# On homogeneous spaces for diagonal ind-groups

Lucas Fresse  Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine,
UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France<EMAIL_ADDRESS>and  Ivan Penkov  Constructor University, 28759 Bremen, Germany
<EMAIL_ADDRESS>
###### Abstract.

We study the homogeneous ind-spaces $\mathrm{GL}(\mathbf{s})/\mathbf{P}$ where
$\mathrm{GL}(\mathbf{s})$ is a strict diagonal ind-group defined by a
supernatural number $\mathbf{s}$ and $\mathbf{P}$ is a parabolic ind-subgroup
of $\mathrm{GL}(\mathbf{s})$. We construct an explicit exhaustion of
$\mathrm{GL}(\mathbf{s})/\mathbf{P}$ by finite-dimensional partial flag
varieties. As an application, we characterize all locally projective
$\mathrm{GL}(\infty)$-homogeneous spaces, and some direct products of such
spaces, which are $\mathrm{GL}(\mathbf{s})$-homogeneous for a fixed
$\mathbf{s}$. The very possibility for a $\mathrm{GL}(\infty)$-homogeneous
space to be $\mathrm{GL}(\mathbf{s})$-homogeneous for a strict diagonal ind-
group $\mathrm{GL}(\mathbf{s})$ arises from the fact that the automorphism
group of a $\mathrm{GL}(\infty)$-homogeneous space is much larger than
$\mathrm{GL}(\infty)$.

###### Key words and phrases:

Diagonal ind-group; generalized flag; embedding of flag varieties

###### 2010 Mathematics Subject Classification:

22E65; 14M15; 14M17

###### Contents

  1. 1 Introduction
  2. 2 The ind-group $\mathrm{GL}(\mathbf{s})$
  3. 3 On embeddings of flag varieties
  4. 4 A review of generalized flags
  5. 5 Embedding of flag varieties arising from diagonal embedding of groups
  6. 6 Ind-varieties of generalized flags as homogeneous spaces of $\mathrm{GL}(\mathbf{s})$
  7. 7 The case of direct products of ind-varieties of generalized flags

##  1\. Introduction

The ind-group
$\mathrm{GL}(\infty)=\lim\limits_{\to}\mathrm{GL}(n)=\bigcup_{n\geq
1}\mathrm{GL}(n)$ is a most natural direct limit algebraic group, and its
locally projective homogeneous spaces are quite well studied by now, see for
instance [3], [4], [7], [10]. A larger class of direct limit algebraic groups
are the so called diagonal ind-groups. A rather obvious such group, non-
isomorphic to $\mathrm{GL}(\infty)$, is the ind-group
$\mathrm{GL}(2^{\infty})=\lim\limits_{\to}\mathrm{GL}(2^{n})$ where
$\mathrm{GL}(2^{n-1})$ is embedded into $\mathrm{GL}(2^{n})$ via the map

$x\mapsto\begin{pmatrix}x&0\\\ 0&x\end{pmatrix}.$

A general definition of a diagonal Lie algebra has been given by A. Baranov
and A. Zhilinskii in [1], and this definition carries over in a
straightforward way to classical Lie groups, producing the class of diagonal
Lie groups.

Locally projective homogeneous ind-spaces of diagonal ind-groups have been
studied much less extensively than those of $\mathrm{GL}(\infty)$, see [4] and
[2]. In this paper, we undertake such a study for a class of diagonal ind-
groups which we call strict diagonal ind-groups of type A. These ind-groups
are characterized by supernatural numbers $\mathbf{s}$, and are denoted
$\mathrm{GL}(\mathbf{s})$. We consider reasonably general parabolic subgroups
$\mathbf{P}\subset\mathrm{GL}(\mathbf{s})$ and describe the homogeneous ind-
space $\mathrm{GL}(\mathbf{s})/\mathbf{P}$ as direct limits of embeddings

$G_{n-1}/P_{n-1}\to G_{n}/P_{n}$

of usual ind-varieties. Our main result is an explicit formula for the so
arising embeddings, and this formula is an analogue of the formula for
standard extensions introduced in [10] (and used in a particular case in [3]).

The class of locally projective homogeneous ind-spaces of strict (and of
general) diagonal ind-groups will require further detailed studies. In the
current paper we restrict ourselves to the following application of the above
explicit formula: we determine which locally projective homogeneous ind-spaces
of $\mathrm{GL}(\infty)$, i.e., ind-varieties of generalized flags [3], are
also $\mathrm{GL}(\mathbf{s})$-homogeneous for a given infinite supernatural
number $\mathbf{s}$. Furthermore, we also characterize explicitly direct
products of ind-varieties of generalized flags which are
$\mathrm{GL}(\mathbf{s})$-homogeneous.

The very possibility of an ind-variety of generalized flags being a
homogeneous space for $\mathrm{GL}(\mathbf{s})$, where $\mathbf{s}$ is an
infinite supernatural number, is an interesting phenomenon, and can be seen as
one possible motivation for our studies of
$\mathrm{GL}(\mathbf{s})$-homogeneous ind-spaces. Indeed, recall the following
fact for a finite-dimensional algebraic group. If $G$ is a centerless simple
algebraic group of classical type and rank at least four and $P$ is a
parabolic subgroup, a well-known result of A. Onishchik [9] implies that the
connected component of unity of the automorphism group of the homogeneous
space $G/P$ coincides with $G$, except in two special cases when $G/P$ is a
projective space and $G$ is a symplectic group, and when $G/P$ is a maximal
orthogonal isotropic grassmannian and $G$ is an orthogonal group of type B.
Consequently, unless $G/P$ is a projective space or a maximal isotropic
grassmannian, $G/P$ cannot be a homogeneous $G^{\prime}$-space for a
centerless algebraic group $G^{\prime}\not\cong G$.

The explanation of why the situation is very different if one replaces $G$ by
the ind-group $\mathrm{GL}(\infty)$, is that, as shown in [7], the
automorphism group of an ind-variety of generalized flags is much larger than
$\mathrm{GL}(\infty)$. In this way, our results provide embeddings of
$\mathrm{GL}(\mathbf{s})$ into such automorphism groups, with the property
that the action of $\mathrm{GL}(\mathbf{s})$ on the respective ind-variety of
generalized flags is transitive. As a corollary we obtain that a “generic”
ind-variety of generalized flags is $\mathrm{GL}(\mathbf{s})$-homogeneous also
for any ind-group $\mathrm{GL}(\mathbf{s})$. This statement is in some sense
opposite to the classical statement in the finite-dimensional case.

The paper is organized as follows. Sections 2, 3, 4 are devoted to
preliminaries. We start by introducing the ind-groups
$\mathrm{GL}(\mathbf{s})$ where $\mathbf{s}$ is a supernatural number. We then
discuss Cartan, Borel, and parabolic ind-subgroups of
$\mathrm{GL}(\mathbf{s})$. In Section 3 we review the notions of linear
embedding of flag varieties and standard extension of flag varieties, and in
Section 4 we recall the necessary results on ind-varieties of generalized
flags.

In Section 5 we prove our explicit formula for embeddings of partial flag
varieties $\mathrm{GL}(n)/Q\hookrightarrow\mathrm{GL}(dn)/P$ induced by pure
diagonal embeddings $\mathrm{GL}(n)\hookrightarrow\mathrm{GL}(dn)$. In Section
6 we use this formula to describe all $\mathrm{GL}(\mathbf{s})$-homogeneous
ind-varieties of generalized flags. Finally, in Section 7 we characterize
direct products of ind-varieties of generalized flags, which are
$\mathrm{GL}(\mathbf{s})$-homogeneous.

### Acknowledgement

The work of I. P. was supported in part by DFG Grant PE 980/8-1.

##  2\. The ind-group $\mathrm{GL}(\mathbf{s})$

###  2.1. Direct systems associated to a supernatural number

Throughout this paper we consider a fixed supernatural number $\mathbf{s}$, in
other words

$\mathbf{s}=\prod_{p\in\mathcal{P}}p^{\alpha_{p}}$

where $\mathcal{P}$ is a (possibly infinite) set of prime numbers and
$\alpha_{p}$ is either a positive integer or $\infty$. Moreover, we suppose
that $\mathbf{s}$ is infinite, hence at least one of the exponents
$\alpha_{p}$ is infinite or the set $\mathcal{P}$ is infinite. By
$\mathcal{D}(\mathbf{s})$ we denote the set of finite divisors of
$\mathbf{s}$.

Let $\mathcal{A}$ be a direct system of sets with injective maps. We say that
$\mathcal{A}$ is associated to the supernatural number $\mathbf{s}$ if the
sets in $\mathcal{A}$

$A(s),\quad s\in\mathcal{D}(\mathbf{s})$

are parametrized by the finite divisors of $\mathbf{s}$, and the injective
maps

$\delta_{s,s^{\prime}}:A(s)\hookrightarrow A(s^{\prime})$

correspond to pairs of divisors $s,s^{\prime}\in\mathcal{D}(\mathbf{s})$ such
that $s|s^{\prime}$. Then, if $L(\mathcal{A})=\lim\limits_{\to}A(s)$, the
resulting map

$\delta_{s}:A(s)\hookrightarrow L(A)$

is injective for every $s\in\mathcal{D}(\mathbf{s})$.

######  Definition 2.1.

We call exhaustion of $\mathbf{s}$ any sequence $\\{s_{n}\\}_{n\geq 1}$ of
integers such that

  * •

$s_{n}\in\mathcal{D}(\mathbf{s})$ for all $n$,

  * •

$s_{n}$ divides $s_{n+1}$ for all $n$,

  * •

any $s\in\mathcal{D}(\mathbf{s})$ is a multiple of $s_{n}$ for some $n$.
$\blacksquare$

######  Lemma 2.2.

Let $\\{s_{n}\\}_{n\geq 1}$ be an exhaustion of $\mathbf{s}$. Then
$L(\mathcal{A})$ coincides with the limit of the inductive system formed by
the sets $A(s_{n})$ and the maps
$\delta_{n}=\delta_{s_{n},s_{n+1}}:A(s_{n})\hookrightarrow A(s_{n+1})$.

###### Proof.

Straightforward. ∎

According to the lemma, the limit $L(\mathcal{A})$ can be described in terms
of an exhaustion

$L(\mathcal{A})=\bigcup_{n}A(s_{n}).$

  * •

In the case where $A(s)$ are vector spaces and the maps
$\delta_{s,s^{\prime}}$ are linear, then $L(\mathcal{A})$ is the direct limit
in the category of vector spaces.

  * •

In the case where $A(s)$ are algebraic varieties and the maps
$\delta_{s,s^{\prime}}$ are closed embeddings, the limit $L(\mathcal{A})$ is
an ind-variety as defined in [11] and [8].

  * •

In the case where $A(s)$ are algebraic groups and the maps
$\delta_{s,s^{\prime}}$ are group homomorphisms, the limit is both an ind-
variety and a group. It is in particular an ind-group111An ind-group is an
ind-variety with a group structure such that the multiplication $(x,y)\mapsto
xy$ and the inversion $x\mapsto x^{-1}$ are morphisms of ind-varieties. .

###  2.2. Definition of the groups $\mathrm{GL}(\mathbf{s})$ and
$\mathrm{SL}(\mathbf{s})$

Whenever $s,s^{\prime}$ are two positive integers such that $s$ divides
$s^{\prime}$, we have a diagonal embedding

$\delta_{s,s^{\prime}}:\mathrm{GL}(s)\to\mathrm{GL}(s^{\prime}),\
x\mapsto\mathrm{diag}(\underbrace{x,\ldots,x}_{\mbox{\scriptsize$\frac{s^{\prime}}{s}$
blocks}}).$

We refer to the embeddings $\delta_{s,s^{\prime}}$ as strict diagonal
embeddings. A more general definition of diagonal embeddings is given, at the
Lie algebra level, in [1].

The groups $\mathrm{GL}(s)$ (for $s\in\mathcal{D}(\mathbf{s})$) and the maps
$\delta_{s,s^{\prime}}$ (for all pairs of integers
$s,s^{\prime}\in\mathcal{D}(\mathbf{s})$ such that $s$ divides $s^{\prime}$)
form a direct system. By definition, the ind-group $\mathrm{GL}(\mathbf{s})$
is the limit of this direct system.

The group $\mathrm{GL}(\mathbf{s})$ can be viewed as the group of infinite
$\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$-matrices consisting of one diagonal
block of size equal to any (finite) divisor $s$ of $\mathbf{s}$, repeated
infinitely many times along the diagonal:

(2.1) $\displaystyle\mathrm{GL}(\mathbf{s})$ $\displaystyle=$
$\displaystyle\left\\{\begin{pmatrix}x&0&\cdots\\\ 0&x&\ddots\\\
\vdots&\ddots&\ddots\\\ \end{pmatrix}:x\in\mathrm{GL}(s),\
s\in\mathcal{D}(\mathbf{s})\right\\}.$

Similarly, we define $\mathrm{SL}(\mathbf{s})$ as the limit of the direct
system formed by the groups $\mathrm{SL}(s)$ and the same maps
$\delta_{s,s^{\prime}}$. In fact, $\mathrm{SL}(\mathbf{s})$ is the derived
group of $\mathrm{GL}(\mathbf{s})$. By $\mathfrak{gl}(\mathbf{s})$ and
$\mathfrak{sl}(\mathbf{s})$, we denote the Lie algebras of
$\mathrm{GL}(\mathbf{s})$ and $\mathrm{SL}(\mathbf{s})$, respectively. Thus
$\mathfrak{sl}(\mathbf{s})=[\mathfrak{gl}(\mathbf{s}),\mathfrak{gl}(\mathbf{s})]$.

######  Remark 2.3.

Lemma 2.2 shows that the group $\mathrm{GL}(\mathbf{s})$ can be obtained
through any exhaustion

$\mathrm{GL}(\mathbf{s})=\bigcup_{n}\mathrm{GL}(s_{n})$

where $\\{s_{n}\\}_{n\geq 1}$ is an exhaustion of $\mathbf{s}$ (see Definition
2.1). However, the ind-group $\mathbf{GL}(\mathbf{s})$ has various other
exhaustions. If we set

$\mathbf{K}(n):=\underbrace{\mathrm{GL}(s_{n})\times\cdots\times\mathrm{GL}(s_{n})}_{\mbox{\scriptsize$\frac{s_{n+1}}{s_{n}}$
factors}}$

and

$\psi_{n}:\mathbf{K}(n)\to\mathbf{K}(n+1),\
(x_{1},\ldots,x_{d_{n}})\mapsto(\underbrace{\mathop{diag}\nolimits(x_{1},\ldots,x_{d_{n}}),\ldots,\mathop{diag}\nolimits(x_{1},\ldots,x_{d_{n}})}_{\frac{s_{n+2}}{s_{n+1}}\mbox{\scriptsize\
terms}}),$

then the direct system
$\\{\mathbf{K}(n)\stackrel{{\scriptstyle\psi_{n}}}{{\to}}\mathbf{K}(n+1)\\}$
intertwines in a natural way with the direct system
$\\{\mathrm{GL}(s_{n})\stackrel{{\scriptstyle\delta_{s_{n},s_{n+1}}}}{{\longrightarrow}}\mathrm{GL}(s_{n+1})\\}$
considered above. This yields an equality

$\mathrm{GL}(\mathbf{s})=\lim_{\to}\mathrm{GL}(s_{n})=\lim_{\to}\mathbf{K}(n).$

$\blacksquare$

We say that two exhaustions
$\mathbf{G}=\bigcup_{n}G_{n}=\bigcup_{n}G^{\prime}_{n}$ of a given ind-group
are equivalent if there are $n_{0}\geq 1$ and a commutative diagram

$\textstyle{G_{n_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{G_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\cdots}$$\textstyle{G^{\prime}_{n_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\prime}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\prime}_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

such that the vertical arrows are isomorphisms of algebraic groups and the
horizontal arrows are the embeddings of the exhaustions.

######  Lemma 2.4.

(a) Any exhaustion of $\mathrm{SL}(\mathbf{s})$ by almost simple, simply
connected algebraic groups is equivalent to
$\\{\mathrm{SL}(s_{n}),\delta_{s_{n},s_{n+1}}\\}_{n\geq 1}$ for an exhaustion
$\\{s_{n}\\}_{n\geq 1}$ of $\mathbf{s}$.

(b) Any exhaustion of $\mathrm{GL}(\mathbf{s})$ by classical groups (i.e. by
groups of the form $\mathrm{GL}(n)$, $\mathrm{SL}(n)$, $\mathrm{SO}(n)$, or
$\mathrm{Sp}(n)$) is equivalent to
$\\{\mathrm{GL}(s_{n}),\delta_{s_{n},s_{n+1}}\\}_{n\geq 1}$ for an exhaustion
$\\{s_{n}\\}_{n\geq 1}$ of $\mathbf{s}$.

###### Proof.

(a) It suffices to prove the claim at the level of Lie algebras. Let
$\mathfrak{sl}(\mathbf{s})=\bigcup_{n}\mathfrak{g}_{n}$ be an exhaustion by
simple Lie algebras, hence of classical type for $n$ large enough. There is a
subsequence $\\{\mathfrak{g}_{k_{n}}\\}_{n\geq 1}$ and an exhaustion
$\\{s_{n}\\}_{n\geq 1}$ of $\mathbf{s}$ such that we have a commutative
diagram of embeddings

$\textstyle{\mathfrak{sl}(s_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{s_{n},s_{n+1}}}$$\scriptstyle{\eta_{n}}$$\textstyle{\mathfrak{sl}(s_{n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{n+1}}$$\textstyle{\cdots}$$\textstyle{\mathfrak{g}_{k_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{n}}$$\textstyle{\mathfrak{g}_{k_{n+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

By [1, Lemma 2.7], the embeddings $\eta_{n}$ and $\xi_{n}$ are diagonal, in
the sense that there is an isomorphism of $\mathfrak{sl}(s_{n})$-modules

$W_{n}\cong V_{n}^{\oplus t}\oplus{V_{n}^{*}}^{\oplus
r}\oplus\mathbb{C}^{\oplus s}$

and an isomorphism of $\mathfrak{g}_{k_{n}}$-modules

$V_{n+1}\cong W_{n}^{\oplus t^{\prime}}\oplus{W_{n}^{*}}^{\oplus
r^{\prime}}\oplus\mathbb{C}^{\oplus s^{\prime}}$

for some triples of nonnegative integers $(t,r,s)$ and
$(t^{\prime},r^{\prime},s^{\prime})$, where $V_{n}$ and $W_{n}$ denote the
natural representations of $\mathfrak{sl}(s_{n})$ and $\mathfrak{g}_{k_{n}}$,
and $\mathbb{C}$ is a trivial representation. Also since
$\delta_{s_{n},s_{n+1}}$ is strict diagonal, we have an isomorphism of
$\mathfrak{sl}(s_{n})$-modules

(2.2) $V_{n+1}\cong\underbrace{V_{n}\oplus\ldots\oplus
V_{n}}_{\mbox{\scriptsize$\frac{s_{n+1}}{s_{n}}$ copies}}.$

Arguing by contradiction, assume that $\mathfrak{g}_{k_{n}}$ is not of type A.
Then [1, Proposition 2.3] implies that $t=r$. Moreover,
$t^{\prime}+r^{\prime}>0$ since otherwise $V_{n+1}$ would be a trivial
representation of $\mathfrak{sl}(s_{n})$. Altogether this implies that
$V_{n}^{*}$ is isomorphic to a direct summand of $V_{n+1}$ considered as an
$\mathfrak{sl}(s_{n})$-module, which is impossible in view of (2.2). We
conclude that $\mathfrak{g}_{k_{n}}$ is of type A for all $n$.

Moreover, from (2.2), we obtain $s=s^{\prime}=1$ and either $r=r^{\prime}=0$
or $t=t^{\prime}=0$. Up to replacing
$\mathfrak{g}_{k_{n}}=\mathfrak{sl}(W_{n})$ by $\mathfrak{sl}(W_{n}^{*})$, we
can assume that $r=r^{\prime}=0$, and so
$\mathfrak{g}_{k_{n}}\cong\mathfrak{sl}(s^{\prime}_{k_{n}})$ for some integer
such that $s_{n}|s^{\prime}_{k_{n}}$, $s^{\prime}_{k_{n}}|s_{n+1}$, and the
embedding $\mathfrak{g}_{k_{n}}\hookrightarrow\mathfrak{g}_{k_{n+1}}$ is
induced by $\delta_{s^{\prime}_{k_{n}},s^{\prime}_{k_{n+1}}}$.

If $k:=k_{n}+1<k_{n+1}$, we get a commutative diagram

$\textstyle{\mathfrak{sl}(s^{\prime}_{k_{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\delta_{s^{\prime}_{k_{n}},s^{\prime}_{k_{n+1}}}}$$\textstyle{\mathfrak{sl}(s^{\prime}_{k_{n+1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathfrak{g}_{k_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{g}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{g}_{k_{n+1}}}$

where the horizontal arrows are embeddings. Relying as above on [1,
Proposition 2.3], we get that $\mathfrak{g}_{k}$ is necessarily of type A, and
up to replacing $\mathfrak{g}_{k}=\mathfrak{sl}(W)$ by $\mathfrak{sl}(W^{*})$,
we can assume that $\mathfrak{g}_{k}\cong\mathfrak{sl}(s^{\prime}_{k})$ for
some $s^{\prime}_{k}$ with $s^{\prime}_{k_{n}}|s^{\prime}_{k}$,
$s^{\prime}_{k}|s^{\prime}_{k_{n+1}}$, and that the embeddings
$\mathfrak{g}_{k_{n}}\hookrightarrow\mathfrak{g}_{k}\hookrightarrow\mathfrak{g}_{k_{n+1}}$
are induced by $\delta_{s^{\prime}_{k_{n}},s^{\prime}_{k}}$ and
$\delta_{s^{\prime}_{k},s^{\prime}_{k_{n+1}}}$.

By iterating the reasoning, we obtain an exhaustion
$\\{s^{\prime}_{n}\\}_{n\geq 1}$ of $\mathbf{s}$ such that the exhaustions
$\mathfrak{sl}(\mathbf{s})=\bigcup_{n}\mathfrak{g}_{n}$ and
$\mathfrak{sl}(\mathbf{s})=\bigcup_{n}\mathfrak{sl}(s^{\prime}_{n})$ are
equivalent. This shows (a).

(b) From (a) it follows that for every $n$, the derived group $(G_{n},G_{n})$
is isomorphic to $\mathrm{SL}(s_{n})$ and, after identifying $(G_{n},G_{n})$
with $\mathrm{SL}(s_{n})$ and $(G_{n+1},G_{n+1})$ with $\mathrm{SL}(s_{n+1})$,
the map $(G_{n},G_{n})\hookrightarrow(G_{n+1},G_{n+1})$ becomes the
restriction of $\delta_{s_{n},s_{n+1}}$. This implies that $G_{n}$ is either
isomorphic to $\mathrm{SL}(s_{n})$ or to $\mathrm{GL}(s_{n})$. For $n\geq 1$
large enough, $G_{n}$ has to contain the center $Z(\mathrm{GL}(\mathbf{s}))$,
which is isomorphic to $\mathbb{C}^{*}$. Since the connected component of the
center of $\mathrm{SL}(s_{n})$ is trivial, this forces
$G_{n}\cong\mathrm{GL}(s_{n})$. Moreover, since $G_{n}=Z(G_{n})(G_{n},G_{n})$
and the embedding $G_{n}\hookrightarrow G_{n+1}$ maps
$Z(G_{n})=Z(\mathrm{GL}(\mathbf{s}))$ into $Z(G_{n+1})$, we deduce that this
embedding $G_{n}\hookrightarrow G_{n+1}$ coincides with
$\delta_{s_{n},s_{n+1}}:\mathrm{GL}(s_{n})\hookrightarrow\mathrm{GL}(s_{n+1})$
after suitably identifying $G_{n}$ with $\mathrm{GL}(s_{n})$ and $G_{n+1}$
with $\mathrm{GL}(s_{n+1})$.. ∎

The following statement is a corollary of the classification of general
diagonal Lie algebras [1]. We give a proof for the sake of completeness.

######  Proposition 2.5.

(a) If $\mathbf{s}$ and $\mathbf{s}^{\prime}$ are two different infinite
supernatural numbers, then the ind-groups $\mathrm{GL}(\mathbf{s})$ and
$\mathrm{GL}(\mathbf{s}^{\prime})$ (resp. $\mathrm{SL}(\mathbf{s})$ and
$\mathrm{SL}(\mathbf{s}^{\prime})$) are not isomorphic.

(b) If $\mathbf{s}$ is an infinite supernatural number, then
$\mathrm{GL}(\mathbf{s})$ is not isomorphic to $\mathrm{GL}(\infty)$, and
$\mathrm{SL}(\mathbf{s})$ is not isomorphic to $\mathrm{SL}(\infty)$.

###### Proof.

(a) Since $\mathrm{SL}(\cdot)$ is the derived group of $\mathrm{GL}(\cdot)$,
it suffices to establish the claim concerning $\mathrm{SL}(\mathbf{s})$ and
$\mathrm{SL}(\mathbf{s}^{\prime})$. Assume there is an isomorphism of ind-
groups $\varphi:\mathrm{SL}(\mathbf{s}^{\prime})\to\mathrm{SL}(\mathbf{s})$.
Then any exhaustion $\\{s^{\prime}_{n}\\}$ of $\mathbf{s}^{\prime}$ yields an
exhaustion
$\mathrm{SL}(\mathbf{s})=\bigcup_{n}\varphi(\mathrm{SL}(s^{\prime}_{n}))$ of
the group $\mathrm{SL}(\mathbf{s})$, and Lemma 2.4 implies
$\mathbf{s}=\mathbf{s}^{\prime}$, a contradiction.

(b) By definition, $\mathrm{SL}(\infty)$ has an exhaustion by the groups
$\mathrm{SL}(n)$ ($n\geq 1$) via the standard embeddings
$\mathrm{SL}(n)\to\mathrm{SL}(n+1)$, $x\mapsto\begin{pmatrix}x&0\\\
0&1\end{pmatrix}$. Clearly this exhaustion is not equivalent to
$\\{\mathrm{SL}(s_{n}),\delta_{s_{n},s_{n+1}}\\}_{n\geq 1}$ for any exhaustion
$\\{s_{n}\\}_{n\geq 1}$ of $\mathbf{s}$. Therefore, the ind-groups
$\mathrm{SL}(\mathbf{s})$ and $\mathrm{SL}(\infty)$ are not isomorphic by
Lemma 2.4 (a). The same argument shows that $\mathrm{GL}(\mathbf{s})$ and
$\mathrm{GL}(\infty)$ are not isomorphic. ∎

###  2.3. Parabolic and Borel subgroups

An ind-subgroup $\mathbf{H}\subset\mathrm{GL}(\mathbf{s})$ is said to be a
(locally splitting) Cartan subgroup if there is an exhaustion
$\mathrm{GL}(\mathbf{s})=\bigcup_{n}G_{n}$ by classical groups such that
$G_{n}\cap\mathbf{H}$ is a Cartan subgroup of $G_{n}$ for all $n$. For
instance, the subgroup of invertible periodic diagonal matrices in the
realization (2.1) is a Cartan subgroup of $\mathrm{GL}(\mathbf{s})$.

If $\mathbf{P}$ is an ind-subgroup of $\mathrm{GL}(\mathbf{s})$, then the
quotient $\mathrm{GL}(\mathbf{s})/\mathbf{P}$ is an ind-variety obtained as
the direct limit of the quotients $\mathrm{GL}(s)/\mathbf{P}(s)$ for
$s\in\mathcal{D}(\mathbf{s})$.

For the purposes of this paper, we say that an ind-subgroup
$\mathbf{P}\subset\mathrm{GL}(\mathbf{s})$ is a parabolic subgroup if there
exists an exhaustion $\mathrm{GL}(\mathbf{s})=\bigcup_{n}G_{n}$ by classical
groups such that $G_{n}\cap\mathbf{P}$ is a parabolic subgroup of $G_{n}$ for
all $n$ (cf. [2]). This implies in particular that the ind-variety
$\mathrm{GL}(\mathbf{s})/\mathbf{P}$ is locally projective as it has an
exhaustion

$\mathrm{GL}(\mathbf{s})/\mathbf{P}=\bigcup_{n}G_{n}/(G_{n}\cap\mathbf{P})$

by projective varieties. If, in addition, the unipotent radical of
$G_{n}\cap\mathbf{P}$ is contained in the unipotent radical of
$G_{n+1}\cap\mathbf{P}$ for every $n$, then we say that $\mathbf{P}$ is a
strong parabolic subgroup.

An ind-subgroup $\mathbf{B}\subset\mathrm{GL}(\mathbf{s})$ is said to be a
Borel subgroup if it is locally solvable and parabolic. This means
equivalently that there is an exhaustion
$\mathrm{GL}(\mathbf{s})=\bigcup_{n}G_{n}$ as above for which
$G_{n}\cap\mathbf{B}$ is a Borel subgroup of $G_{n}$ for all $n$. Note that a
Borel subgroup is necessarily a strong parabolic subgroup.

######  Lemma 2.6.

A subgroup $\mathbf{G}^{\prime}$ of $\mathrm{GL}(\mathbf{s})$ is a Cartan
(respectively, parabolic or Borel) subgroup of $\mathbf{G}$ if and only if
there is an exhaustion $\\{s_{n}\\}_{n\geq 1}$ of $\mathbf{s}$ such that for
every $n$ the intersection $\mathbf{G}^{\prime}\cap\mathrm{GL}(s_{n})$ is a
Cartan (respectively, parabolic or Borel) subgroup of $\mathrm{GL}(s_{n})$.

###### Proof.

This follows from Lemma 2.4. ∎

The following example shows that for a given parabolic subgroup
$\mathbf{P}\subset\mathrm{GL}(\mathbf{s})$, the property that the group
$G_{n}\cap\mathbf{P}$ is a parabolic subgroup of $G_{n}$ may no longer hold
for a refinement of the exhaustion used to define $\mathbf{P}$.

######  Example 2.7.

Let $\mathbf{s}=2^{\infty}$, $s_{n}=2^{2n-2}$, and $s^{\prime}_{n}=2^{n-1}$.
Then both $\\{s_{n}\\}_{n\geq 1}$ and $\\{s^{\prime}_{n}\\}_{n\geq 1}$ are
exhaustions of $\mathbf{s}$, and $\\{s^{\prime}_{n}\\}_{n\geq 1}$ is a
refinement of $\\{s_{n}\\}_{n\geq 1}$. Let $H_{n}\subset\mathrm{GL}(s_{n})$ be
the subgroup of diagonal matrices. We define a Borel subgroup
$B_{n}\subset\mathrm{GL}(s_{n})$ that contains $H_{n}$, by induction in the
following way: $B_{1}:=\mathrm{GL}(1)$, and

$B_{n+1}:=\left(\begin{array}[]{cccc}B_{n}&*&*&*\\\ 0&B_{n}&*&*\\\
0&0&B_{n}&0\\\ 0&0&*&B_{n}\end{array}\right)$

for $n\geq 2$, where all the blocks are square matrices of size $s_{n}$. Then
$B_{n+1}\cap\mathrm{GL}(s_{n})=B_{n}$ for all $n$, which implies that
$\mathbf{B}=\bigcup_{n\geq 1}B_{n}$ is a well-defined Borel subgroup of
$\mathrm{GL}(\mathbf{s})$ arising from the exhaustion $\\{s_{n}\\}_{n\geq 1}$
of $\mathbf{s}$. However, for all $n$,

$\mathbf{B}\cap\mathrm{GL}(s^{\prime}_{2n})=\begin{pmatrix}B_{n}&0\\\
0&B_{n}\end{pmatrix}$

is not a Borel subgroup (nor a parabolic subgroup) of
$\mathrm{GL}(s^{\prime}_{2n})$. $\blacksquare$

##  3\. On embeddings of flag varieties

In this section we review some preliminaries on finite-dimensional (partial)
flag varieties. In particular, we recall the notions of linear embedding and
standard extension introduced in [10].

###  3.1. Grassmannians and (partial) flag varieties

Let $V$ be a finite-dimensional vector space. For an integer $0\leq p\leq\dim
V$, we denote by $\mathrm{Gr}(p;V)$ the grassmannian of $p$-dimensional
subspaces in $V$. This grassmannian can be realized as a projective variety by
the Plücker embedding
$\mathrm{Gr}(p;V)\hookrightarrow\mathbb{P}(\bigwedge^{p}V)$. Moreover, the
Picard group $\mathrm{Pic}(\mathrm{Gr}(p;V))$ is isomorphic to $\mathbb{Z}$
with generator $\mathcal{O}_{\mathrm{Gr}(p;V)}(1)$, the pull-back of the line
bundle $\mathcal{O}(1)$ on $\mathbb{P}(\bigwedge^{p}V)$.

For a sequence of integers $0<p_{1}<\ldots<p_{k-1}<p_{k}<\dim V$, we denote by
$\mathrm{Fl}(p_{1},\ldots,p_{k};V)$ the variety of (partial) flags

$\mathrm{Fl}(p_{1},\ldots,p_{k};V)=\\{(V_{1},\ldots,V_{k})\in\mathrm{Gr}(p_{1};V)\times\cdots\times\mathrm{Gr}(p_{k};V):V_{1}\subset\ldots\subset
V_{k}\\}.$

We have

$\mathrm{Pic}(\mathrm{Fl}(p_{1},\ldots,p_{k};V))\cong\mathbb{Z}^{k}.$

If we let $L_{i}$ be the pull-back

$L_{i}=\mathrm{proj}_{i}^{*}\mathcal{O}_{\mathrm{Gr}(p_{i};V)}(1)$

along the projection

$\mathrm{proj}_{i}:\mathrm{Fl}(p_{1},\ldots,p_{k};V)\to\mathrm{Gr}(p_{i};V)$

(for $i=1,\ldots,k$), then $[L_{1}],\ldots,[L_{k}]$ is a set of generators of
the Picard group.

By embedding of flag varieties we mean a closed immersion

$\varphi:X=\mathrm{Fl}(p_{1},\ldots,p_{k};V)\hookrightarrow
Y=\mathrm{Fl}(q_{1},\ldots,q_{\ell};W).$

If $\mathcal{F}=\\{F_{1},\ldots,F_{k}\\}\in X$ is a variable point, we set

$C_{i}(\varphi)=\bigcap_{\mathcal{F}\in X}\varphi(\mathcal{F})_{i}.$

Then $C_{1}(\varphi)\subset\ldots\subset C_{\ell}(\varphi)$ is a chain of
subspaces of $W$ with possible repetitions. We define the support of $\varphi$
to be the set of indices $i\in\\{1,\ldots,\ell\\}$ such that $\dim
C_{i}(\varphi)<q_{i}$.

###  3.2. Linear embedding

Let

$Q:=\mathrm{Gr}(q_{1};W_{1})\times\cdots\times\mathrm{Gr}(q_{\ell};W_{\ell})$

where $W_{1},\ldots,W_{\ell}$ is a sequence of vector spaces and $0<q_{j}<\dim
W_{j}$ for all $j$. Consider an embedding

$\psi:X=\mathrm{Fl}(p_{1},\ldots,p_{k};V)\to Q.$

We use the notation of the previous section for $X$. The Picard group of $Q$
is isomorphic to $\mathbb{Z}^{\ell}$, with generators associated to the line
bundles
$M_{j}=\mathrm{proj}_{j}^{*}\mathcal{O}_{\mathrm{Gr}(q_{j};W_{j})}(1)$.

######  Definition 3.1.

We say that the embedding $\psi$ is linear if we have

$[\psi^{*}M_{j}]=0\quad\mbox{or}\quad[\psi^{*}M_{j}]\in\\{[L_{1}],\ldots,[L_{k}]\\}$

for all $j\in\\{1,\ldots,\ell\\}$. $\blacksquare$

Let

$\varphi:X=\mathrm{Fl}(p_{1},\ldots,p_{k};V)\hookrightarrow
Y=\mathrm{Fl}(q_{1},\ldots,q_{\ell};W)$

be an embedding of flag varieties. The following definition is equivalent to
[10, Definition 2.1].

######  Definition 3.2.

The embedding $\varphi$ is said to be linear if the composed embedding
$\psi=\pi\circ\varphi$ is linear, where
$\pi:=\prod_{j=1}^{\ell}\mathrm{proj}_{j}:Y\to\prod_{j=1}^{\ell}\mathrm{Gr}(q_{j};W)$.
$\blacksquare$

###  3.3. Standard extension

######  Definition 3.3 ([10]).

(a) The embedding
$\varphi:\mathrm{Fl}(p_{1},\ldots,p_{k};V)\hookrightarrow\mathrm{Fl}(q_{1},\ldots,q_{\ell};W)$
is said to be a strict standard extension if there are

  * •

a decomposition of vector spaces $W=V^{\prime}\oplus Z$ with a linear
isomorphism $\varepsilon:V\stackrel{{\scriptstyle\sim}}{{\to}}V^{\prime}$,

  * •

a chain of subspaces $Z_{1}\subset\ldots\subset Z_{\ell}$ of $Z$ (with
possible repetitions),

  * •

a nondecreasing map $\kappa:\\{1,\ldots,\ell\\}\to\\{0,1,\ldots,k,k+1\\}$,

such that

(3.1)
$\varphi\big{(}\\{V_{1},\ldots,V_{k}\\}\big{)}=\\{\varepsilon(V_{\kappa(1)})+Z_{1},\ldots,\varepsilon(V_{\kappa(\ell-1)})+Z_{\ell-1},\varepsilon(V_{\kappa(\ell)})+Z_{\ell}\\}$

where $V_{0}:=0$ and $V_{k+1}:=V$.

(b) More generally, we say that $\varphi$ is a standard extension if $\varphi$
itself is a strict standard extension or its composition of $\varphi$ with the
duality map $\mathrm{Fl}(q_{1},\ldots,q_{\ell};W)\to\mathrm{Fl}(\dim
W-q_{1},\ldots,\dim W-q_{\ell};W^{*})$ is a strict standard extension.
$\blacksquare$

######  Remark 3.4.

Since the map $\varphi$ of (3.1) is an embedding of flag varieties, the
following conditions must hold: $1,\ldots,k$ have preimages by $\kappa$, and
the map $j\in\\{1,\ldots,\ell\\}\mapsto(\kappa(j),Z_{j})$ is injective and
does not contain $(0,0)$ nor $(k+1,Z)$ in its image. $\blacksquare$

Note that, if $\varphi$ is a strict standard extension, then
$C_{i}(\varphi)=Z_{i}$ for all $i\in\\{1,\ldots,\ell\\}$, and the support of
$\varphi$ is the interval $\kappa^{-1}([1,k])$.

Also, a composition of standard extensions is a standard extension.

######  Example 3.5.

Let $W=V\oplus Z$, where $\dim Z=d$. For $1\leq k_{0}\leq k+1$, we consider
the embeddings

$\displaystyle\varphi:\mathrm{Fl}(p_{1},\ldots,p_{k};V)$
$\displaystyle\hookrightarrow$
$\displaystyle\mathrm{Fl}(q_{1},\ldots,q_{k};W)$
$\displaystyle\\{V_{1},\ldots,V_{k}\\}$ $\displaystyle\mapsto$
$\displaystyle\\{V_{1},\ldots,V_{k_{0}-1},V_{k_{0}}+Z,\ldots,V_{k}+Z\\}$

and

$\displaystyle\bar{\varphi}:\mathrm{Fl}(p_{1},\ldots,p_{k};V)$
$\displaystyle\hookrightarrow$
$\displaystyle\mathrm{Fl}(\bar{q}_{1},\ldots,\bar{q}_{k+1};W)$
$\displaystyle\\{V_{1},\ldots,V_{k}\\}$ $\displaystyle\mapsto$
$\displaystyle\\{V_{1},\ldots,V_{k_{0}-1},V_{k_{0}-1}+Z,V_{k_{0}}+Z,\ldots,V_{k}+Z\\}$

where

$q_{i}=\left\\{\begin{array}[]{ll}p_{i}&\mbox{if $1\leq i<k_{0}$,}\\\
p_{i}+d&\mbox{if $k_{0}\leq i\leq
k$}\end{array}\right.\quad\mbox{and}\quad\bar{q}_{i}=\left\\{\begin{array}[]{ll}p_{i}&\mbox{if
$1\leq i<k_{0}$,}\\\ p_{i-1}+d&\mbox{if $k_{0}\leq i\leq
k+1$.}\end{array}\right.$

Here, we still use the convention that $V_{0}:=0$ and $V_{k+1}:=V$, and we set
accordingly $p_{0}:=0$ and $p_{k+1}:=\dim V$. Then $\varphi$ and
$\bar{\varphi}$ are strict standard extensions, associated with the respective
chains of subspaces

$\underbrace{0\subset\ldots\subset 0}_{\mbox{\scriptsize$k_{0}-1$
times}}\subset\underbrace{Z\subset\ldots\subset
Z}_{\mbox{\scriptsize$k+1-k_{0}$
times}}\quad\mbox{and}\quad\underbrace{0\subset\ldots\subset
0}_{\mbox{\scriptsize$k_{0}-1$ times}}\subset\underbrace{Z\subset\ldots\subset
Z}_{\mbox{\scriptsize$k+2-k_{0}$ times}}$

and respective maps $\kappa$ and $\bar{\kappa}$, where $\kappa(i)=i$ for all
$i$, $\bar{\kappa}(i)=i$ for $i\leq k_{0}-1$, $\bar{\kappa}(i)=i-1$ for $i\geq
k_{0}$. $\blacksquare$

######  Remark 3.6.

Every strict standard extension is the composition of, possibly several, maps
$\varphi$ and $\bar{\varphi}$ as in Example 3.5. $\blacksquare$

##  4\. A review of generalized flags

###  4.1. Generalized flags

Let $V$ be an infinite-dimensional vector space of countable dimension and let
$E=\\{e_{1},e_{2},\ldots\\}$ be a basis of $V$. By $\langle S\rangle$, we
denote the span of vectors in a subset $S\subset V$. Following [3], we call
generalized flag a collection $\mathcal{F}$ of subspaces of $V$ that satisfies
the following conditions:

  * •

$\mathcal{F}$ is totally ordered by inclusion;

  * •

every subspace $F\in\mathcal{F}$ has an immediate predecessor or an immediate
successor in $\mathcal{F}$;

  * •

$V\setminus\\{0\\}=\bigcup_{(F^{\prime},F^{\prime\prime})}(F^{\prime\prime}\setminus
F^{\prime})$, where the union is over pairs of consecutive subspaces in
$\mathcal{F}$.

Moreover, a generalized flag $\mathcal{F}$ is said to be $E$-compatible if
every subspace $F\in\mathcal{F}$ is spanned by elements of $E$. An
$E$-compatible generalized flag $\mathcal{F}$ can be encoded by a (not order
preserving) surjective map $\sigma:\mathbb{Z}_{>0}\to A$ onto a totally
ordered set $(A,\leq)$ such that
$\mathcal{F}=\\{F^{\prime}_{a},F^{\prime\prime}_{a}\\}_{a\in A}$ where
$F^{\prime}_{a}=\langle e_{k}:\sigma(k)<a\rangle$ and
$F^{\prime\prime}_{a}=\langle e_{k}:\sigma(j)\leq a\rangle$. More generally, a
generalized flag $\mathcal{F}$ is said to be weakly $E$-compatible if it is
$E^{\prime}$-compatible for some basis $E^{\prime}$ of $V$ differing from $E$
in finitely many vectors.

Let

$\mathrm{GL}(E)=\\{g\in\mathrm{GL}(V):g(e_{k})=e_{k}\ \mbox{for all but
finitely many $k$}\\}.$

Then $\mathrm{GL}(E)$ is an ind-group, isomorphic to the finitary classical
ind-group $\mathrm{GL}(\infty)$. The group $\mathrm{GL}(E)$ acts on the set of
all weakly $E$-compatible generalized flags. Furthermore, it is established in
[3] that weakly $E$-compatible generalized flags $\mathcal{F}$ of $V$ are in
one-to-one correspondence with splitting parabolic subgroups
$\mathbf{P}\subset\mathrm{GL}(E)$. More precisely, the map

$\mathcal{F}\mapsto\mathbf{P}=\mathrm{Stab}_{\mathrm{GL}(E)}(\mathcal{F})$

is a bijection between these two sets.

By a natural representation of $\mathrm{GL}(\mathbf{s})$ we mean a direct
limit of natural representations of $\mathrm{GL}(s)$ for
$s\in\mathcal{D}(\mathbf{s})$. Two natural representations do not have to be
isomorphic; see [5].

Assume now that $V$ is a natural representation for $\mathrm{GL}(\mathbf{s})$,
$\mathbf{H}\subset\mathrm{GL}(\mathbf{s})$ is a Cartan subgroup such that
there is a basis $E$ of $V$ consisting of eigenvectors of $\mathbf{H}$. The
group $\mathrm{GL}(\mathbf{s})$ acts in a natural way on the generalized flags
in $V$, and a generalized flag is $E$-compatible if and only if it is
$\mathbf{H}$-stable. However, generalized flags are less suited for describing
parabolic subgroups of $\mathrm{GL}(\mathbf{s})$ than for describing parabolic
subgroups of $\mathrm{GL}(\infty)\cong\mathrm{GL}(E)$, since the stabilizer of
a generalized flag in $\mathrm{GL}(\mathbf{s})$ is not always a parabolic
subgroup. Moreover, there are parabolic subgroups of $\mathrm{GL}(\mathbf{s})$
which cannot be realized as stabilizers of generalized flags in a prescribed
natural representation. These observations are illustrated by the following
two examples.

######  Example 4.1.

For every $n\geq 0$, we define inductively a subset
$I_{n}\subset\\{1,\ldots,2^{n+1}\\}$ by setting

$I_{0}:=\\{1\\}\subset\\{1,2\\},\qquad
I_{n}:=I_{n-1}\cup\\{2^{n}+i:i\in\\{1,\ldots,2^{n}\\}\setminus I_{n-1}\\}\ \
\mbox{for $n\geq 1$}.$

Note that $\\{I_{n}\\}_{n\geq 0}$ is a nested sequence of sets, and let
$I:=\bigcup_{n\geq 0}I_{n}$. For $V=\langle e_{1},e_{2},\ldots\rangle$ as
above, put

$W:=\langle e_{i}:i\in I\rangle.$

Thus $\mathcal{F}:=\\{0\subset W\subset V\\}$ is a generalized flag.

By Lemma 2.4 (b), any exhaustion of $\mathrm{GL}(2^{\infty})$ by classical
groups is equivalent to
$\\{\mathrm{GL}(s_{n}),\delta_{s_{n},s_{n+1}}\\}_{n\geq 1}$ for an exhaustion
$\\{s_{n}=2^{k_{n}}\\}_{n\geq 1}$ of $\mathbf{s}$. Every element
$g\in\mathrm{GL}(2^{k_{n}})$ stabilizing $\mathcal{F}$ should be such that the
blockwise diagonal matrix

$\begin{pmatrix}g&0\\\ 0&g\end{pmatrix}$

stabilizes $\langle e_{i}:i\in I_{k_{n}}\rangle=\langle e_{i}:i\in
I_{k_{n}-1}\rangle\oplus\langle
e_{2^{k_{n}-1}+i}:i\in\\{1,\ldots,2^{k_{n}}\\}\setminus I_{k_{n}-1}\rangle$,
hence $g$ should stabilize both subspaces $\langle e_{i}:i\in
I_{k_{n}-1}\rangle$ and $\langle e_{i}:i\in\\{1,\ldots,2^{k_{n}}\\}\setminus
I_{k_{n}-1}\rangle$. This implies that the stabilizer of $W$ in
$\mathrm{GL}(2^{k_{n}})$ is not a parabolic subgroup, for all $n\geq 1$.
Therefore, $\mathrm{Stab}_{\mathrm{GL}(2^{\infty})}(\mathcal{F})$ is not a
parabolic subgroup of $\mathrm{GL}(2^{\infty})$. $\blacksquare$

######  Example 4.2.

(a) Let $V=\bigcup_{n}\mathbb{C}^{2^{n}}$ be seen as a natural representation
of $\mathrm{GL}(2^{\infty})$ where the embedding
$\mathbb{C}^{2^{n}}\cong\mathbb{C}^{2^{n}}\times\\{0\\}^{2^{n}}\subset\mathbb{C}^{2^{n+1}}$
is considered. For $n\geq 1$, let $P_{n}\subset\mathrm{GL}(2^{n})$ be the
stabilizer of $L_{n}:=\\{0\\}^{2^{n}-1}\times\mathbb{C}$, the line spanned by
the $2^{n}$-th vector of the standard basis of $\mathbb{C}^{2^{n}}$. Then
$P_{n+1}\cap\mathrm{GL}(2^{n})=P_{n}$ for all $n\geq 1$, hence
$\mathbf{P}:=\bigcup_{n\geq 1}P_{n}$ is a parabolic subgroup of
$\mathrm{GL}(2^{\infty})$. However, $\mathbf{P}$ acts transitively on the
nonzero vectors of $V$, so that there is no nonzero proper subspace of $V$
which is stable by $\mathbf{P}$. Therefore, $\mathbf{P}$ cannot be realized as
the stabilizer of a generalized flag in $V$.

(b) If in part (a) we replace the embeddings defining the structure of natural
representation on $V$ by
$\mathbb{C}^{2^{n}}\cong\\{0\\}^{2^{n}}\times\mathbb{C}^{2^{n}}\subset\mathbb{C}^{2^{n+1}}$,
then $L_{n}=L_{1}$ for all $n\geq 1$ and the parabolic subgroup $\mathbf{P}$
of (a) becomes the stabilizer of the generalized flag $\\{0\subset
L_{1}\subset V\\}$. $\blacksquare$

###  4.2. Ind-varieties of generalized flags

######  Definition 4.3.

(a) Two generalized flags $\mathcal{F}$ and $\mathcal{G}$ are said to be
$E$-commensurable [3] if $\mathcal{F}$ and $\mathcal{G}$ are weakly
$E$-compatible and there is an isomorphism of totally ordered sets
$\phi:\mathcal{F}\to\mathcal{G}$ and there is a finite-dimensional subspace
$U\subset V$ such that, for all $F\in\mathcal{F}$, $F+U=\phi(F)+U$ and $\dim
F\cap U=\dim\phi(F)\cap U$.

(b) Given an $E$-compatible generalized flag $\mathcal{F}$, we define
$\mathrm{Fl}(\mathcal{F},E)$ as the set of all generalized flags which are
$E$-commensurable with $\mathcal{F}$. $\blacksquare$

Let $\mathcal{F}$ be an $E$-compatible generalized flag. We now recall the
ind-variety structure on $\mathrm{Fl}(\mathcal{F},E)$ [3]. To do this, we
write $E=\\{e_{k}\\}_{k\geq 1}$ and, for $n\geq 1$, set $V_{n}:=\langle
e_{1},\ldots,e_{n}\rangle$. The collection of subspaces $\\{F\cap
V_{n}:F\in\mathcal{F}\\}$ determines a flag $F^{(n)}_{1}\subset\ldots\subset
F^{(n)}_{p_{n}-1}$ in $F_{p_{n}}^{(n)}:=V_{n}$; furthermore we set
$d_{i}^{(n)}:=\dim F^{(n)}_{i}$ and

$X_{n}:=\mathrm{Fl}(d_{1}^{(n)},\ldots,d_{p_{n}-1}^{(n)};V_{n}).$

We define an embedding $\eta_{n}:X_{n}\to X_{n+1}$ in the following way. Let
$i_{0}\in\\{1,\ldots,p_{n+1}\\}$ be minimal such that $e_{n+1}\in
F^{(n+1)}_{i_{0}}$. We have either $p_{n+1}=p_{n}$ or $p_{n+1}=p_{n}+1$. In
the former case we set

$\eta_{n}:\\{M_{1},\ldots,M_{p_{n}-1}\\}\mapsto\\{M_{1},\ldots,M_{i_{0}-1},M_{i_{0}}\oplus\langle
e_{n+1}\rangle,\ldots,M_{p_{n}-1}\oplus\langle e_{n+1}\rangle\\}.$

In the latter case, we define

$\eta_{n}:\\{M_{1},\ldots,M_{p_{n}-1}\\}\mapsto\\{M_{1},\ldots,M_{i_{0}-1},M_{i_{0}-1}\oplus\langle
e_{n+1}\rangle,\ldots,M_{p_{n}-1}\oplus\langle e_{n+1}\rangle\\}.$

######  Proposition 4.4 ([3]).

(a) The maps $\\{\eta_{n}\\}_{n\geq 1}$ are strict standard extensions and
they yield an exhaustion $\mathrm{Fl}(\mathcal{F},E)=\bigcup_{n\geq 1}X_{n}$.
This endows $\mathrm{Fl}(\mathcal{F},E)$ with a structure of locally
projective ind-variety.

(b) If $\mathbf{P}=\mathrm{Stab}_{\mathrm{GL}(E)}(\mathcal{F})$, then there is
a natural isomorphism of ind-varieties
$\mathrm{GL}(E)/\mathbf{P}\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{Fl}(\mathcal{F},E)$.

Note also that, up to isomorphism, the ind-variety
$\mathrm{Fl}(\mathcal{F},E)$ only depends on the type of $\mathcal{F}$, i.e.,
on the isomorphism type of the totally ordered set $(\mathcal{F},\subset)$ and
on the dimensions $\dim F^{\prime\prime}/F^{\prime}$ of the quotients of
consecutive subspaces in $\mathcal{F}$.

##  5\. Embedding of flag varieties arising from diagonal embedding of groups

In this section we study embeddings of flag varieties induced by strictly
diagonal embeddings of general linear groups.

Let us fix the following data:

  * •

positive integers $m<n$ such that $m$ divides $n$, and $d:=\frac{n}{m}$;

  * •

$\mathrm{GL}(m)$ seen as a subgroup of $\mathrm{GL}(n)$ through the diagonal
embedding

$x\mapsto\mathrm{diag}(x,\ldots,x);$

  * •

a decomposition of the natural representation $V:=\mathbb{C}^{n}$ of
$\mathrm{GL}(n)$ as

$V=W^{(1)}\oplus\ldots\oplus W^{(d)}$

where $W^{(i)}:=\\{0\\}^{(i-1)m}\times\mathbb{C}^{m}\times\\{0\\}^{(d-i)m}$;
let $\chi_{i}:W:=\mathbb{C}^{m}\to W^{(i)}$ be the natural isomorphism. For a
subspace $M\subset W$, we write $M^{(i)}:=\chi_{i}(M)$.

###  5.1. Restriction of parabolic subgroup

Let $\\{e_{1},\ldots,e_{n}\\}$ be a basis of $V$ such that
$\\{e_{1},\ldots,e_{m}\\}$ is a basis of $W^{(1)}\cong W$. By
$H=H(n)\subset\mathrm{GL}(n)$ we denote the maximal torus for which
$e_{1},\ldots,e_{n}$ are eigenvectors. Then $H^{\prime}:=H\cap\mathrm{GL}(m)$
is a maximal torus of $\mathrm{GL}(m)$.

A parabolic subgroup $P=P(n)\subset\mathrm{GL}(n)$ that contains $H$ is the
stabilizer of a flag

$\mathcal{F}_{\alpha}=\\{\langle e_{i}:\alpha(i)\leq j\rangle\\}_{j=1}^{p-1}$

for some surjective map $\alpha:\\{1,\ldots,n\\}\to\\{1,\ldots,p\\}$. The
following statement determines under what condition the intersection
$P\cap\mathrm{GL}(m)$ is a parabolic subgroup.

######  Lemma 5.1.

Consider the map

$\beta:\\{1,\ldots,m\\}\to\\{1,\ldots,p\\}^{d},\
r\mapsto(\alpha(r),\alpha(m+r),\ldots,\alpha((d-1)m+r)),$

and denote by $\mathcal{I}$ the image of $\beta$. Let $\leq$ denote the
partial order on $\\{1,\ldots,p\\}^{d}$ such that
$(x_{1},\ldots,x_{d})\leq(y_{1},\ldots,y_{d})$ if $x_{i}\leq y_{i}$ for all
$i$.

(a) The intersection $Q:=P\cap\mathrm{GL}(m)$ is a parabolic subgroup of
$\mathrm{GL}(m)$ if and only if $\leq$ restricts to a total order on
$\mathcal{I}$. Moreover, letting $b_{1},\ldots,b_{q}$ be the elements of
$\mathcal{I}$ written in increasing order, we have

$Q=\mathrm{Stab}_{\mathrm{GL}(m)}(\mathcal{F}_{\beta})$

where

$\mathcal{F}_{\beta}=\\{\langle e_{i}:\beta(i)\leq b_{j}\\}_{j=1}^{q-1}.$

In particular, if $d_{j}=\\#\beta^{-1}(\\{b_{1},\ldots,b_{j}\\})$ then
$\mathrm{GL}(m)/Q$ can be identified with the flag variety
$\mathrm{Fl}(d_{1},\ldots,d_{q-1};W)$.

(b) If $Q$ is a parabolic subgroup, the inclusion $U_{Q}\subset U_{P}$ of
unipotent radicals holds if and only if any two distinct elements
$(x_{1},\ldots,x_{d}),(y_{1},\ldots,y_{d})$ of $\mathcal{I}$ satisfy
$x_{i}\not=y_{i}$ for all $i\in\\{1,\ldots,d\\}$.

###### Proof.

(a) We have a decomposition

$\mathfrak{gl}(n)=\mathfrak{gl}(V)=\mathfrak{h}\oplus\bigoplus_{1\leq
i\not=j\leq n}\mathfrak{g}_{i,j}$

where $\mathfrak{h}=\mathrm{Lie}\,H$ and
$\mathfrak{g}_{i,j}=\mathbb{C}(e_{i}\otimes e_{j}^{*})$. With this notation,

(5.1)
$\mathfrak{p}:=\mathrm{Lie}\,P=\mathfrak{h}\oplus\bigoplus_{\alpha(i)\leq\alpha(j)}\mathfrak{g}_{i,j}\supset\mathfrak{nil}(\mathfrak{p})=\bigoplus_{\alpha(i)<\alpha(j)}\mathfrak{g}_{i,j},$

where $\mathfrak{nil}(\mathfrak{p})$ is the nilpotent radical of
$\mathfrak{p}$.

There is a similar decomposition

$\mathfrak{gl}(m)=\mathfrak{gl}(W)=\mathfrak{h}^{\prime}\oplus\bigoplus_{1\leq
i\not=j\leq m}\mathfrak{g}^{\prime}_{i,j}.$

Set $\mathfrak{q}:=\mathrm{Lie}\,Q$ where $Q=P\cap\mathrm{GL}(m)$ as before.
Since we already know that $\mathfrak{h}^{\prime}\subset\mathfrak{q}$, the
subalgebra $\mathfrak{q}$ is parabolic if and only if

(5.2) $1\leq i\not=j\leq
m\quad\Longrightarrow\quad(\mathfrak{g}^{\prime}_{i,j}\subset\mathfrak{q}\quad\mbox{or}\quad\mathfrak{g}^{\prime}_{j,i}\subset\mathfrak{q}).$

In view of (5.1) and the diagonal embedding
$\mathfrak{gl}(m)\subset\mathfrak{gl}(n)$, whenever $1\leq i\not=j\leq m$ we
have the equivalence

$\displaystyle\mathfrak{g}^{\prime}_{i,j}\subset\mathfrak{q}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\mathfrak{g}_{i+km,j+km}\subset\mathfrak{p}\ \ \forall
k=0,\ldots,d-1$ $\displaystyle\Longleftrightarrow$
$\displaystyle\alpha(i+km)\leq\alpha(j+km)\ \forall k=0,\ldots,d-1$
$\displaystyle\Longleftrightarrow$ $\displaystyle\beta(i)\leq\beta(j).$

Hence, from (5.2) we obtain that $\mathfrak{q}$ is a parabolic subalgebra of
$\mathfrak{gl}(m)$ if and only if

$1\leq i\not=j\leq
m\quad\Longrightarrow\quad(\beta(i)\leq\beta(j)\quad\mbox{or}\quad\beta(j)\leq\beta(i)).$

The condition means that $\leq$ is a total order set on $\mathcal{I}$. We also
have the equality

$\displaystyle\mathfrak{q}=\mathfrak{h}\oplus\bigoplus_{\beta(i)\leq\beta(j)}\mathfrak{g}^{\prime}_{i,j}$
$\displaystyle=$ $\displaystyle\\{X\in\mathfrak{gl}(W):X(\langle
e_{i}:\beta(i)\leq b_{j}\rangle)\subset\langle e_{i}:\beta(i)\leq
b_{j}\rangle\ \forall j\\}$ $\displaystyle=$
$\displaystyle\mathrm{Lie}(\mathrm{Stab}_{\mathrm{GL}(m)}(\mathcal{F}_{\beta}))$

which implies that $Q=\mathrm{Stab}_{\mathrm{GL}(m)}(\mathcal{F}_{\beta})$.

(b) Assume that $Q$ is a parabolic subgroup of $\mathrm{GL}(m)$. The inclusion
$U_{Q}\subset U_{P}$ holds if and only if the similar inclusion holds for the
nilradicals of the Lie algebras. Through the diagonal embedding of
$\mathfrak{gl}(m)$ into $\mathfrak{gl}(n)$, the nilradical of $\mathfrak{q}$
can be described as

$\mathfrak{nil}(\mathfrak{q})=\bigoplus_{\begin{subarray}{c}1\leq i\not=j\leq
m\\\
\beta(i)<\beta(j)\end{subarray}}(\mathfrak{g}^{\prime}_{i,j}\oplus\mathfrak{g}^{\prime}_{i+m,j+m}\oplus\ldots\oplus\mathfrak{g}^{\prime}_{i+(d-1)m,j+(d-1)m}).$

Therefore, the desired inclusion
$\mathfrak{nil}(\mathfrak{q})\subset\mathfrak{nil}(\mathfrak{p})$ holds if and
only if, for all $i,j\in\\{1,\ldots,m\\}$,

$\beta(i)<\beta(j)\quad\Longleftrightarrow\quad\alpha(i+km)<\alpha(j+km)\ \
\forall k\in\\{0,\ldots,d-1\\}.$

This condition is equivalent to the one stated in (b) (knowing that the
partial order $\leq$ restricts to a total order on $\mathcal{I}$, due to (a)).
∎

###  5.2. Diagonal embedding of flag varieties

Assuming that the condition of Lemma 5.1 (a) is fulfilled, we now describe the
embedding of partial flag varieties

(5.3)
$\phi:\mathrm{GL}(m)/Q=\mathrm{Fl}(d_{1},\ldots,d_{q-1};W)\to\mathrm{GL}(n)/P$

obtained in this case. We rely on a combinatorial object, introduced in the
next definition.

######  Definition 5.2.

(a) We call E-graph an unoriented graph with the following features:

  * •

The vertices consist of two sets $\\{l_{1},\ldots,l_{q}\\}$ (“left vertices”)
and $\\{r_{1},\ldots,r_{p}\\}$ (“right vertices”), displayed from top to
bottom in two columns, and two vertices are joined by an edge only if they
belong to different sets.

  * •

The edges display into $d$ subsets $E_{c}$ corresponding to a given colour
$c\in\\{1,\ldots,d\\}$.

  * •

Every vertex is incident with at least one edge, and every vertex is incident
with at most one edge of a given colour. The vertex $l_{q}$ is incident with
exactly $d$ edges (one per colour).

  * •

Two edges of the same colour never cross, that is, if $(l_{i},r_{j})$ and
$(l_{k},r_{\ell})$, with $i<k$, are joined with two edges of the same colour,
then $j<\ell$.

In an E-graph, we call “bounding edges” the edges passing through $l_{q}$, and
we call “ordinary edges” all other edges.

(b) With the notation of Lemma 5.1, we define the E-graph
$\mathcal{G}(\alpha,\beta)$ such that

  * •

we put an edge of colour $k$ between $l_{i}$ and $r_{j}$ whenever
$b_{i}=(x_{1},\ldots,x_{d})$ satisfies $x_{k}=j$ and $i$ is maximal for this
property.

(The conditions given in Lemma 5.1 (a) justify that
$\mathcal{G}(\alpha,\beta)$ is a well-defined E-graph.) $\blacksquare$

In the following statement we describe explicitly the embedding $\phi$ of
(5.3) and its properties in terms of the E-graph $\mathcal{G}(\alpha,\beta)$.

######  Proposition 5.3.

(a) The map $\phi:Y=\mathrm{GL}(m)/Q=\mathrm{Fl}(d_{1},\ldots,d_{q-1};W)\to
X=\mathrm{GL}(n)/P$ is given by

$\phi:\\{F_{1},\ldots,F_{q-1}\\}\mapsto\\{V_{1},\ldots,V_{p-1}\\}$

where for all $j\in\\{1,\ldots,p-1\\}$ we have

(5.4) $V_{j}=V_{j-1}+F_{i_{1}}^{(1)}\oplus\ldots\oplus F_{i_{d}}^{(d)},$

where $V_{0}=F_{0}:=0$, $F_{q}:=W$, and

$i_{k}:=\left\\{\begin{array}[]{ll}i&\mbox{if the vertex $r_{j}$ is incident
with an edge $(l_{i},r_{j})$ of colour $k$ in $\mathcal{G}(\alpha,\beta)$,}\\\
0&\mbox{if there is no edge of colours $k$ passing through
$r_{j}$.}\end{array}\right.$

We have also

$V_{j}=F_{i^{\prime}_{1}}^{(1)}\oplus\ldots\oplus F_{i^{\prime}_{d}}^{(d)},$

where $i^{\prime}_{k}$ is the index of the left end point of the last edge of
colour $k$ arriving at or above $r_{j}$ in $\mathcal{G}(\alpha,\beta)$, with
$i^{\prime}_{k}=0$ if there is no such edge.

(b) Let $([L_{1}],\ldots,[L_{p-1}])$ and $([M_{1}],\ldots,[M_{q-1}])$ denote
the sequences of preferred generators of $\mathrm{Pic}\,X$ and
$\mathrm{Pic}\,Y$, respectively. The map
$\phi^{*}:\mathrm{Pic}\,X\to\mathrm{Pic}\,Y$ is given by

$\phi^{*}[L_{j}]=\sum_{k=1}^{d}[M_{i^{\prime}_{k}}],$

where we set by convention $[M_{0}]=[M_{q}]=0$.

(c) The map $\phi$ is linear if and only if, whenever $r_{j},r_{j^{\prime}}$
with $j<j^{\prime}$ are incident with edges of the same colour $c$ in the
graph $\mathcal{G}(\alpha,\beta)$, every ordinary edge arriving at
$r_{j^{\prime\prime}}$ for $j\leq j^{\prime\prime}<j^{\prime}$ is also of
colour $c$.

(d) The map $\phi$ is a standard extension if and only if all ordinary edges
of $\mathcal{G}(\alpha,\beta)$ are of the same colour. Moreover, in this case,
$\phi$ is a strict standard extension.

###### Proof.

(a) As in Section 5.1, we write $P=\mathrm{Stab}(\mathcal{F}_{\alpha})$ where
$\alpha:\\{1,\ldots,n\\}\to\\{1,\ldots,p\\}$ is surjective. Then we have
$Q=\mathrm{Stab}(\mathcal{F}_{\beta})$ where
$\beta:\\{1,\ldots,m\\}\to\mathcal{I}\subset\\{1,\ldots,p\\}^{d}$ is described
in Lemma 5.1. Let $\hat{\phi}:\mathrm{GL}(m)/Q\to\mathrm{GL}(n)/P$ be the map
given by formula (5.4). Thus we have to show that $\hat{\phi}=\phi$. Since the
maps $\phi$ and $\hat{\phi}$ are $\mathrm{GL}(m)$-equivariant, it suffices to
show that $\hat{\phi}(\mathcal{F}_{\beta})=\mathcal{F}_{\alpha}$. We write
$\mathcal{F}_{\alpha}=\\{F_{\alpha,1},\ldots,F_{\alpha,p-1}\\}$ and
$\mathcal{F}_{\beta}=\\{F_{\beta,1},\ldots,F_{\beta,q-1}\\}$. For
$j\in\\{1,\ldots,p\\}$, we have

(5.5) $F_{\alpha,j}=\langle e_{i}:\alpha(i)\leq
j\rangle=F_{\alpha,j-1}+\langle e_{i}:\alpha(i)=j\rangle$

where $F_{\alpha,0}:=0$. Every $i\in\\{1,\ldots,n\\}$ can be written
$i=(k-1)m+r\in\\{1,\ldots,n\\}$ with $k\in\\{1,\ldots,d\\}$ and
$r\in\\{1,\ldots,m\\}$, so that $e_{i}=\chi_{k}(e_{r})$.

Assume that $\alpha(i)=j$. Then there is
$b_{i^{\prime}}=(x_{1},\ldots,x_{d})\in\mathcal{I}$ with
$i^{\prime}\in\\{1,\ldots,q\\}$ maximal such that $x_{k}=j$. Moreover there is
$s\in\\{r,\ldots,m\\}$ such that $x_{\ell}=\alpha((\ell-1)m+s)$ for all
$\ell\in\\{1,\ldots,d\\}$. This implies that the graph
$\mathcal{G}(\alpha,\beta)$ contains an edge of colour $k$ joining
$b_{i^{\prime}}$ and $j$, and we have

$e_{i}=\chi_{k}(e_{r})\in\chi_{k}(F_{\beta,i^{\prime}})=F_{\beta,i^{\prime}}^{(k)}$

where $F_{\beta,q}:=W$. Conversely, assume that there is an edge of colour $k$
joining $b_{i^{\prime}}$ and $j$. The subspace $F_{\beta,i^{\prime}}^{(k)}$ is
spanned by vectors of the form $\chi_{k}(e_{r})$ with $r\in\\{1,\ldots,m\\}$
such that $\beta(r)=(\alpha((\ell-1)m+r)_{\ell=1}^{d}\leq b_{i^{\prime}}$. The
latter inequality implies $\alpha((k-1)m+r)\leq j$. Hence
$\chi_{k}(e_{r})=e_{(k-1)m+r}\in F_{\alpha,j}$.

Combining these observations with (5.5), we deduce that

$F_{\alpha,j}=F_{\alpha,j-1}+F_{\beta,i_{1}}^{(1)}\oplus\ldots\oplus
F_{\beta,i_{d}}^{(d)}$

where $i_{1},\ldots,i_{d}$ are as defined in the statement of the proposition.
Therefore, the claimed equality
$\hat{\phi}(\mathcal{F}_{\beta})=\mathcal{F}_{\alpha}$ holds.

The second formula stated in (a) is an immediate consequence of (5.4). The
proof of (a) is complete.

Part (b) is a corollary of the second formula in (a), whereas parts (c) and
(d) of the proposition easily follow from parts (a) and (b). The proof of the
proposition is complete. ∎

######  Remark 5.4.

Proposition 5.3 shows how the E-graph $\mathcal{G}(\alpha,\beta)$ describes
the embedding $\phi:Y\to X$. Moreover, the chain of constant spaces
$(C_{j}(\phi))$ is expressed in the following way. We enumerate the colours
$k_{1},\ldots,k_{d}$ so that $i_{1}\leq\ldots\leq i_{d}$ where $r_{i_{j}}$ is
the right end point of the bounding edge of colour $k_{j}$. Then

$C_{j}(\phi)=F_{q}^{(k_{1})}\oplus\ldots\oplus F_{q}^{(k_{j})}\quad\mbox{for
$j=1,\ldots,d$.}$

$\blacksquare$

######  Example 5.5.

(a) Let us consider for instance the graph

$l_{1}$$\bullet$$l_{2}$$\bullet$$l_{3}$$\bullet$$r_{1}$$\bullet$$r_{2}$$\bullet$$r_{3}$$\bullet$$r_{4}$$\bullet$.

It encodes an embedding

$\displaystyle\phi:X=\mathrm{Fl}(d_{1},d_{2};\mathbb{C}^{n})$
$\displaystyle\to$ $\displaystyle
Y=\mathrm{Fl}(d_{1},d_{1}+d_{2},d_{2}+n;\mathbb{C}^{2n}=\mathbb{C}^{n}\oplus\overline{\mathbb{C}^{n}})$
$\displaystyle\\{V_{1},V_{2}\\}$ $\displaystyle\mapsto$
$\displaystyle\\{V_{1},V_{1}\oplus\overline{V_{2}},V_{2}\oplus\overline{\mathbb{C}^{3}}\\}.$

If we denote by $([L_{1}],[L_{2}])$ and $([M_{1}],[M_{2}],[M_{3}])$ the sets
of preferred generators of the Picard groups of $X$ and $Y$ respectively, then
the induced map $\phi^{*}:\mathrm{Pic}\,Y\to\mathrm{Pic}\,X$ is given by

$[M_{1}]\mapsto[L_{1}],\quad[M_{2}]\mapsto[L_{1}]+[L_{2}],\quad[M_{3}]\mapsto[L_{2}].$

Thus $\phi$ is not linear in this case.

(b) Here we consider the graph

$l_{1}$$\bullet$$l_{2}$$\bullet$$\vdots$$l_{i}$$\bullet$$\vdots$$l_{q}$$\bullet$$r_{1}$$\bullet$$r_{2}$$\bullet$$\vdots$$r_{i}$$\bullet$$\vdots$$r_{i+1}$$\bullet$$r_{q+1}$$\bullet$.

There are two colours which means that the embedding is from a flag variety of
a space $V$ to the flag variety of a doubled space $W=V\oplus\overline{V}$:

$\mathrm{Fl}(d_{1},\ldots,d_{q-1};V)\hookrightarrow\mathrm{Fl}(d^{\prime}_{1},\ldots,d^{\prime}_{q};W=V\oplus\overline{V}).$

The embedding has the following explicit form

(5.6)
$\\{F_{1},\ldots,F_{q-1}\\}\mapsto\\{F_{1},\ldots,F_{i-1},F_{i-1}\oplus\overline{V},\ldots,F_{q-1}\oplus\overline{V}\\}.$

Note that $\dim F_{i-1}\oplus\overline{V}/\dim F_{i-1}=\dim V$. The dimensions
of the other quotients are unchanged.

(c) Now consider

$l_{1}$$\bullet$$l_{2}$$\bullet$$\vdots$$l_{i}$$\bullet$$\vdots$$l_{q}$$\bullet$$r_{1}$$\bullet$$r_{2}$$\bullet$$\vdots$$r_{i}$$\bullet$$\vdots$$r_{q}$$\bullet$.

In this case we get an embedding

$\mathrm{Fl}(d_{1},\ldots,d_{q};V)\hookrightarrow\mathrm{Fl}(d^{\prime}_{1},\ldots,d^{\prime}_{q};W=V\oplus\overline{V})$

given by

(5.7)
$\\{F_{1},\ldots,F_{q-1}\\}\mapsto\\{F_{1},\ldots,F_{i-1},F_{i}\oplus\overline{V},\ldots,F_{q-1}\oplus\overline{V}\\}.$

The only quotient whose dimension changes is $F_{i}\oplus\overline{V}/F_{i-1}$
which has dimension $\dim V+\dim F_{i}/F_{i-1}$.

By Proposition 5.3 (d), the embeddings of parts (a) and (b) of this example
are the only possible standard extensions that can come from a diagonal
embedding $\mathrm{GL}(n)\hookrightarrow\mathrm{GL}(2n)$.

(d) In the case of a diagonal embedding of the form
$\mathrm{GL}(n)\hookrightarrow\mathrm{GL}(dn)$, if the embedding of flag
varieties is a standard extension, then it can be described as a composition
of embeddings of the previous form, involving a subspace $\overline{V}$ still
of dimension $n$. $\blacksquare$

######  Remark 5.6.

The fact that $U_{Q}\subset U_{P}$ is equivalent to the following property of
the graph $\mathcal{G}(\alpha,\beta)$: every left vertex is incident with
exactly $d$ edges (one per colour). $\blacksquare$

Proposition 5.3 has the following corollary.

######  Corollary 5.7.

For an embedding $\phi:Y=\mathrm{GL}(n)/Q\to X=\mathrm{GL}(m)/P$ as in
Proposition 5.3 and for every $j\in\\{1,\ldots,q-1\\}$, we have
$\mathrm{Im}\,\phi^{*}\not\subset\langle[M_{i}]:i\in\\{1,\ldots,q-1\\}\setminus\\{j\\}\rangle$.

###  5.3. Application to ind-varieties

######  Definition 5.8.

Let $\\{s_{n}\\}_{n\geq 1}$ be an exhaustion of $\mathbf{s}$. We call
$\mathbf{s}$-graph a graph with infinitely many columns of vertices $B_{n}$,
with $1\leq|B_{n}|\leq s_{n}$ for all $n\geq 1$, such that the subgraph
consisting of $B_{n},B_{n+1}$ and the corresponding edges is an E-graph.
$\blacksquare$

A parabolic subgroup $\mathbf{P}$ of $\mathrm{GL}(\mathbf{s})$ gives rise to
an $\mathbf{s}$-graph. According to the above proposition, this graph encodes
the embeddings of flag varieties in an exhaustion of
$\mathrm{GL}(\mathbf{s})/\mathbf{P}$. Conversely, any $\mathbf{s}$-graph
arises from a parabolic subgroup $\mathbf{P}$ of $\mathrm{GL}(\mathbf{s})$.

##  6\. Ind-varieties of generalized flags as homogeneous spaces of
$\mathrm{GL}(\mathbf{s})$

Our purpose in this section is to characterize ind-varieties of generalized
flags (introduced in Section 4.2) which can be realized as homogeneous spaces
$\mathrm{GL}(\mathbf{s})/\mathbf{P}$ for the given supernatural number
$\mathbf{s}$.

###  6.1. The case of finitely many finite-dimensional subspaces

We start with a special situation which is easier to deal with: let
$\mathbf{X}=\mathrm{Fl}(\mathcal{F},E)$ where
$\mathcal{F}=\\{F^{\prime}_{a},F^{\prime\prime}_{a}\\}_{a\in A}$ is an
$E$-compatible generalized flag, for an arbitrary totally ordered set
$(A,\leq)$, but with the assumption that

(6.1) $\dim F^{\prime\prime}_{a}/F^{\prime}_{a}=+\infty\ \ \mbox{for all but
finitely many $a\in A$.}$

######  Theorem 6.1.

If condition (6.1) holds, then for every supernatural number $\mathbf{s}$,
there is an isomorphism of ind-varieties
$\mathrm{Fl}(\mathcal{F},E)\cong\mathrm{GL}(\mathbf{s})/\mathbf{P}$ for an
appropriate parabolic subgroup $\mathbf{P}\subset\mathrm{GL}(\mathbf{s})$.

###### Proof.

In the situation of the theorem, the ind-variety
$\mathbf{X}=\mathrm{Fl}(\mathcal{F},E)$ has an exhaustion

$X_{1}\hookrightarrow X_{2}\hookrightarrow\cdots\hookrightarrow
X_{n}\stackrel{{\scriptstyle\phi_{n}}}{{\hookrightarrow}}X_{n+1}\hookrightarrow\cdots$

such that $X_{n}$ is a finite-dimensional variety of flags in the space
$\mathbb{C}^{s_{n}}$ for some exhaustion $\\{s_{n}\\}_{n\geq 1}$ of
$\mathbf{s}$ with $s_{1}$ sufficiently large, and $\phi_{n}:X_{n}\to X_{n+1}$
is one of the two maps from Example 5.5 (b) and (c). Using the maps
$\phi_{n}$, one constructs nested parabolic subgroups
$P_{n}\subset\mathrm{GL}(s_{n})$ such that
$X_{n}\cong\mathrm{GL}(s_{n})/P_{n}$ and $P_{n}=\mathrm{GL}(s_{n})\cap
P_{n+1}$ for all $n$. The union $\mathbf{P}=\bigcup_{n\geq 1}P_{n}$ is then a
parabolic subgroup of $\mathrm{GL}(\mathbf{s})$ which satisfies the conditions
of the theorem. ∎

###  6.2. The general case

To treat the general case, we need to start with a definition.

######  Definition 6.2.

Let $\mathcal{F}=\\{F^{\prime}_{a},F^{\prime\prime}_{a}\\}_{a\in A}$ be an
$E$-compatible generalized flag and let $A^{\prime}=\\{a\in A:\dim
F^{\prime\prime}_{a}/\dim F^{\prime}_{a}<+\infty\\}$. We say that the ind-
variety $\mathrm{Fl}(\mathcal{F},E)$ is $\mathbf{s}$-admissible if either
$A^{\prime}$ is finite or $A^{\prime}$ is infinite and there are a exhaustion
$\\{s_{n}\\}_{n\geq 1}$ for $\mathbf{s}$ and a numbering
$A^{\prime}=\\{k_{n}\\}_{n\geq 1}$ (not necessarily compatible with the total
order on $A^{\prime}$) such that, for all $n\geq 0$:

$\frac{\dim
F^{\prime\prime}_{k_{n}}/F^{\prime}_{k_{n}}}{s_{n}}\in\\{1,\ldots,\frac{s_{n+1}}{s_{n}}-1\\}$
and $s_{n}|\dim F^{\prime\prime}_{a}/F^{\prime}_{a}$ for all $a\in
A^{\prime}\setminus\\{k_{1},\ldots,k_{n}\\}$.

$\blacksquare$

######  Theorem 6.3.

The following conditions are equivalent:

  * (i)

The ind-variety $\mathrm{Fl}(\mathcal{F},E)$ is $\mathbf{s}$-admissible.

  * (ii)

There is a parabolic subgroup $\mathbf{P}\subset\mathrm{GL}(\mathbf{s})$ and
an isomorphism of ind-varieties
$\mathrm{Fl}(\mathcal{F},E)\cong\mathrm{GL}(\mathbf{s})/\mathbf{P}$.

###### Proof.

(i)$\Rightarrow$(ii): The ind-variety $\mathrm{Fl}(\mathcal{F},E)$ admits an
exhaustion $\mathrm{Fl}(\mathcal{F},E)=\bigcup_{n}X_{n}$ with embeddings of
the form

(6.2) $\displaystyle\phi_{n}:X_{n}=\mathrm{Fl}(p_{1},\ldots,p_{k_{n}};V_{n})$
$\displaystyle\to$ $\displaystyle
X_{n+1}=\mathrm{Fl}(q_{1},\ldots,q_{\ell_{n}};V_{n}\oplus C_{n}),$
$\displaystyle\\{F_{1},\ldots,F_{k_{n}}\\}$ $\displaystyle\mapsto$
$\displaystyle\\{F_{\tau(1)}\oplus C_{1}^{n},\ldots,F_{\tau(\ell_{n})}\oplus
C^{n}_{\ell_{n}}\\}$

(with $F_{0}:=0$, $F_{k_{n}+1}:=V_{n}$) for a nondecreasing surjective map
$\tau:\\{1,\ldots,\ell_{n}\\}\to\\{0,1,\ldots,k_{n},k_{n}+1\\}$ and a sequence
$C^{n}_{1}\subset\ldots\subset C^{n}_{\ell_{n}}$ (with possible repetitions)
of subspaces of $C_{n}$.

Assume that there is another exhaustion
$\mathrm{Fl}(\mathcal{F},E)=\bigcup_{n}Y_{n}$ for which the embeddings are as
described in Proposition 5.3, where
$Y_{n}=\mathrm{Fl}(r_{1},\ldots,r_{m_{n}};W_{n})$ and $\dim W_{n}=s_{n}$ for
an exhaustion $\\{s_{n}\\}$ of $\mathbf{s}$. Then the two exhaustions
interlace, and there is no loss of generality in assuming that the interlacing
holds for the sequences $(X_{n})$ and $(Y_{n})$, and not only for
subsequences:

$\textstyle{X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{n}}$$\scriptstyle{\xi_{n}}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{n+1}}$$\textstyle{Y_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{n}}$$\scriptstyle{\chi_{n}}$$\textstyle{Y_{n+1}.}$

Claim. The embedding $\xi_{n}$ is a standard extension.

First we show that $\xi_{n}$ is linear. Arguing by contradiction, assume that
there is a generator $[M_{i}]$ among the sequence $[M_{1}],\ldots,[M_{q-1}]$
of preferred generators of $\mathrm{Pic}\,X_{n}$ such that
$\xi_{n}^{*}[M_{i}]$ is neither $0$ nor a preferred generator of
$\mathrm{Pic}\,Y_{n}$. Since $\phi_{n}^{*}=\chi_{n}^{*}\circ\xi_{n+1}^{*}$, we
have the inclusion
$\mathrm{Im}\,\phi_{n}^{*}\subset\mathrm{Im}\,\chi_{n}^{*}$, and due to
Corollary 5.7 we get that there is a generator $[L]\in\mathrm{Pic}\,Y_{n+1}$
such that

$\chi_{n}^{*}[L]=\sum_{j=1}^{q-1}\lambda_{j}[M_{j}]\quad\mbox{with
$\lambda_{i}\not=0$.}$

Since the map $\chi_{n}$ is an embedding, we have $\lambda_{j}\geq 0$ for all
$j$ and in particular $\lambda_{i}\geq 1$. The same argument applied to
$\xi_{n}$ implies that $\xi_{n}^{*}[M_{j}]$ should be a linear combination of
the preferred generators of $\mathrm{Pic}\,Y_{n}$ with nonnegative integer
coefficients. This implies that $\psi_{n}^{*}[L]=\xi_{n}^{*}\chi_{n}^{*}[L]$
is neither $0$ nor a preferred generator of $\mathrm{Pic}\,Y_{n}$,
contradicting the linearity of the standard extension $\psi_{n}$.

Recall that in [10] the notion of an embedding factoring through a direct
product is introduced. Note that $\xi_{n}$ cannot factor through a direct
product: otherwise, $\psi_{n}$ would also factor through a direct product,
which is impossible since this is a standard extension. Consequently,
$\xi_{n}$ is a standard extension, and the claim is established.

Now we can assume that $\xi_{1}$ is a strict standard extension. Since the
maps $\phi_{n}$ are strict standard extensions, by using the formula for
$\psi_{n}$ in Proposition 5.3 we derive that $\xi_{n}$ is a strict standard
extension for all $n\geq 1$.

Due to (6.2) and Proposition 5.3, one has $W_{n}=V_{n}\oplus Z_{n}$ and the
map $\xi_{n}$ has the form

$\xi_{n}:\\{F_{1},\ldots,F_{k_{n}}\\}\mapsto\\{F_{\sigma(1)}\oplus
Z_{1}^{n},\ldots,F_{\sigma(p_{n})}\oplus Z^{n}_{p_{n}}\\}.$

Since this applies likewise to $\xi_{n+1}$, and taking into account the form
of $\phi_{n}$ in (6.2), we see that the map $\psi_{n}$ has the form

$\\{F_{1},\ldots,F_{k_{n}},F^{\prime}_{1},\ldots,F^{\prime}_{p_{n}}\\}\mapsto\\{F_{1},\ldots,F_{k_{n}},R^{n}_{1},\ldots,R^{n}_{\ell_{n}},\Gamma^{n}_{1},\ldots,\Gamma^{n}_{\delta_{n}}\\}$

for an arbitrary map
$\zeta_{n}:\\{F^{\prime}_{1},\ldots,F^{\prime}_{p_{n}}\\}\mapsto\\{R^{n}_{1},\ldots,R^{n}_{\ell_{n}}\\}$
as described in Proposition 5.3, and where $\Gamma^{n}_{i}$ are constant
subspaces which are copies of $W_{n}$ in
$W_{n+1}=\bigoplus_{i=1}^{d_{n}}W_{n}^{(i)}$. This implies $\dim
V_{n}=d^{\prime}_{n}s_{n}$ for some
$d^{\prime}_{n}\in\\{1,\ldots,d_{n}=\frac{s_{n+1}}{s_{n}}\\}$.

Since $\\{V_{n}\oplus Z^{n}_{1},\ldots,V_{n}\oplus
Z^{n}_{p_{n}}\\}=\\{\Gamma^{n-1}_{1},\ldots,\Gamma^{n-1}_{\delta_{n-1}}\\}$,
we must have $p_{n}=\delta_{n-1}$ and the dimension of $Z_{i}^{n}$ is a
multiple of $s_{n-1}$. Therefore $\dim R_{i}^{n}$ is also a multiple of
$s_{n-1}$ for all $i$. Condition (ii) is established.

(ii)$\Rightarrow$(i): Let $d^{\prime}_{n}=\frac{\dim
F_{k_{n}}}{s_{n}}\in\\{1,\ldots,d_{n}\\}$ and set $V_{n}=F_{k_{n}}$. The
conditions imply that we can choose a decomposition $W_{n}=V_{n}\oplus
W_{n-1}^{(1)}\oplus\ldots\oplus W_{n-1}^{(d_{n}-d^{\prime}_{n})}$ where the
$W_{n-1}^{(i)}$’s are copies of $W_{n-1}$ such that the strict standard
extension $\mathrm{Fl}_{n}(\mathcal{F},E)\to\mathrm{Fl}_{n+1}(\mathcal{F},E)$
is given by

$\phi_{n}:\\{F_{1},\ldots,F_{k_{n}}\\}\mapsto\\{F_{k_{1}}+C^{n}_{1},\ldots,F_{k_{n+1}}+C^{n}_{k_{n+1}}\\}$

with $C^{n}_{i}=W_{n-1}^{(1)}\oplus\ldots\oplus
W_{n-1}^{(m_{i})}=C_{i}^{\prime n}\oplus C_{i}^{\prime\prime n}$ for some
nondecreasing sequence $m_{1},\ldots,m_{k_{n+1}}$.

Letting
$\xi_{n}:\mathrm{Fl}_{n}(\mathcal{F},E)\to\mathrm{Fl}(\mathbf{t}_{n};W_{n})$
be given by

$\xi_{n}:\\{F_{1},\ldots,F_{k_{n}}\\}\mapsto\\{F_{k_{1}}+C^{\prime
n}_{1},\ldots,F_{k_{n+1}}+C^{\prime n}_{k_{n+1}}\\},$

and
$\psi_{n}:\mathrm{Fl}(\mathbf{t}_{n};W_{n})\to\mathrm{Fl}(\mathbf{t}_{n+1};W_{n+1})$

$\psi_{n}:\\{F_{1},\ldots,F_{k_{n}}\\}\mapsto\\{F_{1}+C^{\prime\prime
n+1}_{1},\ldots,F_{\ell_{n}+1}+C^{\prime\prime n+1}_{\ell_{n+1}}\\}$

(for suitable types $\mathbf{t}_{n}$), we get exhaustions of
$\mathrm{Fl}(\mathcal{F},E)$ and a homogeneous space for
$\mathrm{GL}(\mathbf{s})$, which interlace. Hence if $\mathcal{F}$ satisfies
the condition above, then we can realize $\mathrm{Fl}(\mathcal{F},E)$ as a
homogeneous space for $\mathrm{GL}(\mathbf{s})$. ∎

######  Remark 6.4.

It is shown in [2, Corollary 5.40] that $\mathrm{GL}(\mathbf{s})/\mathbf{B}$
is never projective when $\mathbf{B}$ is a Borel subgroup. On the other hand,
according to [3, Proposition 7.2], an ind-variety of generalized flags is
projective if and only if the total order on the flag can be induced by a
subset of $(\mathbb{Z},\leq)$, and Theorem 6.3 shows that in many situations
$\mathrm{GL}(\mathbf{s})/\mathbf{P}$ is projective. $\blacksquare$

##  7\. The case of direct products of ind-varieties of generalized flags

In this section, we point out that many direct products of ind-varieties of
generalized flags can be homogeneous spaces for the group
$\mathrm{GL}(\mathbf{s})$.

###  7.1. Direct products of ind-varieties

Let $\mathbf{X}_{i}=\bigcup_{n\geq 1}X_{i,n}$ ($i\in I$) be a collection of
ind-varieties indexed by $\mathbb{Z}_{>0}$ or a finite subset of it. For each
$i\in I$ we pick an element $x_{i}\in X_{i,1}$ and we set
$X_{i,0}=\\{x_{i}\\}$. The direct product in the category of pointed ind-
varieties is then given by

$\prod_{i\in I}\mathbf{X}_{i}:=\bigcup_{n\geq 1}\prod_{i\in
I}X_{i,\phi_{i}(n)}$

for a collection of increasing maps
$\phi_{i}:\mathbb{Z}_{>0}\to\mathbb{Z}_{\geq 0}$ such that for every
$n\in\mathbb{Z}_{>0}$ we have $\phi_{i}(n)=0$ for all but finitely many $i\in
I$ (the definition does not depend essentially on the choice of the maps
$\phi_{i}$).

######  Remark 7.1.

(a) As a set, the direct product can be identified with the set of sequences
$(y_{i})_{i\in I}$ where $y_{i}\in\mathbf{X}_{i}$ for all $i\in I$ and
$y_{i}=x_{i}$ for all but finitely many $i\in I$.

(b) For a finite set of indices $I$, as a set, $\prod_{i\in I}\mathbf{X}_{i}$
coincides with the usual cartesian product, and its structure of ind-variety
is given by the exhaustion $\prod_{i\in I}\mathbf{X}_{i}:=\bigcup_{n\geq
1}X_{i,n}$.

Fixing an index $i_{0}\in I$, there are a canonical projection

$\mathrm{proj}_{i_{0}}:\prod_{i\in I}\mathbf{X}_{i}\to\mathbf{X}_{i_{0}},\
(y_{i})\mapsto y_{i_{0}}$

and an embedding

$\mathrm{emb}_{i_{0}}:\mathbf{X}_{i_{0}}\to\prod_{i\in I}\mathbf{X}_{i},\
x\mapsto(y_{i})\ \mbox{with}\ y_{i}=\left\\{\begin{array}[]{ll}x_{i}&\mbox{if
$i\not=i_{0}$},\\\ x&\mbox{if $i=i_{0}$,}\end{array}\right.$

which are morphisms of ind-varieties.

If the product is endowed with an action of a group $G$, then each ind-variety
$\mathbf{X}_{i}$ inherits an action of $G$ defined through the maps
$\mathrm{proj}_{i}$ and $\mathrm{emb}_{i}$. Conversely, if every ind-variety
$\mathbf{X}_{i}$ is endowed with an action of a group $G$, then we obtain an
action of $G$ on the product defined diagonally provided that the following
condition is fulfilled:

(7.1) every $g\in G$ fixes $x_{i}$ for all but finitely many $i\in I$.

(This condition is automatically satisfied in the case where $I$ is finite.)
Moreover, in both directions, when $G=\mathbf{G}$ is an ind-group, we have
that the obtained action is algebraic provided that the initial one is. The
following lemma is an immediate consequence of this discussion.

######  Lemma 7.2.

Assume that the direct product $\prod_{i\in I}\mathbf{X}_{i}$ is a homogeneous
space for an ind-group $\mathbf{G}$. Then, every ind-variety $\mathbf{X}_{i}$
is also a homogeneous space for $\mathbf{G}$.

Note also that a direct product $\prod_{i\in I}\mathbf{X}_{i}$ is locally
projective if and only if it is the case of $\mathbf{X}_{i}$ for all $i\in I$.

###  7.2. The case of ind-varieties of generalized flags

We start with an example.

######  Example 7.3.

Let $\mathbf{s}=2^{\infty}$. We consider the space $V$ of countable dimension,
endowed with its fixed basis $E=\\{e_{k}\\}_{k\in\mathbb{Z}_{>0}}$, and we set
$V_{n}:=\langle e_{1},\ldots,e_{n}\rangle$. We have the exhaustion
$\mathrm{GL}(\mathbf{s})=\bigcup_{n\geq 1}\mathrm{GL}(V_{2^{n}})$ defined
through the diagonal embedding
$\mathrm{GL}(V_{2^{n}})\hookrightarrow\mathrm{GL}(V_{2^{n+1}})$,
$x\mapsto\mbox{\scriptsize$\begin{pmatrix}x&0\\\ 0&x\end{pmatrix}$}$. Consider
the sequence of parabolic subgroups

$P_{n}:=\mathrm{Stab}_{\mathrm{GL}(V_{2^{n}})}(V_{1},V_{2^{n}-1}),\quad n\geq
2.$

In this way, $P_{n}\cap\mathrm{GL}(V_{2^{n-1}})=P_{n-1}$ for all $n\geq 3$.
Moreover, every quotient $\mathrm{GL}(V_{2^{n}})/P_{n}$ is a flag variety
formed by flags $(F_{1}\subset F_{2}\subset V_{2^{n}})$ of length 2, and we
have an embedding of flag varieties

$\mathrm{GL}(V_{2^{n-1}})/P_{n-1}\to\mathrm{GL}(V_{2^{n}})/P_{n},\
(F_{1},F_{2})\mapsto(F_{1},V_{2^{n-1}}+\overline{F_{2}})$

where as before the map

$V_{2^{n-1}}=\langle
e_{1},\ldots,e_{2^{n-1}}\rangle\to\overline{V_{2^{n}-1}}=\langle
e_{2^{n-1}+1},\ldots,e_{2^{n-1}+2^{n-1}}\rangle,\ v\mapsto\overline{v},$

is the isomorphism from Section 5. For every $n$, this embedding factors
through a direct product of grassmannians

$\mathrm{GL}(V_{2^{n-1}})/P_{n-1}\to\mathrm{Gr}(1;V_{2^{n-1}})\times\mathrm{Gr}(2^{n-1}-1;\overline{V_{2^{n-1}}})\to\mathrm{GL}(V_{2^{n}})/P_{n},$

which allows us to chech that the ind-variety
$\mathrm{GL}(\mathbf{s})/\mathbf{P}$, where $\mathbf{P}=\bigcup_{n}P_{n}$, is
isomorphic as an ind-variety to a direct product of two ind-grassmannians.

The following theorem shows that many homogeneous spaces for
$\mathrm{GL}(\mathbf{s})$ can be isomorphic to direct products of ind-
varieties of generalized flags.

######  Theorem 7.4.

Let $\mathbf{X}=\mathrm{GL}(\mathbf{s})/\mathbf{P}$ be a homogeneous space,
defined by a parabolic subgroup $\mathbf{P}$. Assume that we have an
exhaustion $\mathbf{X}=\bigcup_{n\geq 1}\mathrm{GL}(s_{n})/P_{s_{n}}$
determined by an exhaustion $\\{s_{n}\\}_{n\geq 1}$ of $\mathbf{s}$, where
each embedding
$\mathrm{GL}(s_{n})/P_{s_{n}}\hookrightarrow\mathrm{GL}(s_{n+1})/P_{s_{n+1}}$
is linear. Then, $\mathbf{X}$ is isomorphic as an ind-variety to a direct
product of ind-varieties of generalized flags $\prod_{i\in
I}\mathrm{Fl}(\mathcal{F}^{i},E^{i})$ where $I$ is either $\mathbb{Z}_{>0}$ or
a finite subset of it.

###### Proof.

Fix $n\geq 1$. Let $d_{n}=\frac{s_{n+1}}{s_{n}}$ and fix a decomposition

(7.2) $V_{n}=W_{n}^{(1)}\oplus\ldots\oplus W_{n}^{(d_{n})}$

of the space $V_{n}=\mathbb{C}^{s_{n+1}}$ as in Section 5. Thus we are in the
setting of Proposition 5.3, and the embedding

$\phi_{n}:\mathrm{GL}(s_{n})/P_{s_{n}}\hookrightarrow\mathrm{GL}(s_{n+1})/P_{s_{n+1}}$

can be encoded by an E-graph with $d_{n}$ colours in the sense of Proposition
5.3 (a). The formula therein, combined with the characterization of $\phi_{n}$
given in Proposition 5.3 (c), yields a commutative diagram

---
$\textstyle{\mathrm{GL}(s_{n})/P_{s_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{n}}$$\scriptstyle{\prod_{i}\psi^{(i)}}$$\textstyle{\mathrm{GL}(s_{n+1})/P_{s_{n+1}}}$$\textstyle{\prod_{i=1}^{d_{n}}\mathrm{Fl}(\underline{\ell}^{(i)};W_{n}^{(i)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{n}}$

where $\psi^{(i)}$ is the embedding corresponding to the subgraph of
$\mathcal{G}(\alpha,\beta)$ formed by removing all the ordinary edges which
are not of colour $i$ ($\psi^{(i)}$ is a strict standard extension due to
Proposition 5.3 (c)–(d)), $\underline{\ell}^{(i)}$ is an appropriate dimension
vector, and the embedding $\xi_{n}$ is induced by the decomposition (7.2). The
theorem follows from this construction. ∎

The above proof yields the following sharpening of Theorem 7.4.

######  Corollary 7.5.

In the framework of Theorem 7.4, let $\mathcal{G}$ be the $\mathbf{s}$-graph
corresponding to $\mathbf{P}$ in the sense of Section 5.3. Let
$\mathcal{G}=\bigcup_{i\in\mathcal{I}}\mathcal{G}_{i}$ be a decomposition into
subgraphs so that all ordinary edges of $\mathcal{G}_{i}$ are of the same
colour. Then, the ind-variety $\mathbf{X}$ is isomorphic to a direct product
of ind-varieties of generalized flags $\prod_{i\in\mathcal{I}}\mathbf{X}_{i}$
where $\mathbf{X}_{i}$ has an exhaustion with embeddings encoded by
$\mathcal{G}_{i}$.

## Outlook

We see the results of this paper as a small first step in the study of locally
projective homogeneous ind-spaces of locally reductive ind-groups. One
inevitable question for a future such study is, given two non-isomorphic
locally reductive ind-groups $\mathbf{G}$ and $\mathbf{G}^{\prime}$, when are
two homogeneous spaces $\mathbf{G}/\mathbf{P}$ and
$\mathbf{G}^{\prime}/\mathbf{Q}$ isomorphic as ind-varieties ? A further
natural direction of research could be a comparison of Bott–Borel–Weil type
results on $\mathbf{G}/\mathbf{P}$ and $\mathbf{G}^{\prime}/\mathbf{Q}$. We
finish the paper by pointing out that the reader can verify that Theorem 6.1
remains valid if one replaces $\mathrm{GL}(\mathbf{s})$ by any pure diagonal
ind-group in the terminology of [1].

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