Ind–varieties of generalized flags as homogeneous spaces for classical ind–groups

Ivan Dimitrov and Ivan Penkov

Abstract

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space V (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of homogeneous spaces of the ind–groups SL(∞), SO(∞) and Sp(∞) in terms of generalized flags. Generalized flags in V are chains of subspaces which in general cannot be enumerated by integers. Given a basis E of V, we define a notion of E–commensurability for generalized flags, and prove that the set Fℓ(F, E) of generalized flags E–commensurable with a fixed generalized flag F in V has a natural structure of an ind–variety. In the case when V is the standard representation of G = SL(∞), all homogeneous ind–spaces G/P for parabolic subgroups P containing a fixed splitting Cartan subgroup of G, are of the form Fℓ(F, E). We also consider isotropic generalized flags. The corresponding ind–spaces are homogeneous spaces for SO(∞) and Sp(∞). As an application of the construction, we compute the Picard group of Fℓ(F, E) (and of its isotropic analogs) and show that Fℓ(F, E) is a projective ind–variety if and only if F is a usual, possibly infinite, flag of subspaces in V.

Key words (2000 MSC): Primary 14M17, Secondary 14L35, 14M15.

To Joseph A. Wolf with admiration

Introduction

Flag varieties play a fundamental role both in representation theory and algebraic geometry. There are two standard approaches to flag varieties: the group–theoretic one, where a flag variety is defined as G/P for a classical algebraic group G and a parabolic subgroup P, and the geometric one, where a flag variety is defined as the set of all chains of subspaces of fixed dimensions in a finite dimensional vector space, which in addition are assumed isotropic in the presence of a bilinear form. The very existence of these two approaches is in the heart of the interplay between representation theory and geometry.

The main topic of this paper is a purely geometric construction of homogeneous spaces for the classical ind–groups SL(∞), SO(∞) and Sp(∞). Despite the fact that many phenomena related to inductive limits of classical groups have been studied (see for instance, [O1], [O2], [SV], [VK1], [VK2], many natural questions remain unanswered. In particular, the only approach to homogeneous spaces of classical ind–groups discussed in the literature is a representation–theoretic one, and has been introduced by Joseph A. Wolf and his collaborators, [NRW], [DPW].

The difficulty in the purely geometric approach is that the consideration of flags, i.e. chains of subspaces enumerated by integers, is no longer sufficient. To illustrate the problem, let, more specifically, G denote the ind–group SL(∞) over a field of characteristic 0, and P be a parabolic subgroup of G. By definition, G is the union of a standard system of nested algebraic groups SL(n) and P is the union of parabolic subgroups. If V is the natural representation of G, all P –invariant subspaces in V form a chain C of subspaces of V. In general, the chain C has a rather complicated structure and is not necessarily a flag, i.e. cannot be indexed by integers. We show, however, that C always contains a canonical subchain F of subspaces of V with the property that every element of F is either the immediate predecessor F′ of a subspace F′′ ∈ F or the immediate successor F′′ of a subspace F′ ∈ F, and, in addition, each nonzero vector v ∈ V belongs to a difference F′′\F′. These two properties define generalized flags. (Maximal generalized flags already appeared in [DP] in a related but somewhat different context.) If, in addition, the vector space V is equipped with a non–degenerate bilinear (symmetric or antisymmetric) form, we introduce the notion of an isotropic generalized flag.

Informally we think of two (possibly isotropic) generalized flags being commensurable if they only differ in a finite dimensional subspace of V in which they reduce to flags of the same type. The precise definition is given in Section 4. The main result of this paper is the construction of the ind–varieties of commensurable generalized flags and their identification with homogeneous ind–spaces G/P for classical locally linear ind–groups G isomorphic to SL(∞), SO(∞), or Sp(∞), and corresponding parabolic subgroups P.

The paper is completed by providing two applications: an explicit computation of the Picard group of any ind–variety of commensurable generalized flags X and a criterion for projectivity of X. We show that the Picard group of X admits a description very similar to the classical one; however, X is projective if and only if it is an ind–variety of usual flags.

The “flag realization” of the ind–varieties G/P given in the present paper opens the way for a detailed and explicit study of the geometry of G/P, which should play a role as prominent as the geometric representation theory of the classical algebraic groups.

Acknowledgement We thank Vera Serganova for a detailed and thoughtful critique of the first version of this paper. The second named author thanks the Max Planck Institute for Mathematics in Bonn for support and hospitality.

Conventions ℕ stands for {1, 2, …} and ℤ_{+} = ℕ ∪ {0}. The ground field is a field k of characteristic 0 which will be assumed algebraically closed only when explicitly indicated in the text. As usual, k^{×} is the multiplicative group of k. The superscript ^{∗} denotes dual vector space.

The signs \underrightarrow{lim} and \underleftarrow{lim} stand respectively for direct and inverse limit over a direct or inverse system of morphisms parametrized by ℕ or ℤ_{+}. Γ(X, L) denotes the global sections of a sheaf L on a topological space X. All orders are assumed linear and strict, and all partial orders are assumed to have the additional property that the relation ”neither x ≺ y nor y ≺ x” is an equivalence relation.

1 Preliminaries

An ind–variety (over k) is a set X with a filtration

(1)  X_{0} ⊂ X_{1} ⊂ … ⊂ X_{n} ⊂ X_{n+1} ⊂ …

such that X = ∪_{n∈ℤ_{+}}X_{n}, each X_{n} is a Noetherian algebraic variety, and the inclusions X_{n} ⊂ X_{n+1} are closed immersions of algebraic varieties. An ind–variety X is automatically a topological space: a subset U ⊂ X is open in X if and only if, for each n, U ∩ X_{n} is an open subvariety of X_{n}. The sheaf of regular functions on X, or the structure sheaf O_{X} of X, is the inverse limit O_{X} = \underleftarrow{lim} O_{X_{n}} of the sheaves of regular functions O_{X_{n}} on the X_{n}. An ind–variety X = ∪_{n∈ℤ_{+}}X_{n} = \underrightarrow{lim} X_{n} is proper if and only if all the varieties X_{n} are proper, is affine if and only if all the X_{n} are affine. A morphism from an ind–variety X to an ind– variety Y is a map φ : X → Y such that, for every n ≥ 0, the restriction φ|_{X_{n}} is a morphism of X_{n} into Y_{m} for some m = m(n). An isomorphism of ind–varieties is a morphism which admits an inverse morphism. An ind–subvariety Z of X is a subset Z ⊂ X such that Z ∩ X_{n} is a subvariety of X_{n} for each n. Finally, an ind–group is by definition a group object in the category of ind–varieties. In this paper we consider only ind–groups G which are locally linear, i.e. ind–varieties G with an ind–variety filtration G_{0} ⊂ G_{1} ⊂ … ⊂ G_{n} ⊂ G_{n+1} ⊂ …, such that all G_{n} are linear algebraic groups and the inclusions are group morphisms.

Let V be a vector space of countable dimension. Fix an integer l ≥ 1. The set Gr(l; V) of all l–dimensional subspaces of V has a canonical structure of proper ind–variety: any filtration 0 ⊂ V_{l} ⊂ V_{l+1} ⊂ … ⊂ V = ∪_{r≥0}V_{l+r}, dim V_{l+r} = l + r, induces a filtration

Gr(l; V_{l}) ⊂ Gr(l; V_{l+1}) ⊂ … ⊂ Gr(l; V),

and the associated ind–variety structure on Gr(l; V) is independent of choice of filtration on V. For l = 1, ℙ(V) := Gr(1; V) is by definition the projective ind–space associated to V.

An invertible sheaf on an ind–variety X is a sheaf of O_{X}–modules locally isomorphic to O_{X}. The set of isomorphism classes of invertible sheaves on X is an abelian group (the group structure being induced by the operation of tensor product over O_{X} of invertible sheaves). By definition, the latter is the Picard group Pic X of X. It is an easy exercise to show that Pic X = \underleftarrow{lim} Pic X_{n} for any filtration (1). If X = ℙ(V), then Pic X ≅ ℤ. The preimage of 1 under this isomorphism is the class of the standard sheaf O_{ℙ(V)}(1) where, by definition, O_{ℙ(V)}(1) := \underleftarrow{lim} O_{ℙ(V_{n})}(1).

An invertible sheaf L on a proper ind–variety X is very ample if, for some filtration (1), its restrictions L_{n} on X_{n} are very ample for all n, and all restriction maps Γ(X_{n}; L_{n}) → Γ(X_{n−1}, L_{n−1}) are surjective. A very ample invertible sheaf defines a closed immersion of X into ℙ(\underrightarrow{lim} Γ(X_{n}, L_{n})^{∗}) as for each n the restrictions L_{n} and L_{n−1} define a commutative diagram of closed immersions

X_{n−1} ↪ ℙ(Γ(X_{n−1}, L_{n−1})^{∗})
∩  ∩
X_{n} ↪ ℙ(Γ(X_{n}, L_{n})^{∗}).

Conversely, given a closed immersion X ↪ ℙ(V), the inverse image of O_{ℙ(V)}(1) on X is a very ample invertible sheaf on X. Therefore, a proper ind–variety X is projective, i.e. X admits a closed immersion into a projective ind–space, if and only if it admits a very ample invertible sheaf.

2 Generalized flags: definition and first properties

Let V be a vector space over k. A chain of subspaces in V is a set C of pairwise distinct subspaces of V such that for any pair F′, F′′ ∈ C, either F′ ⊂ F′′ or F′′ ⊂ F′. Every chain of subspaces C is ordered by proper inclusion. Given C, we denote by C′ (respectively, by C′′) the subchain of C which consists of all C ∈ C with an immediate successor (respectively, an immediate predecessor). A generalized flag in V is a chain of subspaces F which satisfies the following conditions:

(i) each F ∈ F has an immediate successor or an immediate predecessor, i.e. F = F′ ∪ F′′;

(ii) V \{0} = ∪_{F′∈F′}F′′\F′, where F′′ ∈ F′′ is the immediate successor of F′ ∈ F′.

Given a generalized flag F and a subspace F′′ ∈ F′′ (respectively, F′ ∈ F′), we will always denote by F′ (resp., by F′′) its immediate predecessor (resp., immediate successor). Furthermore, condition (ii) implies that each nonzero vector v ∈ V determines a unique pair F′_{v} ⊂ F′′_{v} of subspaces in F with v ∈ F′′_{v}\F′_{v}.

Example 1.

(i) We define a flag in V to be a chain of subspaces F satisfying (ii) and which is isomorphic as an ordered set to a subset of ℤ. A flag can be equivalently defined as a chain of subspaces F for which there exists a strictly monotonic map of ordered sets φ : F → ℤ and, in addition, ∩_{F ∈F}F = 0 and ∪_{F ∈F} = V. There are four different kinds of flags: a finite flag of length k F = {0 = F_{1} ⊂ F_{2} ⊂ … ⊂ F_{k−1} ⊂ F_{k} = V }; an infinite accending flag F = {0 = F_{1} ⊂ F_{2} ⊂ F_{3} …}, where ∪_{i≥1}F_{i} = V; an infinite descending flag F = {… ⊂ F_{−3} ⊂ F_{−2} ⊂ F_{−1} = V }, where ∩_{i≤−1} = 0; and a two sided infinite flag F = {… ⊂ F_{−1} ⊂ F_{0} ⊂ F_{1} ⊂ …}, where ∩_{i∈ℤ}F_{i} = 0 and ∪_{i∈ℤ}F_{i} = V.

(ii) One of the simplest examples of a generalized flag in V which is not a flag is a generalized flag with both an infinite accending part and an infinite descending part, i.e. F = {0 = F_{1} ⊂ F_{2} ⊂ F_{3} ⊂ … ⊂ F_{−3} ⊂ F_{−2} ⊂ F_{−1} = V }, where ∪_{i≥1}F_{i} = ∩_{j≤−1}F_{j}.

(iii) Let V be a countable dimensional vector space with basis {e_{q}}_{q∈ℚ}. Set F′(q) := span{e_{r} | r < q} and F′′(q) := span{e_{r} | r ≤ q}. Then F := ∪_{q∈ℚ}{F′(q), F′′(q)} is a generalized flag in V. F is not a flag, and moreover, no F ∈ F has both an immediate predecessor and an immediate successor.

The following Proposition shows that each of the subchains F′ and F′′ reconstructs F.

Proposition 1 Let F be a generalized flag in V. Then

(i) for every F′ ∈ F′, F′ = ∪_{G′′∈F′′,G′′⊂F′′,G′′≠F′′}G′′; (ii) for every F′′ ∈ F′′, F′′ = ∩_{G′∈F′,G′⊃F′,G′≠F′}G′.

Proof. (i) The inclusion F′ ⊃ ∪_{G′′∈F′′,G′′⊂F′′,G′′≠F′′}G′′ is obvious. Assume now that v ∈ F′. Let v ∈ H′′\H′ for some H′ ∈ F′ and its immediate successor H′′ ∈ F′′. Then H′ ⊂ F′ and hence H′′ ⊂ F′, i.e. v ∈ ∪_{G′′∈F′′,G′′⊂F′′,G′′≠F′′}G′′ which proves that F′ ⊂ ∪_{G′′∈F′′,G′′⊂F′′,G′′≠F′′}G′′. Assertion (ii) is proved in a similar way. □

Any chain C of subspaces in V determines the following partition of V:

(2)  V = ⊔_{v∈V} [v]_{C}, where [v]_{C} := {w ∈ V | w ∈ F ⇔ v ∈ F, ∀F ∈ C}.

Consider this correspondence as a map π from the set of chains of subspaces in V into the set of partitions of V. This map is not injective, for π(C′) = π(C) if C′ is obtained from C by adding arbitrary intersections and unions of elements of C. As we show in Proposition 2 below, the notion of a generalized flag provides us with a natural right inverse of π, i.e. with a map γ (defined on the image of π) such that π ∘ γ = id. This explains the special role of generalized flags among arbitrary chains of subspaces in V. Namely, every generalized flag in V is a natural representative of the class of chains of subspaces in V which yield the same partition of V.

Proposition 2 Given a chain C of subspaces in V, there exists a unique generalized flag F in V for which π(C) = π(F).

Proof. To prove the existence, set F′_{v} := ∪_{W ∈C,v∉W} W and F′′_{v} := ∩_{W ∈C,v∈W} W, and put F := ∪_{v∈V \{0}}{F′_{v}, F′′_{v}}. It is obvious from the definition of F that π(C) = π(F). To show that F is a generalized flag, notice that, for any pair of nonzero vectors u, v ∈ V, exactly one of the following three possibilities holds:

F′_{u} = F′_{v}, and hence F′′_{u} = F′′_{v};

F′_{u} ⊂ F′_{v}, and hence F′′_{u} ⊂ F′_{v};
F′_{u} ⊃ F′_{v}, and hence F′_{u} ⊃ F′′_{v}.

Indeed, if, for every W ∈ C, u ∈ W if and only if v ∈ W, then F′_{u} = F′_{v} and F′′_{u} = F′′_{v}. Assume now, that there exists W ∈ C such that u ∈ W but v ∉ W. Then F′′_{u} ⊂ W ⊂ F′_{v}. Similarly, if there exists W ∈ C such that u ∉ W but v ∈ W, we have F′′_{v} ⊂ W ⊂ F′_{u}. The existence of F is now established.

The uniqueness follows from the fact that [v]_{C} = (∩_{W ∈C,v∈W} W)\(∪_{W ∈C,v∉W} W), while, for a generalized flag F, [v]_{F} = F′′_{v}\F′_{v}. □

We now define the map γ by setting γ(π(C)) := F, and put fl := γ ∘ π. In the example below we determine the preimages under fl of the generalized flags introduced in Example 1. The computation is based on the following simple fact: if C̄ is any chain in fl^{−1} (F), then every nonzero subspace C̄ ∈ C̄ is the union of spaces from F.

Example 2. The cases (i), (ii) and (iii) below refer to the corresponding cases in Example 1.

(i) If F is a flag in V then fl^{−1} (F) consists of F and the chains obtained from F by adding 0, V or both, in case 0 and/or V do not belong to F.

(ii) In this case fl^{−1} (F) consists of two chains: F itself and the chain obtained by adding ∪_{i≥1}F_{i} = ∩_{j≤−1}F_{j} to F.

(iii) In this case there are infinitely many chains C̄ with fl(C̄) = F. Set F′(x) := span{e_{r} | r < x} for any x ∈ ℝ, and let C denote the chain {F′(x) | x ∈ ℝ} ∪ {F′′(q) | q ∈ ℚ} ∪ {0, V }. It is easy to check that fl(C) = F and that any chain in fl^{−1} (F) is a subchain of C. To characterize explicitly all chains in fl^{−1} (F), for any subchain C̄ ⊂ C, set ℝC̄ := {x ∈ ℝ | F′(x) ∈ C̄} and ℚC̄:= {q ∈ ℚ | F′′(q) ∈ C̄}. Then fl(C̄) = F if and only if, for any r ∈ ℚ, we have r ∈ ℚC̄ or r = inf{x ∈ ℝC̄ ∪ ℚC̄ | r < x}, and r ∈ ℝC̄ or r = sup{x ∈ ℝC̄ ∪ ℚC̄ | x < r}.

A generalized flag F in V is maximal if it is not properly contained in another generalized flag in V. It is easy to see that the generalized flags introduced in Example 1(ii) and (iii) are maximal. More generally, a generalized flag F is maximal if and only if dim(F′′_{v}/F′_{v}) = 1 for every nonzero v ∈ V. Indeed, assume dim(F′′_{v_{0}}/F′_{v_{0}}) > 1 for some v_{0}. Let F ⊂ V be a subspace with proper inclusions F′_{v_{0}} ⊂ F ⊂ F′′_{v_{0}}. Then the generalized flag F ∪ {F } properly contains F. Conversely, if dim(F′′_{v}/F′_{v}) = 1 for every nonzero v ∈ V, and if G is a generalized flag which contains F, then F′_{v} ⊂ G′_{v} ⊂ G′′_{v} ⊂ F′′_{v}. Hence F′_{v} = G′_{v}, F′′_{v} = G′′_{v}, i.e. F = G.

The map fl establishes a bijection between maximal chains of subspaces in V and maximal generalized flags in V. More precisely, if C is a maximal chain, fl(C) is the unique maximal generalized flag which is a subchain of C. Conversely, C is the unique maximal chain containing fl(C ). These latter statements are essentially equivalent to Theorem 9 in [DP]. For example, if F is the maximal generalized flag from Example 1 (iii), its corresponding maximal chain C is described in Example 2 (iii).

We conclude this section by introducing isotropic generalized flags. Let w : V × V → V be a non–degenerate symmetric or skew–symmetric bilinear form on V. Denote by U^{⊥} the w–orthogonal complement of a subspace U ⊂ V. A generalized flag F in V is w–isotropic if there exists an involution τ : F → F such that (F′)^{⊥} = τ(F)′′ for every F′ ∈ F′. If F is w–isotropic, τ is determined by w and is a decreasing map. If τ has a fixed point, the latter is unique and will be denoted by F_{τ}. If τ has no fixed point, we introduce F′_{τ} ∉ F by setting F′_{τ} := ∪_{F ⊂τ(F)}F. One checks immediately that F′_{τ} = ∩_{τ(F)⊂F} F.

3 Compatible bases

If V is finite dimensional, any ordered basis determines a maximal flag in V. Conversely, a maximal flag in V determines a set of compatible bases in V. More generally, if V is any vector space, F is a generalized flag in V and {e_{α}}_{α∈A} is a basis of V, we say that F and {e_{α}}_{α∈A} are compatible if there exists a strict partial order ≺ on A (satisfying the condition stated in the Conventions) such that F′_{e_{α}} = span{e_{β} | β ≺ α} and F = fl({F′_{e_{α}}}_{α∈A}).

Not every generalized flag admits a compatible basis. Indeed, let V := ℂ[[x]] be the space of formal power series in the indeterminate x and let F denote the flag … ⊂ F_{n} ⊂ F_{n−1} ⊂ … ⊂ F_{1} ⊂ F_{0} = V, where F_{n} := x^{n}V. Clearly, F is a maximal flag in V as dim(F_{n−1}/F_{n}) = 1 for all n > 0. However, as V is uncountable dimensional, no basis of V can be compatible with the countable flag F.

The following proposition shows that the uncountability of dim V is crucial in the above example.

Proposition 3 If V is countable dimensional, every generalized flag F in V admits a compatible basis.

Proof. Assume first that F is a maximal generalized flag in V. Let {l_{i}}_{i∈ℕ} be a basis of V. Define inductively a basis {e_{i}}_{i∈ℕ} of V as follows. Put e_{1} := l_{1}. Assuming that e_{1}, …, e_{n} have been constructed, choose e_{n+1} of the form l_{n+1} + c_{1}e_{1} + … + c_{n}e_{n} so that F′_{e_{n+1}} is not among F′_{e_{1}}, …, F′_{e_{n}}. Then, obviously,

(3)  span{l_{1}, …, l_{n}} = span{e_{1}, …, e_{n}}

for every n and the subspaces F′_{e_{n}} are pairwise distinct. Furthermore, as it is not difficult to check, for every F′ ∈ F′, the set F′′\F′ contains exactly one element of the basis {e_{i}}_{i∈ℕ}, and hence ℕ is linearly ordered in the following way: i ≺ j if and only if F′_{e_{i}} ⊂ F′_{e_{j}}. This proves that F is compatible with {e_{i}}_{i∈ℕ}.

For a not necessarily maximal generalized flag F, it is enough to consider a basis compatible with a maximal generalized flag G containing F. Such a basis is automatically compatible with F. □

Let V be a finite or countable dimensional vector space and w be a non–degenerate symmetric or skew–symmetric bilinear form on V. Define a basis of V of the form {e_{n}, e^{n}} to be of type C if w(e_{i}, e_{j}) = w(e^{i}, e^{j}) = 0 and w(e_{i}, e^{j}) = δ_{i,j} for a skew–symmetric w. A basis of V of the form {e_{0} = e^{0}, e_{n}, e^{n}} (respectively {e_{n}, e^{n}}) is of type B (resp. of type D) if w(e_{i}, e_{j}) = w(e^{i}, e^{j}) = 0 and w(e_{i}, e^{j}) = δ_{i,j} for a symmetric w. For uniformity we will always label a basis of type B, C or D simply as {e_{i}, e^{i}} where we assume that in the case of B, e_{0} = e^{0} and i runs over ℤ_{+} when V is countable dimensional, or over a finite subset of ℤ_{+} when V is finite dimensional, while in the cases of C and D, i runs over ℕ or over a finite subset of ℕ. A w–isotropic basis of V is by definition a basis of V admitting an order which makes it a basis of type B, C or D. If V is finite dimensional, V admits a basis of type B, C or D if and only if w is symmetric and V is odd dimensional, w is skew–symmetric and then V is necessarily even dimensional, or w is symmetric and V is even dimensional respectively. If V is countable dimensional, by an appropriate modification of the Gram– Schmidt orthogonalization process, one shows that if w is symmetric, then V admits both a basis of type B and a basis of type D, and if w is skew–symmetric, then V admits a basis of type C.

In the rest of the paper we assume that the dimension of V is countable.

Proposition 4 Let F be an w–isotropic generalized flag in V.

(i) Assume that w is skew–symmetric. Then V admits a basis of type C compatible with F. In particular, the vector space F′′_{τ}/F′_{τ} is even dimensional or infinite dimensional. (ii) Assume that k is algebraically closed and w is symmetric. If the vector space F′′_{τ}/F′_{τ} is odd dimensional or infinite dimensional, then V admits a basis of type D compatible with F. If F′′_{τ}/F′_{τ} is even dimensional or infinite dimensional, then V admits a basis of type B compatible with F.

Proof. Let {l_{n}}_{n∈ℕ} be a basis of V compatible with F. Set U_{F′} := span{l_{i} | F′_{l_{i}} = F′} for F′ ∈ F′. Then F′ = ⊕_{G′∈F′,G′⊂F′,G′≠F′}U_{G′} and F′′ = F′ ⊕ U_{F′} for every F′ ∈ F′. It is clear therefore that the restriction of w on U_{F′} × U_{τ(F′)} is a non–degenerate bilinear form for every F′ ∈ F′. Furthermore, if τ has a fixed point, the restriction of w on U_{F′_{τ}} × U_{F′_{τ}} is a non-degenerate skew–symmetric (respectively symmetric) bilinear form. If w is skew– symmetric, this implies, in particular, that U_{F′_{τ}}, and hence F′′_{τ}/F′_{τ} is even dimensional or infinite dimensional. Then U_{F′_{τ}} admits a basis of type C, B, or D depending on whether w is skew–symmetric or symmetric and on the dimension of U_{F′_{τ}}. Denote such a basis by {l′_{i}, l′^{i}}. Let, furthermore, {l′′_{i}} be the subset of {l_{n}} consisting of all l_{n} with F′_{l_{n}} ⊂ F′_{τ}. Finally, relabel the set {l′_{i}} ∪ {l′′_{i}} and denote the resulting set by {g_{n}}, where g_{0} := l′_{0} = l′^{0} if {l′_{i}, l′^{i}} is of type B.

We are now ready to construct inductively the desired w–isotropic basis {e_{n}, e^{n}}. Assume that e_{i}, e^{i} have been constructed for i ≤ n. Put e_{n+1} := g_{n+1} − ∑_{i=1}^{n}(w(e_{i}, g_{n+1})e^{i} + w(g_{n+1}, e^{i})e_{i}). There exists g ∈ U_{τ(F′_{g_{n+1}})} such that w(e_{n+1}, g) = 1. Set then e^{n+1} := g − ∑_{i=1}^{n}(w(e_{i}, g)e^{i} + w(g, e^{i})e_{i}). One checks immediately that the basis {e_{n}, e^{n}} is w– isotropic and compatible with F. □

4 Ind–varieties of generalized flags

For a finite dimensional V, two flags belong to the same connected component of the variety of all flags in V if and only if their types coincide, i.e. if the dimensions of the subspaces in the flags coincide. If V is infinite dimensional the notion of type is in general not defined, and flags, or generalized flags, can be compared using a notion of commensurability. Such notions are well–known in the special case of subspaces of V, i.e. of flags of the form 0 ⊂ W ⊂ V, see [T] and Chapter 7 of [PS]. Below we introduce a notion of commensurability for generalized flags which in the case of subspaces reduces to a refinement of Tate’s notion of commensurability, [T].

In the rest of the paper we fix a basis E = {e_{n}} of V. In the presence of a bilinear form w on V we fix a w–isotropic basis E = {e_{n}, e^{n}}, and whenever other bases of V or generalized flags in V are considered they are automatically assumed to be w–isotropic. We call a generalized flag F weakly compatible with E if F is compatible with a basis L of V such that E\(E ∩ L) is a finite set. Furthermore, we define two generalized flags F and G in V to be E–commensurable if both F and G are weakly compatible with E and there exists an inclusion preserving bijection φ : F → G and a finite dimensional subspace U ⊂ V, such that for every F ∈ F

(i) F ⊂ φ(F) + U and φ(F) ⊂ F + U;
(ii) dim(F ∩ U) = dim(φ(F) ∩ U).

It follows immediately from the definition that any two E–commensurable generalized flags are isomorphic as ordered sets, and that two flags in a finite dimensional space are E– commensurable if and only if their types coincide. (In the latter case the condition of weak compatibility with E is empty.) Furthermore, E–commensurability is an equivalence relation. Indeed, it is obviously reflexive and symmetric. It is also transitive. To see this, note first that, in the definition of E–commensurability, one can replace (ii) by

(ii′) dim(F/(F ∩ φ(F))) = dim(φ(F)/(F ∩ φ(F))).

Consider now F, G and H, such that F is E–commensurable with G and G is E–commensurable with H. Let φ : F → G and ψ : G → H be the respective bijections, and U and W be the finite dimensional subspaces of V corresponding to φ and ψ respectively. Then F and H satisfy (i) and (ii′) with ψ ∘ φ : F → H and U + W.

Example 3.

(i) Let F = {0 ⊂ F ⊂ V } and G = {0 ⊂ G ⊂ V }. If F and G are finite dimensional, then F and G are automatically weakly compatible with E. Furthermore, F and G are E– commensurable if and only if dim F = dim G. If, however, F and G are infinite dimensional, the condition that F and G are weakly compatible with E is not automatic. For example, if F = span{e_{2}, e_{3}, …} and G = {e_{2} − e_{1}, e_{3} − e_{1}, …}, then F is weakly compatible with E but G is not, and consequently, F and G are not E–commensurable. Finally, if F and G are both of finite codimension in V, and F and G are weakly E–compatible, then F and G are E–commensurable if and only if codim_{V} F = codim_{V} G.

(ii) Let F = {0 = F_{1} ⊂ F_{2} ⊂ F_{3} ⊂ …} and G = {0 = G_{1} ⊂ G_{2} ⊂ G_{3} ⊂ …} be two finite or infinite accending flags in V compatible with E. If all subspaces F_{i} and G_{i} are finite dimensional, then F and G are E–commensurable if and only if dim F_{i} = dim G_{i} for every i, and F_{n} = G_{n} for large enough n. If, however, there are infinite dimensional spaces among F_{i} and G_{i}, the above conditions are still necessary for F and G to be E–commensurable but they are not always sufficient. The exact sufficient conditions can be derived as a consequence of the proof of Proposition 5 below.

Given a generalized flag F weakly compatible with E, we denote by Fℓ(F, E) the set of all generalized flags in V E–commensurable with F. For the rest of the paper we fix the following notations: E_{n} = {e_{i}}_{i≤n}, V_{n} = span E_{n}, E_{n}^{c} := {e_{i}}_{i>n} and V_{n}^{c} := span E_{n}^{c}. If F is a w–isotropic generalized flag in V, Fℓ(F, w, E) stands for the set of all w–isotropic generalized flags E–commensurable with F. If G is an isotropic generalized flag in V, the involution τ is an order reversing involution on G considered as an ordered set. In this case E_{n} = {e_{i}, e^{i}}_{i≤n}, V_{n} = span E_{n}, E_{n}^{c} := {e_{i}, e^{i}}_{i>n} and V_{n}^{c} := span E_{n}^{c}. Since all generalized flags in Fℓ(F, w, E) are isomorphic as ordered sets, we will use the same letter τ to denote the involution on any G ∈ Fℓ(F, w, E).

Proposition 5 Fℓ(F, E), as well as Fℓ(F, w, E), has a natural structure of an ind–variety.

Proof. We present the proof in the case of Fℓ(F, E) only. The reader will supply a similar proof for Fℓ(F, w, E). For any G ∈ Fℓ(F, E) choose a positive integer n_{G} such that F and G are compatible with bases containing E_{n_{G}}^{c}, and V_{n_{G}} contains a finite dimensional subspace U which (together with the corresponding φ) makes F and G E–commensurable. Obviously we can pick n_{F} so that n_{F} ≤ n_{G} for every G ∈ Fℓ(F, E). Set also

(4)  G_{n} := {G ∩ V_{n} | G ∈ G}

for n ≥ n_{G}.

The type of the flag F_{n} yields a sequence of integers 0 = d_{n,0} < d_{n,1} < … < d_{n,s_{n−1}} < d_{n,s_{n}} = n, and Fℓ(F_{n}, E_{n}) is the usual flag variety Fℓ(d_{n}; V_{n}) of type d_{n} = (d_{n,1}, …, d_{n,s_{n}−1}) in V_{n}. Notice that s_{n+1} = s_{n} or s_{n+1} = s_{n} + 1. Furthermore, in both cases an integer j_{n} is determined as follows: in the former case, d_{n+1,i} = d_{n,i} for 0 ≤ i < j_{n} and d_{n+1,i} = d_{n,i} + 1 for j_{n} ≤ i < s_{n}, and in the latter case d_{n+1,i} = d_{n,i} for 0 ≤ i < j_{n} and d_{n+1,i} = d_{n,i−1} + 1 for j_{n} ≤ i < s_{n}.

Now we define a map ι_{n} : Fℓ(d_{n}; V_{n}) → Fℓ(d_{n+1}, V_{n+1}) for every n ≥ n_{F}. Given G_{n} = {0 = G_{0}^{n} ⊂ G_{1}^{n} ⊂ … ⊂ G_{s_{n}}^{n} = V_{n}} ∈ Fℓ(d_{n}; V_{n}), put ι_{n}(G_{n}) = G_{n+1} := {0 = G_{0}^{n+1} ⊂ G_{1}^{n+1} ⊂ … ⊂ G_{s_{n+1}}^{n+1} = V_{n+1}}, where

(5)  G_{i}^{n+1} := \{ \begin{matrix}G_{i}^{n} & if & 0 ≤ i < j_{n} \\ G_{i}^{n} ⊕ ke_{n+1} & if & j_{n} ≤ i ≤ s_{n+1} & and & s_{n+1} = s_{n} \\ G_{i−1}^{n} ⊕ ke_{n+1} & if & j_{n} ≤ i ≤ s_{n+1} & and & s_{n+1} = s_{n} + 1. \end{matrix}

It is clear that ι_{n} is a closed immersion of algebraic varieties, and hence \underrightarrow{lim} Fℓ(d_{n}; V_{n}) is an ind–variety. Let ψ_{n} : Fℓ(d_{n}; V_{n}) → \underrightarrow{lim} Fℓ(d_{n}; V_{n}) denote the canonical embedding corresponding to the direct system {ι_{n}}.

To endow Fℓ(F, E) with an ind–variety structure we construct a bijection Fℓ(F, E) → \underrightarrow{lim} Fℓ(d_{n}; V_{n}). Set

θ : Fℓ(F, E) → \underrightarrow{lim} Fℓ(d_{n}; V_{n}), θ(G) := \underrightarrow{lim} G_{n},

see (4). Checking that θ is injective is straightforward. To check that θ is surjective, fix G̃ = \underrightarrow{lim} G̃_{n} ∈ \underrightarrow{lim} Fℓ(d_{n}; V_{n}), an integer ñ and a flag G̃ñ ∈ Fℓ(dñ; Vñ) with ψñ(G̃ñ) = G̃. Denote by φñ the inclusion preserving bijection φñ : Fñ → G̃ñ. For every F ∈ F, put φ(F) := φñ(F ∩ Vñ) ⊕ (F ∩ V c ; ñ). It is clear that G := {φ(F)}_{F ∈F} is a generalized flag in V E–commensurable with F via φ and Vñ. Furthermore, using (5), one verifies that θ(G) = G̃. Hence θ is surjective. □

Example 4. Let F = {0 ⊂ F ⊂ V }, see Example 3 (i). If F is a finite dimensinal subspace of V of dimension l, then, regardless of E, Fℓ(F, E) is nothing but the ind–variety Gr(l; V) introduced in Section 1. If F is infinite dimensional subspace of V of codimension l, then as a set Fℓ(F, E) depends on the choice of E. However, the isomorphisms between Gr(l; V_{n}) and Gr(n − l; V_{n}) extend to an ind–variety isomorphism between Fℓ(F, E) and Gr(l; V) which depends on E. The latter isomorphism is a particular case of the following general duality. Let F be an arbitrary generalized flag in V. Assume that E is compatible with F, and for every F ∈ F set F^{c} := span{e ∈ E | e ∉ F }. Then F^{c} := {F^{c} | F ∈ F} is a generalized flag in V compatible with E and Fℓ(F^{c}, E) is isomorphic to Fℓ(F, E).

We complete this section by defining big cells in Fℓ(F, E) and Fℓ(F, w, E). Let L = {l_{n}}_{n∈ℕ} be a basis of V compatible with F and such that E\(E ∩ L) is a finite set, and let U_{F′} = span{l ∈ L | F′_{l} = F′} for any F′ ∈ F′. Denote by Φ = {Φ_{F′}}_{F′∈F′} a set of linear maps of finite rank Φ_{F′} : F′ → U_{F′}, such that Φ_{F′} ≠ 0 for finitely many subspaces F′_{1} ⊂ … ⊂ F′_{p} only. Given Φ, define

\begin{matrix}Γ_{F′} : F′ → F′′, & Γ_{F′}(v) := v + Φ_{F′}(v), \\ Γ : V → V, & Γ(v) := Γ_{F′_{p}} ∘ … ∘ Γ_{F′_{i+1}}(v), \end{matrix}

where i is the largest integer with v ∉ F′_{i}. Put Φ(F) := fl({Γ (F′)}_{F′∈F′}). Then define the big cell C(F, E; L) of Fℓ(F, E) corresponding to the basis L by setting

C(F, E; L) := {Φ(F) | for all possible Φ}.

To define the big cell C(F, w, E; L) in Fℓ(F, w, E), we start with a w–isotropic basis L = {l_{n}, l^{n}} of V compatible with F and such that E\(E ∩ L) is a finite set, and repeat the above construction of Γ(F′) for all F′ ∈ F′ with F′ ⊂ τ(F′). As a result we obtain subspaces Γ(F′) for F′ ⊂ τ(F′) and set Φ(F) := fl({Γ (F′), (Γ (F′))^{⊥}}_{F′∈F,F′⊂τ(F′)}). Then, we set

C(F, w, E; L) := {Φ(F) | for all possible Φ}.

Note that the role of F in defining big cells is not special and that big cells C(G, E; L), or C(G, w, E; L), are well defined for every G ∈ Fℓ(F, E), or respectively G ∈ Fℓ(F, w, E).

Proposition 6

(i) The big cell C(F, E; L) (respectively, C(F, w, E; L)) is an affine open ind–subvariety of Fℓ(F, E) (resp. of Fℓ(F, w, E)).

(ii) We have

(6)  Fℓ(F, E) = ∪_{L} C(F, E; L)

and

(7)  Fℓ(F, w, E) = ∪_{L} C(F, w, E; L),

where the unions run over all bases (respectively w–isotropic bases) L of V compatible with F and such that E\(E ∩ L) is a finite set.

Proof. We discuss the case of Fℓ(F, E) only. The argument for the case of Fℓ(F, w, E) is similar. Put L_{n} := {l_{i}}_{i≤n} and W_{n} := span L_{n}. Let F_{n} and Fℓ(d_{n}; W_{n}) be as in the proof of Proposition 5 above. Set C(d_{n}; W_{n}; L_{n}) := {Φ(F)_{n} | for all Φ such that, for every F′ ∈ F′, Φ_{F′}(W_{n}) ⊂ W_{n} and Φ_{F′}(l_{i}) = 0 for i > n}. Obviously, C(d_{n}; W_{n}; L_{n}) is a big cell in Fℓ(d_{n}; W_{n}), and hence is an affine open subset. Therefore, the inclusion ι_{n}(C(d_{n}; W_{n}; L_{n})) ⊂ C(d_{n+1}; W_{n+1}; L_{n+1}) and the equality \underrightarrow{lim} C(d_{n}; W_{n}; L_{n}) = C(F, E; L) show that C(F, E; L) is an affine open ind–subvariety of Fℓ(F, E). The fact that the set of cells {C(F, E; L) | L is a basis of V compatible with F, such that L\(E ∩L) is a finite set} is a covering of Fℓ(F, E) is an easy consequence of the definition of E–commensurability. □

5 Ind–varieties of generalized flags as homogeneous ind– spaces

Let G(E) be the group of automorphisms g of V such that g(e) = e for all but finitely many e ∈ E and in addition det g = 1. Recall that E_{n} = {e_{i}}_{i≤n} and V_{n} = span E_{n}. The natural inclusion

G(E_{n}) ⊂ G(E_{n+1}), g ↦ κ_{n}(g),

where κ_{n}(g)_{|V_{n}} = g and κ_{n}(g)(e) = e for e ∈ E_{n+1}\E_{n}, is a closed immersion of algebraic groups. Furthermore, G(E) = ∪_{n∈ℕ}G(E_{n}). In particular G(E) is a locally linear ind–group, and G(E) = G(L) for any basis L of V such that E\(E ∩ L) is a finite set.

Similarly, when E is a w–isotropic basis of V, let G^{w}(E) := {g ∈ G(E) | w(g(u), g(v)) = w(u, v) for any u, v ∈ V }. There are natural closed immersions G^{w}(E_{n}) ⊂ G^{w}(E_{n+1}), and G^{w}(E) = ∪_{n∈ℕ}G^{w}(E_{n}), where in this case E_{n} := {e_{i}, e^{i}}_{i≤n}.

The ind–group G(E) (respectively, G^{w}(E)) is immediately seen to be isomorphic to the classical ind–group A(∞) (resp., B(∞), C(∞) or D(∞) if E is a w–isotropic basis of type B, C or D). The ind–groups A(∞), B(∞), C(∞) and D(∞) are discussed in detail in [DPW]. An alternative notation for A(∞) is SL(∞), and B(∞) ≅ D(∞) and C(∞) are also denoted respectively by SO(∞) and Sp(∞).

In the rest of the paper the letter G will denote one of the groups G(E) or G^{w}(E), and G_{n} will denote respectively G(E_{n}) or G^{w}(E_{n}). The basis E equips G with a subgroup H, consisting of all diagonal automorphisms of V in G, i.e. of the elements g ∈ G such that g(e) ∈ ke for every e ∈ E. We call H a splitting Cartan subgroup (in the terminology of [DPW], H is a Cartan subgroup of G). Following [DPW], for the purposes of the present paper, we define a parabolic (respectively, a Borel) subgroup of G to be an ind–subgroup P (resp., B) of G such that its intersection with G_{n} for every n is a parabolic (resp., a Borel) subgroup of G_{n} for some, or equivalently any, order on E.

If F is a generalized flag in V compatible with E (and w–isotropic, whenever E is w– isotropic), we denote by P_{F} the stabilizer of F in G.

Proposition 7

(i) P_{F} is a parabolic subgroup of G containing H. (ii) The map F ↦ P_{F} establishes a bijection between generalized flags in V compatible with E and parabolic subgroups of G containing H.

Proof. The inclusion H ⊂ P_{F} follows directly from the definition of H and P_{F}. Furthermore, P_{F} ∩ G_{n} is a parabolic subgroup of G_{n} as it is the stabilizer of F_{n} in G_{n}. Hence P_{F} is a parabolic subgroup of G. If, conversely, P = ∪_{n}P_{n} is a parabolic subgroup of G containing H, denote by F(n) the flag in V_{n} whose stabilizer is P_{n}. Note that F(n) maps into F(n+1). More precisely, for G = G(E), F(n + 1) = ι_{n}(F(n)), see (5); and for G = G^{w}(E), the corresponding map is the w–isotropic analog of ι_{n} which we leave to the reader to reconstruct. In both cases we define F as θ^{−1}(\underrightarrow{lim} F(n)). A direct checking shows that P = P_{F}. □

Proposition 7 further justifies our consideration of generalized flags, see the discussion before Proposition 2 above. Indeed, it is clear that if P ⊂ G is the stabilizer of a chain C of subspaces in V, then P depends only the partition π(C), see (2), and not on C itself. Therefore, the generalized flag F emerges as a representative of the class of all chains C which have P as a stabilizer in G. Moreover, Proposition 3 together with Proposition 7 (respectively, Proposition 4 and Proposition 7 for w–isotropic flags), imply that the stabilizer in G of any generalized flag (resp. isotropic generalized flag) compatible with E is a parabolic subgroup of G. Finally, maximal generalized flags in V correspond to Borel subgroups under the above bijection.

Note that, for any order on E and for any generalized flag F compatible with E, G/P_{F} = ∪_{n}(G_{n}/P_{n}), where P_{n} := P_{F} ∩ G_{n}. In particular, G/P_{F} is an ind–variety. Moreover, any other order on E, for which E is isomorphic to ℕ as an ordered set, defines an isomorphic ind–variety. We are now ready to exhibit the homogeneous ind–space structure on Fℓ(F, E) and Fℓ(F, w, E).

Theorem 1 For any E and F as above there is a respective isomorphism of ind–varieties Fℓ(F, E) ≅ G/P_{F} or Fℓ(F, w, E) ≅ G/P_{F}.

Proof. Given G ∈ Fℓ(F, E) (or, respectively, G ∈ Fℓ(F, w, E)), let U ⊂ V be the corresponding to G finite dimensional subspace. We may assume that U = V_{n} = span E_{n} for some n. Since F_{n} and G_{n} are flags of the same type in the finite dimensional space V_{n}, there exists g_{n} ∈ G_{n}, so that g(F_{n}) = G_{n}. We extend g_{n} to an element g ∈ G by setting g(e) = e for e ∈ E\E_{n}. Now

f : Fℓ(F, E) → G/P_{F} (or, respectively, f : Fℓ(F, w, E) → G/P_{F}), f(G) := gP

is a well–defined map and it is easy to check that it is an isomorphism of ind–varieties. □

6 Picard group and projectivity

The interpretation of Fℓ(F, E) and Fℓ(F, w, E) as homogeneous ind–spaces G/P_{F} provides us with a representation theoretic description of the Picard groups of Fℓ(F, E) and Fℓ(F, w, E). Namely, Pic Fℓ(F, E), as well as Pic Fℓ(F, w, E), is naturally isomorphic to the group of integral characters of the Lie algebra of the ind–group P_{F}.

Consider Fℓ(F, E). There is a canonical isomorphism of abelian groups Pic Fℓ(F, E) = Hom(P_{F}, k^{×}). To see this, notice that Pic Fℓ(F, E) = \underleftarrow{lim} Pic Fℓ(d_{n}; V_{n}) = \underleftarrow{lim} Pic G_{n}/(P_{F})_{n}. It is a classical fact that Pic G_{n}/(P_{F})_{n} = Hom((P_{F})_{n}, k^{×}) for every n, and an immediate verification shows that the diagram

(8)

Pic(G_{n+1}/(P_{F})_{n+1}) ≅ Hom((P_{F})_{n+1}, k^{×})
↓  ↓
Pic(G_{n}/(P_{F})_{n}) ≅ Hom((P_{F})_{n}, k^{×})

is commutative. Hence Pic Fℓ(F, E) ≅ Hom(P, k^{×}), and Hom(P_{F}, k^{×}) is nothing but the group of integral characters of the Lie algebra of P_{F}. In the case of Fℓ(F, w, E) the desired isomorphism is established by replacing Hom((P_{F})_{n}, k^{×}) and Hom((P_{F})_{n+1}, k^{×}) in diagram (8) with the groups of integral characters of the Lie algebras of P_{F_{n}} and P_{F_{n+1}} respectively.

In the rest of this section we give a purely geometric description of Pic Fℓ(F, E) and Pic Fℓ(F, w, E). Consider the corresponding covering (6) or (7). Let L and M be two bases compatible with F for which E\(E ∩ L) and E\(E ∩ M) are finite sets. Denote by g_{L,M} the automorphism of V such that g_{L,M}(l_{i}) = m_{i} for Fℓ(F, E), and g_{L,M}(l_{i}) = m_{i}, g_{L,M}(l^{i}) = m^{i} for Fℓ(F, w, E). It has a well–defined determinant and, moreover, it induces an automorphism of F′′/F′. Denote the determinant of this latter automorphism by det_{L,M}(F′′/F′). In this way we obtain an invertible sheaf L_{F′} with transition functions det_{L,M}(F′′/F′) on C(F, E; L) ∩ C(F, E; M) or C(F, w, E; L) ∩ C(F, w, E; M) respectively. Finally, let γ_{F′} ∈ Pic Fℓ(d_{n}; V_{n}), respectively γ_{F′} ∈ Pic Fℓ(d_{n}, w; V_{n}), denote the class of L_{F′}.

Proposition 8 There are canonical isomorphisms of abelian groups Pic Fℓ(F, E) ≅ (Π_{F′∈F′}(ℤγ_{F′}))/(ℤ Π_{F′∈F′} γ_{F′}) and Pic Fℓ(F, w, E) ≅ Π_{F′∈F′,F′⊂τ(F′),F′≠F′_{τ}} (ℤγ_{F′}).

Proof. Consider the case of Fℓ(F, E) first. Let γ_{F′,n} the class of the restriction (L_{F′})_{n} of L_{F′} to Fℓ(d_{n}; V_{n}). Then γ_{F′,n} = 0 unless F′′ ∩ V_{n} ≠ F′ ∩ V_{n}. Define the group homomorphism φ_{n} : Π_{F′∈F′}(ℤγ_{F′}) → Pic Fℓ(d_{n}; V_{n}) via φ_{n}(Π_{F′∈F′}m_{F′}γ_{F′}) := ∑_{F′∈F′} m_{F′}γ_{F′,n}. The sum ∑_{F′∈F′} m_{F′}γ_{F′,n} makes sense because γ_{F′,n} = 0 for all but finitely many F′ ∈ F′. Clearly φ_{n} = r_{n} ∘ φ_{n+1}, where r_{n} : Pic Fℓ(d_{n+1}; V_{n+1}) → Pic Fℓ(d_{n}; V_{n}) is the restriction map. Therefore, by the universality property of \underleftarrow{lim}, there is a homomorphism

φ : Π_{F′∈F′}(ℤγ_{F′}) → Pic Fℓ(F, E) = \underleftarrow{lim} Pic Fℓ(d_{n}; V_{n}).

Furthermore φ is surjective as φ_{n} is surjective for each n.

To compute ker φ, note that ker φ = ∩ ker φ_{n}. We have ker φ_{n} = (ℤ(Π_{F′∈F′}γ_{F′})) × Π_{F′∈F′,F′∩V_{n}=F′′∩V_{n}}(ℤγ_{F′}) and therefore ker φ = ℤ(Π_{F′∈F′}γ_{F′}), i.e. Pic Fℓ(F, E) ≅ (Π_{F′∈F′}(ℤγ_{F′}))/(ℤ Π_{F′∈F′} γ_{F′}).

In the case of Fℓ(F, w, E) homomorphisms φ_{n} : Π_{F′∈F′}(ℤγ_{F′}) → Pic Fℓ(d_{n}, w; V_{n}) and φ : Π_{F′∈F′}(ℤγ_{F′}) → Pic Fℓ(F, w, E) are defined in a similar way. Here ker φ_{n} = Π_{F′∈F′,F′⊂τ(F′)} (ℤ(γ_{F′} + γ_{τ(F′)′})) × Π_{F′∈F′,F′⊂τ(F′),F′∩V_{n}=F′′∩V_{n}}(ℤγ_{F′}), and consequently ker φ = Π_{F′∈F′,F′⊂τ(F′)} (ℤ(γ_{F′} + γ_{τ(F′)′})) i.e. Pic Fℓ(F, w, E) ≅ Π_{F′∈F′,F′⊂τ(F′),F′≠F′_{τ}} (ℤγ_{F′}). □

We complete this paper by an explicit criterion for the projectivity of Fℓ(F, E) and Fℓ(F, w, E). The following Proposition is a translation of Proposition 15.1 in [DPW] into the language of generalized flags.

Proposition 9 Fℓ(F, E) or Fℓ(F, w, E) is projective if and only if F is a flag.

Proof. Consider the case of Fℓ(F, E) (the case of Fℓ(F, w, E) is similar). Fℓ(F, E) is projective if and only if it admits a very ample invertible sheaf. An immediate verification shows that an invertible sheaf L, whose class in Pic Fℓ(F, E) is the image of Π_{F′∈F′}m_{F′}γ_{F′}, is very ample if and only if the map c : F′ → ℤ, F′ ↦ m_{F′} is strictly increasing. Indeed, Fℓ(F, E) = \underrightarrow{lim} Fℓ(d_{n}; V_{n}), and L is very ample if and only if its restrictions L_{n} onto Fℓ(d_{n}; V_{n}) are very ample for all n. Consider the map c_{n} : F′_{n} → ℤ, defined via c_{n}((F_{n})′_{v}) := c((ψ_{n}(F_{n}))′_{v}) for every nonzero v ∈ V_{n}. (As the reader will check, c_{n} is well defined, i.e. if (F_{n})′_{v_{1}} = (F_{n})′_{v_{2}}, then (ψ_{n}(F_{n}))′_{v_{1}} = (ψ_{n}(F_{n}))′_{v_{2}}.) According to the classical Bott–Borel–Weil Theorem, L_{n} is very ample if and only if the map c_{n} is strictly increasing. Hence, L is very ample if and only if c is strictly increasing. This enables us to conclude that Fℓ(F, E) is projective if and only if there exists a strictly increasing map F′ → ℤ, i.e. if and only if F is a flag. □

Propositions 8 and 9 allow us to make some initial remarks concerning the isomorphism classes of the ind–varieties Fℓ(F, E) and Fℓ(F, w, E). For example, if F is a flag of finite length in V, and G is a flag (or generalized flag) in V of length different from the length of F (finite or infinite), then Fℓ(F, E) and Fℓ(G, L) are not isomorphic because their Picard groups are not isomorphic. Furthermore, if F is a flag in V but G is not, then Fℓ(F, E) and Fℓ(G, L) are not isomorphic because the former ind–variety is projective and the latter is not. Finally, a recent result of J. Donin and the second named author, [DoP], implies that if F = {0 ⊂ F ⊂ V } with F both infinite dimensional and of infinite codimension in V, then the “ind–grassmannian” Fℓ(F, E) is not isomorphic to Gr(l; V) for any l, cf. Example 4.

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[VK2] A. M. Vershik and S. V. Kerov, Locally semisimple algebras. Combinatorial theory and the K_{0}-functor, (Russian) Current problems in mathematics. Newest results, Vol. 26, 3–56, 260, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985.

I.D.:  I.P.:
Department of Mathematics and Statistics  Department of Mathematics
Queen’s University  University of California
Kingston, Ontario, K7L 3N6  Riverside, CA 92521
Canada  USA

dimitrov@mast.queensu.ca                       penkov@math.ucr.edu