# Automorphism groups of ind-varieties of generalized flags

Mikhail Ignatev and Ivan Penkov

###### Abstract

We compute the group of automorphisms of an arbitrary ind-variety of (possibly
isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space
for one of the ind-groups $SL(\infty)$, $O(\infty)$ or $Sp(\infty)$. We show
that the respective automorphism groups are much larger than $SL(\infty)$,
$O(\infty)$ or $Sp(\infty)$, and present the answer in terms of Mackey groups.
The latter are groups of automorphisms of nondegenerate pairings of (in
general infinite-dimensional) vector spaces. An explicit matrix form of the
automorphism group of an arbitrary ind-variety of generalized flags is also
given. The case of the Sato grassmannian is considered in detail, and its
automorphism group is the projectivization of the connected component of unity
in the group known as Japanese $GL(\infty)$.

Keywords: ind-variety, ind-group, generalized flag, homogeneous space,
automorphism group, Makey group, Sato grassmannian, Japanese $GL(\infty)$.

AMS subject classification: 14L30, 14M15, 14M17, 14J50.

Introduciton

Given a homogenous space $X$, it is a natural problem to compute its
automorphism group $\mathrm{Aut}\,X$. In the case when $X$ is a complex flag
variety, that is, $X=G/P$ for a connected reductive complex algebraic group
and a parabolic subgroup $P\subset G$, the automorphism group of $X$ is well
known. Moreover, it is a classical result that here the connected component
$\mathrm{Aut}^{0}\,X$ of the identity equals the projectivized group $PG$,
except in some special cases as described in [O].

In this paper, we would like to pose and solve the problem of computing
$\mathrm{Aut}\,X$ for a class of homogeneous ind-varieties $X$. This is the
class of ind-varieties of generalized flags introduced by that name in [DP]
but also considered earlier in several works, see for instance [DPW], [NRW].
These ind-varieties can be defined simply as $G/P$ where $G$ is one of the
ind-groups $SL(\infty)=\varinjlim SL(n)$, $SO(\infty)=\varinjlim SO(n)$,
$Sp(\infty)=\varinjlim Sp(2n)$ and $P$ is a splitting parabolic subgroup,
i.e., a subgroup for which the intersections $P\cap SL(n)$, $P\cap SO(2n)$,
$P\cap SO(2n+1)$, $P\cap Sp(2n)$ are parabolic subgroups of $SL(n)$, $SO(2n)$,
$SO(2n+1)$, $Sp(2n)$ for all $n$, respectively. The definition from [DP] can
be considered as a flag realization of the ind-varieties $G/P$ as above, and
is recalled in Section Automorphism groups of ind-varieties of generalized
flags below. The main idea of that approach is that one designates certain
chains of subspaces in the natural representation $V$ of $SL(\infty)$ as
generalized flags, and then defines an ind-variety of generalized flags as the
ind-variety of generalized flags which differ only “slightly” from a fixed
generalized flag $W$ in $V$. For the exact definition see Section Automorphism
groups of ind-varieties of generalized flags. One then shows that the so
obtained ind-variety is isomorphic to $G/P$ for $G=SL(\infty)$ and some
splitting parabolic subgroup $P\subset G$.

An ind-grassmannian is an ind-variety of generalized flags for which the fixed
generalized flag consists of a single proper subspace $W\subset V$. For $\dim
W=\mathrm{codim}\,_{V}W=\infty$ the ind-grassmannian is isomorphic to the Sato
grassmannian. This has been pointed out for instance in [GS].

In the cases of the groups $SO(\infty)$ and $Sp(\infty)$ we consider ind-
varieties of isotropic generalized flags, as stated in Section Automorphism
groups of ind-varieties of generalized flags.

Our main result is the explicit determination of the group $\mathrm{Aut}\,X$
for an arbitrary ind-variety of, possibly isotropic, generalized flags. A
notable feature is that the answer is very different from the ind-groups
$PGL(\infty)$, $PO(\infty)$, or $PSp(\infty)$, and we present it in the
language of Mackey groups. Such a group is defined in terms of a nondegenerate
pairing of vector spaces $T\times R\rightarrow\mathbb{C}$, and is a subgroup
of the group of all linear operators $\varphi\colon T\rightarrow T$ for which
the dual operator $\varphi^{*}$ determines a well-defined automorphism
$\overline{\varphi}\colon R\rightarrow R$. This definition of Mackey group is
inspired by G. Mackey’s dissertation [M]. If $T$ and $R$ are finite
dimensional, then the Mackey group is nothing but $GL(T)\simeq GL(R)$. The
group known as Japanese $GL(\infty)$ is a Mackey group and plays a crusial
role in our work. In the Appendix we discuss the structure of this group in
detail.

The precise statement of our main result, Theorem Automorphism groups of ind-
varieties of generalized flags, is presented in Section Automorphism groups of
ind-varieties of generalized flags. The consideration of the isotropic case is
postponed to Section Automorphism groups of ind-varieties of generalized
flags. The proof of Theorem Automorphism groups of ind-varieties of
generalized flags is divided into two parts: the case of an ind-grassmannian
and the case of an arbitrary generalized flag. For the Sato grassmannian
(which is the most interesting ind-grassmannian) our result implies that its
automorphism group is isomorphic to the projectivization of the connected
component of the identity in the group Japanese $GL(\infty)$. In Section
Automorphism groups of ind-varieties of generalized flags we give a matrix
realization of the group of automorphisms of an arbitrary ind-variety of
generalized flags. In the isotropic case such a realization is given in
Corollary Automorphism groups of ind-varieties of generalized flags.

We would like to point out that $\mathrm{Aut}\,X$ depends essentially on the
ind-variety $X$, despite the fact that all $X$ are homogeneous spaces for the
same group $SL(\infty)$ (or, respectively, $SO(\infty)$, $Sp(\infty)$). This
is in contrast with the finite-dimensional case in which the connected
component of the identity in the automorphism group of a variety $SL(n)/P$
(respectively, $SO(2n)/P$, $SO(2n+1)/P$ or $Sp(2n)/P$) depends only on $n$ and
not on the choice of $P$. Further research should be carried out to compare
the isomorphism classes of ind-varieties of generalized flags with the
isomorphism classes of their automorphism groups.

Our possible application of the results of the present paper would be the
study of locally reductive ind-groups $\widetilde{G}$ different from
$G=SL(\infty)$, $SO(\infty)$, $Sp(\infty)$ for which $G/P$ is a homogeneous
$\widetilde{G}$-space.

In conclusion of this short introduction, we should mention that some
particular cases of the automorphism groups of ind-varieties of generalized
flags have been considered in [P] and [T].

Acknowledgements.

I.P. thanks Francesco Esposito for inspiring discussion on topics related to
the present paper. M.I. was supported by the Foundation for the Advancement of
Theoretical Physics and Mathematics “BASIS” grant no. 18–1–7–2–1, and by RFBR
grant no. 20–01–00091–a. I.P. was supported in part by DFG grant PE 980/8–1.

We are grateful to two referees for their sharp comments. One of the referees
found several inaccuracies and made helpful suggestions for improving the
text. Another referee suggested to mention that in the works [H1], [H2] of
Alan Huckleberry the classical result of [O] has been extended to the non-
compact case of flag domains.

1\. Brief background and statement of the main result

The ground field is $\mathbb{C}$. If $R$ is a vector space, we set
$GL(R)=\\{\varphi\in\mathrm{Hom}_{\mathbb{C}}(R,R)\mid\varphi\text{ is
invertible}\\}$ and $R^{*}=\mathrm{Hom}_{\mathbb{C}}(R,\mathbb{C})$. We also
use the superscript $\cdot^{*}$ to denote the dual of a vector bundle, as well
as the pullback functor for vector bundles along a morphism of varieties. In
what follows we consider infinite matrices, in particular, infinite rows and
columns. We call such matrices, rows or columns _finitary_ if they have at
most finitely many nonzero entries. By $\langle\cdot\rangle_{\mathbb{C}}$ we
denote the linear span over $\mathbb{C}$.

We fix a countable-dimensional vector space $V$. A _chain of subspaces_
$W=\\{W_{\alpha}\\}$ is a set of subspaces $W_{\alpha}\subseteq V$,
parameterized by some index set with elements $\alpha$ such that for
$\alpha\neq\alpha^{\prime}$ we have $W_{\alpha}\varsubsetneq
W_{\alpha^{\prime}}$ or $W_{\alpha^{\prime}}\varsubsetneq W_{\alpha}$. The
relation of inclusion induces a total order on the set of indices of a chain.
A chain of subspaces $W=\\{W_{\alpha}\\}$ is a _generalized flag in_ $V$ if
every index $\alpha$ has either an immediate predecessor or an immediate
successor, and every nonzero vector $v$ of $V$ is contained in some difference
$W_{\alpha^{\prime\prime}}\setminus W_{\alpha^{\prime}}$, where
$\alpha^{\prime\prime}$ is the immediate successor of $\alpha^{\prime}$. For a
more detailed discussion of generalized flags, and for an introduction to ind-
varieties of generalized flags, see, e.g., [DP], [IP] and [PT2].

We say that a generalized flag $W$ is _compatible with a basis_
$\widetilde{E}$ of $V$ (or that $W$ is $\widetilde{E}$-_compatible_) if any
space $W_{\alpha}$ of $W$ is spanned by elements of $\widetilde{E}$, i.e.,
$W_{\alpha}=\langle W_{\alpha}\cap\widetilde{E}\rangle_{\mathbb{C}}$ for any
$\alpha$. We set $E_{\alpha}=\widetilde{E}\cap W_{\alpha}$. Then $E_{\alpha}$
is a basis of $W_{\alpha}$. By $(W_{\alpha})_{*}$ we denote the span of the
system of linear functions $\widetilde{E}^{*}$ dual to the basis $E_{\alpha}$.
We have $(W_{\alpha})_{*}\subseteq W^{*}_{\alpha}$. We also let $V_{*}$ equal
the span of the system of linear functions dual to the basis $\widetilde{E}$.
The group $GL(\widetilde{E},V)$ is the subgroup of $GL(V)$ consisting of all
invertible operators $\varphi\colon V\rightarrow V$ each of which acts as the
identity on all but finitely elements of $\widetilde{E}$.

In what follows, we fix a basis $\widetilde{E}$ and a generalized flag $W$ in
$V$ compatible with $\widetilde{E}$. The set $Fl(W,\widetilde{E},V)$ is the
set of all generalized flags $W^{\prime}=\\{W^{\prime}_{\alpha}\\}$ which are
$\widetilde{E}$_-commensurable_ with $W$. This latter requirement spells out
as the following three conditions for each element $W^{\prime}$ of
$Fl(W,\widetilde{E},V)$:

  * •

the index set which parameterizes the generalized flag $W^{\prime}$ is the
same as the index set of $W$;

  * •

there exists a finite-dimensional subspace $Z\subsetneq V$ depending on
$W^{\prime}$, such that for any $\alpha$ we have
$W^{\prime}_{\alpha}+Z=W_{\alpha}+Z$ and $\dim(W_{\alpha}\cap
Z)=\dim(W^{\prime}_{\alpha}\cap Z)$;

  * •

the generalized flag $W^{\prime}$ is compatible with a basis
$\widetilde{E}^{\prime}$ of $V$, depending on $W^{\prime}$, such that
$\widetilde{E}^{\prime}$ differs from $\widetilde{E}$ by finitely many
vectors.

The set $Fl(W,\widetilde{E},V)$ has a natural structure of ind-variety. This
is explained in detail in [DP] (and in [IP] and [PT2]). Briefly, the ind-
variety structure on $Fl(W,\widetilde{E},V)$ arises as follows. Enumerate the
basis $\widetilde{E}$ by the set $\mathbb{Z}_{>0}$ and put $V_{i}:=\langle
e_{1},\ldots,e_{i}\rangle_{\mathbb{C}}$ for $i\in\mathbb{Z}_{>0}$. Each
intersection $W\cap V_{i}$ is a flag in $V_{i}$ of certain type
$\underline{d}_{i}=(d_{i}^{1},\ldots,d_{i}^{k_{i}})$, and this ordering of the
basis $\widetilde{E}$ induces embeddings

$Fl(\underline{d}_{i},V_{i})\lhook\joinrel\longrightarrow
Fl(\underline{d}_{i+1},V_{i+1}),$ (1)

called _strict standard extensions_ , such that
$Fl(W,\widetilde{E},V)=\varinjlim Fl(\underline{d}_{i},V_{i})$. The embeddings
(1) endow $Fl(W,\widetilde{E},V)$ with an ind-variety structure. In Section
Automorphism groups of ind-varieties of generalized flags below we recall the
definition of a strict standard extension.

Next we recall that if $T$ and $R$ are two (in general, infinite-dimensional)
vector spaces endowed with a non-degenerate pairing $\mathbf{p}\colon T\times
R\rightarrow\mathbb{C}$, then the Mackey group $G(T,R)$ is defined as

$G(T,R)=\\{\varphi\in GL(T)\mid\varphi^{*}(R)=R\\},$ (2)

see [M]. Here $\varphi^{*}\colon T^{*}\rightarrow T^{*}$ is the operator dual
(adjoint) to the operator $\varphi\colon T\rightarrow T$, and $R$ is
considered as a subspace of $T^{*}$ via the embedding $R\hookrightarrow T^{*}$
induced by the pairing $\mathbf{p}$. Equivalently, $G(T,R)$ can be defined as
the group

$\\{\psi\in GL(R)\mid\psi^{*}(T)=T\\}$ (3)

where $T$ is considered as a subspace of $R^{*}$ via $\mathbf{p}$. The
correspondence

$\varphi\mapsto\left(\varphi^{*}|_{R}\right)^{-1}$

is a canonical isomorphism between the groups (2) and (3). In what follows,
when writing $\varphi\in G(T,R)$ we will assume that $\varphi\in GL(T)$, and
will denote the operator $\varphi^{*}|_{R}$ by $\overline{\varphi}$. Note
that, given a subspace $A$ of $T$, one has
$\varphi(A)=\overline{\varphi}^{-1}(A^{\perp})$, where $A^{\perp}$ is the
annihilator of $A$ in $R$.

Consider again the ind-variety of generalized flags $Fl(W,\widetilde{E},V)$.
Define the spaces $V^{W}_{\widetilde{E}}$ and $V^{W}_{*\widetilde{E}}$ as

$\begin{split}V^{W}_{\widetilde{E}}&:=\bigcap\limits_{\alpha}\left(\left(\left(W_{\alpha}\right)_{*}\right)^{*}\oplus
U_{\alpha}\right),\\\
V^{W}_{*\widetilde{E}}&:=\bigcap\limits_{\alpha}\left(\left(W_{\alpha}\right)_{*}\oplus
U^{*}_{\alpha}\right),\end{split}$

where $\alpha$ runs over the indices parameterizing the generalized flag $W$,
and the spaces $U_{\alpha}$ are direct complements of the spaces $W_{\alpha}$,
i.e., $V=W_{\alpha}\oplus U_{\alpha}$, with the assumption that
$U_{\alpha}\cap\widetilde{E}$ is a basis of $U_{\alpha}$. Note that the spaces
$V^{W}_{\widetilde{E}}$, $V^{W}_{*\widetilde{E}}$ are necessarily infinite-
dimensional and there is a canonical non-degenerate pairing

$V^{W}_{\widetilde{E}}\times V^{W}_{*\widetilde{E}}\rightarrow\mathbb{C},$

therefore the group $G(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}})$ is well
defined.

An essential observation is that the spaces $V^{W}_{\widetilde{E}}$ and
$V^{W}_{*\widetilde{E}}$ depend only on the ind-variety
$Fl(W,\widetilde{E},V)$ and not on the specific point $W\in
Fl(W,\widetilde{E},V)$. This follows from the fact that, for each $\alpha$,
the spaces $((W_{\alpha})_{*})^{*}\oplus U_{\alpha}$ and
$(W_{\alpha})_{*}\oplus U^{*}_{\alpha}$ do not change when $W_{\alpha}$ is
replaced by a subspace $W_{\alpha}^{\prime}\subset W$ which is
$\widetilde{E}$-commensurable with $W_{\alpha}$, and $U_{\alpha}$ is replaced
by a direct complement $U_{\alpha}^{\prime}$ of $W_{\alpha}^{\prime}$
containing all but finitely many vectors from $\widetilde{E}\cap U_{\alpha}$.

Moreover, $GL(\widetilde{E},V)$ is a subgroup of
$G(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}})$. To see this, consider a
linear operator $\varkappa:V\to V$, $\varkappa\in GL(\widetilde{E},V)$, and
fix $\alpha$. There exist subspaces $W_{\alpha}^{\prime}\subset W_{\alpha}$,
$U_{\alpha}^{\prime}\subset U_{\alpha}$, such that
${\left.\kern-1.2pt\varkappa\right|_{W_{\alpha}^{\prime}}}=\mathrm{id}_{W_{\alpha}^{\prime}}$,
${\left.\kern-1.2pt\varkappa\right|_{U_{\alpha}^{\prime}}}=\mathrm{id}_{U_{\alpha}^{\prime}}$,
and $V=W_{\alpha}^{\prime}\oplus K\oplus U_{\alpha}^{\prime}$ for some
$\varkappa$-invariant finite-dimensional subspace $K\subset V$. Then
$(W_{\alpha}^{\prime})_{*}$, $(U_{\alpha}^{\prime})^{*}$ and $K^{*}$ are
$\varkappa^{*}$-invariant subspaces of $V^{*}$, and $(W_{\alpha})_{*}\oplus
U_{\alpha}^{*}=(W_{\alpha}^{\prime})_{*}\oplus
K^{*}\oplus(U_{\alpha}^{\prime})^{*}$ is a $\varkappa^{*}$-invariant subspace
of $V^{*}$. This shows that
$V^{W}_{*\widetilde{E}}=\bigcap_{\alpha}((W_{\alpha})_{*}\oplus
U_{\alpha}^{*})$ is a $\varkappa^{*}$-invariant subspace of $V^{*}$. The same
argument applied to the $\varkappa^{*}$-invariant subspace $V_{*}$ of $V^{*}$
implies that $V^{W}_{\widetilde{E}}$ is a
$({\left.\kern-1.2pt\varkappa^{*}\right|_{V_{*}}})^{*}$-invariant subspace of
$(V_{*})^{*}$. This allows to consider $\varkappa$ as an element of
$G(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}})$.

Next, if $W^{\prime}$ is any chain of subspaces in $V$ and $\varphi\colon
V^{W}_{\widetilde{E}}\rightarrow V^{W}_{\widetilde{E}}$ is any linear operator
from the group $G(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}})$, then

$\overline{\varphi}^{-1}\left({W^{\prime}}^{\perp}\right)^{\perp}\cap
V=\varphi((W^{\prime\perp})^{\perp})\cap V$ (4)

is a chain of subspaces in $V$. Here $W^{\prime\perp}$ is the chain in
$V^{W}_{*\widetilde{E}}$ consisting of the annihilators in
$V^{W}_{*\widetilde{E}}\subseteq V^{*}$ of the spaces $W_{\alpha}^{\prime}$,
and similarly $\overline{\varphi}^{-1}\left(W^{\prime\perp}\right)^{\perp}$,
$W^{\prime\perp\perp}$ are chains in $V^{W}_{\widetilde{E}}\subseteq V^{**}$.

In what follows, we use the notation ${W^{\prime}}^{\perp}$ for chains
perpendicular to $W^{\prime}$ also in appropriate subspaces of $V^{*}$
different from $V^{W}_{*\widetilde{E}}$, and indicate the respective subspace
as necessary. A similar convention applies to the notation
${W^{\prime}}^{\perp\perp}$. Moreover, we call the generalized flag $W$
_symmetric_ if the chain $W^{\perp}\subset V_{*}$ is the image of $W$ under a
linear isomorphism $V_{*}\simeq V$ sending $\widetilde{E}^{*}$ to
$\widetilde{E}$.

Theorem 1.1. __

  * a)

_If_ $W$ is not symmetric, then the group
$\mathrm{Aut}\,Fl(W,\widetilde{E},V)$ is isomorphic to
$P(GL(\widetilde{E},V)\cdot St_{W})$, where $St_{W}$ is the stabilizer of the
generalized flag $W$ in the group
$G(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}})$ under the action (4). Here
the product $\cdot$ is taken inside
$G(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}})$, and $P\cdot$ indicates
passage to the quotient modulo scalar operators.

  * b)

_If_ $W$ is symmetric, then the group $\mathrm{Aut}\,Fl(W,\widetilde{E},V)$ is
isomorphic to $P(GL(\widetilde{E},V)\cdot St_{W})\rtimes\mathbb{Z}_{2}$.

In Section Automorphism groups of ind-varieties of generalized flags we
present an explicit matrix realization of the group $GL(\widetilde{E},V)\cdot
St_{W}$. Let’s also point out that, since Theorem Automorphism groups of ind-
varieties of generalized flags implies that $GL(\widetilde{E},V)\cdot St_{W}$
is a group, we have $GL(\widetilde{E},V)\cdot St_{W}=St_{W}\cdot
GL(\widetilde{E},V)$.

Remark 1.2. In the case of a finite-dimensional flag variety $X$, every
automorphism of $X$ belonging to the connected component of unity in the
automorphism group has a fixed point on $X$. This no longer holds in the
generality of Theorem Automorphism groups of ind-varieties of generalized
flags. Indeed, if $X$ is the projective ind-space $Fl(W,E,V)$ for $\dim W=1$,
then $\mathrm{Aut}X=PGL(V)$ and it is well known that not every invertible
linear automorphism of $V$ has an eigenvector.$\bigcirc$

2\. Examples

Before we embark on proving Theorem Automorphism groups of ind-varieties of
generalized flags, we present five examples in which we compute the respective
group $GL(\widetilde{E},V)\cdot St_{W}$ from Theorem Automorphism groups of
ind-varieties of generalized flags. In all five cases our claims follow from
Theorem Automorphism groups of ind-varieties of generalized flags below, which
provides a matrix form of the group $GL(\widetilde{E},V)\cdot St_{W}$ in the
general case.

2.1. The case of an ind-grassmannian. Let’s consider the case where the
generalized flag $W$ has the form $0\subsetneq W\subsetneq V$, where $W$ is a
single proper subspace of $V$ (we slightly abuse notation by using the same
letter $W$ for a flag and a subspace). There are three cases: $\dim W<\infty$,
or $\dim W=\mathrm{codim}\,_{V}W=\infty$, or $\mathrm{codim}\,_{V}W<\infty$.
If $\dim W<\infty$, then $Gr(W,\widetilde{E},V)$ does not depend on the basis
$\widetilde{E}$, and the points of $Gr(W,\widetilde{E},V)$ are all subspaces
of $V$ of the same dimension as $W$. In this case we may write
$Gr(W,\widetilde{E},V)=Gr(\dim W,V)$. If $\mathrm{codim}\,_{V}W<\infty$ then
$Gr(W,\widetilde{E},V)$ depends as a set on the choice of the basis
$\widetilde{E}$, but up to isomorphism $Gr(W,\widetilde{E},V)$ depends only on
$\mathrm{codim}\,_{V}W$. Moreover, as an ind-variety $Gr(W,\widetilde{E},V)$
is isomorphic to $Gr(\mathrm{codim}\,_{V}W,V)$. If $\dim
W=\mathrm{codim}\,_{V}W=\infty$, the ind-variety $Gr(W,\widetilde{E},V)$ does
not depend up to isomorphism on the choice of both $W$ and $\widetilde{E}$. It
is known, see for instance [GS], that in this case $Gr(W,\widetilde{E},V)$ is
isomorphic to the Sato grassmannian introduced in [S].

If $\dim W<\infty$, then $V^{W}_{\widetilde{E}}=V$,
$V^{W}_{*\widetilde{E}}=V^{*}$ and $G(V,V^{*})=GL(V)$. Since $W$ is not
symmetric, Theorem Automorphism groups of ind-varieties of generalized flags
asserts that $\mathrm{Aut}\,Gr(W,\widetilde{E},V)\cong
P(GL(\widetilde{E},V)\cdot St_{W})$, and we note that here

$P(GL(\widetilde{E},V)\cdot St_{W})\cong PGL(V).$ (5)

Indeed, the action of $G(V,V^{*})=GL(V)$ on $Gr(W,\widetilde{E},V)$ via the
formula (4) is easily checked to coincide with the obvious action of $GL(V)$
on subspaces of $V$, and the isomorphism (5) is a consequence of the
transitive action of the group $GL(\widetilde{E},V)$ on finite-dimensional
subspaces of fixed dimension in $V$.

If $\mathrm{codim}\,_{V}W<\infty$, then $V^{W}_{\widetilde{E}}=(V_{*})^{*}$,
$V^{W}_{*\widetilde{E}}=V_{*}$ and $P(GL(\widetilde{E},V)\cdot St_{W})\cong
PGL(V_{*})$, i.e.,

$\mathrm{Aut}\,Gr(W,\widetilde{E},V)\cong PGL(V_{*}).$

In the case where $\dim W=\mathrm{codim}\,_{V}W=\infty$, we prove in the
Appendix that the group $G(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}})$ can
be represented as invertible
$(\mathbb{Z}\setminus\\{0\\})\times(\mathbb{Z}\setminus\\{0\\})$-matrices
which together with their inverses satisfy the condition: in the block
structure

$\left(\begin{array}[]{c|c}A&B\\\ \hline\cr C&D\end{array}\right)$ (6)

induced by the equality
$\mathbb{Z}\setminus\\{0\\}=\mathbb{Z}_{<0}\sqcup\mathbb{Z}_{>0}$, the matrix
$A$ has finitary rows (no restriction on the columns), the matrix $D$ has
finitary columns (no restriction on the rows) and the matrix $C$ is finitary.
The group $GL(\widetilde{E},V)\cdot St_{W}$ consists of matrices $M$ such that
$M$ and $M^{-1}$ have the form (6) and satisfy the additional condition
$\mathrm{rk}\,C=\mathrm{rk}\,C^{\prime}$ where

$M^{-1}=\left(\begin{array}[]{c|c}A^{\prime}&B^{\prime}\\\ \hline\cr
C^{\prime}&D^{\prime}\end{array}\right).$

Moreover, in this case $W$ is symmetric.

2.2. The case of $Fl(W,\widetilde{E},V)$, where $W=\\{W_{n}\\}$, $\dim
W_{n}=n$ for $n\in\mathbb{Z}_{>0}$, $\bigcup_{n}W_{n}=V$. In this case
$V^{W}_{\widetilde{E}}=V$, $V^{W}_{*\widetilde{E}}=V^{*}$, and the group
$GL(\widetilde{E},V)\cdot St_{W}$ can be identified with all invertible
$\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$-matrices with finitely many nonzero
entries below the main diagonal, cf. [P].

2.3. The case of $Fl(W,\widetilde{E},V)$, where
$\widetilde{E}=\\{e_{i}\\}_{i\in\mathbb{Z}}$ and $W=\\{W_{n}=\langle
e_{i},~{}i\leq n\rangle_{\mathbb{C}},~{}n\in\mathbb{Z}\\}$. Here

$V^{W}_{\widetilde{E}}=\left(\left(W_{0}\right)_{*}\right)^{*}\oplus\langle
e_{i},~{}i>0\rangle_{\mathbb{C}},~{}V^{W}_{*\widetilde{E}}=(W_{0})_{*}\oplus\left(\langle
e_{i},~{}i>0\rangle_{\mathbb{C}}\right)^{*}.$

In coordinate form, the vectors from $V^{W}_{\widetilde{E}}$ are columns
$(a_{j})_{j\in\mathbb{Z}}$ with $a_{j}=0$ for $j\gg 0$, and
$GL(\widetilde{E},V)\cdot St_{W}$ consists of all invertible
$\mathbb{Z}\times\mathbb{Z}$-matrices $M$ which, together with their inverses,
have finitely many nonzero entries below the main diagonal and satisfy the
condition $\mathrm{rk}\,C=\mathrm{rk}\,C^{\prime}$, where $C$ and $C^{\prime}$
are respectively the strictly lower-triangular parts of $M$ and $M^{-1}$.

2.4. The case of $Fl(W,\widetilde{E},V)$, where $W=\\{0\subset W_{1}\subset
W_{-1}\subset V\\}$, $\dim W_{1}=1$, $\mathrm{codim}\,_{V}W_{-1}=1$. Here
$\widetilde{E}$ can be ordered by any countable ordered set $I$ with a minimal
and a maximal element. We have $V^{W}_{\widetilde{E}}=V$,
$V^{W}_{*\widetilde{E}}=V_{*}$, and $GL(\widetilde{E},V)\cdot St_{W}$ consists
of all invertible $I\times I$-matrices which, together with their inverses,
satisfy the condition that each row and column is finitary.

2.5. The case of $Fl(W,\widetilde{E},V)$ for
$\widetilde{E}=\\{e_{i}\\}_{i\in\mathbb{Z}_{>0}\sqcup\mathbb{Z}_{<0}}$ and
$W=\\{W_{n}\\}$, where $W_{n}=\langle e_{1},\ldots,e_{n}\rangle_{\mathbb{C}}$
if $n\in\mathbb{Z}_{>0}$ and
$W_{n}=\langle\ldots,e_{n-2},e_{n-1},e_{1},e_{2},\ldots\rangle_{\mathbb{C}}$
if $n\in\mathbb{Z}_{<0}$. Then $V^{W}_{\widetilde{E}}=V$,
$V^{W}_{*\widetilde{E}}=V_{*}$, $I=\mathbb{Z}_{>0}\sqcup\mathbb{Z}_{<0}$ with
$k<l$ if $k\in\mathbb{Z}_{>0}$, $l\in\mathbb{Z}_{<0}$, and
$GL(\widetilde{E},V)\cdot St_{W}$ consists of all invertible $I\times
I$-matrices which, together with their inverses, satisfy the condition: each
row and column is finitary, and there are at most finitely many nonzero
entries below the main diagonal.

Note that in the cases Automorphism groups of ind-varieties of generalized
flags, Automorphism groups of ind-varieties of generalized flags, and
Automorphism groups of ind-varieties of generalized flags the generalized flag
$W$ is symmetric, while in the case Automorphism groups of ind-varieties of
generalized flags $W$ is not symmetric.

3\. More background

We need to recall some facts about linear embeddings of finite-dimensional
grassmannians and flag varieties. If $T$ is a finite-dimensional space and
$\underline{d}=\\{d^{1},\ldots,d^{i}\\}$ is a vector of positive integers
satisfying $d^{k}<d^{l}<\dim T$ for $k<l$, then $Fl(\underline{d},T)$ denotes
the variety of all flags of subspaces $T_{1}\subsetneq\ldots\subsetneq
T_{i}\subsetneq T$, where $\dim T_{j}=d^{j}$. If $\underline{d}$ consists of
one integer $d$, we write simply $Gr(d,T)$. If $T$ is endowed with a non-
degenerate symmetric or antisymmetric (symplectic) form, we write respectively
$FlO(\underline{d},T)$ and $FlS(\underline{d},T)$ for the varieties of
isotropic flags in $T$ with respect to the fixed form. We also write
$GrO(d,T)$ and $GrS(d,T)$. An isotropic flag has always length less or equal
$\frac{\dim T}{2}$ but, for convenience, in this paper by an _isotropic flag_
we will mean a flag of the form

$W_{1}\subset W_{2}\subset\ldots\subset W_{k}\subset
W^{\perp}_{k}\subset\ldots\subset W^{\perp}_{1},$

where the spaces $W_{1},~{}\ldots,~{}W_{k}$ are isotropic and the spaces
$W^{\perp}_{k},~{}\ldots,~{}W^{\perp}_{1}$ are coisotropic. All flag varieties
$FlO(\underline{d},T)$ and $FlS(\underline{d},T)$ are connected, except
$GrO(d,T)$ for $\dim T=2d$. In what follows, by $FlO(\underline{d},T)$ or
$GrO(\underline{d},T)$ we always denote a connected component.

The Picard group of any grassmannian or ind-grassmannian $Z$ is isomorphic to
$\mathbb{Z}$ except in the case of $GrO\left(\frac{\dim T}{2}-1,~{}T\right)$
for $2<\dim T\in 2\mathbb{Z}_{>0}$, and $\mathcal{O}_{Z}(1)$ always denotes
the ample generator of $\mathrm{Pic}\,Z$. In the case of $GrO\left(\frac{\dim
T}{2}-1,~{}T\right)$ for $2<\dim T\in 2\mathbb{Z}_{>0}$ we have
$\mathrm{Pic}\,GrO\left(\frac{\dim
T}{2}-1,~{}T\right)\simeq\mathbb{Z}\times\mathbb{Z}$.

The automorphism groups of the flag varieties $Fl(\underline{d},~{}T)$,
$FlO(\underline{d},~{}T)$, $FlS(\underline{d},~{}T)$ have been known for long
time. The fact that the automorphism group of the projective space
$\mathbb{P}^{n}$ is $PGL(n+1)$ goes back to the nineteenth century. Wei-Liang
Chow [C] extended this result to grassmannians in 1949. For a general flag
variety $X=Fl(\underline{d},~{}T)$, $FlO(\underline{d},~{}T)$,
$FlS(\underline{d},~{}T)$ the connected component of the identity in the
automorphism group $\mathrm{Aut}\,X$ is the respective group $PGL(T)$,
$SO(T)$, or $Sp(T)$, except in several cases listed by A.L. Onishchik in [O].
These special cases are $GrS(1,T)\simeq\mathbb{P}(T)$, the five dimensional
quadric $GrO(1,T)$ for $\dim T=7$, and $GrO\left(\frac{\dim
T-1}{2},~{}T\right)$ for $\dim T\in 2\mathbb{Z}_{>0}+1$.

In all cases, see for instance [A, Section 3.3], the full automorphism group
$G$ is always a semidirect product of its connected component of unity $G^{0}$
and a finite group of automorphisms of the Dynkin diagram of the Lie algebra
$\mathfrak{g}=\mathrm{Lie}\,G^{0}$ which keep fixed the simple roots of the
Lie algebra of the isotropy subgroup of a point on the respective flag
variety. In the present paper we only consider classical groups of large
enough rank, hence we can summarize the relevant part of this result as
follows:

  * •

for $\dim T\geq 3$,

$\mathrm{Aut}\,Fl(\underline{d},T)\simeq\begin{cases}PGL(T)\rtimes\mathbb{Z}_{2}\text{
if
}\underline{d}=(d^{0}=0,~{}d^{1},~{}d^{2},~{}\ldots,~{}d^{s},~{}d_{n}^{s+1}=\dim
T)\text{ satisfies }\\\ \text{ the condition
}d^{i}-d^{i-1}=d^{s+2-i}-d^{s+1-i}\text{ for all }1\leq s\leq n+1\\\
PGL(T)\text{ in all other cases. }\end{cases}$

  * •

for $\dim T\geq 8$, $\mathrm{Aut}\,FlO(\underline{d},T)\simeq O(T)$, except
for $GrO(\frac{\dim T}{2},T)$ where $\mathrm{Aut}\,GrO(\frac{\dim
T}{2},T)\simeq SO(T)$, and for $GrO(\frac{\dim T-1}{2},T)$ where
$\mathrm{Aut}\,GrO(\frac{\dim T-1}{2},T)\simeq SO(T^{\prime})$ for $\dim
T^{\prime}=\dim T+1$.

  * •

for $\dim T\geq 4$, $\mathrm{Aut}\,FlS(\underline{d},T)\simeq Sp(T)$, except
for $GrS(1,T)$ where $\mathrm{Aut}\,GrS(1,T)\simeq PGL(T)$.

Based on the above exception concerning $GrO\left(\frac{\dim
T-1}{2},~{}T\right)$ for $\dim T\in 2\mathbb{Z}_{>0}+1$, in what follows we
will automatically assume that this case is excluded from consideration. This
leads to no loss of generality as $GrO\left(\frac{\dim T-1}{2},~{}T\right)$ is
isomorphic to $GrO\left(\frac{\dim T^{\prime}}{2},~{}T^{\prime}\right)$ where
$T^{\prime}$ is an orthogonal space of dimension $\dim T+1$.

A nice class of embeddings of flag varieties
$Fl(\underline{d}_{1},T_{1})\hookrightarrow Fl(\underline{d}_{2},T_{2})$ for
$\dim T_{1}<\dim T_{2}$ is the class of standard extensions. Embeddings of
ind-grassmannians are discussed in detail in [PT1], and of arbitrary flag
varieties in [PT2]. Here we just recall a definition and a basic fact needed
to understand our arguments in Section Automorphism groups of ind-varieties of
generalized flags, Automorphism groups of ind-varieties of generalized flags,
Automorphism groups of ind-varieties of generalized flags.

Definition 3.1.

  * a)

An embedding

$\eta\colon Fl(d^{1}_{1},\ldots,d^{k}_{1},T)\hookrightarrow
Fl(d^{1}_{2},\ldots,d^{l}_{2},T^{\prime}),$

respectively,

$\eta\colon FlO(d^{1}_{1},\ldots,d^{k}_{1},T)\hookrightarrow
FlO(d^{1}_{2},\ldots,d^{l}_{2},T^{\prime}),$

respectively,

$\eta\colon FlS(d^{1}_{1},\ldots,d^{k}_{1},T)\hookrightarrow
FlS(d^{1}_{2},\ldots,d^{l}_{2},T^{\prime})$

is a _strict standard extension_ if there exists a surjection

$p\colon\\{0,1,\ldots,l,l+1\\}\to\\{0,1,\ldots,k,k+1\\}$

satisfying $p(i)\leq p(j)$ for $i<j$, together with an isomorphism

$V^{\prime}=V\oplus\widehat{W},$ (7)

satisfying $\widehat{W}=V^{\bot}$ in the orthogonal and symplectic case, and
subspaces $W_{i}\in\widehat{W}$ for $1\leq i\leq l$ with $W_{i}\subset W_{j}$
for $i<j$ such that $\eta$ has the form

$\eta\big{(}\\{0\\}=V_{p(0)}\subset V_{p(1)}\subset\ldots\subset
V_{p(l)}\subset V\big{)}=\big{(}\\{0\\}=V_{p(0)}\subset V_{p(1)}\oplus
W_{1}\subset\ldots\subset V_{p(l)}\oplus W_{l}\subset V^{\prime}\big{)}.$ (8)

Here the spaces $W_{1},\ldots,W_{l}$ are not required to be pairwise distinct,
while the spaces in the right-hand side of (8) are pairwise distinct by
definition. In the orthogonal or symplectic case we require that for each $i$
there is $j$ such that $W_{j}=W_{i}^{\perp}$.

  * b)

An embedding

$\eta\colon
Fl(d^{1}_{1},~{}\ldots,~{}d^{k}_{1},~{}T)\lhook\joinrel\longrightarrow
Fl(d^{1}_{2},~{}\ldots,~{}d^{l}_{2},~{}T^{\prime})$

is a _standard extension_ if after composing with one of the duality
isomorphisms

$Fl(d^{1}_{1},~{}\ldots,~{}d^{k}_{1},~{}T)\simeq Fl(\dim
T-d^{k}_{1},~{}\ldots,~{}\dim T-d^{1}_{1},~{}T^{*}),$
$Fl(d^{1}_{2},~{}\ldots,~{}d^{l}_{2},~{}T^{\prime})\simeq Fl(\dim
T^{\prime}-d^{l}_{2},~{}\ldots,~{}\dim
T^{\prime}-d^{1}_{2},~{}{T^{\prime}}^{*})$

$\eta$ becomes a strict standard extension. For varieties of isotropic flags,
standard extension and strict standard extension are synonyms.

The following theorem follows directly from Corollary 4.4 in [PT2].

Theorem 3.2. _Let $X_{1}\hookrightarrow X_{2}$ and $Y_{1}\hookrightarrow
Y_{2}$ be embeddings of flag varieties or of isotropic flag varieties, such
that $Y_{1}$ is the image of $X_{1}$ under some isomorphism $\varphi\colon
X_{2}\rightarrow Y_{2}$. Then the embedding $Y_{1}\hookrightarrow Y_{2}$ is a
standard extension whenever the embedding $X_{1}\hookrightarrow X_{2}$ is a
standard extension._

We need to recall also the notion of an ind-variety of isotropic generalized
flags. There are several cases. If a symmetric non-degenerate form
$(\cdot,\cdot)$ on $V$ is given, then there are two types of relevant bases
$\widetilde{E}$ we consider:

$\\{e_{i}\\}_{i\in\mathbb{Z}}\text{ with }(e_{i},~{}e_{-i})=1\text{ for all
}i,~{}(e_{i},~{}e_{k})=0\text{ for }k\neq-i,$

or

$\\{e_{i}\\}_{i\in\mathbb{Z}\setminus\\{0\\}}\text{ with
}(e_{i},~{}e_{-i})=1\text{ for all
}i\in\mathbb{Z}_{>0},~{}(e_{i},~{}e_{k})=0\text{ for }k\neq-i.$ (9)

In the case of a symplectic non-degenerate form on $V$, we consider bases
satisfying (9) (here $(e_{-i},e_{i})=-1$ for $i\in\mathbb{Z}_{>0}$). We refer
to bases as above as _isotropic bases_ of $V$. An _isotropic generalized flag_
in $V$ is by definition a generalized flag $W=\\{W_{\alpha}\\}$ in $V$ such
that each space $W_{\alpha}$ is either isotropic or coisotropic and
$W_{\alpha}$ belongs to $W$ if and only if
$W^{\perp}_{\alpha}=\\{w^{\prime}\in V\mid(w^{\prime},w)=0\text{ for all }w\in
W_{\alpha}\\}$ belongs to $W$. If $W$ is an $E$-compatible isotropic
generalized flag in $V$, then by definition, $FlO(W,\widetilde{E},V)$ in the
case of a symmetric form, or $FlS(W,\widetilde{E},V)$ in the case of a
symplectic form, consists of all isotropic generalized flags in $V$ which are
$\widetilde{E}$-commensurable with $W$. In all cases, $FlO(W,\widetilde{E},V)$
or, respectively, $FlS(W,\widetilde{E},V)$ is a direct limit of finite-
dimensional varieties of isotropic flags under standard extensions.

Finally, we should point out that in order to follow the proof of our main
results in Sections Automorphism groups of ind-varieties of generalized
flags–Automorphism groups of ind-varieties of generalized flags the readers
should first familiarize themselves with the results of the Appendix.

4\. Proof of Theorem Automorphism groups of ind-varieties of generalized flags
for ind-grassmannians

We start by proving Theorem Automorphism groups of ind-varieties of
generalized flags under the assumption that the generalized flag $W$ has
exactly one proper subspace, which we also denote by $W$. In what follows, we
write most of the time $Gr(W,E,V)$ instead of $Gr(W,\widetilde{E},V)$, where
$E=\widetilde{E}\cap W$. The set $E$ is a basis of $W$, and the ind-variety
$Gr(W,\widetilde{E},V)$ depends only on $E$ and not on the entire basis
$\widetilde{E}$. We feel that this notation makes the argument more
transparent. Also, the space $W$ is fixed and we write $V_{E}$ and $V_{*E}$
instead of $V^{W}_{\widetilde{E}}$ and $V^{W}_{*\widetilde{E}}$, respectively.

Recall that $Gr(W,E,V)$ is defined as the direct limit of strict standard
extensions

$Gr(d_{n},V_{n})\lhook\joinrel\longrightarrow Gr(d_{n+1},V_{n+1})$

for some $d_{n}$ and some subspaces $V_{n}\subset V$, $\dim V_{n}\geqslant n$,
$\varinjlim V_{n}=V$. If $d_{n}$ stabilizes at $k\in\mathbb{Z}_{>0}$ for large
$n$, then $\varinjlim Gr(d_{n},V_{n})=Gr(k,V)$ is the ind-grassmannian of all
$k$-dimensional subspaces in $V$. If $\dim{V_{n}}-d_{n}$ stabilizes at $k>0$,
then $Gr(W,E,V)$ is isomorphic to $Gr(k,V)$ as an ind-variety via the map

$\delta\colon Gr(W,E,V)\rightarrow Gr(k,V_{*}),$ $W^{\prime}\longmapsto
W^{\prime\perp}\subset V_{*},$

where $W^{\prime}$ denotes a variable point of $Gr(W,E,V)$ and
${W^{\prime}}^{\perp}:=\\{\alpha\in
V_{*}\mid\alpha(w^{\prime})=0~{}\forall~{}w^{\prime}\in W^{\prime}\\}$.

As we already mentioned, the automorphism groups of (finite-dimensional)
grassmannians have been described in the classical paper [C]. This description
implies that if

$\alpha\colon
Gr(d_{n},T_{n})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}Gr(d_{n},T^{\prime}_{n})$
(10)

is any isomorphism of grassmannians, where $\dim T_{n}=\dim T^{\prime}_{n}$,
then the pullback $\alpha^{*}S^{\prime}_{n}$ of the tautological bundle
$S^{\prime}_{n}$ on $Gr(d_{n},T^{\prime}_{n})$ is isomorphic to the
tautological bundle $S_{n}$ on $Gr(d_{n},T_{n})$, or to the bundle
$(\widetilde{T}_{n}/S_{n})^{*}$ in case $2d_{n}=n$, where $\widetilde{T}_{n}$
is the trivial bundle on $Gr(d_{n},T_{n})$ with fiber $T_{n}$. Moreover, if
$\alpha^{*}S^{\prime}_{n}\simeq S_{n}$, then the isomorphism (10) is
determined by the linear operator $\eta\colon(T^{\prime}_{n})^{*}\rightarrow
T^{*}_{n}$ which it induces via pullback: we have
$\alpha(T_{d_{n}})=\eta^{*}(T_{d_{n}})$ where $T_{d_{n}}\in Gr(d_{n},T_{n})$
and the operator $\eta^{*}$ is dual to $\eta$. Recall also that any global
endomorphism of the bundle $S_{n}$ or $(\widetilde{T}/S_{n})^{*}$ is scalar.

Set $X_{n}=Gr(d_{n},V_{n})$. Then $Gr(W,E,V)=\varinjlim X_{n}$. In the rest of
the argument we assume in addition that $\dim W=\mathrm{codim}\,_{V}W=\infty$
and that $d_{n}=n$, $\dim V_{n}=2n$. This is the case of the Sato
grassmannian. The remaining cases where $\dim W<\infty$ or
$\mathrm{codim}\,_{V}W<\infty$ have been considered in [P], and it has been
proved there that $\mathrm{Aut}\,Gr(W,E,V)\cong PGL(V)$ for $\dim W<\infty$,
and $\mathrm{Aut}\,Gr(W,E,V)\cong PGL(V_{*})$ for $\mathrm{codim}\,W<\infty$.
This is in agreement with Theorem Automorphism groups of ind-varieties of
generalized flags, as in these two cases $W$ is not symmetric and there is an
isomorphism

$GL(\widetilde{E},V)\cdot St_{W}\cong\begin{cases}GL(V)\text{ for }\dim
W<\infty\\\ GL(V_{*})\text{ for }\mathrm{codim}\,_{V}W<\infty.\end{cases}$
(11)

We leave the proof of (11) as an exercise to the reader.

Our first step will be to prove that the group of automorphisms of the ind-
variety $Gr(W,E,V)$ is a subgroup of $P(GL(\widetilde{E},V)\cdot
St_{W})\rtimes\mathbb{Z}_{2}$. Let $\widetilde{\varphi}\colon
Gr(W,E,V)\rightarrow Gr(W,E,V)$ be an arbitrary automorphism. Denote by
$Y_{n}$ the image of $X_{n}$, that is, $\widetilde{\varphi}(X_{n})=Y_{n}$.
Clearly, $Y_{n}$ is a grassmannian isomorphic to $X_{n}$. Moreover, the
embeddings $Y_{n}\hookrightarrow Y_{n+1}$ can be assumed to be strict standard
extensions by Theorem Automorphism groups of ind-varieties of generalized
flags.

Next, we have two possibilities: for some $n$, the isomorphism

$\widetilde{\varphi}_{n}:=\widetilde{\varphi}|_{X_{n}}\colon
X_{n}\longrightarrow Y_{n}$

has the property

$\widetilde{\varphi}^{*}_{n}S_{Y_{n}}\cong\left(\widetilde{V}_{n}/S_{n}\right)^{*},$
(12)

where $S_{n}$ is the tautological bundle on $X_{n}$ and $S_{Y_{n}}$ is the
tautological bundle on $Y_{n}$, or the property

$\widetilde{\varphi}^{*}_{n}S_{Y_{n}}\cong S_{n}$ (13)

for all $n$. Since our chains of embeddings $X_{n}\hookrightarrow X_{n+1}$ and
$Y_{n}\hookrightarrow Y_{n+1}$ are strict standard extensions, if the
isomorphism (12) holds for some $n$, it must hold for all $n$. However, if
this happens we can compose our automorphism $\widetilde{\varphi}$ with the
following automorphism which represents an element of $\mathbb{Z}_{2}$ in the
semidirect product from the statement of the theorem:

$\delta:Gr(W,\widetilde{E},V)\longmapsto
Gr(W^{\perp},\widetilde{E}^{*},V_{*})\longrightarrow Gr(W,\widetilde{E},V),$

where the left arrow sends $W$ to $W^{\perp}\subset V_{*}$ and the right arrow
is induced by an appropriate linear isomorphism

$V_{*}\longrightarrow V$

which maps $W^{\perp}$ to $W$. Option (13) certainly holds for the composition
$\delta\circ\widetilde{\varphi}$, so without loss of generality we can assume
in the rest of the argument that (13) holds.

The latter assumption implies

$\widetilde{\varphi}^{*}_{n}S^{*}_{Y_{n}}=S^{*}_{n}$ (14)

for all $n$. We write equality, as such an isomorphism is determined up to a
scalar $c_{n}$, and we assume that the scalars $c_{n}$ are chosen in a way
compatible with the restriction maps

${\widetilde{\varphi}^{*}_{n}S^{*}_{Y_{n}}|_{X_{n-1}}}$${\widetilde{\varphi}^{*}_{n-1}S^{*}_{Y_{n-1}}}$${S^{*}_{n}|_{X_{n-1}}}$${S^{*}_{n-1}.}$

Then, by our above remark that isomorphisms of grassmannians are encoded by
linear operators, the isomorphisms $\widetilde{\varphi}_{n}\colon
X_{n}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}Y_{n}$ are recovered by a
choice of compatible invertible linear operators

$\varphi^{*}_{n}\colon(V^{\prime}_{n})^{*}=\mathrm{H}^{0}(Y_{n},S^{*}_{Y_{n}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{H}^{0}(X_{n},S^{*}_{n})=V^{*}_{n}.$

The operators $\varphi^{*}_{n}$ are dual to unique operators
$\varphi_{n}=(\varphi^{*}_{n})^{*}\colon V_{n}\rightarrow V^{\prime}_{n}$
which we will also consider.

Since both chains of embeddings $X_{n}\hookrightarrow X_{n+1}$ and
$Y_{n}\hookrightarrow Y_{n+1}$ are strict standard extensions, we have
$\varinjlim V^{\prime}_{n}=V=\varinjlim V_{n}$, and consequently,
$\varprojlim(V^{\prime}_{n})^{*}=V^{*}=\varprojlim V^{*}_{n}$. Therefore, the
operators $\varphi^{*}_{n}$ induce a linear operator

$\Phi\colon V^{*}\rightarrow V^{*},$

and more precisely, a commutative diagram

${\begin{array}[]{c}V^{*}\\\ \vdots\end{array}}$${\begin{array}[]{c}V^{*}\\\
\vdots\end{array}}$${V^{*}_{n}}$${(V^{\prime}_{n})^{*}}$${V^{*}_{n-1}}$${(V^{\prime}_{n-1})^{*}}$${\vdots}$${\vdots}$$\scriptstyle{\Phi}$$\scriptstyle{\varphi^{*}_{n}}$$\scriptstyle{\varphi^{*}_{n-1}}$
(15)

The diagram (15) encodes the automorphism $\widetilde{\varphi}$ in the
following way. Let $W^{\prime}=\varinjlim(W^{\prime}\cap V_{n})$ be a point of
$Gr(W,E,V)$. Then

$\widetilde{\varphi}(W^{\prime})=\varinjlim\varphi_{n}(W^{\prime}\cap
V_{n})=\varinjlim\left(\left(\varphi_{n}^{*}\right)^{-1}\left((W^{\prime}\cap
V_{n})^{\perp}\right)^{\perp}\right),$ (16)

where the orthogonal to $W^{\prime}\cap V_{n}$ is taken in $V^{*}_{n}$ and the
orthogonal to $\left(\varphi_{n}^{*}\right)^{-1}\left(\left(W^{\prime}\cap
V_{n}\right)^{\perp}\right)$ is taken in $V^{\prime}_{n}$.

Next, it is essential to observe that the subspace $V_{*E}\subset V^{*}$ is
nothing but the subspace of global sections $\mu$ of the sheaf $\varprojlim
S^{*}_{n}$ satisfying the condition: the value of $\mu$ at any point
$W^{\prime}\in Gr(W,E,V)$ is a linear function on $W^{\prime}$ which belongs
to the subspace $W^{\prime}_{*}\subset{W^{\prime}}^{*}$. Here $W^{\prime}_{*}$
is defined in terms of a basis of $W^{\prime}$ which differs from $E$ by
finitely many vectors ($W^{\prime}_{*}$ is the span of the system of linear
functionals dual to such a basis). Note that, for each $W^{\prime}\in
Gr(W,E,V)$, the subspace $W^{\prime}_{*}\subset W^{\prime*}$ is determined
solely by the set $Gr(W,E,V)$, and hence the above subspace of global sections
$\mu$, i.e., the space $V_{*E}$, must be invariant under the operator $\Phi$.
Next, the subspace $V_{E}$ is the counterpart of the space $V_{*E}$ for the
ind-grassmannian $Gr(W^{\perp},\widetilde{E}^{*},V_{*})$, where $V_{*}$ is
defined by the fixed extension $\widetilde{E}$ of $E$ to a basis of $V$, and
$E^{*}$ are the linear functions in $\widetilde{E}^{*}$ which do not vanish on
$W$. Since $Gr(W^{\perp},\widetilde{E}^{*},V_{*})$ is isomorphic to
$Gr(W,E,V)$, the space $V_{E}$ is also invariant under the linear map
$\Phi^{\prime}\colon(V_{*})^{*}\rightarrow(V_{*})^{*}$ induced by the
automorphism $\widetilde{\varphi}$ of $Gr(W,E,V)$.

We have shown that any automorphism $\widetilde{\varphi}\colon
Gr(W,E,V)\rightarrow Gr(W,E,V)$ satisfying (13) induces a pair of invertible
operators $\varphi:=\Phi^{\prime}|_{V_{E}}\colon V_{E}\rightarrow V_{E}$ and
$\overline{\varphi}:=\Phi|_{V_{*E}}\colon V_{*E}\rightarrow V_{*E}$ which
determine an element of the Mackey group $G(V_{E},V_{*E})$. Moreover, if
$\mathrm{Aut}^{0}Gr(W,E,V)$ stands for the group of automorphisms of the ind-
variety $Gr(W,E,V)$ satisfying (13), then the assignment
$\widetilde{\varphi}\longmapsto\varphi$, or equivalently
$\widetilde{\varphi}\longmapsto\overline{\varphi}^{-1}$, defines an injective
group homomorphism

$\varepsilon\colon\mathrm{Aut}^{0}Gr(W,E,V)\hookrightarrow PG(V_{E},V_{*E}).$

We now check that the action of the image in $PG(V_{E},V_{*E})$ of
$\mathrm{Aut}^{0}Gr(W,E,V)$ is given by the formula

$\widetilde{\varphi}\left(W^{\prime}\right)=\overline{\varphi}^{-1}\left(W^{\prime\perp}\right)^{\perp}\cap
V,$

where $W^{\prime\perp}\subset V_{*E}$ and
$\overline{\varphi}^{-1}\left(W^{\prime\perp}\right)^{\perp}\subset(V_{*E})^{*}$
(clearly, $V\subset(V_{*E})^{*}$). To do this, recall that
$\widetilde{\varphi}(W^{\prime})$ is given by formula (16). Therefore, we need
to verify that

$\varinjlim\left(\left(\varphi^{*}_{n}\right)^{-1}\left(\left(W^{\prime}\cap
V_{n}\right)^{\perp}\right)^{\perp}\right)=\overline{\varphi}^{-1}\left(\left(\varprojlim\left(\left(W^{\prime}\cap
V_{n}\right)^{\perp}\right)\right)^{\perp}\right)\cap V.$ (17)

However, formula (17) follows from the observation that both its left-hand and
right-hand sides coincide with the subspace of vectors in $V$ which vanish on
$\left(\varphi^{*}_{n}\right)^{-1}\left(\left(W^{\prime}\cap
V_{n}\right)^{\perp}\right)$ whenever they belong to $V_{n}$; we consider here
vectors in $V_{n}$ as linear functions on $V^{*}_{n}$.

In conclusion, the image of $\mathrm{Aut}^{0}Gr(W,E,V)$ in $PG(V_{E},V_{*E})$
is a subgroup of $PG(V_{E},V_{*E})$ which acts on $Gr(W,E,V)$ via the formula
(17). Next, we note that since $GL(\widetilde{E},V)$ acts transitively on
$Gr(W,E,V)$, for any $\widetilde{\varphi}\in\mathrm{Aut}^{0}Gr(W,E,V)$ there
are $\varkappa\in GL(\widetilde{E},V)$ and
$\widetilde{\varphi}_{W}\in\mathrm{Aut}^{0}Gr(W,E,V)$ such that
$\widetilde{\varphi}_{W}(W)=W$ and
$\widetilde{\varphi}=\varkappa\circ\widetilde{\varphi}_{W}$. Indeed if
$\widetilde{\varphi}(W)=W^{\prime}$ for $W^{\prime}\in Gr(W,E,V)$, then
$\widetilde{\varphi}=\varkappa^{-1}\circ\varkappa\circ\widetilde{\varphi}$
where $\varkappa\in GL(\widetilde{E},V)$ satisfies $\varkappa(W^{\prime})=W$.
Consequently, $\varphi=\varkappa^{-1}\varphi_{W}$ for $\varphi_{W}\in St_{W}$,
in other words, the image of $\varepsilon$ lies in $P(GL(\widetilde{E},V)\cdot
St_{W})$.

To complete the proof, we need to show that any operator $\varphi\in
GL(\widetilde{E},V)\cdot St_{W}$ determines a well-defined automorphism of
$Gr(W,E,V)$. In the Appendix we introduce the degree $d(\varphi)$ of an
operator $\varphi$ and show that the space $\varphi\cdot
W^{\prime}=\overline{\varphi}^{-1}({W^{\prime}}^{\perp})^{\perp}\cap V$ is
$\widetilde{E}$-commensurable with $W$ for any $\varphi\in G(V_{E},V_{*E})$
such that $d(\varphi)=0$. Denote by $G^{0}(V_{E},V_{*E})$ the group of all
operators $\varphi\in G(V_{E},V_{*E})$ with $d(\varphi)=0$.

We have to convince ourselves that the action of $G^{0}(V_{E},V_{*E})$ on
$Gr(W,E,V)$ is by automorphisms of ind-varieties, and not merely by bijections
of the set $Gr(W,E,V)$. Let $V=W\oplus U$ where $\widetilde{E}\cap U$ spans
$U$. Recall that our fixed nested finite-dimensional spaces
$V_{1}\subset\ldots\subset V_{n}\subset V_{n+1}\subset\ldots$ are spanned by
elements of $\widetilde{E}$ and $Gr(W,E,V)=\varinjlim Gr(d_{n},V_{n})$ where
$d_{n}=\dim V_{n}\cap W$. The embeddings $X_{n}=Gr(d_{n},V_{n})\hookrightarrow
Gr(d_{n+1},V_{n+1})=X_{n+1}$ are strict standard extensions

$F_{d_{n}}\longmapsto F_{d_{n}}\oplus W_{n|n+1},$

where $W_{n+1}=W\cap V_{n+1}=W_{n}\oplus W_{n|n+1}$ for $W_{n}=W\cap V_{n}$,
and $W_{n|n+1}\cap\widetilde{E}$ spans $W_{n|n+1}$. Therefore we have a
decomposition $V_{*E}={\overline{W}_{n}}_{*}\oplus V^{*}_{n}\oplus U^{*}_{n}$,
where ${\overline{W}_{n}}_{*}$ is the $\widetilde{E}$-compatible direct
complement of $V^{*}_{n}$ within $W_{*}+V^{*}_{n}$ and $U^{*}_{n}$ is the dual
of the $\widetilde{E}$-compatible direct complement $U_{n}$ of $W+V_{n}$ in
$V$. Any invertible linear operator $\zeta\colon V_{*E}\rightarrow V_{*E}$
induces a decomposition

$V_{*E}=\zeta({\overline{W}_{n}}_{*})\oplus\zeta(V^{*}_{n})\oplus\zeta(U^{*}_{n}),$

and hence an operator

$\zeta_{n}\colon V^{*}_{n}:=V_{*E}/({\overline{W}_{n}}_{*}\oplus
U^{*}_{n})\longrightarrow
V_{*E}/(\zeta({\overline{W}_{n}}_{*})\oplus\zeta(U^{*}_{n}))=\zeta_{n}(V^{*}_{n}).$

Moreover, we have $\zeta=\left(\varprojlim\zeta_{n}\right)|_{V_{*E}}.$

Each linear operator $\zeta_{n}$ induces an isomorphism of grassmannians

$Gr(d_{n},\zeta_{n}(V^{*}_{n})^{*})\longrightarrow Gr(d_{n},V_{n}),$

and the varieties $Gr(d_{n},\zeta_{n}(V^{*}_{n})^{*})$ form an ind-variety
isomorphic to $Gr(W,E,V)$. We conclude that, if $\varinjlim
Gr(d_{n},\zeta_{n}(V^{*}_{n})^{*})=Gr(W,E,V)$ then $\zeta$ induces an
automorphism of the ind-variety $Gr(W,E,V)$. Set now
$\zeta:=\overline{\varphi}^{-1}$ for $\varphi\in G^{0}(V_{E},V_{*E})$. Then
$\zeta_{n}=\left(\varphi^{*}_{n}\right)^{-1}$ and
$\zeta_{n}(V^{*}_{n})^{*}=\varphi_{n}(V_{n})$. By the above mentioned result
from the Appendix, we know that $\varphi\cdot W^{\prime}\in Gr(W,E,V)$
whenever $\varphi\in G^{0}(V_{E},V_{*E})$ and $W^{\prime}\in Gr(W,E,V)$.
Therefore $\varinjlim Gr(d_{n},\varphi_{n}(V_{n}))=Gr(W,E,V)$, and we have
shown that $PG^{0}(V_{E},V_{*E})\subset\mathrm{im}\,\varepsilon$.

Finally, Theorem A.Automorphism groups of ind-varieties of generalized flags
(i) from the Appendix implies that $St_{W}\subset G^{0}(V_{E},V_{*E})$, and
hence that also $GL(\widetilde{E},V)\cdot St_{W}\subset G^{0}(V_{E},V_{*E})$.
Consequently, $GL(\widetilde{E},V)\cdot St_{W}=G^{0}(V_{E},V_{*E})$ and
$GL(\widetilde{E},V)\cdot St_{W}$ is a group. In particular,
$GL(\widetilde{E},V)\cdot St_{W}=St_{W}\cdot GL(\widetilde{E},V)$. The proof
is now complete as we have shown that
$\mathrm{im}\,\varepsilon=P(GL(\widetilde{E},V)\cdot St_{W}).$$\square$

Corollary 4.1. _If $\dim W=\mathrm{codim}\,_{V}W=\infty$, the group
$\mathrm{Aut}Gr(W,E,V)$ is isomorphic to the projectivization of the connected
component of unity in the group Japanese $GL(\infty)$._

Proof. We proved that $\mathrm{Aut}Gr(W,E,V)\cong PG^{0}(V_{E},V_{*E})$. As
pointed out in the Appendix, the group $G^{0}(V_{E},V_{*E})$ is isomorphic to
the connected component of unity in the group Japanese $GL(\infty)$.$\square$

5\. Proof of Theorem Automorphism groups of ind-varieties of generalized flags
in the general case

_Step 1. Reduction to the case of an automorphism which preserves all inverse
limits of dual tautological bundles._ Consider the ind-variety
$Fl(W,\widetilde{E},V)$ for our fixed generalized flag $W=\\{W_{\alpha}\\}$
compatible with the fixed basis $\widetilde{E}$ of $V$. Fix an exhaustion of
$Fl(W,\widetilde{E},V)$ as a direct limit $\varinjlim
Fl(\underline{d}_{n},V_{n})$ of strict standard extensions of finite-
dimensional flag varieties $Fl(\underline{d}_{n},V_{n})$. If the generalized
flag $W$ is symmetric, then the exhaustion can be chosen so that each flag
variety $Fl(\underline{d}_{n},V_{n})$ is symmetric, i.e., the vector
$(d_{n}^{0}=0,d^{1}_{n},d^{2}_{n},\ldots,d^{s}_{n},d_{n}^{s+1}=\dim V_{n})$
satisfies the condition

$d^{i}_{n}-d^{i-1}_{n}=d^{s+2-i}_{n}-d^{s+1-i}_{n}$

for all $1\leq s\leq n+1$. If $W$ is not symmetric, then infinitely many
vectors $\underline{d}_{n}$ are not symmetric, so (by passing to a subsequence
of the sequence $\\{n\\}$) we can assume that all vectors $\underline{d}_{n}$
are not symmetric.

Let $\widetilde{\varphi}\colon Fl(W,\widetilde{E},V)\rightarrow
Fl(W,\widetilde{E},V)$ be an automorphism. Set
$X_{n}:=Fl(\underline{d}_{n},V_{n})$ and let
$Y_{n}=\widetilde{\varphi}(X_{n})$. The varieties are finite-dimensional flag
varieties, and there are fixed isomorphisms
$\widetilde{\varphi}_{n}:=\widetilde{\varphi}|_{X_{n}}\colon
X_{n}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}Y_{n}$. Moreover, Theorem
Automorphism groups of ind-varieties of generalized flags implies that the
embeddings $Y_{n}\hookrightarrow Y_{n+1}$ are standard extensions. By
replacing $Y_{n}$ or $Y_{n+1}$ by its dual flag variety we can further assume
that the embeddings $Y_{n}\hookrightarrow Y_{n+1}$ are strict standard
extensions.

Denote by $S^{j}_{n}$ and $S^{j}_{Y_{n}}$ the tautological bundles of rank
$d^{j}_{n}$ on $X_{n}$ and $Y_{n}$, respectively. There are two possibilities:
either

$\widetilde{\varphi}^{*}_{n}S^{j}_{Y_{n}}\simeq S^{j}_{n}$ (18)

for all $n$, or

$\widetilde{\varphi}^{*}_{n}S^{j}_{Y_{n}}\simeq(\widetilde{V}_{n}/S^{s+1-j}_{n})^{*}$
(19)

for some $n=n_{0}$ and all $j$, $1<j\leqslant s$. Case (19) can occur only if
the vector $\underline{d}_{n_{0}}$ is symmetric. Moreover, then (19) will
necessarily hold for all $n>n_{0}$ due to the assumption that all embeddings
$X_{n}\hookrightarrow X_{n+1}$ and $Y_{n}\hookrightarrow Y_{n+1}$ are strict
standard extensions. In that case, we may as well assume that (19) holds for
all $n$.

Similarly to the case of an ind-grassmannian, if (19) holds for all $n$ we can
compose $\widetilde{\varphi}$ with an automorphism

$\delta:Fl(W,\widetilde{E},V)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}Fl(W^{\perp},\widetilde{E}^{*},V_{*})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}Fl(W,\widetilde{E},V)$

which maps $W$ first to $W^{\perp}$ and then maps $W^{\perp}$ to a point of
$Fl(W,\widetilde{E},V)$ under the isomorphism
$Fl(W^{\perp},\widetilde{E}^{*},V_{*})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}Fl(W,\widetilde{E},V)$
induced by an appropriate linear isomorphism $V_{*}\rightarrow V$ mapping
$\widetilde{E}^{*}$ to $\widetilde{E}$. Then the composition
$\delta\circ\widetilde{\varphi}$ satisfies the condition (18) for all $n$.
Therefore, in order to prove Theorem Automorphism groups of ind-varieties of
generalized flags, it suffices to prove that

$\mathrm{Aut}^{0}Fl(W,\widetilde{E},V)\simeq P(GL(\widetilde{E},V)\cdot
St_{W})$

where $\mathrm{Aut}^{0}Fl(W,\widetilde{E},V)$ denotes the group of
automorphisms of $Fl(W,\widetilde{E},V)$ satisfying (18) for all $n$.

_Step 2. From automorphisms to linear operators._ Since (18) holds, the
automorphism

$\widetilde{\varphi}\colon Fl(W,\widetilde{E},V)\rightarrow
Fl(W,\widetilde{E},V)$

induces automorphisms

$\widetilde{\varphi}_{\alpha}\colon Gr(W_{\alpha},E_{\alpha},V)\rightarrow
Gr(W_{\alpha},E_{\alpha},V)$ (20)

for each subspace $W_{\alpha}$ in $W$. In turn, the automorphisms (20) induce
linear operators, defined up to scalar multiples,

$\Phi_{\alpha}:V^{*}\rightarrow V^{*}$

as explained in Section Automorphism groups of ind-varieties of generalized
flags.

We now point out that the operators $\Phi_{\alpha}$ can be chosen to coincide,
i.e., to define a single operator

$\Phi:V^{*}\rightarrow V^{*}$

not depending on $\alpha$. This observation is justified as follows. Denote by
$S^{*}_{\alpha}$ the pullback to $Fl(W,\widetilde{E},V)$ of the inverse limit
of the tautological bundles $S^{*}_{n}$ on $Gr(W_{\alpha},E_{\alpha},V)$. Let
$\alpha^{\prime}<\alpha$ be two indices in the chain $W=\\{W_{\alpha}\\}$.
Then the morphism of inverse limits $S^{*}_{\alpha}\rightarrow
S^{*}_{\alpha^{\prime}}$, arising from the respective morphism of inverse
systems, induces a commutative diagram

${\mathrm{H}^{0}(Fl(W,\widetilde{E},V),S^{*}_{\alpha})}$${\mathrm{H}^{0}(Fl(W,\widetilde{E},V),S^{*}_{\alpha^{\prime}})}$${V^{*}}$${V^{*},}$$\scriptstyle{id}$
(21)

the vertical equalities being the identifications $V^{*}=\varprojlim
V^{*}_{n}=\varprojlim\mathrm{H}^{0}(Gr(d^{\gamma}_{n},V_{n}),S^{*}_{n})$
where$Gr(W_{\gamma},E_{\gamma},V)=\varinjlim Gr(d^{\gamma}_{n},V_{n})$ for
$\gamma=\alpha$ and $\gamma=\alpha^{\prime}$, respectively. Therefore, for a
fixed $\alpha$ and all $\alpha^{\prime}<\alpha$, the spaces of the form
$\mathrm{H}^{0}(S^{*}_{\alpha^{\prime}})$ are identified with $V^{*}$ in a way
compatible with the upper horizontal arrows of the diagrams (21). Next, the
following diagram is commutative

${\mathrm{H}^{0}(Fl(W,\widetilde{E},V),S^{*}_{\alpha})}$${\mathrm{H}^{0}(Fl(W,\widetilde{E},V),S^{*}_{\alpha^{\prime}})}$${V^{*}}$${V^{*}}$${V^{*}}$${V^{*}}$${\mathrm{H}^{0}(Fl(W,\widetilde{E},V),\widetilde{\varphi}^{*}S^{*}_{\alpha})}$${\mathrm{H}^{0}(Fl(W,\widetilde{E},V),\widetilde{\varphi}^{*}S^{*}_{\alpha^{\prime}}),}$$\scriptstyle{id}$$\scriptstyle{\Phi_{\alpha}}$$\scriptstyle{\Phi_{\alpha^{\prime}}}$$\scriptstyle{id}$

and this implies $\Phi_{\alpha}=\Phi_{\alpha^{\prime}}$.

_Step 3. Injective homomorphism
$\mathrm{Aut}^{0}Fl(W,\widetilde{E},V)\rightarrow
PG\left(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}}\right)$._ Note first that
each homomorphism

$\mathrm{Aut}^{0}Fl(W,\widetilde{E},V)\longrightarrow\mathrm{Aut}^{0}Gr(W_{\alpha},\widetilde{E},V)$
$\widetilde{\varphi}\longmapsto\widetilde{\varphi}_{\alpha}$

is injective, since $\widetilde{\varphi}_{\alpha}$ recovers
$\widetilde{\varphi}$ through the formula

$\widetilde{\varphi}\left(W^{\prime}\right)_{\alpha^{\prime}}=\Phi^{-1}\left({W^{\prime}_{\alpha^{\prime}}}^{\perp}\right)^{\perp}\cap
V$

for any $\alpha^{\prime}$, where
${W^{\prime}_{\alpha^{\prime}}}^{\perp}\subset
V^{W_{\alpha^{\prime}}}_{*E_{\alpha^{\prime}}}$,
$\Phi^{-1}\left({W^{\prime}_{\alpha^{\prime}}}^{\perp}\right)^{\perp}\subset
V^{W_{\alpha^{\prime}}}_{E_{\alpha^{\prime}}}$ and, as explained above,
$\Phi=\Phi_{\alpha}$.

Moreover, the automorphism $\Phi:V^{*}\rightarrow V^{*}$ induces pairs of
automorphisms

$\varphi_{\alpha}=\Phi^{\prime}_{\alpha}|_{V^{W_{\alpha}}_{E_{\alpha}}}\colon
V^{W_{\alpha}}_{E_{\alpha}}\longrightarrow
V^{W_{\alpha}}_{E_{\alpha}},~{}\overline{\varphi}_{\alpha}\colon
V^{W_{\alpha}}_{*E_{\alpha}}\longrightarrow V^{W_{\alpha}}_{*E_{\alpha}},$
(22)

compatible with all inclusions of the form
$V^{W_{\alpha^{\prime}}}_{E_{\alpha^{\prime}}}\subset
V^{W_{\alpha}}_{E_{\alpha}}$, $V^{W_{\alpha}}_{*E_{\alpha}}\subset
V^{W_{\alpha^{\prime}}}_{*E_{\alpha^{\prime}}}$ for $\alpha^{\prime}<\alpha$,
and such that $\varphi_{\alpha}\in
G\left(V^{W_{\alpha}}_{E_{\alpha}},V^{W_{\alpha}}_{*E_{\alpha}}\right)$. The
compatibility of $\varphi_{\alpha}$ and $\varphi_{\alpha^{\prime}}$ is clear,
while the compatibility of $\overline{\varphi}_{\alpha}$ and
$\overline{\varphi}_{\alpha^{\prime}}$ follows from the inclusion

$\varphi_{\alpha^{\prime}}^{-1}(W_{\alpha^{\prime}}^{\perp})^{\perp}\subset\varphi_{\alpha}^{-1}(W_{\alpha}^{\perp})^{\perp},$

where $W_{\alpha}^{\perp}\subset V^{W_{\alpha}}_{*E_{\alpha}}$,
$W_{\alpha^{\prime}}^{\perp}\subset
V^{W_{\alpha^{\prime}}}_{*E_{\alpha^{\prime}}}$,
$\varphi_{\alpha}^{-1}(W_{\alpha}^{\perp})^{\perp}\subset
V^{W_{\alpha}}_{E_{\alpha}}$,
$\varphi_{\alpha^{\prime}}^{-1}(W_{\alpha^{\prime}}^{\perp})^{\perp}\subset
V^{W_{\alpha^{\prime}}}_{E_{\alpha^{\prime}}}$. Therefore, we conclude that
the system of linear operators (22) determines a unique element in
$G\left(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}}\right)$ for
$V^{W}_{\widetilde{E}}=\bigcap\limits_{\alpha}V^{W_{\alpha}}_{E_{\alpha}}$ and
$V^{W}_{*\widetilde{E}}=\bigcap\limits_{\alpha}V^{W_{\alpha}}_{*E_{\alpha}}$,
and we obtain a homomorphism

$\varepsilon:\mathrm{Aut}^{0}Fl(W,\widetilde{E},V)\longrightarrow
PG\left(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}}\right).$

The fact that $GL(\widetilde{E},V)$ acts transitively on
$Fl(W,\widetilde{E},V)$ shows, by the same argument as in Section Automorphism
groups of ind-varieties of generalized flags, that the image of $\varepsilon$
lies in the subset $P\left(GL(\widetilde{E},V)\cdot St_{W}\right)$ of
$PG\left(V^{W}_{\widetilde{E}},V^{W}_{*\widetilde{E}}\right)$.

We have to prove that $\varepsilon$ is injective, i.e., that the image of
$\varepsilon$ determines all operators $\varphi_{\alpha}$ and
$\overline{\varphi}_{\alpha}$ as above. We will do this by recalling that each
operator $\varphi_{\alpha}:V^{W_{\alpha}}_{E_{\alpha}}\rightarrow
V^{W_{\alpha}}_{E_{\alpha}}$ admits a matrix as described in the Appendix. The
key point is that if an invertible operator on $V^{W_{\alpha}}_{E_{\alpha}}$
admits such a matrix, then this matrix is unique. Now the compatibility of the
operators $\varphi_{\alpha}$ under all inclusions
$V^{W_{\alpha^{\prime}}}_{E_{\alpha^{\prime}}}\hookrightarrow
V^{W_{\alpha}}_{E_{\alpha}}$ for $\alpha^{\prime}<\alpha$ implies that the
matrices of all operators $\varphi_{\alpha}$ coincide. Since each homomorphism

$\widetilde{\varphi}_{\alpha}\longmapsto\varphi_{\alpha}$

is injective according to Section Automorphism groups of ind-varieties of
generalized flags, we conclude that $\varepsilon$ is injective.

_Step 4. The image of $\varepsilon$._ As a final step of the proof, we need to
show that the image of $\varepsilon$ coincides with the set
$P(GL(\widetilde{E},V)\cdot St_{W})$. For this it suffices to prove that
$P(GL(\widetilde{E},V)\cdot St_{W})$ belongs to the image of $\varepsilon$,
i.e., that $GL(\widetilde{E},V)\cdot St_{W}$ acts on the ind-variety
$Fl(W,\widetilde{E},V)$ via the formula (4).

Pick an operator $\varphi\in GL(\widetilde{E},V)\cdot St_{W}$ and a flag
$W^{\prime}\in Fl(W,\widetilde{E},V)$. Since $W$ and $W^{\prime}$ are
$\widetilde{E}$-commensurable, one has $GL(\widetilde{E},V)\cdot
St_{W}=GL(\widetilde{E},V)\cdot St_{W^{\prime}}$. Therefore
$\varphi=\varkappa^{-1}\varphi_{W^{\prime}}$ for some $\varphi_{W^{\prime}}\in
St_{W^{\prime}}$ and some $\varkappa\in GL(\widetilde{E},V)$. Consequently,
$\varphi(W^{\prime})=\varkappa^{-1}(W^{\prime})$, i.e.
$\varphi^{-1}(W^{\prime})$ is $\widetilde{E}$-commensurable with $W$. In
conclusion, $\mathrm{im}\varepsilon=P(GL(\widetilde{E},V)\cdot St_{W})$, and
since $GL(\widetilde{E},V)\cdot St_{W}$ is a group we have also
$\mathrm{im}\varepsilon=P(St_{W}\cdot GL(\widetilde{E},V))$. The proof is
complete.

6\. An explicit matrix form of the group $GL(\widetilde{E},V)\cdot St_{W}$

Now we would like to characterize the product $GL(\widetilde{E},V)\cdot
St_{W}$ in terms of matrices. We start by describing a matrix form of the
group $St_{W}$.

Choose a linear order on $\widetilde{E}$ such that $e_{j}\in
W_{\alpha}\setminus W_{\alpha^{\prime}}$, $e_{k}\in W_{\alpha^{\prime}}$ for
$\alpha^{\prime}<\alpha$ implies $k<j$. It follows from the Appendix that for
each space $W_{\alpha}$ the stabilizer of $W_{\alpha}$ in
$G\left(V^{W_{\alpha}}_{E_{\alpha}},V^{W_{\alpha}}_{*E_{\alpha}}\right)$ under
the action (4) can be represented by infinite matrices (with rows and columns
ordered by the fixed order on $\widetilde{E}$) which, together with their
inverses, have the form

$\left(\begin{array}[]{c|c}A&B\\\ \hline\cr 0&D\end{array}\right)$ (23)

where $A$ has finitary rows, $D$ has finitary columns, and there are no
restrictions on the rows and columns of $B$. Certainly, the splitting (23)
depends on the space $W_{\alpha}$.

Next, the fact that the operators $\varphi_{\alpha}$ and
$\overline{\varphi}_{\alpha}$ form a system compatible with the inclusions
$V^{W_{\alpha^{\prime}}}_{E_{\alpha}}\subset V^{W_{\alpha}}_{E_{\alpha}}$,
$V^{W_{\alpha}}_{*E_{\alpha}}\subset V^{W_{\alpha^{\prime}}}_{*E_{\alpha}}$
for $\alpha^{\prime}<\alpha$, implies that all operators $\varphi_{\alpha}$,
respectively $\overline{\varphi}_{\alpha}$, are represented by the same
matrix. This means that all matrices (23) are just one matrix which satisfies
the above conditions for all spaces $W_{\alpha}$. Consequently, $St_{W}$
consists of matrices $M$ which, together with their inverses, have the form

$M=\begin{pmatrix}\ddots&&&&\\\ 0&\boxed{A_{\alpha^{\prime}}}&&&\\\
&&\ddots&&\\\ 0&0&&\boxed{A_{\alpha}}&&\\\ 0&0&&0&\ddots\\\ \end{pmatrix},$
(24)

the rows and columns of $M$ being ordered by the ordered set which orders the
elements of $\widetilde{E}$ as above, and the diagonal blocks being of size
$\dim(W_{\alpha_{2}}/W_{\alpha_{1}})\times\dim(W_{\alpha_{2}}/W_{\alpha_{1}})$
where $\alpha$ is one of the indices $\alpha_{1}$ and $\alpha_{2}$, and
$\alpha_{1}$ is the immediate predecessor of $\alpha_{2}$. This datum induces
a block structure on the entire matrix $M$, and all strictly lower-triangular
blocks are set to equal zero. Furthermore, since $M$ and $M^{-1}$ are subject
to the above additional conditions for all splittings arising from spaces
$W_{\alpha}$ of $W$, the matrices $M$ and $M^{-1}$ satisfy the following:

  * •

if there exists a rightmost highest block $L$ (this depends on the order on
$\widetilde{E}$, i.e., ultimately on the structure of the generalized flag
$W$), then there are no conditions on the rows and columns of $L$, all columns
of $M$ which intersect $L$ have at most finitely many nonzero entries outside
of $L$, all rows of $L$ which intersect $L$ have at most finitely many nonzero
entries outside of $L$, and all other rows and columns of $M$ have at most
finitely many nonzero entries in the region in which the first index is
greater then the first index of any entry of $L$ and the second index is
smaller than the second index of any entry of $L$;

  * •

if $M$ has no rightmost highest block, then all rows have at most finitely
many nonzero entries in direction to the left (from any point on) and all
columns have at most finitely many nonzero entries in the downward direction
(from any point on).

In order to pass to the full group $GL(\widetilde{E},V)\cdot St_{W}$, consider
matrices of the form (24) and replace the zeros in the lower-triangular part
by finitely many nonzero entries. Let’s refer to such matrices as
$W$-_aligned_. If $M$ is a $W$-aligned matrix, then every space $W_{\alpha}$
induces a splitting of $M$ into four blocks

$\left(\begin{array}[]{c|c}A&B\\\ \hline\cr C&D\end{array}\right)$ (25)

where $C$ is a finitary matrix. A $W$-aligned matrix M is _eligible_ if
$M^{-1}$ is also $W$-aligned, and for any $\alpha$ the splittings

$\left(\begin{array}[]{c|c}A&B\\\ \hline\cr C&D\end{array}\right)\text{ and
}\left(\begin{array}[]{c|c}A^{\prime}&B^{\prime}\\\ \hline\cr
C^{\prime}&D^{\prime}\end{array}\right)$

of $M$ and $M^{-1}$ respectively, satisfy the condition

$\mathrm{rk}\,C=\mathrm{rk}\,C^{\prime}.$

We leave it as an exercise to the reader to check that the condition of
eligibility is empty (i.e., it is automatically satisfied) if the ordered set
parameterizing the rows and columns of $A$ or $D$ is finite.

Our result in this section states as follows.

Theorem 6.1. _The group $GL(\widetilde{E},V)\cdot St_{W}$ is isomorphic to the
group of all eligible $W$-aligned matrices._

Proof. Any matrix in $GL(\widetilde{E},V)\cdot St_{W}$ has the form $M_{f}M$
where $M_{f}$ is an element of $GL(\widetilde{E},V)$ and $M$ is a matrix from
$St_{W}$ having the form (24). The necessary and sufficient condition for
$M_{f}M$ to lie in $GL(\widetilde{E},V)\cdot St_{W}$ is that the splitting
(25) of $M_{f}M$ for any space $W_{\alpha}$ in $W$ satisfies the condition of
Theorem A.Automorphism groups of ind-varieties of generalized flags. For all
$\alpha$ these conditions amount precisely to the requirement that the matrix
$M_{f}M$ be eligible.$\square$

7\. The isotropic case

Theorem 7.1. _Let $X=FlO(W,\widetilde{E},V)$ or $FlS(W,\widetilde{E},V)$ for
some isotropic generalized flag $W$ compatible with an isotropic basis
$\widetilde{E}$ of $V$. We assume in addition that $W$ does not have the form
$0\subset W_{1}\subset W^{\perp}_{1}\subset V$ where $\dim W_{1}=1$ in the
symplectic case, and that $\dim W^{\perp}_{1}/W_{1}\neq 2$ in the orthogonal
case if the basis $\widetilde{E}$ has the form (9). Then_

$\mathrm{Aut}X\simeq P(O(\widetilde{E},V)\cdot St^{O}_{W})$

_or_

$\mathrm{Aut}X\simeq P(Sp(\widetilde{E},V)\cdot St^{Sp}_{W}),$

_where $St^{O}_{W}$ and $St^{Sp}_{W}$ denote respectively the stabilizer of
$W$ in $O(V)$ and $Sp(V)$. The action of $O(V)$ on$FlO(W,\widetilde{E},V)$, or
respectively of $Sp(V)$ on $FlS(W,\widetilde{E},V)$, is induced by the linear
action of $O(V)$ or $Sp(V)$ on $V$._

Proof. _Step 1. The case of an isotropic ind-grassmannian._ The argument
starts in the same way as for ordinary ind-grassmannians. The embeddings
$X_{n}\hookrightarrow X_{n+1}$ are assumed to be standard extensions of
isotropic grassmannians, and by Theorem Automorphism groups of ind-varieties
of generalized flags the embeddings
$Y_{n}:=\varphi(X_{n})\xhookrightarrow{\eta_{n}}Y_{n+1}:=\varphi(X_{n+1})$ are
also standard extensions.

Next, in the isotropic case the isomorphisms (14) must hold, so we arrive to
diagram (15) or, equivalently, to a commutative diagram

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The isomorphism $V=\varinjlim V^{\prime}_{n}$ holds since the embeddings
$Y_{n}\hookrightarrow Y_{n+1}$ are standard extensions, and hence any
$V^{\prime}_{n}$ is a subspace of $V_{N}$ for some $N>n$. Furthermore, the
restriction of the fixed (symmetric or antisymmetric) form on $V$ to
$V_{n}^{\prime}$ is a form defining the corresponding isotropic grassmannian
in $V_{n}^{\prime}$. (Such a form is unique up to a scalar if $W$ satisfies
the conditions of the theorem.) Recall (from Section Automorphism groups of
ind-varieties of generalized flags) that the maps $\varphi_{n}$ are defined up
to compatible scalars $c_{n}$. The key observation is that there is a unique
choice of these scalars such that the maps $\varphi_{n}$ are isomorphisms of
orthogonal or, respectively, symplectic vector spaces. This follows from the
fact that the automorphism groups of our finite-dimensional isotropic
grassmannians $X_{n}$ and $Y_{n}$ are the respective orthogonal or symplectic
groups.

Set

$\varphi:=\varinjlim\varphi_{n}\colon V\rightarrow V.$

Then by construction $\varphi$ is an orthogonal or, respectively, symplectic
operator, and $\varphi$ determines our automorphism $\widetilde{\varphi}$
which acts on a point $W^{\prime}$ by formula (16). However, in the case
considered we have

$\varinjlim\left(\varphi_{n}\left(\left(W^{\prime}\cap
V_{n}\right)^{\perp}\right)^{\perp}\right)=\varinjlim\left(\varphi_{n}\left(W^{\prime}\cap
V_{n}\right)\right)=\varphi(W^{\prime}),$ (26)

hence the action of $\widetilde{\varphi}$ on $X$ is simply induced by the
action of $\varphi$ as a linear automorphism of $V$. This implies that there
is an injective homomorphism

$\varepsilon:\mathrm{Aut}GrO(W,\widetilde{E},V)\hookrightarrow PO(V)$

or, respectively,

$\varepsilon:\mathrm{Aut}GrS(W,\widetilde{E},V)\hookrightarrow PSp(V),$
$\widetilde{\varphi}\longmapsto\varphi$

and that the action of the image of $\varepsilon$ on $GrO(W,\widetilde{E},V)$
or $GrS(W,\widetilde{E},V)$ is induced by the linear action of $O(V)$ or,
respectively, $Sp(V)$ on $V$.

In the case of a general ind-variety of isotropic generalized flags, an
injective homomorphism $\varepsilon$ is constructed exactly as in Step 3 of
the proof of Theorem Automorphism groups of ind-varieties of generalized
flags. Here the images of all homomorphisms $\varepsilon_{\alpha}$ lie in
$PO(V)$ or, respectively, $PSp(V)$, therefore $\varepsilon$ is just the
homomorphism into the intersection of all images of $\varepsilon_{\alpha}$.
Moreover, the image of $\varepsilon$ coincides respectively with
$P(O(\widetilde{E},V)\cdot St^{O}_{W})$ or $P(Sp(\widetilde{E},V)\cdot
St^{Sp}_{W})$. Indeed, since $O(\widetilde{E},V)$ or, respectively,
$Sp(\widetilde{E},V)$, acts transitively on $FlO(W,\widetilde{E},V)$ or
$FlS(W,\widetilde{E},V)$, the image of $\varepsilon$ must be a subgroup of
$P(O(\widetilde{E},V)\cdot St^{O}_{W})$ or, respectively,
$P(Sp(\widetilde{E},V)\cdot St^{Sp}_{W})$. On the other hand, by the same
argument as in Section Automorphism groups of ind-varieties of generalized
flags, $O(\widetilde{E},V)\cdot St^{O}_{W}$ or, respectively,
$Sp(\widetilde{E},V)\cdot St^{Sp}_{W}$ acts on the ind-variety
$FlO(W,\widetilde{E},V)$ or $FlS(W,\widetilde{E},V)$ via the formula (16).
Since in our case the equality (26) holds, the proof is complete.$\square$

Corollary 7.2. _The group $O(\widetilde{E},V)\cdot St^{O}_{W}$, or
$Sp(\widetilde{E},V)\cdot St^{Sp}_{W}$, is isomorphic to the group of all
invertible $W$-aligned matrices $M$ with finitary rows and columns satisfying
$\overline{M}=\pm M^{-1}$, where $\overline{\,\,\cdot\,\,}$ denotes reflection
along the antidiagonal, plus corresponds to the case of $O$ and minus
corresponds to the case of $Sp$. _

Proof. The group $O(\widetilde{E},V)\cdot St^{O}_{W}$ or
$Sp(\widetilde{E},V)\cdot St^{Sp}_{W}$ is clearly the intersection of the
group $GL(\widetilde{E},V)\cdot St_{W}$ with $O(V)$ or, respectively, with
$Sp(V)$, and this implies the claim. Note that the condition $\overline{M}=\pm
M^{-1}$ makes $M$ eligible automatically.$\square$

## Appendix: on the structure of the group $G(V_{E},V_{*E})$

In this Appendix we have collected some basic statements about the Mackey
group $G(V_{E},V_{*E})$. The isomorphism of this group a the group sometimes
referred to as Japanese $GL(\infty)$ is stated at the end.

A.1. A lemma from linear algebra.

Lemma A.1. _Let $A$ be a countable-dimensional vector space, and $\mu\colon
A^{*}\to A$, $\nu\colon A^{*}\to A$ be linear maps such that
$\alpha(\mu(\beta))=\beta(\nu(\alpha))$ for all $\alpha,~{}\beta\in A^{*}$.
Then the dimensions of the images of $\mu$ and $\nu$ are finite and equal. _

Proof. We first check that $\dim\mathrm{im}\,\mu<\infty$. This will imply that
$\dim\mathrm{im}\,\nu$ is also finite because the conditions on $\mu$ and
$\nu$ are symmetric. Pick a basis $\\{f_{1},~{}f_{2},~{}\ldots\\}$ of $A$, and
let $\\{f_{1}^{*},~{}f_{2}^{*},~{}\ldots\\}$ be the dual system in $A^{*}$.
Assume that $\mathrm{im}\,\mu$ is infinite dimensional. Then there exist
linear functions $\lambda_{n}\in A^{*}$ for $n\in\mathbb{Z}_{>0}$, such that
$\dim\langle\mu(\lambda_{1}),~{}\ldots,~{}\mu(\lambda_{n})\rangle_{\mathbb{C}}=n$.
It is easy to see that these linear functions can be chosen so that
$\lambda_{n}(f_{i})=0$ for $i<n$ and $\lambda_{n}(f_{n})\neq 0$. Let
$B:=\langle\lambda_{n}|n\in\mathbb{Z}_{>0}\rangle_{\mathbb{C}}$. The formula
$(a,b)\mapsto b(a)$ for $a\in A$, $b\in B$ defines a non-degenerate pairing
between $A$ and $B$. By a result of G. Mackey [M, Lemma on p. 171], we may
assume without loss of generality that $\lambda_{n}=f_{n}^{*}$ for all $n$.

Since $\dim\mathrm{im}\,\mu=\infty$, for any $n\in\mathbb{Z}_{>0}$ there
exists $i_{n}\in\mathbb{Z}_{>0}$ such that $\mu(f_{i_{n}}^{*})$ is not
contained in $A_{n}:=\langle f_{1},\ldots,f_{n}\rangle_{\mathbb{C}}$. In other
words, there exist two infinite sequences of integers $1\leq
i_{1}<i_{2}<\ldots$ and $2\leq k_{1}<k_{2}<\ldots$ so that
$\mu(f_{i_{n}}^{*})\in A_{k_{n}}\setminus A_{k_{n}-1}$ for all $n\geq 1$.

Now we define a sequence $c_{1},~{}c_{2},~{}\ldots$ of complex numbers
inductively as follows. Set $c_{1}$ to be an arbitrary nonzero scalar. For
$n>1$ we let $c_{n}$ to be an arbitrary complex number such that

$\sum_{j=1}^{n}c_{j}f_{k_{j}}^{*}(\mu(f_{i_{n}}^{*}))=c_{n}f_{k_{n}}^{*}(\mu(f_{i_{n}}^{*}))+\sum_{j=1}^{n-1}c_{j}f_{k_{j}}^{*}(\mu(f_{i_{n}}^{*}))\neq
0.$

(Such a number exists because $f_{k_{n}}^{*}(\mu(f_{i_{n}}^{*}))\neq 0$.)
Finally, we define $\alpha\in A^{*}$ by the formula

$\alpha(f_{j})=\begin{cases}c_{n}&\text{if }j=k_{n}\text{ for some }n\geq
1,\\\ 0&\text{otherwise}.\end{cases}$

Then

$\begin{split}f_{i_{n}}^{*}(\nu(\alpha))=\alpha(\mu(f_{i_{n}}^{*}))=\sum_{j=1}^{n}c_{j}f_{k_{j}}^{*}(\mu(f_{i_{n}}^{*}))\neq
0\end{split}$

for any $n\in\mathbb{Z}_{>0}$. This means that $\nu(\alpha)\notin A_{i_{n}-1}$
for all $n\geq 1$, i.e., that $\nu(\alpha)$ can not be expressed as a finite
linear combination of $f_{1},~{}f_{2},~{}\ldots$, and this is a contradiction.

Hence, $\dim\mathrm{im}\,\mu<\infty$, and also $\dim\mathrm{im}\,\nu<\infty$.
It remains to check that $\dim\mathrm{im}\,\mu=\dim\mathrm{im}\,\nu$. Since
$\dim\mathrm{im}\,\mu<\infty$, there exist $n\in\mathbb{Z}_{>0}$ and linear
functions $\alpha_{1},~{}\ldots,~{}\alpha_{n}\in A^{**}$ such that
$\mu(a)=\sum_{i=1}^{n}\alpha_{i}(a)f_{i}$ for all $a\in A^{*}$. We note that
in fact $\alpha_{i}\in A$ for all $i$. Indeed,

$f_{j}^{*}(\mu(a))=f_{j}^{*}\left(\sum_{i=1}^{n}\alpha_{i}(a)f_{i}\right)=\alpha_{j}(a)=a(\nu(f_{j}^{*})),$

so $\alpha_{j}=\nu(f_{j}^{*})\in A$ for all $j$. It follows that there exist
$m\in\mathbb{Z}_{>0}$ and scalars $\alpha_{ij}\in\mathbb{C}$ so that
$\alpha_{j}=\sum_{i=1}^{m}\alpha_{ij}f_{i}$ for all $1\leq j\leq n$. Clearly,
$\dim\mathrm{im}\,\mu=\mathrm{rk}\,(\alpha_{ij})_{i,j=1}^{m,n}$. Similarly,
there exist $n^{\prime},~{}m^{\prime}\in\mathbb{Z}_{>0}$, vectors
$\beta_{j}\in A$, and scalars $\beta_{ij}\in\mathbb{C}$ so that
$\nu(b)=\sum_{j=1}^{n^{\prime}}\beta_{j}(b)f_{j}$ and
$\beta_{j}=\sum_{i=1}^{m^{\prime}}\beta_{ij}f_{i}$ for all $1\leq j\leq
n^{\prime}$. Then
$\dim\mathrm{im}\,\nu=\mathrm{rk}\,(\beta_{ij})_{i,j=1}^{m^{\prime},n^{\prime}}$.
However,
$\alpha_{ij}=f_{i}^{*}(\alpha_{j})=f_{i}^{*}(\nu(f_{j}^{*}))=f_{j}^{*}(\mu(f_{i}^{*}))=f_{j}^{*}(\beta_{i})=\beta_{ji}$
for all $i,~{}j$. This means that $m^{\prime}=n$, $n^{\prime}=m$, and the
matrices $(\alpha_{ij})_{i,j=1}^{m,n}$ and $(\beta_{ij})_{i,j=1}^{n,m}$ are
transpose to each other. The result follows.$\square$

A.2. A grading on the group $G(V_{E},V_{*E})$. We work in the setup of Section
Automorphism groups of ind-varieties of generalized flags. Recall that we
consider a generalized flag of the form $\\{0\\}\subset W\subset V$, where
$\dim W=\mathrm{codim}\,_{V}W=\infty$, and that $E=\widetilde{E}\cap W$ is a
basis of $W$. Let $U=\langle\widetilde{E}\setminus E\rangle_{\mathbb{C}}$. We
have $V_{E}=(W_{*})^{*}\oplus U$, $V_{*E}=W_{*}\oplus U^{*}$ and there is a
natural nondegenerate pairing $V_{E}\times V_{*E}\rightarrow\mathbb{C}$. The
group $G(V_{E},V_{*E})$ consists of invertible linear operators $\varphi\colon
V_{E}\rightarrow V_{E}$ such that
$\overline{\varphi}={\left.\kern-1.2pt\varphi^{*}\right|_{V_{*E}}}\colon
V_{*E}\rightarrow V_{*E}$ is a well-defined isomorphism.

Choose an ordering of $E$ via $\mathbb{Z}_{<0}$ and an ordering of
$\widetilde{E}\setminus E$ via $\mathbb{Z}_{>0}$. Set

$E^{k}:=\begin{cases}E\setminus\\{e_{k},\ldots,e_{-1}\\}\text{ for }k<0\\\
E\cup\\{e_{1},\ldots,e_{k}\\}\text{ for }k>0\end{cases}$

and $W^{k}:=\langle E^{k}\rangle_{\mathbb{C}}$. The disjoint union
$\bigsqcup\limits_{k\in\mathbb{Z}}Gr(W^{k},E^{k},V)$, where $E=E^{0}$,
$W=W^{0}$, is an ind-variety which we denote by $X$.

Proposition A.2. _The formula(4) defines an action of $G(V_{E},V_{*E})$ on $X$
by ind-variety automorphisms._

Proof. We first prove that $\varphi\cdot W^{\prime}\in X$ for any
$W^{\prime}\in X$ and any $\varphi\colon V_{E}\rightarrow V_{E}$, $\varphi\in
G(V_{E},V_{*E})$. Since in the definition of $X$, $W$ can be replaced by
$W^{\prime}$, it suffices to check that $\varphi\cdot W\in X$. Denote by
$\pi_{U}\colon V_{E}\to U$ and $\pi_{W_{*}}\colon V_{*E}\to W_{*}$ the
canonical projections onto the corresponding direct summands. Set also
$\mu:=\pi_{U}\circ\left(\varphi|_{(W_{*})^{*}}\right)$ and
$\nu:=\pi_{W_{*}}\circ(\overline{\varphi}|_{U^{*}})$.

We claim that $\dim\mathrm{im}\,\mu$ and $\dim\mathrm{im}\,\nu$ are finite
(and equal). Fix an isomorphism of vector spaces $\eta\colon W_{*}\to U$ and
the dual isomorphism $\eta^{*}\colon U^{*}\to(W_{*})^{*}$. Consider the linear
maps $\mu^{\prime}=\mu\circ\eta^{*}$ and $\nu^{\prime}=\eta\circ\nu$ from
$U^{*}$ to $U$. Then, for all $\alpha,~{}\beta\in U^{*}$, we have

$\begin{split}\alpha(\mu^{\prime}(\beta))&=\alpha(\mu(\eta^{*}(\beta)))=\alpha(\pi_{U}(\varphi(\eta^{*}(\beta))))=\alpha(\varphi(\eta^{*}(\beta)))=\overline{\varphi}(\alpha)(\eta^{*}(\beta))\\\
&=\pi_{W_{*}}(\overline{\varphi}(\alpha))(\eta^{*}(\beta))=\nu(\alpha)(\eta^{*}(\beta))=\eta(\nu(\alpha))(\beta)=\beta(\nu^{\prime}(\alpha)).\end{split}$

Hence, $\dim\mathrm{im}\,\mu^{\prime}=\dim\mathrm{im}\,\nu^{\prime}<\infty$
(and, consequently, $\dim\mathrm{im}\,\mu=\dim\mathrm{im}\,\nu<\infty$) by
Lemma A.Automorphism groups of ind-varieties of generalized flags.

Now, denote by $A_{0}$ the preimage of zero under the restriction of $\pi_{U}$
to $\varphi((W_{*})^{*})$. Obviously,
$A_{0}=(W_{*})^{*}\cap\varphi((W_{*})^{*})$. According to the proof of Lemma
A.Automorphism groups of ind-varieties of generalized flags, there exist
$n\in\mathbb{Z}_{>0}$ and $\alpha_{1},~{}\ldots,~{}\alpha_{n}\in W_{*}$ such
that

$\mu(\omega)=\pi_{U}(\varphi|_{(W_{*})^{*}}(\omega))=\sum\nolimits_{i=1}^{n}\omega(\alpha_{i})e_{i}$

for all $\omega\in(W_{*})^{*}$. A vector $\omega$ of $(W_{*})^{*}$ belongs to
$A_{0}$ if and only if $\mu(\omega)=0$, i.e, if $\omega(\alpha_{i})=0$ for
$1\leq i\leq n$. It follows immediately that $A_{0}\cap W$ has finite
codimension in $W$ and, moreover, given $w\in W$, one has $w\in A_{0}$ if and
only if $\alpha_{i}(w)=0$ for all $1\leq i\leq n$. Hence, $A_{0}\cap W$
contains all but finitely many vectors of $E$.

Recall that $\varphi\cdot W=\overline{\varphi}^{-1}(W^{\perp})^{\perp}\cap V$.
Clearly, $W^{\perp}=U^{*}$, and since
$\overline{\varphi}^{-1}(B)^{\perp}=\varphi(B^{\perp})$ for any subspace $B$
of $V_{*E}$, we conclude that

$\overline{\varphi}^{-1}(U^{*})^{\perp}=\varphi((U^{*})^{\perp})=\varphi((W_{*})^{*}).$

Hence $\varphi\cdot W=\varphi((W_{*})^{*})\cap V$. Since
$A_{0}\subset\varphi((W_{*})^{*})$, we have $A_{0}\cap W\subset\varphi\cdot
W$, and so $\varphi\cdot W$ contains all but finitely many vectors of $E$. On
the other hand,

$\varphi((W_{*})^{*})\subset(W_{*})^{*}\oplus\pi_{U}(\varphi((W_{*})^{*}))=(W_{*})^{*}\oplus\mathrm{im}\,\mu.$

This implies $\varphi\cdot W\subset W\oplus\mathrm{im}\,\mu$, and consequently
$\varphi\cdot W\in X$.

Next, a trivial modification of the argument at the end of the proof of
Theorem Automorphism groups of ind-varieties of generalized flags for ind-
grassmannians shows that $\varphi$ induces an automorphism of ind-varieties

$\widetilde{\varphi}:X\rightarrow X.$

Finally, the fact that
$\widetilde{\varphi^{\prime}\circ\varphi^{\prime\prime}}=\widetilde{\varphi^{\prime}}\circ\widetilde{\varphi}^{\prime\prime}$
follows from the realization of $\tilde{\varphi}$ as an inverse limit of
morphisms of finite-dimensional grassmannians where the property
$\widetilde{\varphi_{n}^{\prime}\circ\varphi_{n}^{\prime\prime}}=\widetilde{\varphi}_{n}^{\prime}\circ\widetilde{\varphi}_{n}^{\prime\prime}$
is obvious. $\square$

The ind-variety $X$ is the disjoint union of the ind-grassmannians
$Gr(W^{k},E^{k},V)$ for $k\in\mathbb{Z}$. These closed ind-subvarieties are
not stable with respect to this action.

Example A.3.

Let a _reverse sequence_ be a set parameterized by $\mathbb{Z}_{<0}$. The
vectors in $V_{E}$ and $V_{*E}$ can be written respectively as
$x=\sum\limits_{i>0}\omega_{i}e_{-i}+\sum\limits_{i>0}u_{i}e_{i}\in V_{E}$,
$y=\sum\limits_{i>0}w_{i}e^{*}_{-i}+\sum\limits_{i>0}\nu_{i}e^{*}_{i}\in
V_{*E}$ where $\omega=(\ldots,~{}\omega_{-2},~{}\omega_{-1})$ and
$w=(\ldots,~{}w_{-2},~{}w_{-1})$ are reverse sequences of complex numbers, and
$u=(u_{1},~{}u_{2},~{}\ldots)$, $\nu=(\nu_{1},~{}\nu_{2},~{}\ldots)$ are usual
sequences of complex numbers. In addition, $u$ and $w$ are finitary, i.e. have
at most finitely many nonzero entries. In what follows, we write simply
$(\omega,u)$ for vectors in $V_{E}$, and $(w,\nu)$ for vectors in $V_{*E}$.

Consider the shift (linear) operator $Sh\colon V_{E}\to V_{E}$

$Sh((\omega,~{}u))=((\ldots,~{}\omega_{-3},~{}\omega_{-2}),(\omega_{-1},~{}u_{1},~{}u_{2}\ldots)).$

One checks immediately that $Sh$ is an element of the group $G(V_{E},V_{*E})$
with $\overline{Sh}$ having the form

$\overline{Sh}((w,~{}\nu))=((\ldots,~{}w_{-2},~{}w_{-1},\nu_{1}),(\nu_{2},~{}\nu_{3},~{}\ldots)).$

Moreover, $Sh\cdot W$ belongs to $Gr(W^{1},E^{1},V)$, and in fact $Sh\cdot
Gr(W,E,V)=Gr(W^{1},E^{1},V)$. In addition, we note that

$\dim\pi_{U}(Sh((W_{*})^{*}))-\dim\pi_{U}(Sh^{-1}((W_{*})^{*}))=\dim\pi_{W_{*}}(\overline{Sh}(U^{*}))-\dim\pi_{W_{*}}(\overline{Sh}^{-1}(U^{*}))=1.$

$\bigcirc$

This example motivates the following.

Definition A.4. Let $\varphi\in G(V_{E},V_{*E})$. We define the integer

$d(\varphi)=d(\overline{\varphi}):=\dim\pi_{U}(\varphi((W_{*})^{*}))-\dim\pi_{U}(\varphi^{-1}((W_{*})^{*}))=\dim\pi_{W_{*}}(\overline{\varphi}(U^{*}))-\dim\pi_{W_{*}}(\overline{\varphi}^{-1}(U^{*}))$

to be the _degree_ of $\varphi$. (The latter equality follows immediately from
the fact that
$\dim\mathrm{im}\,(\pi_{U}\circ{\left.\kern-1.2pt\varphi\right|_{(W_{*})^{*}}})=\dim\mathrm{im}\,(\pi_{W_{*}}\circ{\left.\kern-1.2pt\overline{\varphi}\right|_{U^{*}}})$
checked in the proof of Proposition A.Automorphism groups of ind-varieties of
generalized flags.)

Proposition A.5. _Given $\varphi\in G(V_{E},V_{*E})$ and $k\in\mathbb{Z}$, one
has $\varphi\cdot Gr(W,E,V)=Gr(W^{k},E^{k},V)$ if and only if $d(\varphi)=k$._

Proof. It suffices to check that the condition $\varphi\cdot W\in
Gr(W^{k},E^{k},V)$ is equivalent to $d(\varphi)=k$. Since $\varphi\cdot W$
belongs to $X$, the condition $\varphi\cdot W\in Gr(W^{k},E^{k},V)$ is
equivalent to the equality

$\mathrm{codim}\,_{\varphi\cdot W}((\varphi\cdot W)\cap
W)-\mathrm{codim}\,_{W}((\varphi\cdot W)\cap W)=k.$

We have $\varphi\cdot W=\varphi((W_{*})^{*})\cap V$ and $W\subset(W_{*})^{*}$,
hence

$(\varphi\cdot W)\cap W\subset(\varphi((W_{*})^{*})\cap
V)\cap(W_{*})^{*}=A_{0}\cap V=A_{0}\cap W$

where $A_{0}=(W_{*})^{*}\cap\varphi((W_{*})^{*})$. Moreover, the opposite
inclusion is clear, so $(\varphi\cdot W)\cap W=A_{0}\cap W$. Therefore, we
need to prove that

$\mathrm{codim}\,_{\varphi((W_{*})^{*})\cap V}(A_{0}\cap
W)-\mathrm{codim}\,_{W}(A_{0}\cap W)=d(\varphi).$

As was shown in the proof of Proposition A.Automorphism groups of ind-
varieties of generalized flags, there exist $n\in\mathbb{Z}_{>0}$ and
$\alpha_{1},~{}\ldots,~{}\alpha_{n}\in W_{*}$ such that
$\pi_{U}(\varphi|_{(W_{*})^{*}}(\omega))=\sum_{i=1}^{n}\omega(\alpha_{i})e_{i}$
for all $\omega\in(W_{*})^{*}$, and
$A_{0}=\\{\omega\in(W_{*})^{*}\mid\omega(\alpha_{i})=0,~{}1\leq i\leq n\\}$.
Hence, $A_{0}\cap W=\\{w\in W\mid\alpha_{i}(w)=0,~{}1\leq i\leq n\\}$ and

$\mathrm{codim}\,_{(W_{*})^{*}}A_{0}=\mathrm{codim}\,_{W}(A_{0}\cap
W)=\dim\pi_{U}(\varphi((W_{*})^{*})).$ (27)

It remains to check that $\mathrm{codim}\,_{\varphi((W_{*})^{*})\cap
V}(A_{0}\cap W)=\dim\pi_{U}(\varphi^{-1}((W_{*})^{*})$. This is done
essentially by the same argument as above. Indeed, the argument in the proof
of Proposition A.Automorphism groups of ind-varieties of generalized flags now
shows that there exist $m\in\mathbb{Z}_{>0}$ and $\widetilde{\alpha}_{i}\in
W_{*}$, $1\leq i\leq m$, such that
$\pi_{U}(\varphi^{-1}|_{(W_{*})^{*}}(\omega))=\sum_{i=1}^{m}\omega(\widetilde{\alpha}_{i})e_{i}$
for $\omega\in(W_{*})^{*}$. This, together with fact that
$\dim\pi_{U}(\varphi^{-1}((W_{*})^{*}))<\infty$, implies that, given
$(\omega,u)\in V_{E}$, the condition $\varphi^{-1}((\omega,u))\in(W_{*})^{*}$
is equivalent to a finite system of linear equations on finitely many of the
coordinates of $(\omega,u)$. Thus,

$\begin{split}\mathrm{codim}\,_{\varphi((W_{*})^{*})\cap V}(A_{0}\cap
W)&=\mathrm{codim}\,_{\varphi((W_{*})^{*})}A_{0}=\mathrm{codim}\,_{\varphi((W_{*})^{*})}(\varphi((W_{*})^{*})\cap(W_{*})^{*})\\\
&=\mathrm{codim}\,_{(W_{*})^{*}}((W_{*})^{*}\cap\varphi^{-1}((W_{*})^{*}))=\dim\pi_{U}(\varphi^{-1}((W_{*})^{*})).\end{split}$

The proof is now complete.$\square$

Corollary A.6. _The definition of $d(\varphi)$ does not depend on the choice
of $W$ in the following sense: in the definition of $d(\varphi)$ one can
replace $W$ by any subspace $W^{\prime}\subset V$ which is
$\widetilde{E}$-commensurable with $W$, and $U$ — by any direct complement
$U^{\prime}$ of $W^{\prime}$ in $V$ such that
$\mathrm{codim}\,_{U^{\prime}}\langle
U^{\prime}\cap\widetilde{E}\rangle_{\mathbb{C}}<\infty$._

Proof. The claim follows directly from Proposition A.Automorphism groups of
ind-varieties of generalized flags as the ind-varieties $Gr(W^{k},E^{k},V)$
remain unchanged under a replacement $W\leadsto W^{\prime}$, $U\leadsto
U^{\prime}$.$\square$

Now we are ready to describe the structure of the $G(V_{E},V_{*E})$-action on
$X$ in more detail. In particular, we prove that the degree defines a grading
on the group $G(V_{E},V_{*E})$. In the next theorem we set
$X(k)=Gr(W^{k},E^{k},V)$ for $k\in\mathbb{Z}$. Then $X(0)=Gr(W,E,V)$ and
$X=\bigsqcup\limits_{k\in\mathbb{Z}}X(k)$.

Theorem A.7. _Given $d\in\mathbb{Z}$, set $G^{d}(V_{E},V_{*E}):=\\{\varphi\in
G(V_{E},V_{*E})\mid d(\varphi)=d\\}$. Then_

$\begin{split}&\text{{i)} $\varphi\cdot X(k)=X(k+d(\varphi))$ for all
$\varphi\in G(V_{E},V_{*E})$, $k\in\mathbb{Z}$};\\\ &\text{{ii)}
$d(\varphi\circ\varphi^{\prime})=d(\varphi)+d(\varphi^{\prime})$ for all
$\varphi,~{}\varphi^{\prime}\in G(V_{E},V_{*E})$};\\\ &\text{{iii)}
$G^{0}(V_{E},V_{*E})$ is a normal subgroup of $G(V_{E},V_{*E})$ whose cosets
are $G^{d}(V_{E},V_{*E})$, $d\in\mathbb{Z}$}.\\\ \end{split}$

Proof. i) We have $d(\varphi^{-1})=-d(\varphi)$ by the definition of degree.
This implies that if $\varphi\cdot X(0)=X(d)$ then $\varphi^{-1}\cdot
X(0)=X(-d)$. In particular, since $Sh\cdot X(n)=X(n+1)$ and consequently
$Sh^{k}\cdot X(n)=X(n+k)$, we obtain $Sh^{-k}\cdot X(n)=X(n-k)$ for all
$k,~{}n\in\mathbb{Z}$.

We claim that if $\varphi\cdot X(0)=X(0)$ then $\varphi\cdot X(k)=X(k)$ for
all $k\in\mathbb{Z}$. Indeed, assume that $\varphi\cdot X(0)=X(0)$ and pick an
integer $k$. We have $(Sh^{-k}\circ\varphi)\cdot
X(0)=Sh^{-k}\cdot(\varphi\cdot X(0))=Sh^{-k}\cdot X(0)=X(-k)$. By our
observation in the first paragraph, we conclude that
$(Sh^{-k}\circ\varphi)^{-1}\cdot X(0)=X(k),$ i.e,

$(\varphi^{-1}\cdot Sh^{-k})\cdot X(0)=\varphi^{-1}\cdot(Sh^{k}\cdot
X(0))=\varphi^{-1}\cdot X(k)=X(k).$

This clearly implies $\varphi\cdot X(k)=X(k)$.

Now, assume that $\varphi\cdot X(0)=X(d)$, i.e., that $d(\varphi)=d$. Then
$(Sh^{-d}\circ\varphi)\cdot X(0)=Sh^{-d}\cdot X(d)=X(0)$. Thus, by the
previous paragraph, for all $k\in\mathbb{Z}$ we have
$(Sh^{-d}\circ\varphi)\cdot X(k)=X(k)$, i.e.,

$\varphi\cdot X(k)=Sh^{d}\cdot X(k)=X(k+d).$

ii) Follows immediately from (i).

iii) Follows immediately from (ii).$\square$

A.3. Matrix realization. Consider the spaces $V_{E}=(W_{*})^{*}\oplus U$,
$V_{*E}=W_{*}\oplus U^{*}$, and assume that the isomorphism
$\eta:W_{*}\rightarrow U$ maps $e^{*}_{-i}$ to $e_{i}$ for
$i\in\mathbb{Z}_{>0}$. Then the pairing $V_{E}\times
V_{*E}\rightarrow\mathbb{C}$ is identified with the pairing

$(U^{*}\oplus U)\times(U^{*}\oplus
U)\to\mathbb{C}\colon((\gamma,~{}y),~{}(\varkappa,z))\mapsto\langle(\gamma,~{}y),~{}(\varkappa,z)\rangle=\gamma(z)+\varkappa(y),~{}y,~{}z\in
U,~{}\gamma,~{}\varkappa\in U^{*}.$

This allows us to denote the group $G(V_{E},V_{*E})=G(U^{*}\oplus
U,U^{*}\oplus U)$ simply by $G_{U}$.

We now present an explicit matrix realization of the group $G_{U}$. We have a
fixed basis $\\{e_{1},~{}e_{2},~{}\ldots\\}$ of $U$, and we identify
$U^{*}\oplus U$ with the space $\\{(\gamma,~{}y)\\}$ where
$\gamma=(\ldots,~{}\gamma_{-2},~{}\gamma_{-1})$ are reverse sequences and
$y=(y_{1},~{}y_{2},~{}\ldots)$ are finitary usual sequences. We consider
matrices whose rows and columns are parameterized by the ordered set
$\mathbb{Z}\setminus\\{0\\}=\mathbb{Z}_{<0}\sqcup\mathbb{Z}_{>0}$. Such a
matrix $M$ naturally splits into four blocks

$M=\left(\begin{array}[]{c|c}A&B\\\ \hline\cr C&D\end{array}\right).$

Let $J$ be the subset of such matrices satisfying the following conditions:

$\begin{split}&\text{$\bullet$ each row of $A$ and each column of $D$ is
finitary};\\\ &\text{$\bullet$ $C$ is a finitary matrix}.\\\ \end{split}$ (28)

Matrices from $J$ act on $U^{*}\oplus U$ via left multiplication: one
considers vectors of $U^{*}\oplus U$ as columns

$(\gamma,y)^{T}=\begin{pmatrix}\gamma^{T}\\\ y^{T}\end{pmatrix},$

and we have $(\gamma,y)^{T}\longmapsto\psi(\gamma,y)^{T}$ for $\psi\in J$.

Given a matrix $\psi\in J$, we let $\overline{\psi}$ denote the reflection of
$\psi$ with respect to its antidiagonal. Clearly,
$\overline{\overline{\psi}}=\psi$. One checks immediately that
$\overline{\psi}$ belongs to $J$, and that
$\overline{\psi_{1}\psi_{2}}=\overline{\psi_{2}}~{}\overline{\psi_{1}}$.
Furthermore, if
$\overline{\gamma}:=(\gamma_{-1},~{}\gamma_{-2},~{}\gamma_{-3},~{}\ldots)$ for
a reverse sequence
$\gamma=(\ldots,~{}\gamma_{-3},~{}\gamma_{-2},~{}\gamma_{-1})$ and
$\overline{y}:=(\ldots,~{}y_{3},~{}y_{2},~{}y_{1})$ for a finitary sequence
$y=(y_{1},~{}y_{2},~{}y_{3},~{}\ldots)$, then we have
$\overline{\psi(\gamma,~{}y)^{T}}=\overline{(\gamma,~{}y)^{T}}~{}\overline{\psi}$
for all $(\gamma,~{}y)\in U^{*}\oplus U$, where
$\overline{(\gamma,~{}y)^{T}}=(\overline{y},~{}\overline{\gamma})$.

Let $\widetilde{G}_{U}$ be the group of automorphisms of $V_{E}$ with matrices
$\psi$ such that $\psi$, $\psi^{-1}\in J$.

Theorem A.8. _The group $G_{U}$ coincides with $\widetilde{G}_{U}$._

Proof. First, we prove that $G_{U}$ is contained in $\widetilde{G}_{U}$. Pick
an operator $\varphi\in G_{U}$. Since
$\pi_{U}\circ({\left.\kern-1.2pt\varphi\right|_{U}})$ is a well-defined
operator on the countable-dimensional space $U$, it is represented by a
(unique) matrix $D=(d_{i,j})_{i,j\in\mathbb{Z}_{>0}}$ with finitary columns.

Next, Lemma A.Automorphism groups of ind-varieties of generalized flags
implies that
$\dim\mathrm{im}\,(\pi_{U}\circ({\left.\kern-1.2pt\varphi\right|_{U^{*}}}))<\infty$.
Moreover, according to the proof of Lemma A.Automorphism groups of ind-
varieties of generalized flags there exist $n\in\mathbb{Z}_{\geq 0}$ and
vectors $c_{1},~{}\ldots,~{}c_{n}\in U$ such that
$\pi_{U}({\left.\kern-1.2pt\varphi\right|_{U^{*}}}(\gamma))=\sum_{i=1}^{n}\gamma(c_{i})e_{i}$
for all $\gamma\in U^{*}$. This means that
$\pi_{U}\circ({\left.\kern-1.2pt\varphi\right|_{U^{*}}})$ can be represented
as a finitary matrix $C=(c_{i,-j})_{i,j\in\mathbb{Z}_{>0}}$, where
$c_{i}=\sum_{j\geq 1}c_{i,-j}e_{j}$.

Let $\pi_{U^{*}}$ be the projection operator $U^{*}\oplus U\rightarrow U^{*}$.
The operator $\pi_{U^{*}}\circ({\left.\kern-1.2pt\varphi\right|_{U}})$ is an
operator from the space $U$ of finitary sequences to the space $U^{*}$ of
arbitrary reverse sequences, and is therefore given by a matrix
$B=(b_{-i,j})_{i,j\in\mathbb{Z}_{>0}}$.

Finally, given $\gamma\in U^{*}$, we have
$\pi_{U^{*}}({\left.\kern-1.2pt\varphi\right|_{U^{*}}}(\gamma))(e_{i})=\varphi(\gamma)(e_{i})=\gamma(\overline{\varphi}(e_{i}))=\gamma(\pi_{U}(\overline{\varphi}(e_{i})))$.
There exist linear functions $\beta_{j}\in U^{*}$, $j\in\mathbb{Z}_{>0}$, such
that
$\pi_{U}(\overline{\varphi}(y))=(\beta_{1}(y),~{}\beta_{2}(y),~{}\ldots)$.
Since $\pi_{U}(\overline{\varphi}(e_{i}))$ belongs to $U$, there exists
$k(i)\in\mathbb{Z}_{>0}$ such that $\beta_{j}(e_{i})=0$ for $j>k(i)$, i.e.,
$\pi_{U}(\overline{\varphi}(e_{i}))=\sum_{j=1}^{k(i)}\beta_{j}(e_{i})e_{j}$.
Thus, $\pi_{U^{*}}\circ({\left.\kern-1.2pt\varphi\right|_{U^{*}}})$ can be
represented as a matrix with finitary rows
$A=(a_{-i,-j})_{i,j\in\mathbb{Z}_{>0}}$, where $a_{-i,-j}=\beta_{j}(e_{i})$.

It follows that $\varphi\in G_{U}$ can be represented by invertible matrices
from $L$. This conclusion applies also to the inverse operator $\varphi^{-1}$
by the same argument.

In order to verify the inclusion $\widetilde{G}_{U}\subset G_{U}$, let now
$\psi$ be a matrix from $L$ such that $\psi^{-1}\in L$. Then the linear
operator on $U^{*}\oplus U$ with matrix $\overline{\psi}$ is the restriction
to $U^{*}\oplus U$ of the operator dual to the operator defined by $\psi$.
Indeed,

$\begin{split}\langle\overline{\psi}(\gamma,y)^{T},(\gamma^{\prime},y^{\prime})^{T}\rangle&=\overline{\overline{\psi}(y,\gamma)^{T}}(\gamma^{\prime},y^{\prime})^{T}=\overline{(\gamma,y)^{T}}~{}\overline{\overline{\psi}}(\gamma^{\prime},y^{\prime})^{T}\\\
&=\overline{(\gamma,y)^{T}}\psi(\gamma^{\prime},y^{\prime})^{T}=\langle(\gamma,y)^{T},\psi(\gamma^{\prime},y^{\prime})^{T}\rangle,\end{split}$
(29)

where the three middle terms are products of two or respectively three
matrices. Moreover, the invertibility of $\overline{\psi}$ follows from the
equality $\overline{\psi}^{-1}=\overline{\psi^{-1}}$ which is a consequence of
the relation
$\overline{\psi_{1}\psi_{2}}=\overline{\psi}_{2}\overline{\psi}_{1}$. This
shows that $\psi$ determines an operator from $G_{U}$. $\square$

Remark A.9. Finally, the reader will notice that if in the definition of $J$
we replace the ordered set
$\mathbb{Z}\setminus\\{0\\}=\mathbb{Z}_{<0}\sqcup\mathbb{Z}_{>0}$ by any
ordered set which is the disjoint union $O_{1}\sqcup O_{2}$ of two linearly
ordered countable sets with the condition $r<s$ for $r\in O_{1}$, $s\in
O_{2}$, and define a group by imposing the conditions
(LABEL:formula:matrix_group), we will obtain a group isomorphic to $G_{U}$. Of
course, in such a setting the matrix of the operator
$\overline{\varphi}=\varphi^{*}|_{U^{*}\oplus U}$, for a given $\varphi\in
G_{U}$ with matrix $\psi$, will not be $\overline{\psi}$ as defined above, and
its form will depend on the choice of $O_{1}$ and $O_{2}$. Moreover, there
will also be analogues of the operator $Sh$, and we leave it to the reader to
define one.$\bigcirc$

The above remark is used in the proof of Theorem Automorphism groups of ind-
varieties of generalized flags presented in Section Automorphism groups of
ind-varieties of generalized flags.

Finally, using the matrix form of the group $G_{U}$ given by Theorem
A.Automorphism groups of ind-varieties of generalized flags, it is
straightforward to check that $G_{U}$ is nothing but the group of continuous
automorphisms of $U^{*}\oplus U$ as a Tate space, see, for instance, [K]. In
other words, $G_{U}$ is the group known as Japanese $GL(\infty)$, and
$G^{0}(U,U^{*})$ is simply the connected component of the identity in
$G(U,U^{*})=G_{U}$.

Conflict of interest statement. The authors declare that they have no conflict
of interest.

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Mikhail Ignatyev: Samara National Research University, Ak. Pavlova 1,
Samara443011, Russia

_E-mail address_<EMAIL_ADDRESS>
Ivan Penkov: Jacobs University Bremen, Campus Ring 1, Bremen 2759, Germany

_E-mail address_<EMAIL_ADDRESS>