On homogeneous spaces for diagonal ind-groups

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

The ind-group GL(∞) = lim → GL(n) = n≥1 GL(n) is a most natural direct limit algebraic group, and its locally projective homogeneous spaces are quite well studied by now, see for instance [3], [4], [7], [10].A larger class of direct limit algebraic groups are the so called diagonal ind-groups.A rather obvious such group, non-isomorphic to GL(∞), is the ind-group GL(2 ∞ ) = lim map x → x 0 0 x .
A general definition of a diagonal Lie algebra has been given by A. Baranov and A. Zhilinskii in [1], and this definition carries over in a straightforward way to classical Lie groups, producing the class of diagonal Lie groups.
Locally projective homogeneous ind-spaces of diagonal ind-groups have been studied much less extensively than those of GL(∞), see [4] and [2].In this paper, we undertake such a study for a class of diagonal ind-groups which we call strict diagonal ind-groups of type A. These ind-groups are characterized by supernatural numbers s, and are denoted GL(s).We consider reasonably general parabolic subgroups P ⊂ GL(s) and describe the homogeneous ind-space GL(s)/P as direct limits of embeddings G n−1 /P n−1 → G n /P n of usual ind-varieties.Our main result is an explicit formula for the so arising embeddings, and this formula is an analogue of the formula for standard extensions introduced in [10] (and used in a particular case in [3]).
The class of locally projective homogeneous ind-spaces of strict (and of general) diagonal ind-groups will require further detailed studies.In the current paper we restrict ourselves to the following application of the above explicit formula: we determine which locally projective homogeneous ind-spaces of GL(∞), i.e., ind-varieties of generalized flags [3], are also GL(s)-homogeneous for a given infinite supernatural number s.Furthermore, we also characterize explicitly direct products of ind-varieties of generalized flags which are GL(s)-homogeneous.
The very possibility of an ind-variety of generalized flags being a homogeneous space for GL(s), where s is an infinite supernatural number, is an interesting phenomenon, and can be seen as one possible motivation for our studies of GL(s)-homogeneous ind-spaces.Indeed, recall the following fact for a finite-dimensional algebraic group.If G is a centerless simple algebraic group of classical type and rank at least four and P is a parabolic subgroup, a well-known result of A. Onishchik [9] implies that the connected component of unity of the automorphism group of the homogeneous space G/P coincides with G, except in two special cases when G/P is a projective space and G is a symplectic group, and when G/P is a maximal orthogonal isotropic grassmannian and G is an orthogonal group of type B. Consequently, unless G/P is a projective space or a maximal isotropic grassmannian, G/P cannot be a homogeneous G ′ -space for a centerless algebraic group G ′ ∼ = G.
The explanation of why the situation is very different if one replaces G by the ind-group GL(∞), is that, as shown in [7], the automorphism group of an ind-variety of generalized flags is much larger than GL(∞).In this way, our results provide embeddings of GL(s) into such automorphism groups, with the property that the action of GL(s) on the respective ind-variety of generalized flags is transitive.As a corollary we obtain that a "generic" ind-variety of generalized flags is GL(s)-homogeneous also for any ind-group GL(s).This statement is in some sense opposite to the classical statement in the finite-dimensional case.
The paper is organized as follows.Sections 2, 3, 4 are devoted to preliminaries.We start by introducing the ind-groups GL(s) where s is a supernatural number.We then discuss Cartan, Borel, and parabolic ind-subgroups of GL(s).In Section 3 we review the notions of linear embedding of flag varieties and standard extension of flag varieties, and in Section 4 we recall the necessary results on ind-varieties of generalized flags.
In Section 5 we prove our explicit formula for embeddings of partial flag varieties GL(n)/Q ֒→ GL(dn)/P induced by pure diagonal embeddings GL(n) ֒→ GL(dn).In Section 6 we use this formula to describe all GL(s)-homogeneous ind-varieties of generalized flags.Finally, in Section 7 we characterize direct products of ind-varieties of generalized flags, which are GL(s)-homogeneous.
Acknowledgement.The work of I. P. was supported in part by DFG Grant PE 980/8-1.
2. The ind-group GL(s) 2.1.Direct systems associated to a supernatural number.Throughout this paper we consider a fixed supernatural number s, in other words where P is a (possibly infinite) set of prime numbers and α p is either a positive integer or ∞.Moreover, we suppose that s is infinite, hence at least one of the exponents α p is infinite or the set P is infinite.By D(s) we denote the set of finite divisors of s.
Let A be a direct system of sets with injective maps.We say that A is associated to the supernatural number s if the sets in A A(s), s ∈ D(s) are parametrized by the finite divisors of s, and the injective maps Definition 2.1.We call exhaustion of s any sequence {s n } n≥1 of integers such that • s n ∈ D(s) for all n, • s n divides s n+1 for all n, • any s ∈ D(s) is a multiple of s n for some n.
Lemma 2.2.Let {s n } n≥1 be an exhaustion of s.Then L(A) coincides with the limit of the inductive system formed by the sets A(s n ) and the maps δ n = δ sn,s n+1 : A(s n ) ֒→ A(s n+1 ). Proof.Straightforward.
According to the lemma, the limit L(A) can be described in terms of an exhaustion • In the case where A(s) are vector spaces and the maps δ s,s ′ are linear, then L(A) is the direct limit in the category of vector spaces.• In the case where A(s) are algebraic varieties and the maps δ s,s ′ are closed embeddings, the limit L(A) is an ind-variety as defined in [11] and [8].• In the case where A(s) are algebraic groups and the maps δ s,s ′ are group homomorphisms, the limit is both an ind-variety and a group.It is in particular an ind-group1 .
2.2.Definition of the groups GL(s) and SL(s).Whenever s, s ′ are two positive integers such that s divides s ′ , we have a diagonal embedding ).
We refer to the embeddings δ s,s ′ as strict diagonal embeddings.A more general definition of diagonal embeddings is given, at the Lie algebra level, in [1].
The groups GL(s) (for s ∈ D(s)) and the maps δ s,s ′ (for all pairs of integers s, s ′ ∈ D(s) such that s divides s ′ ) form a direct system.By definition, the ind-group GL(s) is the limit of this direct system.
The group GL(s) can be viewed as the group of infinite Z >0 × Z >0 -matrices consisting of one diagonal block of size equal to any (finite) divisor s of s, repeated infinitely many times along the diagonal: Similarly, we define SL(s) as the limit of the direct system formed by the groups SL(s) and the same maps δ s,s ′ .In fact, SL(s) is the derived group of GL(s).By gl(s) and sl(s), we denote the Lie algebras of GL(s) and SL(s), respectively.Thus sl(s) = [gl(s), gl(s)].
Remark 2.3.Lemma 2.2 shows that the group GL(s) can be obtained through any exhaustion where {s n } n≥1 is an exhaustion of s (see Definition 2.1).However, the ind-group GL(s) has various other exhaustions.If we set −→ GL(s n+1 )} considered above.This yields an equality We say that two exhaustions G = n G n = n G ′ n of a given ind-group are equivalent if there are n 0 ≥ 1 and a commutative diagram  Lemma 2.7], the embeddings η n and ξ n are diagonal, in the sense that there is an isomorphism of sl(s n )-modules and an isomorphism of g kn -modules for some triples of nonnegative integers (t, r, s) and (t ′ , r ′ , s ′ ), where V n and W n denote the natural representations of sl(s n ) and g kn , and C is a trivial representation.Also since δ sn,s n+1 is strict diagonal, we have an isomorphism of sl(s n )-modules Arguing by contradiction, assume that g kn is not of type A. Then [1, Proposition 2.3] implies that t = r.Moreover, t ′ + r ′ > 0 since otherwise V n+1 would be a trivial representation of sl(s n ).Altogether this implies that V * n is isomorphic to a direct summand of V n+1 considered as an sl(s n )-module, which is impossible in view of (2.2).We conclude that g kn is of type A for all n.
Moreover, from (2.2), we obtain s = s ′ = 1 and either r = r ′ = 0 or t = t ′ = 0. Up to replacing g kn = sl(W n ) by sl(W * n ), we can assume that r = r ′ = 0, and so g kn ∼ = sl(s ′ kn ) for some integer such that s n |s ′ kn , s ′ kn |s n+1 , and the embedding g kn ֒→ g k n+1 is induced by .By iterating the reasoning, we obtain an exhaustion {s ′ n } n≥1 of s such that the exhaustions sl(s) = n g n and sl(s) = n sl(s ′ n ) are equivalent.This shows (a).(b) From (a) it follows that for every n, the derived group (G n , G n ) is isomorphic to SL(s n ) and, after identifying (G n , G n ) with SL(s n ) and (G n+1 , G n+1 ) with SL(s n+1 ), the map (G n , G n ) ֒→ (G n+1 , G n+1 ) becomes the restriction of δ sn,s n+1 .This implies that G n is either isomorphic to SL(s n ) or to GL(s n ).For n ≥ 1 large enough, G n has to contain the center Z(GL(s)), which is isomorphic to C * .Since the connected component of the center of SL(s n ) is trivial, this forces ), we deduce that this embedding G n ֒→ G n+1 coincides with δ sn,s n+1 : GL(s n ) ֒→ GL(s n+1 ) after suitably identifying G n with GL(s n ) and G n+1 with GL(s n+1 ).. The following statement is a corollary of the classification of general diagonal Lie algebras [1].We give a proof for the sake of completeness.
Proposition 2.5.(a) If s and s ′ are two different infinite supernatural numbers, then the ind-groups GL(s) and GL(s ′ ) (resp.SL(s) and SL(s ′ )) are not isomorphic.
(b) If s is an infinite supernatural number, then GL(s) is not isomorphic to GL(∞), and SL(s) is not isomorphic to SL(∞).
Proof.(a) Since SL(•) is the derived group of GL(•), it suffices to establish the claim concerning SL(s) and SL(s ′ ).Assume there is an isomorphism of ind-groups ϕ : SL(s ′ ) → SL(s).Then any exhaustion {s ′ n } of s ′ yields an exhaustion SL(s) = n ϕ(SL(s ′ n )) of the group SL(s), and Lemma 2.4 implies s = s ′ , a contradiction.

2.3.
Parabolic and Borel subgroups.An ind-subgroup H ⊂ GL(s) is said to be a (locally splitting) Cartan subgroup if there is an exhaustion GL(s) = n G n by classical groups such that G n ∩ H is a Cartan subgroup of G n for all n.For instance, the subgroup of invertible periodic diagonal matrices in the realization (2.1) is a Cartan subgroup of GL(s).
If P is an ind-subgroup of GL(s), then the quotient GL(s)/P is an ind-variety obtained as the direct limit of the quotients GL(s)/P(s) for s ∈ D(s).
For the purposes of this paper, we say that an ind-subgroup P ⊂ GL(s) is a parabolic subgroup if there exists an exhaustion GL(s) = n G n by classical groups such that G n ∩P is a parabolic subgroup of G n for all n (cf.[2]).This implies in particular that the indvariety GL(s)/P is locally projective as it has an exhaustion GL(s)/P = n G n /(G n ∩ P) by projective varieties.If, in addition, the unipotent radical of G n ∩ P is contained in the unipotent radical of G n+1 ∩ P for every n, then we say that P is a strong parabolic subgroup.
An ind-subgroup B ⊂ GL(s) is said to be a Borel subgroup if it is locally solvable and parabolic.This means equivalently that there is an exhaustion GL(s) = n G n as above for which G n ∩ B is a Borel subgroup of G n for all n.Note that a Borel subgroup is necessarily a strong parabolic subgroup.
Lemma 2.6.A subgroup G ′ of GL(s) is a Cartan (respectively, parabolic or Borel) subgroup of G if and only if there is an exhaustion {s n } n≥1 of s such that for every n the intersection G ′ ∩GL(s n ) is a Cartan (respectively, parabolic or Borel) subgroup of GL(s n ).
Proof.This follows from Lemma 2.4.
The following example shows that for a given parabolic subgroup P ⊂ GL(s), the property that the group G n ∩ P is a parabolic subgroup of G n may no longer hold for a refinement of the exhaustion used to define P. Example 2.7.Let s = 2 ∞ , s n = 2 2n−2 , and s ′ n = 2 n−1 .Then both {s n } n≥1 and {s ′ n } n≥1 are exhaustions of s, and {s ′ n } n≥1 is a refinement of {s n } n≥1 .Let H n ⊂ GL(s n ) be the subgroup of diagonal matrices.We define a Borel subgroup B n ⊂ GL(s n ) that contains H n , by induction in the following way: B 1 := GL(1), and for n ≥ 2, where all the blocks are square matrices of size s n .Then B n+1 ∩ GL(s n ) = B n for all n, which implies that B = n≥1 B n is a well-defined Borel subgroup of GL(s) arising from the exhaustion {s n } n≥1 of s.However, for all n,

On embeddings of flag varieties
In this section we review some preliminaries on finite-dimensional (partial) flag varieties.In particular, we recall the notions of linear embedding and standard extension introduced in [10].
3.1.Grassmannians and (partial) flag varieties.Let V be a finite-dimensional vector space.For an integer 0 ≤ p ≤ dim V , we denote by Gr(p; V ) the grassmannian of p-dimensional subspaces in V .This grassmannian can be realized as a projective variety by the Plücker embedding Gr(p; V ) ֒→ P( p V ).Moreover, the Picard group Pic(Gr(p; V )) is isomorphic to Z with generator O Gr(p;V ) (1), the pull-back of the line bundle O(1) on P( p V ).
Then C 1 (ϕ) ⊂ . . .⊂ C ℓ (ϕ) is a chain of subspaces of W with possible repetitions.We define the support of ϕ to be the set of indices i ∈ {1, . . ., ℓ} such that dim C i (ϕ) < q i .

Linear embedding.
Let where W 1 , . . ., W ℓ is a sequence of vector spaces and 0 < q j < dim W j for all j.Consider an embedding ψ : We use the notation of the previous section for X.The Picard group of Q is isomorphic to Z ℓ , with generators associated to the line bundles M j = proj * j O Gr(q j ;W j ) (1).Definition 3.1.We say that the embedding ψ is linear if we have for all j ∈ {1, . . ., ℓ}.
Note that, if ϕ is a strict standard extension, then C i (ϕ) = Z i for all i ∈ {1, . . ., ℓ}, and the support of ϕ is the interval κ Also, a composition of standard extensions is a standard extension. where Here, we still use the convention that V 0 := 0 and V k+1 := V , and we set accordingly p 0 := 0 and p k+1 := dim V .Then ϕ and φ are strict standard extensions, associated with the respective chains of subspaces and respective maps κ and κ, where Remark 3.6.Every strict standard extension is the composition of, possibly several, maps ϕ and φ as in Example 3.5.

A review of generalized flags
4.1.Generalized flags.Let V be an infinite-dimensional vector space of countable dimension and let E = {e 1 , e 2 , . ..} be a basis of V .By S , we denote the span of vectors in a subset S ⊂ V .Following [3], we call generalized flag a collection F of subspaces of V that satisfies the following conditions: • F is totally ordered by inclusion; • every subspace F ∈ F has an immediate predecessor or an immediate successor in F ; , where the union is over pairs of consecutive subspaces in F .Moreover, a generalized flag F is said to be E-compatible if every subspace F ∈ F is spanned by elements of E. An E-compatible generalized flag F can be encoded by a (not order preserving) surjective map σ : Z >0 → A onto a totally ordered set (A, ≤) such that F = {F ′ a , F ′′ a } a∈A where F ′ a = e k : σ(k) < a and F ′′ a = e k : σ(j) ≤ a .More generally, a generalized flag F is said to be weakly E-compatible if it is E ′ -compatible for some basis E ′ of V differing from E in finitely many vectors.
Let GL(E) = {g ∈ GL(V ) : g(e k ) = e k for all but finitely many k}.Then GL(E) is an ind-group, isomorphic to the finitary classical ind-group GL(∞).The group GL(E) acts on the set of all weakly E-compatible generalized flags.Furthermore, it is established in [3] that weakly E-compatible generalized flags F of V are in one-to-one correspondence with splitting parabolic subgroups P ⊂ GL(E).More precisely, the map is a bijection between these two sets.
By a natural representation of GL(s) we mean a direct limit of natural representations of GL(s) for s ∈ D(s).Two natural representations do not have to be isomorphic; see [5].
Assume now that V is a natural representation for GL(s), H ⊂ GL(s) is a Cartan subgroup such that there is a basis E of V consisting of eigenvectors of H.The group GL(s) acts in a natural way on the generalized flags in V , and a generalized flag is E-compatible if and only if it is H-stable.However, generalized flags are less suited for describing parabolic subgroups of GL(s) than for describing parabolic subgroups of GL(∞) ∼ = GL(E), since the stabilizer of a generalized flag in GL(s) is not always a parabolic subgroup.Moreover, there are parabolic subgroups of GL(s) which cannot be realized as stabilizers of generalized flags in a prescribed natural representation.These observations are illustrated by the following two examples.
the line spanned by the 2 n -th vector of the standard basis of C 2 n .Then P n+1 ∩ GL(2 n ) = P n for all n ≥ 1, hence P := n≥1 P n is a parabolic subgroup of GL(2 ∞ ).However, P acts transitively on the nonzero vectors of V , so that there is no nonzero proper subspace of V which is stable by P. Therefore, P cannot be realized as the stabilizer of a generalized flag in V .
(b) If in part (a) we replace the embeddings defining the structure of natural representation on V by C 2 n ∼ = {0} 2 n × C 2 n ⊂ C 2 n+1 , then L n = L 1 for all n ≥ 1 and the parabolic subgroup P of (a) becomes the stabilizer of the generalized flag {0 ⊂ L 1 ⊂ V }.

4.2.
Ind-varieties of generalized flags.Definition 4.3.(a) Two generalized flags F and G are said to be E-commensurable [3] if F and G are weakly E-compatible and there is an isomorphism of totally ordered sets φ : F → G and there is a finite-dimensional subspace U ⊂ V such that, for all F ∈ F , (b) Given an E-compatible generalized flag F , we define Fl(F , E) as the set of all generalized flags which are E-commensurable with F .
Let F be an E-compatible generalized flag.We now recall the ind-variety structure on Fl(F , E) [3].To do this, we write E = {e k } k≥1 and, for n ≥ 1, set V n := e 1 , . . ., e n .The collection of subspaces {F ∩ V n : F ∈ F } determines a flag We define an embedding η n : X n → X n+1 in the following way.Let i 0 ∈ {1, . . ., p n+1 } be minimal such that e n+1 ∈ F (n+1) i 0 .We have either p n+1 = p n or p n+1 = p n + 1.In the former case we set In the latter case, we define Note also that, up to isomorphism, the ind-variety Fl(F , E) only depends on the type of F , i.e., on the isomorphism type of the totally ordered set (F , ⊂) and on the dimensions dim F ′′ /F ′ of the quotients of consecutive subspaces in F .

Embedding of flag varieties arising from diagonal embedding of groups
In this section we study embeddings of flag varieties induced by strictly diagonal embeddings of general linear groups.
Let us fix the following data: • positive integers m < n such that m divides n, and d := n m ; • GL(m) seen as a subgroup of GL(n) through the diagonal embedding x → diag(x, . . ., x); • a decomposition of the natural representation V := C n of GL(n) as where W (i) := {0} (i−1)m × C m × {0} (d−i)m ; let χ i : W := C m → W (i) be the natural isomorphism.For a subspace M ⊂ W , we write M (i) := χ i (M).5.1.Restriction of parabolic subgroup.Let {e 1 , . . ., e n } be a basis of V such that {e 1 , . . ., e m } is a basis of W (1) ∼ = W .By H = H(n) ⊂ GL(n) we denote the maximal torus for which e 1 , . . ., e n are eigenvectors.Then H ′ := H ∩ GL(m) is a maximal torus of GL(m).
A parabolic subgroup P = P (n) ⊂ GL(n) that contains H is the stabilizer of a flag (a) The intersection Q := P ∩ GL(m) is a parabolic subgroup of GL(m) if and only if ≤ restricts to a total order on I.Moreover, letting b 1 , . . ., b q be the elements of I written in increasing order, we have In particular, if d j = #β −1 ({b 1 , . . ., b j }) then GL(m)/Q can be identified with the flag variety Fl(d 1 , . . ., d q−1 ; W ).
(b) If Q is a parabolic subgroup, the inclusion U Q ⊂ U P of unipotent radicals holds if and only if any two distinct elements (x 1 , . . ., x d ), (y 1 , . . ., y d ) of I satisfy x i = y i for all i ∈ {1, . . ., d}.

Proof. (a)
We have a decomposition where h = Lie H and g i,j = C(e i ⊗ e * j ).With this notation, (5.1) where nil(p) is the nilpotent radical of p.
There is a similar decomposition Set q := Lie Q where Q = P ∩ GL(m) as before.Since we already know that h ′ ⊂ q, the subalgebra q is parabolic if and only if (5.2) 1 ≤ i = j ≤ m =⇒ (g ′ i,j ⊂ q or g ′ j,i ⊂ q).In view of (5.1) and the diagonal embedding gl(m) ⊂ gl(n), whenever 1 ≤ i = j ≤ m we have the equivalence Hence, from (5.2) we obtain that q is a parabolic subalgebra of gl(m) if and only if The condition means that ≤ is a total order set on I.We also have the equality The inclusion U Q ⊂ U P holds if and only if the similar inclusion holds for the nilradicals of the Lie algebras.Through the diagonal embedding of gl(m) into gl(n), the nilradical of q can be described as Therefore, the desired inclusion nil(q) ⊂ nil(p) holds if and only if, for all i, j ∈ {1, . . ., m}, This condition is equivalent to the one stated in (b) (knowing that the partial order ≤ restricts to a total order on I, due to (a)).

Diagonal embedding of flag varieties.
Assuming that the condition of Lemma 5.1 (a) is fulfilled, we now describe the embedding of partial flag varieties obtained in this case.We rely on a combinatorial object, introduced in the next definition.

Definition 5.2. (a)
We call E-graph an unoriented graph with the following features: • The vertices consist of two sets {l 1 , . . ., l q } ("left vertices") and {r 1 , . . ., r p } ("right vertices"), displayed from top to bottom in two columns, and two vertices are joined by an edge only if they belong to different sets.• The edges display into d subsets E c corresponding to a given colour c ∈ {1, . . ., d}.
• Every vertex is incident with at least one edge, and every vertex is incident with at most one edge of a given colour.The vertex l q is incident with exactly d edges (one per colour).• Two edges of the same colour never cross, that is, if (l i , r j ) and (l k , r ℓ ), with i < k, are joined with two edges of the same colour, then j < ℓ.In an E-graph, we call "bounding edges" the edges passing through l q , and we call "ordinary edges" all other edges.
(b) With the notation of Lemma 5.1, we define the E-graph G(α, β) such that • we put an edge of colour k between l i and r j whenever b i = (x 1 , . . ., x d ) satisfies x k = j and i is maximal for this property.(The conditions given in Lemma 5.1 (a) justify that G(α, β) is a well-defined E-graph.) In the following statement we describe explicitly the embedding φ of (5.3) and its properties in terms of the E-graph G(α, β).
where for all j ∈ {1, . . ., p − 1} we have i d , where V 0 = F 0 := 0, F q := W , and i k := i if the vertex r j is incident with an edge (l i , r j ) of colour k in G(α, β), 0 if there is no edge of colours k passing through r j .
We have also where i ′ k is the index of the left end point of the last edge of colour k arriving at or above ) denote the sequences of preferred generators of Pic X and Pic Y , respectively.The map φ * : Pic X → Pic Y is given by where we set by convention The map φ is linear if and only if, whenever r j , r j ′ with j < j ′ are incident with edges of the same colour c in the graph G(α, β), every ordinary edge arriving at r j ′′ for j ≤ j ′′ < j ′ is also of colour c.
(d) The map φ is a standard extension if and only if all ordinary edges of G(α, β) are of the same colour.Moreover, in this case, φ is a strict standard extension.
The second formula stated in (a) is an immediate consequence of (5.4).The proof of (a) is complete.
Part (b) is a corollary of the second formula in (a), whereas parts (c) and (d) of the proposition easily follow from parts (a) and (b).The proof of the proposition is complete.
Remark 5.4.Proposition 5.3 shows how the E-graph G(α, β) describes the embedding φ : Y → X.Moreover, the chain of constant spaces (C j (φ)) is expressed in the following way.We enumerate the colours k 1 , . . ., k d so that i 1 ≤ . . .≤ i d where r i j is the right end point of the bounding edge of colour k j .Then Example 5.5.(a) Let us consider for instance the graph

It encodes an embedding
) the sets of preferred generators of the Picard groups of X and Y respectively, then the induced map Thus φ is not linear in this case.
(b) Here we consider the graph There are two colours which means that the embedding is from a flag variety of a space V to the flag variety of a doubled space W = V ⊕ V : The embedding has the following explicit form The dimensions of the other quotients are unchanged.
The only quotient whose dimension changes is By Proposition 5.3 (d), the embeddings of parts (a) and (b) of this example are the only possible standard extensions that can come from a diagonal embedding GL(n) ֒→ GL(2n).
(d) In the case of a diagonal embedding of the form GL(n) ֒→ GL(dn), if the embedding of flag varieties is a standard extension, then it can be described as a composition of embeddings of the previous form, involving a subspace V still of dimension n.
Remark 5.6.The fact that U Q ⊂ U P is equivalent to the following property of the graph G(α, β): every left vertex is incident with exactly d edges (one per colour).Proposition 5.3 has the following corollary.
6. Ind-varieties of generalized flags as homogeneous spaces of GL(s) Our purpose in this section is to characterize ind-varieties of generalized flags (introduced in Section 4.2) which can be realized as homogeneous spaces GL(s)/P for the given supernatural number s.
6.1.The case of finitely many finite-dimensional subspaces.We start with a special situation which is easier to deal with: let X = Fl(F , E) where F = {F ′ a , F ′′ a } a∈A is an E-compatible generalized flag, for an arbitrary totally ordered set (A, ≤), but with the assumption that (6.1) dim F ′′ a /F ′ a = +∞ for all but finitely many a ∈ A.
Theorem 6.1.If condition (6.1) holds, then for every supernatural number s, there is an isomorphism of ind-varieties Fl(F , E) ∼ = GL(s)/P for an appropriate parabolic subgroup P ⊂ GL(s).
Proof.In the situation of the theorem, the ind-variety X = Fl(F , E) has an exhaustion such that X n is a finite-dimensional variety of flags in the space C sn for some exhaustion {s n } n≥1 of s with s 1 sufficiently large, and φ n : X n → X n+1 is one of the two maps from Example 5.5 (b) and (c).Using the maps φ n , one constructs nested parabolic subgroups P n ⊂ GL(s n ) such that X n ∼ = GL(s n )/P n and P n = GL(s n ) ∩ P n+1 for all n.The union P = n≥1 P n is then a parabolic subgroup of GL(s) which satisfies the conditions of the theorem.
6.2.The general case.To treat the general case, we need to start with a definition.Definition 6.2.Let F = {F ′ a , F ′′ a } a∈A be an E-compatible generalized flag and let We say that the ind-variety Fl(F , E) is s-admissible if either A ′ is finite or A ′ is infinite and there are a exhaustion {s n } n≥1 for s and a numbering A ′ = {k n } n≥1 (not necessarily compatible with the total order on A ′ ) such that, for all n ≥ 0: (ii) There is a parabolic subgroup P ⊂ GL(s) and an isomorphism of ind-varieties Fl(F , E) ∼ = GL(s)/P.
Proof.(i)⇒(ii): The ind-variety Fl(F , E) admits an exhaustion Fl(F , E) = n X n with embeddings of the form Assume that there is another exhaustion Fl(F , E) = n Y n for which the embeddings are as described in Proposition 5.3, where Y n = Fl(r 1 , . . ., r mn ; W n ) and dim W n = s n for an exhaustion {s n } of s.Then the two exhaustions interlace, and there is no loss of generality in assuming that the interlacing holds for the sequences (X n ) and (Y n ), and not only for subsequences: Claim.The embedding ξ n is a standard extension.
First we show that ξ n is linear.Arguing by contradiction, assume that there is a generator Since the map χ n is an embedding, we have λ j ≥ 0 for all j and in particular λ i ≥ 1.
The same argument applied to ξ n implies that ξ * n [M j ] should be a linear combination of the preferred generators of Pic Y n with nonnegative integer coefficients.This implies that ψ is neither 0 nor a preferred generator of Pic Y n , contradicting the linearity of the standard extension ψ n .
Recall that in [10] the notion of an embedding factoring through a direct product is introduced.Note that ξ n cannot factor through a direct product: otherwise, ψ n would also factor through a direct product, which is impossible since this is a standard extension.Consequently, ξ n is a standard extension, and the claim is established.Now we can assume that ξ 1 is a strict standard extension.Since the maps φ n are strict standard extensions, by using the formula for ψ n in Proposition 5.3 we derive that ξ n is a strict standard extension for all n ≥ 1.
Due to (6.2) and Proposition 5.3, one has W n = V n ⊕ Z n and the map ξ n has the form The conditions imply that we can choose a decomposition ℓ n+1 } (for suitable types t n ), we get exhaustions of Fl(F , E) and a homogeneous space for GL(s), which interlace.Hence if F satisfies the condition above, then we can realize Fl(F , E) as a homogeneous space for GL(s).Remark 6.4.It is shown in [2,Corollary 5.40] that GL(s)/B is never projective when B is a Borel subgroup.On the other hand, according to [3, Proposition 7.2], an indvariety of generalized flags is projective if and only if the total order on the flag can be induced by a subset of (Z, ≤), and Theorem 6.3 shows that in many situations GL(s)/P is projective.

The case of direct products of ind-varieties of generalized flags
In this section, we point out that many direct products of ind-varieties of generalized flags can be homogeneous spaces for the group GL(s).7.1.Direct products of ind-varieties.Let X i = n≥1 X i,n (i ∈ I) be a collection of ind-varieties indexed by Z >0 or a finite subset of it.For each i ∈ I we pick an element x i ∈ X i,1 and we set X i,0 = {x i }.The direct product in the category of pointed indvarieties is then given by i∈I for a collection of increasing maps φ i : Z >0 → Z ≥0 such that for every n ∈ Z >0 we have φ i (n) = 0 for all but finitely many i ∈ I (the definition does not depend essentially on the choice of the maps φ i ).
Remark 7.1.(a) As a set, the direct product can be identified with the set of sequences (y i ) i∈I where y i ∈ X i for all i ∈ I and y i = x i for all but finitely many i ∈ I.
(b) For a finite set of indices I, as a set, i∈I X i coincides with the usual cartesian product, and its structure of ind-variety is given by the exhaustion i∈I X i := n≥1 X i,n .
Fixing an index i 0 ∈ I, there are a canonical projection proj i 0 : i∈I X i → X i 0 , (y i ) → y i 0 and an embedding emb i 0 : X i 0 → i∈I X i , x → (y i ) with which are morphisms of ind-varieties.
If the product is endowed with an action of a group G, then each ind-variety X i inherits an action of G defined through the maps proj i and emb i .Conversely, if every ind-variety X i is endowed with an action of a group G, then we obtain an action of G on the product defined diagonally provided that the following condition is fulfilled: (7.1) every g ∈ G fixes x i for all but finitely many i ∈ I.
(This condition is automatically satisfied in the case where I is finite.)Moreover, in both directions, when G = G is an ind-group, we have that the obtained action is algebraic provided that the initial one is.The following lemma is an immediate consequence of this discussion.
Lemma 7.2.Assume that the direct product i∈I X i is a homogeneous space for an ind-group G.Then, every ind-variety X i is also a homogeneous space for G.
Note also that a direct product i∈I X i is locally projective if and only if it is the case of X i for all i ∈ I. 7.2.The case of ind-varieties of generalized flags.We start with an example.
Example 7.3.Let s = 2 ∞ .We consider the space V of countable dimension, endowed with its fixed basis E = {e k } k∈Z >0 , and we set V n := e 1 , . . ., e n .We have the exhaustion GL(s) = n≥1 GL(V 2 n ) defined through the diagonal embedding GL(V 2 n ) ֒→ GL(V 2 n+1 ), x → x 0 0 x .Consider the sequence of parabolic subgroups In this way, P n ∩ GL(V 2 n−1 ) = P n−1 for all n ≥ 3.Moreover, every quotient GL(V 2 n )/P n is a flag variety formed by flags (F 1 ⊂ F 2 ⊂ V 2 n ) of length 2, and we have an embedding of flag varieties GL(V 2 n−1 )/P n−1 → GL(V 2 n )/P n , (F 1 , F 2 ) → (F 1 , V a direct product of ind-varieties of generalized flags i∈I X i where X i has an exhaustion with embeddings encoded by G i .

Outlook
We see the results of this paper as a small first step in the study of locally projective homogeneous ind-spaces of locally reductive ind-groups.One inevitable question for a future such study is, given two non-isomorphic locally reductive ind-groups G and G ′ , when are two homogeneous spaces G/P and G ′ /Q isomorphic as ind-varieties ?A further natural direction of research could be a comparison of Bott-Borel-Weil type results on G/P and G ′ /Q.We finish the paper by pointing out that the reader can verify that Theorem 6.1 remains valid if one replaces GL(s) by any pure diagonal ind-group in the terminology of [1].
of flag varieties arising from diagonal embedding of groups 13 6.Ind-varieties of generalized flags as homogeneous spaces of GL(s) 19 7. The case of direct products of ind-varieties of generalized flags 22

5. 3 .
Application to ind-varieties.Definition 5.8.Let {s n } n≥1 be an exhaustion of s.We call s-graph a graph with infinitely many columns of vertices B n , with 1 ≤ |B n | ≤ s n for all n ≥ 1, such that the subgraph consisting of B n , B n+1 and the corresponding edges is an E-graph.
It suffices to prove the claim at the level of Lie algebras.Let sl(s) = n g n be an exhaustion by simple Lie algebras, hence of classical type for n large enough.There is a subsequence {g kn } n≥1 and an exhaustion {s n } n≥1 of s such that we have a commutative neither 0 nor a preferred generator of Pic Y n .Since φ * n = χ * n • ξ * n+1 , we have the inclusion Im φ * n ⊂ Im χ * n , and due to Corollary 5.7 we get that there is a generator F σ(pn) ⊕ Z n pn }.Since this applies likewise to ξ n+1 , and taking into account the form of φ n in (6.2), we see that the map ψ n has the form i are constant subspaces which are copies of W n