox{${\mathfrak{l}}$})$ of
$\mbox{${\mathfrak{k}}$}\cap w(\mbox{${\mathfrak{l}}$})$.

Equation (4.9) implies that

(4.10) $m(s_{\alpha})*Q\cong B_{n-1}\times_{B_{n-1}\cap w(R)}\mathcal{O},$

where $\mathcal{O}$ is the open $B_{n-1}\cap w(R)$-orbit in the variety
$(P_{\alpha}\cap w(L)){\rm
Ad}(\dot{w}\ell)(\mbox{${\mathfrak{b}}$}\cap\mbox{${\mathfrak{l}}$})\subset\mathcal{B}_{w(\mbox{${\mathfrak{l}}$})}.$
Now $B_{n-1}\cap w(R)$ acts on $\mathcal{B}_{w(\mbox{${\mathfrak{l}}$})}$
through its image in the quotient
$w(R)/w(U)\cong\mathcal{B}_{w(\mbox{${\mathfrak{l}}$})}$. In the proof of
Theorem 3.5, we observed that this image is the Borel subgroup $B_{n-1}\cap
w(L)$ of $w(L)\cap K$ (see Equation (3.12)). Equation (4.6) now follows by
translating by $\dot{w}^{-1}$, as is prescribed in the last comments of the
proof of Theorem 3.5, and from Equations (3.9) and (4.1).

We now suppose that $\alpha$ is not a root of $\mbox{${\mathfrak{k}}$}\cap
w(\mbox{${\mathfrak{l}}$})$. Then $P_{\alpha}\cap w(L)=B_{n-1}\cap w(L)$.
Thus, we can rewrite Equation (4.8) as

$\mathcal{Y}\cong P_{\alpha}\times_{P_{\alpha}\cap{\rm
Ad}(\dot{w})R}(B_{n-1}\cap w(L)){\rm
Ad}(\dot{w}\ell)\mbox{${\mathfrak{b}}$}=P_{\alpha}\times_{P_{\alpha}\cap
w(R)}{\rm Ad}(\dot{w})Q_{\mbox{${\mathfrak{l}}$}},$

where the last equality uses the definition of $Q_{\mbox{${\mathfrak{l}}$}}.$
Therefore, the $B_{n-1}$-orbits in $\mathcal{Y}$ are in one-to-one
correspondence with the $B_{n-1}$-orbits on the base $P_{\alpha}wR/R\cong
P_{\alpha}w(K\cap R)/(K\cap R)$. Thus,

$m(s_{\alpha})*Q=\pi_{\mbox{${\mathfrak{r}}$}}|_{\mathcal{Y}}^{-1}(m(s_{\alpha})*Q_{\mbox{${\mathfrak{r}}$}})=\mathcal{O}(m(s_{\alpha})*Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}}),$

yielding (4.7).

Q.E.D.

To understand how the bundle structure of $Q\subset Q_{K}$ behaves with
respect to the right monoid action by roots
$\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$, we need some preparation.

######  Proposition 4.8.

Let $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$. Suppose that $Q\subset Q_{K}$.
Then

(4.11) $m(s_{\alpha})*Q\subset m(s_{\alpha})*Q_{K}$

In particular, if $m(s_{\alpha})*Q_{K}\neq Q_{K}$, then $m(s_{\alpha})*Q\neq
Q$.

###### Proof.

Let $Q=B_{n-1}xB_{n}/B_{n}$. It follows from definitions that
$m(s_{\alpha})*Q_{K}$ is the unique open $K$-orbit in the variety
$KxP_{\alpha}/B_{n}$. Since $B_{n-1}xP_{\alpha}/B_{n}\subset
KxP_{\alpha}/B_{n}$, the subvariety $m(s_{\alpha})*Q_{K}\cap
B_{n-1}xP_{\alpha}/B_{n}$ is non-empty and open in $B_{n-1}xP_{\alpha}/B_{n}$.
Since $B_{n-1}xP_{\alpha}/B_{n}$ is irreducible, it follows from Equation
(4.2) that $m(s_{\alpha})*Q\cap m(s_{\alpha})*Q_{K}\neq\emptyset.$ This
implies equation (4.11).

Q.E.D.

By Proposition 4.8, if $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$ and
$m(s_{\alpha})*Q_{K}=Q_{K}$, then $m(s_{\alpha})*Q\subset Q_{K}$. For such
$\alpha$, we study the monoid action by $m(s_{\alpha})$ in terms of the fibre
bundle description of the orbit
${\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})$ of
$Q$. But first we need to determine the roots
$\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$ for which
$m(s_{\alpha})*Q_{K}=Q_{K}$. The following results are well-known and partly
appear in tables on page 92 of [Col85]. For the case of
$\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n)$, proofs are given in Examples 4.16
and 4.30 of [CE14] and for the type $B$ and $D$ cases, see Propositions 2.23
and 2.24 of [CE].

######  Proposition 4.9.

  1. (1)

Let $\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n)$. First, consider the case when
$Q_{K}=Q_{i}$ is a closed $K$-orbit as in Part (2) of Proposition 2.8. Then
the roots $\alpha_{i-1}$ and $\alpha_{i}$ are non-compact for $Q_{K}$ and all
other simple roots are compact. Next, consider the case when $Q_{K}=Q_{i,j}$
as in Part (3) of Proposition 2.8. Then the roots $\alpha_{i-1}$ and
$\alpha_{j}$ are complex stable for $Q_{K}$. All other simple roots
$\alpha_{k}$ with $k\neq i,j-1$ are compact for $Q_{K}$. If $j=i+1$, then the
root $\alpha_{i}$ is real for $Q_{K}$. If $j>i+1$, the roots $\alpha_{i}$ and
$\alpha_{j-1}$ are complex unstable.

  2. (2)

Let $\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l+1)$. If $Q_{K}$ is one of the
two closed $K$-orbits given in Part (2) of Proposition 2.9, then the root
$\alpha_{l}$ is non-compact for $Q_{K}$, and all other simple roots are
compact. Consider the orbit $Q_{K}=Q_{i}$ as in Part (3) of Proposition 2.9.
Then the root $\alpha_{i}$ is complex stable for $Q_{K}$. The root
$\alpha_{i+1}$ is complex unstable for $Q_{K}$ unless $i=l-1$ in which case it
is real. All other simple roots are compact for $Q_{K}$.

  3. (3)

Let $\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l)$. In the case $Q_{K}=Q_{+}$ is
the unique closed orbit, then the roots $\alpha_{l-1}$ and $\alpha_{l}$ are
complex stable for $Q_{+}$ and all other simple roots are compact. In the case
$Q_{K}=Q_{i}$ as in Part (3) of Proposition 2.10, then the roots
$\alpha_{i-1}$ and $\alpha_{i}$ are complex for $Q_{i}$ with $\alpha_{i-1}$
stable and $\alpha_{i}$ unstable. All other simple roots are compact for
$Q_{i}$, unless $i=l-1$ in which case $\alpha_{l}$ is also complex unstable.

We use Proposition 4.9 to study the relation between roots
$\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$ such that $m(s_{\alpha})*Q_{K}=Q_{K}$
and the special parabolic subalgebras
$\mbox{${\mathfrak{r}}$}\subset\mbox{${\mathfrak{g}}$}$ constructed in Theorem
3.1. Let $\tilde{w}\in W$ be given as follows:

(4.12) $\tilde{w}=\left\\{\begin{array}[]{ccc}w_{i}=(n,n-1,\dots,i+1,i)\mbox{
if }\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n),&\mbox{ and }&Q_{K}=Q_{i,j}\mbox{
or }Q_{i}\\\ \mbox{ id }\mbox{ if
}\mbox{${\mathfrak{g}}$}=\mathfrak{so}(n),&\mbox{ and }&Q_{K}=Q_{i}\mbox{ or
}\,Q_{+}\\\ s_{\alpha_{l}}\mbox{ if
}\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l+1),&\mbox{ and
}&Q_{K}=Q_{-}\end{array}\right\\},$

and let $\dot{\tilde{w}}$ be a fixed representative of $\tilde{w}$. Then it
follows from the construction of the special parabolic subalgebra
${\mathfrak{r}}$ in Theorem 3.1 that
$\mbox{${\mathfrak{r}}$}={\tilde{w}}(\mbox{${\mathfrak{p}}$}_{S})$, where
$\mbox{${\mathfrak{p}}$}_{S}$ is a standard parabolic subalgebra of
${\mathfrak{g}}$ defined by the subset $S\subset\Pi_{\mbox{${\mathfrak{g}}$}}$
given by:

(4.13) $S=\left\\{\begin{array}[]{cc}\\{\alpha_{i},\dots,\alpha_{j-1}\\}\mbox{
for }\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n),&Q_{K}=Q_{i,j}\\\
\\{\alpha_{i},\dots,\alpha_{l}\\}\mbox{ for
}\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l),&Q_{K}=Q_{i}\\\
\\{\alpha_{i+1},\dots,\alpha_{l}\\}\mbox{ for
}\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l+1),&Q_{K}=Q_{i}\end{array}\right\\},$

and where $S=\emptyset$ and ${\mathfrak{r}}$ is a Borel if $Q_{K}$ is closed.
Proposition 4.9 implies the following observation.

######  Corollary 4.10.

Let $S\subset\Pi_{\mbox{${\mathfrak{g}}$}}$ be the subset of simple roots
given in (4.13). Then

  1. (1)

If $\alpha\in S$, $m(s_{\alpha})*Q_{K}=Q_{K}$.

  2. (2)

If $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}\setminus S$ and
$m(s_{\alpha})*Q_{K}=Q_{K}$, then $\alpha$ is compact for $Q_{K}$.

We can now describe the right monoid action on a $B_{n-1}$-orbit $Q\subset
Q_{K}$ by a simple root $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$ with
$m(s_{\alpha})*Q_{K}=Q_{K}$. Recalling notation from Theorem 4.7, we let
$Q=\mathcal{O}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})=B_{n-1}\cdot{\rm
Ad}(\dot{w}){\rm Ad}(\ell\hat{v})\mbox{${\mathfrak{b}}$}_{+}$, where $w\in
W_{K}^{K\cap R}$ and $\ell\in K\cap L$, with
$Q_{\mbox{${\mathfrak{r}}$}}=B_{n-1}\cdot w(\mbox{${\mathfrak{r}}$})$ and
$Q_{\mbox{${\mathfrak{l}}$}}=(B_{n-1}\cap L)\cdot{\rm
Ad}(\ell)(\mbox{${\mathfrak{b}}$}_{\hat{v}}\cap\mbox{${\mathfrak{l}}$})$, and
$\hat{v}$ is given in Propositions 2.8, 2.9, 2.10

######  Theorem 4.11.

Suppose $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$.

  1. (1)

If $\alpha\in S$, then

(4.14)
$m(s_{\alpha})*\mathcal{O}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})=\mathcal{O}(Q_{\mbox{${\mathfrak{r}}$}},m(s_{\tilde{w}(\alpha)})*Q_{\mbox{${\mathfrak{l}}$}}).$

  2. (2)

If $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}\setminus S$ and
$m(s_{\alpha})*Q_{K}=Q_{K}$, then
$\tilde{w}(\alpha)\in\Pi_{\mbox{${\mathfrak{k}}$}}$ is a standard simple root
strongly orthogonal to the roots of $S$, and

(4.15)
$m(s_{\alpha})*\mathcal{O}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})=\mathcal{O}(m(s_{\tilde{w}(\alpha)})*Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}}),$

where $m(s_{\tilde{w}(\alpha)})*Q_{\mbox{${\mathfrak{r}}$}}$ is the right
monoid action of the simple root $\tilde{w}(\alpha)$ on the $B_{n-1}$-orbit
$Q_{\mbox{${\mathfrak{r}}$}}$ in the partial flag variety
$Q_{K,\mbox{${\mathfrak{r}}$}}=K/K\cap R$ of ${\mathfrak{k}}$ defined in Part
(2) of Remark 4.5.

###### Proof.

First, suppose that $\alpha\in S$. Then by Corollary 4.10,
$m(s_{\alpha})*Q_{K}=Q_{K}$. Recall that by Equation (4.2) $m(s_{\alpha})*Q$
is the open $B_{n-1}$-orbit in the variety
$\mathcal{Z}:=B_{n-1}\dot{w}\ell\hat{v}P_{\alpha}/B_{+}$. Since $\alpha\in S$,
$P_{\alpha}\subset P_{S}$. Further, the representative
$\hat{v}=\dot{\tilde{w}}p$, where $\dot{\tilde{w}}$ is the above
representative of $\tilde{w}$ given by (4.12) and $p\in M_{S}$ (see Equations
(2.8), (2.11), and (2.13)). It follows that the special parabolic subalgebra
$\mbox{${\mathfrak{r}}$}={\rm Ad}(\hat{v})\mbox{${\mathfrak{p}}$}_{S}$. Thus,
$\pi_{\mbox{${\mathfrak{r}}$}}:G/B\to G/R$ maps $\mathcal{Z}$ to $B_{n-1}\cdot
w(\mbox{${\mathfrak{r}}$})=Q_{\mbox{${\mathfrak{r}}$}}$ and hence
$\pi_{\mbox{${\mathfrak{r}}$}}:\mathcal{Z}\to Q_{\mbox{${\mathfrak{r}}$}}$ is
a $B_{n-1}$-equivariant fibre bundle. We call the fibre $F$, and using
Equation (3.8) and Remark 3.7, we can identify $F=(B_{n-1}\cap w(L))\cdot{\rm
Ad}(\dot{w}\ell\hat{v})(P_{\alpha}\cdot\mbox{${\mathfrak{b}}$}_{+})$. Using
Equation (3.9), we see $F$ is isomorphic to

(4.16) $(B_{n-1}\cap L)\cdot{\rm
Ad}(\ell\hat{v})(P_{\alpha}\cdot\mbox{${\mathfrak{b}}$}_{+})\cong(B_{n-1}\cap
L)\ell\hat{v}(P_{\alpha}\cap M_{S})/(B_{+}\cap M_{S}).$

The open $B_{n-1}$-orbit in $\mathcal{Z}$ fibres over
$Q_{\mbox{${\mathfrak{r}}$}}$ with fibre isomorphic to the open $B_{n-1}\cap
L$-orbit in the variety $F$. Using the decomposition
$\hat{v}=\dot{\tilde{w}}p$, we can rewrite $F$ as

(4.17) $(B_{n-1}\cap L)\ell\tilde{w}p\tilde{w}^{-1}(P_{\tilde{w}(\alpha)}\cap
L)/({\rm Ad}(\tilde{w})B_{+}\cap L).$

But the open $B_{n-1}\cap L$-orbit in (4.17) is exactly
$m(s_{\tilde{w}(\alpha)})*Q_{\mbox{${\mathfrak{l}}$}}$, using Equation (4.2).
Equation (4.14) follows.

Let $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}\setminus S$ with
$m(s_{\alpha})*Q_{K}=Q_{K}$. By (4.2), $m(s_{\alpha})*Q_{K}$ is the open
$B_{n-1}$-orbit in the variety

(4.18) $\mathcal{Y}=B_{n-1}\cdot{\rm
Ad}(\dot{w}\ell\hat{v})(P_{\alpha}\cdot\mbox{${\mathfrak{b}}$}_{+})=B_{n-1}\cdot{\rm
Ad}(\dot{w}\ell)(P_{{\rm Ad}(\hat{v})\alpha}\cdot\mbox{${\mathfrak{b}}$}).$

We begin by analyzing the homogeneous space $P_{{\rm Ad}(\hat{v})\alpha}/B$.
First, consider the root ${\rm Ad}(\hat{v})\alpha$ of ${\rm
Ad}(\hat{v})\mbox{${\mathfrak{h}}$}_{n}$. We claim

(4.19) ${\rm
Ad}(\hat{v})\alpha=\tilde{w}(\alpha)\in\Pi_{\mbox{${\mathfrak{k}}$}},$

where $\tilde{w}\in W$ is given by (4.12). We show (4.19) in the case where
$\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n)$. If $Q_{K}=Q_{i}$ is a closed
$K$-orbit as in Part (2) of Proposition 2.8, then $\hat{v}=w_{i}=\tilde{w}$.
Since $m(s_{\alpha})*Q_{K}=Q_{K}$, then by Proposition 4.9,
$\alpha=\alpha_{j},$ $j\neq i-1,\,i$. It is routine to check that
$\tilde{w}(\alpha_{j})=w_{i}(\alpha_{j})=\alpha_{j-1}$ for $i<j\leq n$ and
$\tilde{w}(\alpha_{j})=\alpha_{j}$ for $j<i-1.$ In either case,
$\tilde{w}(\alpha)\in\Pi_{\mbox{${\mathfrak{k}}$}}=\\{\alpha_{1},\dots,\alpha_{n-2}\\}$.
Now suppose $Q_{K}=Q_{i,j}$ is not closed as in Part (3) of Proposition 2.8.
Then $\hat{v}=w_{i}u_{\alpha_{i+1}}s_{\alpha_{i+1}}\dots s_{\alpha_{j-1}}$ by
Equation (2.8). Then since $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}\setminus S$
with $m(s_{\alpha})*Q_{K}=Q_{K}$, then it follows from Definition 4.2 and
Proposition 4.9 that $\alpha$ is strongly orthogonal to the roots of
$S=\\{\alpha_{i},\dots,\alpha_{j-1}\\}$ (see Equation (4.13)). In particular,
$\alpha$ is orthogonal to the roots of $S$, so ${\rm
Ad}(\hat{v})\alpha=\tilde{w}u_{\alpha_{i}}(\alpha)$. Further, since $\alpha$
is strongly orthogonal to $\alpha_{i}$, Proposiiton 6.72 of [Kna02] implies
that $u_{\alpha_{i}}(\alpha)=\alpha$, and we conclude that ${\rm
Ad}(\hat{v})\alpha=\tilde{w}(\alpha)$. The remainder of the argument proceeds
as in the case where $Q_{K}=Q_{i}$ is closed, and the claim in (4.19) is
established in the $\mathfrak{gl}(n)$ case. The orthogonal cases are simpler
and can be handled in an analogous fashion.

Now it follows from Part (2) of Corollary 4.10 that $\alpha$ is compact for
$Q_{K}$. It then follows from our discussion in Section 2.3 that the space
$P_{\tilde{w}(\alpha)}/B$ is a single orbit of a Levi factor of
$P_{\tilde{w}(\alpha)}$. Let $M_{\alpha}\subset P_{\alpha}$ be the
$H_{n}$-stable Levi factor, so that $M_{\tilde{w}(\alpha)}:={\rm
Ad}(\tilde{w})M_{\alpha}\subset K$ is a Levi factor of
$P_{\tilde{w}(\alpha)}$. Thus, the variety $\mathcal{Y}$ in Equation (4.18)
equals $\mathcal{Y}=B_{n-1}\cdot{\rm
Ad}(\dot{w}\ell)(M_{\tilde{w}(\alpha)}\cdot\mbox{${\mathfrak{b}}$}$). Since
$\alpha$ is strongly orthogonal to the roots of $S$, it follows that
$\tilde{w}(\alpha)$ is strongly orthogonal to the roots of ${\mathfrak{l}}$.
Since $L\cap K\subset L$, it follows that the groups $M_{\tilde{w}(\alpha)}$
and $L\cap K$ commute. Therefore,

(4.20)
$\pi_{\mbox{${\mathfrak{r}}$}}(\mathcal{Y})=B_{n-1}wM_{\tilde{w}(\alpha)}(K\cap
R)/(K\cap R).$

If $J\subset\Pi_{\mbox{${\mathfrak{k}}$}}$, we let $S_{J}\subset K$ be the
standard parabolic subgroup of $K$ given by the subset $J$ of
$\Pi_{\mbox{${\mathfrak{k}}$}}$. Now we claim that

(4.21)
$\pi_{\mbox{${\mathfrak{r}}$}}(\mathcal{Y})=B_{n-1}wS_{\Pi_{\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$}}\cup\tilde{w}(\alpha)}(K\cap
R)/(K\cap R),$

Indeed,

$\begin{split}B_{n-1}wS_{\Pi_{\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$}}\cup\tilde{w}(\alpha)}(K\cap
R)/(K\cap R)&=B_{n-1}wS_{\tilde{w}(\alpha)}(K\cap R)/(K\cap R)\\\
&=B_{n-1}wM_{\tilde{w}(\alpha)}(K\cap R)/(K\cap R)\\\
&=\pi_{\mbox{${\mathfrak{r}}$}}(\mathcal{Y}).\end{split}$

Now by Equation (4.4), $m(s_{\tilde{w}(\alpha)})*Q_{\mbox{${\mathfrak{r}}$}}$
is the open $B_{n-1}$-orbit in the variety
$B_{n-1}wS_{\Pi_{\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$}}\cup\tilde{w}(\alpha)}(K\cap
R)/(K\cap R)=\pi_{\mbox{${\mathfrak{r}}$}}(\mathcal{Y})$. Since the group
$M_{\tilde{w}(\alpha)}$ acts trivially on
$\mbox{${\mathfrak{l}}$}\cap\mbox{${\mathfrak{k}}$}$, the variety
$\mathcal{Y}$ has the structure of a $B_{n-1}$-equivariant fibre bundle with
smooth base $\pi_{\mbox{${\mathfrak{r}}$}}(\mathcal{Y})$ and fibre
$Q_{\mbox{${\mathfrak{l}}$}}$. It follows that
$\pi_{\mbox{${\mathfrak{r}}$}}|_{\mathcal{Y}}$ is a flat morphism and is
therefore open by Exercise III.9.1 of [Har77]. Thus,
$\pi_{\mbox{${\mathfrak{r}}$}}(m(s_{\alpha})*Q)=m(s_{\tilde{w}(\alpha)})*Q_{\mbox{${\mathfrak{r}}$}}$
and Equation (4.15) now follows.

Q.E.D.

It follows from Theorems 4.7 and 4.11 that to understand the monoid action on
$Q={\mathcal{O}}(Q_{\mbox{${\mathfrak{r}}$}},Q_{\mbox{${\mathfrak{l}}$}})\subset
Q_{K}$ by roots
$\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}\cup\Pi_{\mbox{${\mathfrak{k}}$}}$ such
that $m(s_{\alpha})*Q_{K}=Q_{K}$, it suffices to understand the monoid action
on $B_{n-1}$-orbits in $K/(K\cap R)$ discussed in Remark 4.5 as well as the
monoid action on $Q_{\mbox{${\mathfrak{l}}$}}$. The former is well understood,
and in the next result, we use the correspondence $Q\mapsto Q^{op}$ in
Equation (3.14) to understand the latter. For the statement of the next
result, we recall the particular representative
$\tilde{\mbox{${\mathfrak{b}}$}}$ of the open $K$-orbit that is used in the
proof of Theorem 3.8 to establish the correspondence in (3.14). We let
${\widetilde{Q}}_{K}=K\cdot\tilde{\mbox{${\mathfrak{b}}$}}$ where
$\tilde{\mbox{${\mathfrak{b}}$}}$ is the Borel subalgebra of ${\mathfrak{g}}$
which stabilizes the flag $\tilde{\mathcal{F}}$:

(4.22) $\tilde{\mathcal{F}}=\left\\{\begin{array}[]{cc}(e_{n-1}+e_{n}\subset
e_{1}\subset e_{2}\subset\dots\subset e_{n-2}\subset e_{n})&\mbox{ if
}\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n),\\\ (e_{l}\subset e_{1}\subset
e_{2}\subset\dots\subset e_{l-1})&\mbox{ if
}\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l)\\\ (\imath e_{l}+\sqrt{2}\imath
e_{0}-\imath e_{-l}\subset e_{1}\subset e_{2}\subset\dots\subset
e_{l-1})&\mbox{ if
}\mbox{${\mathfrak{g}}$}=\mathfrak{so}(2l+1)\end{array}\right\\},$

######  Theorem 4.12.

Let $Q=B_{n-1}\cdot{\rm Ad}(k)\tilde{\mbox{${\mathfrak{b}}$}}$ be a
$B_{n-1}$-orbit in the open $K$-orbit ${{\widetilde{Q}}}_{K}$ on
$\mathcal{B}_{n}$, and let $Q^{op}=B_{n-2}\cdot{\rm
Ad}(k^{-1})\mbox{${\mathfrak{b}}$}_{n-1}$ be the corresponding $B_{n-2}$-orbit
on $\mathcal{B}_{n-1}$ as in Remark 3.9.

Let $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$, then

(4.23) $(m(s_{\alpha})*Q)^{op}=m(s_{\alpha})*Q^{op}\subset\mathcal{B}_{n-1},$

where $m(s_{\alpha})*Q$ denotes the left monoid action of $s_{\alpha}$ on the
$B_{n-1}$-orbit $Q$ in $\mathcal{B}_{n}$, and $m(s_{\alpha})*Q^{op}$ denotes
the right monoid action of $s_{\alpha}$ on the $B_{n-2}$-orbit $Q^{op}$ in
$\mathcal{B}_{n-1}$.

Let $\alpha=\alpha_{j}\in\Pi_{\mbox{${\mathfrak{g}}$}}$ be a simple root which
is compact for ${\tilde{Q}}_{K}$. Then

(4.24) $(m(s_{\alpha_{j}})*Q)^{op}=m(s_{\alpha_{j-1}})*Q^{op},$

where $m(s_{\alpha_{j}})*Q$ is the right monoid action of $s_{\alpha_{j}}$ on
$Q$ and $m(s_{\alpha_{j-1}})*Q^{op}$ is the left monoid action of
$s_{\alpha_{j-1}}$ on $Q^{op}$ and
$\alpha_{j-1}\in\Pi_{\mbox{${\mathfrak{k}}$}_{n-2}}$ is a standard simple
root.

Moreover, the correspondences in Equations (4.23) and (4.24) preserve root
types. That is to say that $\alpha$ is complex stable, complex unstable, non-
compact, etc for $Q$ if and only if $\alpha$ is complex stable, unstable, non-
compact, etc for $Q^{op}$ in (4.23) and similarly for $\alpha_{j}$ and
$\alpha_{j-1}$ in (4.24).

###### Proof.

Let $\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$. Then by Equation 4.1, we know
$m(s_{\alpha})*Q$ is the open $B_{n-1}$-orbit in the variety
$\mathcal{Y}_{Q}:=P_{\alpha}kB_{n-2}/B_{n-2}\subset\tilde{Q}$. Under the
correspondence in Equation (3.16), we have

$\mathcal{Y}_{Q}^{op}=B_{n-2}k^{-1}P_{\alpha}/B_{n-1}.$

Since $m(s_{\alpha})*Q^{op}$ is the open $B_{n-2}$-orbit in
$B_{n-2}k^{-1}P_{\alpha}/B_{n-1}$, Equation (4.23) now follows from Equation
(4.2).

Now we consider the right monoid action. Let
$\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$ be compact for the open $K$-orbit
${\widetilde{Q}}_{K}=K\cdot\tilde{\mbox{${\mathfrak{b}}$}}$. Let
$\tilde{\mbox{${\mathfrak{b}}$}}={\rm Ad}(g)\mbox{${\mathfrak{b}}$}_{+}$. By
Equation (4.2), $m(s_{\alpha})*Q$ is the open $B_{n-1}$-orbit in the variety
$B_{n-1}\cdot{\rm Ad}(kg)(P_{\alpha}\cdot\mbox{${\mathfrak{b}}$}_{+})$. Since
$\alpha$ is compact for ${\widetilde{Q}}_{K}$, $gP_{\alpha}/B_{+}$ is a single
$K\cap{\rm Ad}(g)P_{\alpha}$-orbit, so that $gP_{\alpha}/B_{+}\cong(K\cap{\rm
Ad}(g)P_{\alpha})/(K\cap{\rm Ad}(g)B_{+}).$ Therefore,

(4.25) $B_{n-1}\cdot{\rm
Ad}(kg)(P_{\alpha}\cdot\mbox{${\mathfrak{b}}$}_{+})\cong B_{n-1}\cdot{\rm
Ad}(k)(({\rm Ad}(g)P_{\alpha}\cap K)\cdot\mbox{${\mathfrak{b}}$}_{n-2}).$

Using (3.16) and (4.25), it follows that

(4.26) $(B_{n-1}\cdot{\rm
Ad}(kg)(P_{\alpha}\cdot\mbox{${\mathfrak{b}}$}_{+}))^{op}=({\rm
Ad}(g)P_{\alpha}\cap K)\cdot{\rm Ad}(k^{-1})\mbox{${\mathfrak{b}}$}_{n-1}.$

We now claim that the group ${\rm Ad}(g)P_{\alpha}\cap K$ in (4.26) is the
simple parabolic subgroup of $K_{n-2}$ corresponding to the simple root
$\alpha_{j-1}$. Equation (4.24) now follows from (4.26). We verify the claim
in the case where $\mbox{${\mathfrak{g}}$}=\mathfrak{gl}(n)$. Suppose
$\alpha=\alpha_{j}$ is compact for ${\widetilde{Q}}_{K}=Q_{1,n}$. By
Proposition 4.9, it follows that $j\neq 1,n-1$. The image of the flag from
Equation (4.22) is the partial flag $\mathcal{P}\mathcal{F}_{z}$ denoted
$e_{n-1}+e_{n}\subset e_{1}\subset\dots\subset
e_{j-2}\subset\\{e_{j-1},e_{j}\\}\subset e_{j+1}\subset\dots\subset
e_{n-2}\subset e_{n}$. As in the proof of Theorem 3.8, we compute the
stabilizer $G_{n-1,z}$ of this partial flag in $G_{n-1}$. For $g$ in
$G_{n-1,z}$, since $g$ stabilizes the line spanned by $e_{n-1}+e_{n}$, we see
that $g\cdot e_{n-1}=e_{n-1}$. Recalling the notation setting $U_{k}$ equal to
the span of $e_{1},\dots,e_{k}$, we see as before that $g\cdot U_{k}\subset
U_{k}+\mbox{${\mathbb{C}}$}(e_{n-1}+e_{n})$ for $k=1,\dots,n-2$ and
$k\not=j-1$, and for $k=j-1$, $g\cdot U_{j-1}\subset
U_{j}+\mbox{${\mathbb{C}}$}(e_{n-1}+e_{n}).$ As in Theorem 3.8, since $g\in
G_{n-1}$ and hence stabilizes the subspace $U_{n-1}$, we see that $g\in
G_{n-2}$ and the claim follows since the subgroup of $G_{n-2}$ stablilizing
the partial flag
$(e_{1}\subset\dots\subset\\{e_{j-1},e_{j}\\}\subset\dots\subset e_{n-2})$ is
the simple parabolic subgroup of $G_{n-2}$ corresponding to $\alpha_{j-1}$.
The proofs of the claim in the orthogonal cases are similar, and are left to
the reader.

To see the assertion about root types, consider the variety
$\mathcal{Y}_{Q}=p^{-1}(K\cdot(Q,eB_{n-1}))$, where $p:G/B\times K/B_{n-1}\to
G/P_{\alpha}\times K/B_{n-1}$ in case $\alpha\in\Pi_{\mbox{${\mathfrak{g}}$}}$
and $p:G/B\times K/B_{n-1}\to G/B\times K/P_{n-1,\alpha}$ in case
$\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$. For
$\alpha\in\Pi_{\mbox{${\mathfrak{k}}$}}$, then $\alpha$ is complex stable for
$Q$ if and only if $\mathcal{Y}_{Q}$ consists of two double cosets with
$K\cdot(Q,eB_{n-1})$ connected and codimension 1 in the closure of the other
double coset. This is the same as the condition for $\alpha$ to be complex
stable for $Q^{op}$. The remaining cases are similar.

Q.E.D.

######  Remark 4.13.

As a consequence of the above results, we have completely understood the
monoid action for simple roots of ${\mathfrak{k}}$, and for simple roots of
${\mathfrak{g}}$ in the subset $S$ associated to the orbit $Q_{K}$ by (4.13),
and for simple roots $\alpha$ of ${\mathfrak{g}}$ not in $S$ when
$m(s_{\alpha})*Q_{K}=Q_{K}$. We will treat the remaining cases in a sequel to
this paper.

## References

  * [BH00] Michel Brion and Aloysius G. Helminck, _On orbit closures of symmetric subgroups in flag varieties_ , Canad. J. Math. 52 (2000), no. 2, 265–292.
  * [CE] Mark Colarusso and Sam Evens, _Eigenvalue coincidences and multiplicity free spherical pairs_ , arXiv:1410.3901, October 15, 2014, 36 pages.
  * [CE14] Mark Colarusso and Sam Evens, _The Gelfand-Zeitlin integrable system and $K$-orbits on the flag variety_, Symmetry: representation theory and its applications, Progr. Math., vol. 257, Birkhäuser/Springer, New York, 2014, pp. 85–119.
  * [CE19] Mark Colarusso and Sam Evens, _The Complex Orthogonal Gelfand–Zeitlin System_ , International Mathematics Research Notices 2021 (2019), no. 4, 2682–2722.
  * [Col85] David H. Collingwood, _Representations of rank one Lie groups_ , Research Notes in Mathematics, vol. 137, Pitman (Advanced Publishing Program), Boston, MA, 1985.
  * [GP18] Jacopo Gandini and Guido Pezzini, _Orbits of strongly solvable spherical subgroups on the flag variety_ , J. Algebraic Combin. 47 (2018), no. 3, 357–401.
  * [GW98] Roe Goodman and Nolan R. Wallach, _Representations and invariants of the classical groups_ , Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998.
  * [Har77] Robin Hartshorne, _Algebraic geometry_ , Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52.
  * [Has04] Takashi Hashimoto, _$B_{n-1}$ -orbits on the flag variety ${\rm GL}_{n}/B_{n}$_, Geom. Dedicata 105 (2004), 13–27.
  * [Hum75] James E. Humphreys, _Linear algebraic groups_ , Springer-Verlag, New York, 1975, Graduate Texts in Mathematics, No. 21.
  * [Kna02] Anthony W. Knapp, _Lie groups beyond an introduction_ , second ed., Progress in Mathematics, vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002\.
  * [RS90] R. W. Richardson and T. A. Springer, _The Bruhat order on symmetric varieties_ , Geom. Dedicata 35 (1990), no. 1-3, 389–436.
  * [Spr85] T. A. Springer, _Some results on algebraic groups with involutions_ , Algebraic groups and related topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 525–543.