Multiple flag ind-varieties with finitely many orbits

Let $G$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$, and $P_1,\dots, P_l$ be an arbitrary set of $l$ splitting parabolic subgroups of $G$. We determine all such sets with the property that $G$ acts with finitely many orbits on the ind-variety $X_1\times\dots\times X_l$ where $X_i=G/P_i$. In the case of a finite-dimensional classical linear algebraic group $G$, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar-Weyman-Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for $l=2$, the condition that $G$ acts on $X_1\times X_2$ with finitely many orbits is a rather restrictive condition on the pair $P_1,P_2$. We describe this condition explicitly. Using this result, we tackle the most interesting case where $l=3$, and present the answer in the form of a table. For $l\geq 4$, there always are infinitely many G-orbits on $X_1\times \dots\times X_l$.


Introduction
The following is a fundamental question in the theory of group actions: given a linear reductive algebraic group G, on which direct products X 1 × X 2 × · · · × X ℓ of compact G-homogeneous spaces does G act with finitely many orbits? The problem is non-trivial only for ℓ > 2, since it is a classical fact that G always acts with finitely many orbits on X 1 × X 2 (parabolic Schubert decomposition of a partial flag variety). It has turned out that the problem is most interesting for ℓ = 3, as for ℓ ≥ 4 the group G always acts with infinitely many orbits.
In the special case where one of the factors is a full flag variety, e.g. X 1 = G/B, the above problem is equivalent to finding whether there are finitely many B-orbits on X 2 ×X 3 ; this special case is solved in [6] and [13]. In this situation, the theory of spherical varieties is an effective tool. In particular, the existence of a dense B-orbit is sufficient for ensuring that there are finitely many B-orbits. The problem is also related to studying the complexity of a direct product of two HV-varieties, i.e. closures of G-orbits of highest weight vectors in irreducible G-modules; this problem is considered in [12].
If no factor X i is a full flag variety, the problem is considered in the classical cases in [7,8] (types A and C, through the theory of quiver representations) and in [9,10] (types B and D). For exceptional groups, the general question has been considered in [1].
We also mention the references [5] and [11], where the authors study the double flag varieties of the form G/P × K/Q with a finite number of K-orbits for a symmetric subgroup K of G. The problem of finitely many G-orbits on X 1 × X 2 × X 3 is recovered if K is taken to be the diagonal embedding of G into G × G.
In the present paper we address the above general problem in a natural infinitedimensional setting. We let G be one of the classical (or finitary) ind-groups GL(∞), Ø(∞), Sp(∞) and ask the same question, where each X i is now a locally compact Ghomogeneous ind-space. The latter are known as ind-varieties of generalized flags and have been studied in particular in [2] and [4]; see also [3] and the references therein.
For these ind-varieties, our question becomes interesting already for ℓ = 2. Indeed, for which direct products X 1 × X 2 of ind-varieties of generalized flags, does G act with finitely many orbits on X 1 × X 2 ? We prove that this is a quite restrictive property of the ind-variety X 1 × X 2 . More precisely, we show that G acts with finitely many orbits on X 1 × X 2 only if the stabilizers P 1 and P 2 of two respective (arbitrary) points on X 1 and X 2 have each only finitely many invariant subspaces in the natural representation V of G. In addition, it is required that the invariant subspaces of one of the groups, say P 1 , are only of finite dimension or finite codimension. The precise result is Theorem 1.4, where we introduce adequate terminology: we call the parabolic ind-subgroup P 1 large, and the parabolic ind-subgroup P 2 semilarge.
Having settled the case ℓ = 2 in this way, we saw ourselves strongly motivated to solve the problem for any ℓ ≥ 3. The case ℓ ≥ 4 is settled by a general statement, Lemma 4.2, claiming roughly that in the direct limit case the number of orbits can only increase. Hence for ℓ ≥ 4 there are infinitely many orbits on X 1 × · · · × X ℓ . The case ℓ = 3 is the most intriguing. Here we prove that X 1 × X 2 × X 3 has finitely many G-orbits, if and only if the same is true for all products X 1 × X 2 , X 2 × X 3 and X 1 × X 3 , and in addition X 1 × X 2 × X 3 can be exhausted by triple flag varieties with finitely many orbits over the corresponding finite-dimensional groups. Those triple flag varieties have been classified by Magyar-Weymann-Zelevinsky for SL(n) and Sp(2n) [7,8], and by Matsuki for Ø(2n + 1) and Ø(2n) [9,10]. In this way, we settle the problem completely for the classical ind-groups GL(∞), Ø(∞), Sp(∞).
To describe G as an ind-group, we need to take a basis of V . If E is a basis of V , we denote by E * = {φ e : e ∈ E} ⊂ V * the dual family of linear forms defined by We call E admissible if, according to type, the following is satisfied: (A) the dual family E * spans the subspace V * ; (BCD) E is endowed with an involution i E : E → E, with at most one fixed point, such that ω(e, e ′ ) = 0 if and only if e ′ = i E (e).
If E is admissible in the sense of type (BCD), then it is a fortiori admissible in the sense of type (A). Note that in type (C), the involution i E cannot have a fixed point. Whenever E is an admissible basis, the group G can be described as G = {g ∈ GL(V ) : g(e) = e for almost all e ∈ E} in type (A) (where "almost all" means "all but finitely many"), and G = {g ∈ GL(V ) : g(e) = e for almost all e ∈ E, and g preserves ω} in type (BCD). If we take any filtration E = n≥1 E n by finite subsets, such that E n is i E -stable in type (BCD), we get an exhaustion of G, Here the notation · stands for the linear span, and GL( E n ) is viewed as a subgroup of GL(V ) in the natural way. The subgroups G(E n ) are finite-dimensional algebraic groups isomorphic to GL m (C), Ø m (C), or Sp m (C), depending on whether G is GL(∞), Ø(∞), or Sp(∞). This exhaustion provides G with a structure of ind-group, which is independent of the chosen admissible basis.
Splitting parabolic subgroups can be fully classified in terms of so-called generalized flags; see Section 2.2. A splitting Borel subgroup is a splitting parabolic subgroup which is minimal (equivalently this is the stabilizer of a generalized flag which is maximal). We consider two types of splitting parabolic subgroups which are not Borel subgroups: Definition 1.1. We say that a splitting parabolic subgroup P is semilarge if it has only finitely many invariant subspaces in V . This is equivalent to the requirement that P be the stabilizer in G of a finite sequence of subspaces ). We say that P is large if, moreover, each subspace F k is either finite dimensional or finite codimensional.
Given a splitting parabolic subgroup P ⊂ G, the quotient set G/P has a structure of ind-variety, which is given by the exhaustion Each quotient G(E n )/P(E n ) is a flag variety for the group G(E n ), hence a projective variety. Thus G/P is locally projective, but in general it is not projective, i.e. does not admit an embedding as a closed ind-subvariety in the infinite-dimensional projective space P ∞ (C). However, if P is large or semilarge, such an embedding does exist (see [2,Proposition 7.2]). It is worth to note that, for any splitting parabolic subgroup P, the ind-variety G/P can be realized as an ind-variety of generalized flags; see Section 2.3.
Contrary to the finite-dimensional situation, any two splitting Cartan subgroups of G do not have to be conjugate; see Example 2.3. In this paper, a source of difficulty is that we are considering splitting parabolic subgroups which do not a priori have a splitting Cartan subgroup in common, even up to conjugacy. The following characterization of large splitting parabolic subgroups will be useful in this respect. Proposition 1.2 (see Proposition 3.1). Let P ⊂ G be a splitting parabolic subgroup. The following conditions are equivalent.
(i) P is large; (ii) For every splitting Cartan subgroup H ⊂ P, there is g ∈ G such that gHg −1 ⊂ P.
The proof is given in Section 3.

Main results.
We consider a product of ind-varieties of the form (1.1) X = G/P 1 × · · · × G/P ℓ where P 1 , . . . , P ℓ G are splitting parabolic subgroups of G. The ind-variety X is equipped with the diagonal action of G. Our purpose is to solve the following problem: Problem 1.3. Characterize all ℓ-tuples (P 1 , . . . , P ℓ ) such that X has a finite number of G-orbits.
Of course if ℓ = 1, then X has only one G-orbit. If ℓ = 2, the number of orbits in X is infinite in general, and our first main result claims the following. Theorem 1.4. If ℓ = 2, then X (of (1.1)) has a finite number of G-orbits if and only if one of the subgroups P 1 , P 2 is large and the other one is semilarge. Corollary 1.5. Let P be a splitting parabolic subgroup of G. Then the ind-variety G/P has a finite number of P-orbits if and only if P is large.
By Theorem 1.4, if X has a finite number of G-orbits then all three splitting parabolic subgroups P 1 , P 2 , P 3 are semilarge and at least two of them are large. Moreover, it follows from Proposition 1.2 that, up to replacing the parabolic subgroups by conjugates, there is no loss of generality in assuming that P 1 , P 2 , and P 3 contain the same splitting Cartan subgroup H = H(E) for some admissible basis E. This assumption guarantees that the construction of Section 1.2 can be done simultaneously for each factor G/P i (i ∈ {1, 2, 3}). Hence, by considering a filtration E = n E n as in Section 1.2, we obtain an exhaustion is a triple flag variety for the group G(E n ). See Section 4.1 for more details.
Our main result regarding the case ℓ = 3 can be stated as follows.
Theorem 1.6. If ℓ = 3, then X has a finite number of G-orbits only if the splitting parabolic subgroups P 1 , P 2 , P 3 are semilarge and two of them are large. Moreover, in this situation, X has a finite number of G-orbits if and only if, for all n, the finite-dimensional triple flag variety X(E n ) has a finite number of G(E n )-orbits.
Finally, X has a finite number of G-orbits if and only if the triple (P 1 , P 2 , P 3 ) appears, up to permutation, in Table 1.
In Table 1, we use the following notation in the case of a semilarge parabolic subgroup P as in Definition 1.1. We denote by |P| := m the length of P. We denote by Λ(P) the list of values dim F k /F k−1 (k = 1, . . . , m) written in nonincreasing order; in case of j repetitions of the same value a, we write a j . Note that this list always starts with ∞, and P is large if and only if the list contains a unique occurrence of ∞. Table 1. Classification of triples (P 1 , P 2 , P 3 ), up to permutation, such that G/P 1 × G/P 2 × G/P 3 is of finite type.

GL(∞) case:
Finally, it is not surprising that, like in the case of finite-dimensional multiple flag varieties, the following holds. Theorem 1.7. If ℓ ≥ 4, then X has an infinite number of G-orbits.
The rest of the paper is structured as follows. In Section 2 we summarise some existing results on the classical ind-groups and their homogeneous ind-varieties. In Section 3 we show the characterization of large parabolic subgroups stated in Proposition 1.2. In Section 4 we explain the construction of the exhaustion of (1.2) in more detail, and prove Lemma 4.2 which claims that whenever the multiple ind-variety X of (1.1) has an exhaustion as in (1.2), we get an embedding of orbit sets This lemma plays a key role in the proof of our main results. Theorem 1.4 is proved in Sections 5-6, Theorem 1.6 is proved in Section 7, and finally the proof of Theorem 1.7 appears in the very short Section 8. hence G acts by conjugation on the set of splitting Cartan subgroups. Proof. First, we note that I and I ′ have the same cardinality, equal to the codimension of E 0 in V . In type (A) we take g ∈ GL(V ) such that g(I) = I ′ and g(e) = e for all e ∈ E 0 , hence g(E) = E ′ . This element g actually belongs to GL(∞) = G, and we get H(E ′ ) = H(g(E)) = gH(E)g −1 .
In type (BCD), up to considering larger I and I ′ if necessary, we may assume that I and I ′ are stable by the involutions i E and i E ′ , respectively. Since I and I ′ have the same cardinality and the involutions i E and i E ′ have at most one fixed point, we can write either . Up to replacing the vectors of I and I ′ by scalar multiples (which does not change the splitting Cartan subgroups) we may assume that ω(e i , e * i ) = ω(e ′ i , e ′ * i ) = 1 for 1 ≤ i ≤ k, ω(e 0 , e 0 ) = ω(e ′ 0 , e ′ 0 ) = 1. Then, by letting g(e) = e for e ∈ E 0 and g(e i ) = e ′ i , g(e * j ) = e ′ * j for all i, j, we get an element g ∈ G such that H(E ′ ) = H(g(E)) = gH(E)g −1 .
The converse statement follows by observing that, if E is an admissible basis such that H = H(E), and g ∈ G satisfies H ′ = gHg −1 , then H ′ = H(g(E)), where g(E) is an admissible basis which differs from E by finitely many vectors.

Remark 2.2. (a) In type (A)
, if E is an admissible basis of V , then any basis E ′ which differs from E by finitely many vectors is admissible. Indeed, in this case we have an element g ∈ G such that E ′ = g(E).
(b) In type (BCD), part (a) of the remark does not hold, but we point out the following construction of admissible basis. Let an orthogonal decomposition V = V 1 ⊕ V 2 be given.
(c) In type (BCD) any admissible basis can be written as . By replacing the vectors by scalar multiples (which does not change the splitting Cartan subgroup H(E)), we can transform E into a basis with ω(e m , e * n ) = δ m,n for all m, n. In [2], a basis which satisfies this property is called ω-isotropic. If ω is symplectic, an ω-isotropic basis is said to be a basis of type (C). If ω is symmetric, an ω-isotropic basis is called of type (B) or (D) depending on whether i E has a fixed point or no fixed point.
In type (BD), bases of both types (B) and (D) do exist in V and their corresponding splitting Cartan subgroups cannot be conjugate.
The following example shows that, in any type, there are splitting Cartan subgroups which are not conjugate. In fact, using the construction made in this example, it is easy to show that there are infinitely many conjugacy classes of splitting Cartan subgroups. Example 2.3. Let H = H(E) ⊂ G be a splitting Cartan subgroup, associated to an admissible basis. Let I = {e n , e ′ n } n≥1 be a double infinite sequence of (pairwise distinct) vectors of E, moreover in type (BCD) we assume that these vectors are pairwise orthogonal, that is, {e n , e ′ n } n≥1 spans an isotropic subspace of V . We construct a splitting Cartan subgroup H ′ ⊂ G such that The subgroup H ′ cannot be conjugate to H: for every g ∈ G, we have g(e n ) = e n and g(e ′ n ) = e ′ n whenever n ≥ 1 is large enough, and (2.
For constructing H ′ , we construct an admissible basisẼ of V which contains the vectors e n := e n + e ′ n andẽ ′ n := e n − e ′ n , for all n ≥ 1, and then we define H ′ as the subgroup H(Ẽ) ⊂ G of all elements which are diagonal in the basisẼ. This subgroup fulfills (2.2), since for all n ≥ 1 we can find h ∈ H ′ such that h(ẽ n ) =ẽ n and h(ẽ ′ n ) = −ẽ ′ n . The construction ofẼ is done as follows. In type (A), we takẽ The dual familyẼ * = {φ e : e ∈Ẽ} consists of the linear functions E} is the dual family of E. Hence Ẽ * = E * , and this shows thatẼ is admissible.
In type (BCD), we note first that condition (2.1) implies that the double sequence I and its image by the involution i E : E → E are pairwise disjoint. Then we set .
It is easy to check that the basisẼ so-obtained is admissible, with involution iẼ :Ẽ →Ẽ Splitting parabolic subgroups and generalized flags. The notion of splitting parabolic subgroup can be described in a more handy way by using a model from linear algebra, based on the following definition. • The inclusion relation ⊂ is a total order on F ; moreover every subspace F ∈ F has an immediate predecessor or an immediate successor in F with respect to ⊂. • For every nonzero vector v ∈ V , there is a pair of consecutive subspaces F ′ , F ′′ ∈ F such that v ∈ F ′′ \ F ′ . • In type (BCD) we require a generalized flag to be isotropic in the following sense: for every F ∈ F we have F ⊥ ∈ F , and the map i F : F → F ⊥ is an involution of F . Note that the group G acts on the respective set of generalized flags in a natural way.
This is equivalent to requiring that each subspace F ∈ F is spanned by a subset of E.
(c) We say that F is weakly E-compatible if it is compatible with a basis E ′ of V such that E ′ \ E and E \ E ′ are finite (that is, E and E ′ differ by finitely many vectors).
(b) Let F ⊂ V be a subspace (taken isotropic in type (BCD)). In type (A), we set F F := {0 ⊂ F ⊂ V }; this is the minimal generalized flag which contains the subspace F . In type (BCD), we set F F := {0 ⊂ F ⊂ F ⊥ ⊂ V }; this is the minimal isotropic generalized flag which contains F . In each case we call F F the generalized flag associated to F .
is a generalized flag (in type (A)) if n F n = 0 and n F n = V . It is a generalized flag in type (BCD) if and only if there is some n 0 such that F ⊥ n = F n 0 −n for all n. (e) Let E = {e x } x∈Q be a basis of V indexed by the rational numbers. For x ∈ Q let F ′ x := e y : y < x and F ′′ x := e y : y ≤ x . Then F := {F ′ x , F ′′ x } x∈Q is an E-compatible generalized flag (in type (A)) such that each subspace lacks either an immediate predecessor or an immediate successor.
Proposition 2.6 ([2]). Let E be an admissible basis and F be an E-compatible generalized flag in V . Then the subgroup is a splitting parabolic subgroup of G that contains the Cartan subgroup H(E). Moreover, every parabolic subgroup of G that contains H(E) is obtained in this way.
In addition, P F is a splitting Borel subgroup if and only if the generalized flag F is maximal, in the sense that dim F ′′ /F ′ = 1 for each pair of consecutive subspaces {F ′ , F ′′ } in F . Remark 2.7. Contrary to the finite-dimensional situation, two splitting Borel subgroups of G are not necessarily G-conjugate, even if they contain the same splitting Cartan subgroup. This observation follows from Proposition 2.6 and from the fact that two Ecompatible maximal generalized flags F , G do not belong to the same G-orbit in general. For instance, F and G certainly belong to different G-orbits if they are not isomorphic as totally ordered sets. However, even if they are isomorphic as totally ordered sets, the generalized flags F and G do not have to be G-conjugate. For instance if F = { e 1 , . . . , e n } n≥0 and G = { e 1 , . . . , e 2n , e 1 , . . . , e 2n , e 2n+2 } n≥0 (where E = {e k } k≥1 ), then Stab G (F ) and Stab G (G) are two splitting Borel subgroups of G = GL(∞) which are not conjugate. Indeed every g ∈ G satisfies g( e 1 , . . . , e n ) = e 1 , . . . , e n for large n, hence G / ∈ G · F .

2.3.
Ind-varieties of generalized flags. Fix an admissible basis E of V and a generalized flag F = {F α } in V , compatible with E. Let P = P F ⊂ G be the splitting parabolic subgroup obtained as the stabilizer of F , like in Proposition 2.6. In this section, we describe the homogeneous space G/P = G · F as an ind-variety of generalized flags. See [2] for more details.
is parameterized by the same ordered set as F , and in addition satisfies the following conditions: Let Fl(F , E, V ) denote the set of all E-commensurable with F generalized flags G. Let Fl ω (F , E, V ) denote the subset of all such generalized flags which are isotropic. Then the homogeneous space G/P F = G · F coincides with the set Fl(F , E, V ) in type (A), respectively with Fl ω (F , E, V ) in type (BCD). (Note that the notion of commensurability is the same whatever type is considered.) In Section 1.2 we notice that the quotient G/P F has the structure of an ind-variety, obtained by considering a filtration E = n≥1 E n by finite subsets; in type (BCD) the basis is endowed with the involution i E : E → E (with at most one fixed point) and we require the subsets E n to be i E -stable, so that the restriction of the form ω to each subspace E n is nondegenerate.
The ind-structure on Fl(F , E, V ) and Fl ω (F , E, V ) is given via the identification with a direct limit where Fl(F , E n ) and Fl ω (F , E n ) are varieties of partial flags of the space V n := E n defined in the following way. The generalized flag F gives rise to a flag in the finite-dimensional subspace E n , namely let F (n) be the collection of subspaces Let d(F , n) denote the corresponding dimension vector The (finite-dimensional) algebraic variety can be viewed as the set of collections of subspaces of E n For each n ≥ 1, we have the embedding for all F ∈ F . Finally, the ind-variety G/P F = G · F = Fl(F , E, V ), respectively Fl ω (F , E, V ), is obtained as the limit of the inductive system This yields (2.3).
In type (A), the embedding φ F (n) corresponds to an embedding of partial flag varieties obtained in the following way. Assume that E n+1 = E n ∪ {e} for simplicity (in the general case, i n is obtained as a composition of mappings of the following type). Then for some k ≤ s, ifd(F , n + 1) is a longer sequence than d (F , n), for some k < s, ifd(F , n + 1) is a sequence of the same length asd(F , n).
The map i n is now defined via the respective formula The embeddings i n have been introduced in [2] in different notation. In type (BCD) the construction is similar.

A characterization of large splitting parabolic subgroups
Proposition 1.2 is incorporated in the following more complete statement. (ii) F is weakly E-compatible with every admissible basis E; (iii) for every admissible basis E, the ind-variety G/P = G · F contains a generalized flag which is E-compatible; (iv) for every splitting Cartan subgroup H ⊂ G, there is g ∈ G such that gHg −1 ⊂ P.
Proof. (i)⇒(ii): Assume that P is a large splitting parabolic subgroup. Then the generalized flag F has the form and moreover, there is a unique index k ∈ {1, . . . , m} for which the space F k /F k−1 is infinite dimensional. The subspace F k−1 is finite dimensional and the subspace F k is finite codimensional. Moreover, since the generalized flag F is compatible with an admissible basis (Proposition 2.6), there is a finite subset Φ ⊂ V * such that F k = φ∈Φ ker φ.
Let E ⊂ V be an admissible basis, hence the dual family E * = {φ e : e ∈ E} spans the subspace V * ⊂ V * . We can find a finite subset I ⊂ E such that • F k−1 ⊂ V 1 := I ; • Φ ⊂ φ e : e ∈ I , so that V 2 := E \ I ⊂ F k . In type (BCD), up to choosing I larger if necessary, we may assume that I is stable by the involution i E : E → E and contains the fixed point of i E if it exists. Hence the restriction of ω to V 1 and V 2 is nondegenerate and the decomposition The sequence where ⊥ 1 indicates the orthogonal space with respect to the restriction of ω to V 1 , hence we may assume that the basis E 1 is isotropic with respect to the restriction of ω.
We obtain a basis E ′ := E 1 ∪ (E \ I) of V which differs from E by finitely many vectors, and is in addition admissible (see Remark 2.2 (a)-(b)). For all j we have (ii)⇒(iii): Assume that F is weakly E-compatible. This means that there is an admissible basis E ′ which differs from E by finitely many vectors, and such that F is E ′compatible. The latter fact is equivalent to saying that F is a fixed point of the splitting Cartan subgroup H(E ′ ). By Lemma 2.1, we can find g ∈ G satisfying H(E) = gH(E ′ )g −1 . This implies that the generalized flag gF is a fixed point of H(E), and therefore gF is E-compatible.
(iii)⇒(iv): Every splitting Cartan subgroup of G is of the form H(E) for an admissible basis E. By (iii) we can find g ∈ G so that gF is E-compatible, that is, fixed by H(E). This yields g −1 H(E)g ⊂ Stab G (F ) = P.
(iv)⇒(i): Assume that P is not large. This implies that one of the following cases occurs.
(a) F contains a subspace F 0 that is both infinite dimensional and infinite codimensional, or (b) F contains infinitely many subspaces, in particular, an increasing chain F 1 ⊂ F 2 ⊂ · · · ⊂ F k ⊂ · · · or a decreasing chain F 1 ⊃ F 2 ⊃ · · · ⊃ F k ⊃ · · · . Moreover, in type (BCD), since we have F ⊥ ∈ F whenever F ∈ F , we may assume that F 0 is an isotropic subspace of V in case (a), and that {F k } k≥1 is a chain of isotropic subspaces of V in case (b).
Let E be an admissible basis of V such that the generalized flag F is E-compatible. We claim that there is a double infinite sequence ∀n ≥ 1, ∃F ∈ F such that e n ∈ F and e ′ n / ∈ F and moreover (3.3) {e n , e ′ n } n≥1 span an isotropic subspace of V (in type (BCD)). The construction of the double sequence {e n , e ′ n } n≥1 can be done as follows. In case (b), for each k ≥ 2 we take a vector ε k ∈ E lying in F k \ F k−1 in the case of an increasing chain, respectively in F k−1 \ F k in the case of a decreasing chain. Then we set (e n , e ′ n ) := (ε 2n , ε 2n+1 ), respectively (e n , e ′ n ) := (ε 2n+1 , ε 2n ), and we have (3.2). In type (BCD), since each subspace F k is isotropic, we get (3.3).
In case (a), in type (A), relying on the fact that F 0 is infinite dimensional and infinite codimensional, we take an infinite subset {e n } n≥1 ⊂ E of vectors which belong to F 0 and an infinite subset {e ′ n } n≥1 ⊂ E of vectors which do not belong to F 0 . The soobtained double sequence clearly satisfies (3.2). In type (BCD), we first take an infinite subset {ε k } k≥1 ⊂ E of vectors which belong to F 0 , then we set e n := ε 2n and e ′ n := i E (ε 2n+1 ). Since the subspace F 0 is isotropic, the vectors e ′ n do not belong to F 0 , hence (3.2) holds. Finally (3.3) holds due to the definition of the involution i E . This completes the construction of the double sequence of (3.1).
Let H ′ ⊂ G be the splitting Cartan subgroup associated to the double sequence {e n , e ′ n } n≥1 as in Example 2.3. For every g ∈ G, since we have g(e) = e for only finitely many e ∈ E, there is n ≥ 1 such that g(e n ) = e n and g(e ′ n ) = e ′ n . Then (2.2) and (3.2) yield a subspace F ∈ F and an element h ∈ H ′ such that e n ∈ F and ghg −1 (e n ) = e ′ n / ∈ F, hence ghg −1 (F ) = F.
This establishes that, for all g ∈ G, we have gH ′ g −1 ⊂ Stab G (F ) = P. The proof of the proposition is complete.

Exhaustion of X and a key lemma
In this section we consider an ind-variety X of the form (1.1). Our analysis is based on the following assumption: the splitting parabolic subgroups P 1 , . . . , P ℓ−1 are large.

Exhaustion.
Here we explain how to construct a natural exhaustion of the indvariety X. The notation introduced here is used in the subsequent sections. For every i ∈ {1, . . . , ℓ} there is a generalized flag F i such that P i = Stab G (F i ) (see Proposition 2.6), hence X i := G/P i = G · F i . Let E be an admissible basis of V such that F ℓ is E-compatible. By Proposition 3.1, for every i ∈ {1, . . . , ℓ − 1} we can find an element h i ∈ G such that h i F i is E-compatible. In view of the isomorphisms that is, the splitting Cartan subgroup H(E) is contained in P 1 , . . . , P ℓ .
Take a filtration of the basis E = n≥1 E n by finite subsets (stabilized by i E in type (BCD)). As in Sections 1.1 and 1.2, we let which are respectively a classical algebraic group and a parabolic subgroup. In fact, since we are dealing with a single admissible basis E, it is harmless to avoid the reference to E in the notation; we set for simplicity G(n) := G(E n ) and P i (n) := P i (E n ). For each i ∈ {1, . . . , ℓ}, we follow the construction made in Section 2.3: the generalized flag F i gives rise to a flag in the finite-dimensional subspace E n , namely let F i (n) be the collection of subspaces The (finite-dimensional) algebraic variety can be viewed as the set X F i (n) of collections of subspaces of E n described in Section 2.3. (It is isomorphic in a natural way to a partial flag variety of the space E n ; see Remark 2.9.) For each n ≥ 1, we have the embedding Finally, for each i ∈ {1, . . . , ℓ}, the ind-variety X i = G/P i = G · F i is obtained as the limit of the inductive system thus we have an exhaustion Altogether, we have the following exhaustion of the ind-variety X: where for each n we consider the embedding

Key lemma.
We still assume (4.1). Our key lemma is as follows.

In type (A), we may suppose that
Hence M := E n is a hyperplane of N := E n+1 = M ⊕ e . Every element of the variety X(n) (respectively X(n + 1)) consists of a collection of subspaces {M i } i∈I of M (respectively {N i } i∈I of N), and the map φ(n) : X(n) → X(n + 1) is of the form for some subset I 0 ⊂ I. Therefore, in type (A) Lemma 4.2 follows from the following lemma from linear algebra.
The lemma is proved in this case.
Thus it remains to consider the case where g(L) = L.
Let e ∈ L, e = 0. We distinguish two cases. For all i ∈ I \ I 0 , we have M ′ i ⊕ L = g(M i ⊕ L), therefore the subspace M ′ i ⊕ L contains L + g(L) = K. By (4.3), and since η(K) = K, this implies that the subspace Since η(g(e)) = e, i.e. η • g(L) = L, this brings us back to the situation treated at the beginning of the proof. For In view of (4.4), and since Since η(g(L)) = L, again we are brought back to the situation already treated at the beginning of the proof. The proof of the lemma is now complete.
In type (BCD), with the notation of Lemma 4.2, we have already the implications φ(n)(F ) and φ(n)(F ′ ) belong to the same G(n + 1)-orbit ⇒ φ(n)(F ) and φ(n)(F ′ ) belong to the same GL( E n+1 )-orbit ⇒ F and F ′ belong to the same GL( E n )-orbit, where the last implication is valid since we have already proved Lemma 4.2 in type (A). For completing the proof of Lemma 4.2 in type (BCD), we have to show that F and F ′ belong to the same G(n)-orbit. This conclusion is deduced from the following general fact.
Lemma 4.4. Let M be a finite-dimensional linear space, endowed with a nondegenerate orthogonal or symplectic bilinear form ω. We consider the group G(M, ω) = {g ∈ GL(M) : g preserves ω}. Let I be a set equipped with an involution i → i * , and let F = {M i } i∈I and F ′ = {M ′ i } i∈I be two collections of subspaces satisfying We define X to be the set of collections of subspaces {M i } i∈I with dim M i = d i for all i, and we consider the action of G := GL(M) on X given by Note that X is endowed with the involution Let X σ be the fixed point set of this involution. Then F and F ′ are elements of X σ . Let u * denote the adjoint morphism of an endomorphism u ∈ End(M) with respect to the form ω. Thus G is also endowed with an involution given by G → G, g → g σ := (g * ) −1 , and G(M, ω) coincides with the subgroup G σ of fixed points of this involution. Then the claim made in the statement follows once we show that two elements of X σ are G σconjugate whenever they are G-conjugate. This is exactly [ Arguing indirectly, assume that (P 1 , P 2 ) is a pair of splitting parabolic subgroups which does not satisfy the condition of Theorem 1.4, namely, up to exchanging the roles of P 1 and P 2 , we may assume that Case 1: P 1 is not semilarge, or Case 2: P 1 , P 2 are semilarge but not large. Considering generalized flags F 1 , F 2 such that P 1 = Stab G (F 1 ) and P 2 = Stab G (F 2 ), the condition of Case 1 means that F 1 has an infinite number of subspaces.
The condition of Case 2 implies that in each generalized flag F 1 and F 2 , there is at least one subspace which is both infinite dimensional and infinite codimensional. We will show that X = G/P 1 × G/P 2 has infinitely many G-orbits. Since the map is a bijection between the set of P 1 -orbits on G/P 2 and the set of G-orbits on X, it suffices to show that G/P 2 has infinitely many P 1 -orbits.
Let F 2 ∈ F 2 such that 0 F 2 V . In Case 2, we assume that F 2 is infinite dimensional and infinite codimensional. In type (BCD) we assume that F 2 ⊂ F ⊥ 2 . Recall from Example 2.5 (b) that the generalized flag associated to F 2 is given by . By replacing the parabolic subgroup P 2 by the larger splitting parabolic subgroupP 2 := Stab G (F F 2 ), we may assume that F 2 = F F 2 .
We fix an admissible basis E such that F 2 is E-compatible. Then the ind-variety G/P 2 = G · F 2 consists of generalized flags which are E-commensurable with F 2 , in particular are of the form F F for F ⊂ V . Our aim is to construct an infinite sequence of such generalized flags which belong to pairwise distinct P 1 -orbits. By a slight abuse of terminology, we say that a subspace F is weakly E-compatible if its associated generalized flag F F is weakly E-compatible. Also, we say that In type (BCD) we assume in addition that the vector v is isotropic and belongs to (F ∩ ker φ) ⊥ . Then Proof. Clearly, a subspace is weakly E-compatible if and only if it has a finite-codimensional subspace spanned by a subset of E. Let I ⊂ E be such that F contains I as a finitecodimensional subspace. There is a finite set J ⊂ E with φ ∈ φ e : e ∈ J . Then F ′ contains I \ J as a finite-codimensional subspace. Hence F ′ is weakly E-compatible.
Let a vector v ′ satisfy F = (F ∩ ker φ) ⊕ Cv ′ . Then, we see that F F and F F ′ are Ecommensurable by considering any finite-dimensional subspace U ⊂ V such that v, v ′ ∈ U in type (A) and which satisfies in addition Lemma 5.2. Let L M and F = 0 be subspaces of V , with F weakly E-compatible. In type (BCD) we assume that these subspaces are isotropic. In type (A) we assume that L, M are of the form

isotropic in type (BCD), and such that
Proof. (a) Since M ⊂ F we can find a vector v ∈ M \ F . In type (A), since F ⊂ M, we can find φ ∈ Ψ such that F ⊂ ker φ. In type (BCD), either v / ∈ F ⊥ and we take φ = ω(v, ·), or v ∈ F ⊥ and we take φ = ω(v ′ , ·) for any v ′ ∈ M ⊥ \ F ⊥ . In all the cases, it follows from Lemma 5.1 that the subspace is E-commensurable, isotropic in type (BCD), and satisfies

Moreover, the linear projection
(b) In type (A) we take a vector v ∈ V \ (F + M) and we find φ ∈ Φ such that F ∩ M ⊂ ker φ. By Lemma 5.1, the subspace is E-commensurable and satisfies In type (BCD) the construction is adapted as follows.
By Lemma 5.1, the subspace F ′ of (5.1) corresponding to this choice of φ and v is isotropic and fulfills the required conditions. (c) This time the conditions are fulfilled by the subspace F ′ := (F ∩ ker φ) ⊕ v , where v ∈ M \ L and φ is any element of V * such that F ⊂ ker φ.
Claim 1: If there is F 1 ∈ F 1 (isotropic in type (BCD)) such that F 1 ∩ F 2 has infinite codimension in F 1 and F 2 , then there is a sequence {F (n) } n≥0 of (respectively isotropic) subspaces which are E-commensurable with F 2 and satisfy Proof of Claim 1. The sequence {F (n) } n≥0 is constructed by induction: we let F (0) = F 2 and, once F (n) is defined, we let F (n+1) be the subspace F ′ obtained by applying Lemma 5.2 (a) with F = F (n) and M = F 1 (each time F 1 ∩ F (n) still has infinite codimension in F 1 and F (n) hence the conditions for applying Lemma 5.2 (a) are fulfilled).
Claim 2: If F 1 ∈ F 1 (isotropic in type (BCD)) is such that F 1 ∩ F 2 has finite codimension in F 1 or F 2 , then there is a (respectively isotropic) subspace F which is Ecommensurable with F 2 and such that F ⊂ F 1 or F 1 ⊂ F .
Proof of Claim 2. Let m i := codim F i F 1 ∩ F 2 and m := min{m 1 , m 2 }. By applying m times Lemma 5.2 (a) in the same way as in the proof of Claim 1, we obtain a (respectively isotropic) subspace F which is E-commensurable with F 2 and such that Case 1 can now be adressed as follows. If there is F 1 ∈ F 1 (respectively isotropic) such that F 1 ∩ F 2 has infinite codimension in F 1 and F 2 , then Claim 1 yields an infinite sequence {F F (n) } n≥0 of elements of G/P 2 which belong to pairwise distinct orbits of P 1 = Stab G (F 1 ).
It remains to consider the case where F 1 ∩ F 2 has finite codimension in F 1 or F 2 for all (respectively isotropic) F 1 ∈ F 1 . Invoking Claim 2, and using that F 1 contains an infinite number of subspaces, for all n ≥ 1 we can find a sequence F 1,0 F 1,1 · · · F 1,n of subspaces of F 1 such that ∀k ∈ {0, . . . , n}, ∃F F ∈ G/P 2 such that F 1,k ⊂ F or ∀k ∈ {0, . . . , n}, ∃F F ∈ G/P 2 such that F ⊂ F 1,k . In the former case, by Lemma 5.2 (b), for each k ∈ {1, . . . , n} there is a (respectively isotropic) subspace F (k) , E-commensurable with F 2 , such that F 1,k−1 ⊂ F (k) , F 1,k ⊂ F (k) . In the latter case, invoking Lemma 5.2 (c), for each k ∈ {1, . . . , n} we find this time F (k) such that F (k) ⊂ F 1,k , F (k) ⊂ F 1,k−1 . In both cases, the so obtained generalized flags F F (1) , . . . , F F (n) ∈ G/P 2 belong to pairwise distinct orbits of P 1 . Since n is arbitrarily large, we conclude that there are infinitely many P 1 -orbits in G/P 2 .
In Case 2, we assume that F 2 ∈ F 2 has infinite dimension and infinite codimension in V , and that there is F 1 ∈ F 1 with the same property. In type (BCD) these subspaces are also assumed to be isotropic. In the case where F 1 ∩ F 2 has infinite codimension in F 1 and F 2 , Claim 1 yields infinitely many elements F F (n) ∈ G/P 2 which belong to pairwise distinct P 1 -orbits.
It remains to consider the case where F 1 ∩ F 2 has finite codimension in F 1 or F 2 . This implies that F 1 ∩ F has infinite dimension and F 1 + F has infinite codimension in V whenever F is E-commensurable with F 2 . By applying Lemma 5.2 (b) with (L, M) = (0, F 1 ), we get a sequence {F (n) } n≥0 of (respectively isotropic) subspaces which are Ecommensurable with F 2 and satisfy Therefore, the associated generalized flags F F (n) (for n ≥ 0) are points of G/P 2 that belong to pairwise distinct P 1 -orbits. We again conclude that G/P 2 has an infinite number of P 1 -orbits. The proof of the direct implication in Theorem 1.4 is now complete.
6. Proof of the inverse implication in Theorem 1.4 We assume that P 1 is large and P 2 is semilarge. Hence assumption (4.1) is fulfilled, and we can find an exhaustion as in Section 4.1. By Lemma 4.2, we have inclusions of orbit sets X(n)/G(n) ֒→ X(n + 1)/G(n + 1) for all n ≥ 1 and the orbit set X/G is the direct limit To show that X has a finite number of G-orbits, it is sufficient to estimate the number s n of G(n)-orbits on X(n) and prove that the sequence {s n } n≥1 is bounded.
In type (BCD), X(n) is the set of ordered pairs of isotropic flags of a given type, in a finite-dimensional space M endowed with the nondegenerate bilinear form ω, and G(n) is the group G(M, ω) of transformations which preserve ω. According to Lemma 4.4, we have s n ≤ s A n where s A n stands for the number of GL(M)-orbits on the set of ordered pairs of (not necessarily isotropic) flags of the same type. Thus for showing that the sequence {s n } n≥1 is bounded, it suffices to show that the sequence {s A n } n≥1 is bounded. Therefore, it is enough to deal with the type (A) case.
Since P 1 is large, the generalized flag F 1 is a finite chain such that c 1,k := dim F 1,k / dim F 1,k−1 is finite for all k = p 0 . The generalized flag F 2 is a finite chain as P 2 is semilarge. For n ≥ 1 large, X(n) is a double flag variety of the form X := Fl(c 1 , . . . , c p ) × Fl(d 1 , . . . , d q ), whose elements are ordered pairs of flags We have c k > 0, d ℓ > 0 and c 1 + . . . + c p = d 1 + . . . + d q = m := dim E n .
In addition, c k = c 1,k for all k = p 0 , and thus c p 0 = m − k =p 0 c 1,k . We must show that the number s n of G := GL( E n )-orbits on X can be bounded by a constant which depends only on the numbers c k (for k = p 0 ), p, and q. By the Bruhat decomposition, s n is the cardinality of the double coset An element of the quotient S m /S d 1 ×· · ·×S dq can be viewed as a map τ : {1, . . . , m} → {1, . . . , q} such that τ −1 (j) has d j elements for all j. Every such map τ belongs to the S c 1 × · · · × S cp -orbit of a map τ 0 which is in addition nondecreasing on each interval [c k−1 + 1,c k ] withc k := c 1 + . . . + c k . Such a map τ 0 is completely determined by its restriction to {i ∈ [1, m] : i ≤c p 0 −1 or i >c p 0 }. This restriction is a map where the set on the left-hand side has C := k =p 0 c k elements. There are q C maps between these two sets. Therefore, we conclude that The proof of the theorem is complete. 7. Proof of Theorem 1.6 Let X = X 1 × X 2 × X 3 , where each factor X i = G/P i (i = 1, 2, 3) is the quotient by a splitting parabolic subgroup P i ⊂ G and can be viewed as an ind-variety of generalized flags (Section 2.3). Evidently, X can have a finite number of G-orbits only if every product of two factors X i × X j (1 ≤ i < j ≤ 3) has a finite number of G-orbits. In view of Theorem 1.4, this property holds only if all three parabolic subgroups P 1 , P 2 , P 3 are semilarge and two of them are large. This justifies the first claim in Theorem 1. 6.
In what follows, we assume that P 1 , P 2 are large and P 3 is semilarge.
In this way, condition (4.1) is satisfied and we may consider the construction of Section 4.1. Namely, we may choose an admissible basis E such that X 1 , X 2 , and X 3 contain an element which is E-compatible. Relying on a filtration E = n≥1 E n as in Section 4.1, we get exhaustions such that X i (n) (for i = 1, 2, 3) is a (finite-dimensional) flag variety for the algebraic group G(n).
We have to show that X has a finite number of G-orbits if and only if X 1 (n) × X 2 (n) × X 3 (n) has a finite number of G(n)-orbits for all n. The direct implication follows from Lemma 4.2. In the rest of this section, we assume that (7.1) X 1 (n) × X 2 (n) × X 3 (n) has finitely many G(n)-orbits for all n, and we need to show that X has finitely many G-orbits. For every n ≥ 1, we denote V n = E n and V ′ n = E \ E n . Then V = V n ⊕ V ′ n . In type (BCD), the subspaces V n and V ′ n are orthogonal with respect to the form ω. Note that the restrictions of ω to V n and V ′ n , respectively denoted by ω n and ω ′ n , are nondegenerate. Let π n : V → V n and π ′ n : V → V ′ n be the projections determined by the above decomposition.
Let G(V ′ n ) stand for the subgroup of elements g ∈ G such that g(V ′ n ) = V ′ n and g(e) = e for all e ∈ E n . Note that G(V ′ n ) can be viewed as a subgroup of GL(V ′ n ), and it is an ind-group of the same type as G.
We point out two preliminary facts.
Lemma 7.1. Let P ⊂ G be a large splitting parabolic subgroup. Then there is an integer n 0 ≥ 1 such that G(V ′ n 0 ) ⊂ P. Proof. Since P is large, it is the stabilizer of a generalized flag such that dim F k /F k−1 < +∞ for all k = p 0 . In particular, F p 0 −1 is finite dimensional hence there is n 1 ≥ 1 with F p 0 −1 ⊂ V n 1 . By Proposition 3.1, the generalized flag F is weakly E-compatible. Since F p 0 is finite codimensional, we have e ∈ F p 0 for all but finitely many vectors e ∈ E, hence there is n 2 ≥ 1 with F p 0 ⊃ V ′ n 2 . Then the integer n 0 := max{n 1 , n 2 } is as required in the statement.
Lemma 7.2. Let P ⊂ G be a semilarge splitting parabolic subgroup which contains the splitting Cartan subgroup H(E). For all m ≥ 1, there is an integer n 1 ≥ 1 such that for all m-dimensional subspace M ⊂ V , we can find an element g ∈ P with g(M) ⊂ V n 1 .
Proof. Being semilarge, P is the stabilizer of a generalized flag This generalized flag being E-compatible, for all k ∈ {1, . . . , p} we have a subset E ′ k ⊂ E with F k = F k−1 ⊕ E ′ k . The subgroup acts on the set of vectors p k=1 E ′ k with finitely many orbits, hence there is an integer ϕ(0) ≥ 1 with p k=1 E ′ k ⊂ P ′ · V ϕ(0) .
For all n ≥ 1, by applying the same property to the intersection P ∩ G(V ′ n ), which is a semilarge splitting parabolic subgroup of G(V ′ n ) that contains H(E \ E n ), we get an integer ϕ(n) > n satisfying p k=1 E ′ k ∩ V ′ n ⊂ (P ′ ∩ G(V ′ n )) · V ϕ(n) .
We will show that S has a finite number of orbits on X 1 .
Fix n 0 ≥ 1, large enough so that F 2 and F 3 belong to X 2 (n 0 ) and X 3 (n 0 ), respectively, and such that G(V ′ n 0 ) ⊂ Stab G (F 2 ) (see Lemma 7.1). Then S ′ := S ∩ G(V ′ n 0 ) = Stab G (F 3 ) ∩ G(V ′ n 0 ). Since F 3 belongs to X 2 (n 0 ), the chain of subspaces F 3 (V ′ n 0 ) := {F ∩ V ′ n 0 : F ∈ F 3 } is an (E \ E n 0 )-compatible generalized flag in the space V ′ n 0 . Note that in type (BCD) this generalized flag is ω ′ n 0 -isotropic. Moreover, the fact that F 3 belongs to X 3 (n 0 ) guarantees that each subspace F ∈ F 3 satisfies F = (F ∩ V n 0 ) ⊕ (F ∩ V ′ n 0 ). We deduce that ). Consequently, S ′ is a semilarge splitting parabolic subgroup of G(V ′ n 0 ) that contains the splitting Cartan subgroup H(E \ E n 0 ). By (7.1) we know that for every n ≥ n 0 , the (finite-dimensional) subvariety X 1 (n) intersects only finitely many S-orbits. For completing the proof of Theorem 1.6, it suffices to prove the following claim.
Claim 1: There is n 1 ≥ n 0 such that, for all F ∈ X 1 , there is g ∈ S with gF ∈ X 1 (n 1 ).
The combination of Claims 2 and 3 below yields Claim 1, and will make the proof of the theorem complete.
Since P 1 is a large parabolic subgroup, every point F ∈ X 1 is a finite flag of the form for p 0 ≤ k ≤ p. These observations show that the image of the map F ∈X 1 → F (V ′ n 1 ) is contained in a unionX ′ 1 ∪ . . . ∪X ′ r where eachX ′ j is an ind-variety of generalized flags in the space V ′ n 1 corresponding to a large splitting parabolic subgroup of G(V ′ n 1 ). Since n 1 ≥ n 0 , arguing in the same way as for S ′ , we see that the subgroup S ′ := S ∩ G(V ′ n 1 ) is a semilarge splitting parabolic subgroup of G(V ′ n 1 ). By Theorem 1.4,S ′ has a finite number of orbits on everyX ′ j . It follows that the set {F (V ′ n 1 ) : F ∈X 1 } intersects finitely manyS ′ -orbits. Hence we can find n 2 ≥ n 1 such that for every F = {F 0 , . . . , F p } ∈X 1 , there is g ∈S ′ with g(F p 0 ∩ V ′ n 1 ) ⊃ V ′ n 2 . Whence g(F p 0 ) ⊃ V ′ n 2 . The conclusion that gF = {g(F k )} p k=0 belongs to X 1 (n 2 ) is obtained by observing that Let ℓ ≥ 4. Theorem 1.4 implies that X has infinitely many G-orbits whenever at least two of the splitting parabolic subgroups P 1 , . . . , P ℓ are not large. Hence we may assume that P 1 , . . . , P ℓ−1 are large and consider the construction of Section 4.1.
Since ℓ ≥ 4, it follows from the results in [7], [8], [9], [10] that every (finite-dimensional) multiple flag variety X(n) has infinitely many G(n)-orbits whenever n ≥ 1 is large enough. By Lemma 4.2 we infer that X has infinitely many G-orbits, completing the proof of Theorem 1.7.