A Bott–Borel–Weil Theory for Direct Limits of Algebraic Groups

Ivan Dimitrov^{∗}, Ivan Penkov^{†} & Joseph A. Wolf^{‡}

January 2001

Abstract

We develop a Bott–Borel–Weil theory for direct limits of algebraic groups. Some of our results apply to locally reductive ind–groups G in general, i.e. to arbitrary direct limits of connected reductive linear algebraic groups. Our most explicit results concern root–reductive ind–groups G, the locally reductive ind–groups whose Lie algebras admit root decomposition. Given a parabolic subgroup P of G and a rational irreducible P –module, we consider the irreducible Gsheaves O_{G/P} (E) and their duals O_{G/P} (E^{∗}). These sheaves are locally free, in general of infinite rank. We prove a general analog of the Bott–Borel–Weil Theorem for O_{G/P} (E^{∗}), namely that H^{q}(G/P; O_{G/P} (E^{∗})) is nonzero for at most one index q = q_{0} and that H^{q_{0}}(G/P; O_{G/P} (E^{∗})) is isomorphic to the dual of a rational irreducible G–module V. For q_{0} > 0 we show that (in contrast to the finite dimensional case) that V need not admit an irreducible P –submodule. There, however, one has a larger parabolic subgroup ^{w}P ⊃ P, constructed from P and a Weyl group element w of length q_{0} , such that V is generated by an irreducible ^{w}P –submodule. Consequently certain G–modules V can appear only for q_{0} > 0, never for q_{0} = 0. For O_{G/P} (E) we show that there is no analog of Bott’s vanishing theorem, more precisely that O_{G/P} (E) can have arbitrarily many nonzero cohomology groups. Finally, we give an explicit criterion for the projectivity of the ind–variety G/P, showing that G/P is in general not projective.

∗Research partially supported by NSF Grant DMS 97 09820 and the Max Planck Institute for Mathematics, Bonn.

†Research partially supported by NSF Grant DMS 97 09820, the University of Hamburg, and the Max Planck Institute for Mathematics, Bonn.

‡Research partially supported by NSF Grants DMS 97 09820 and 97 05709 and the Alexander von Humboldt Foundation.

Key words (2000 MSC): Primary 22E65, Secondary 20G05.

Contents

§0. Introduction
Part I.  Ind–varieties and Ind–groups
§1. Ind-varieties
§2. Projective and locally projective ind–varieties
§3. Ind–groups and direct limit Lie algebras
§4. Root–reductive ind–groups and parabolic subgroups
§5. Diagonal ind–groups and beyond
Part II.  Representations
§6. Rational and pro–rational G–modules
§7. 𝔤–modules
§8. Irreducible rational G–modules
§9. Combinatorics of locally irreducible locally finite 𝔭–modules
Part III.  Cohomology
§10. Homogeneous spaces and G–sheaves
§11. Statement of main results on cohomology of O_{X}(E^{∗})
§12. Proof and discussion of (i) and (ii)
§13. Proof and discussion of (iii)
§14. The situation when E has a highest weight
§15. Projectivity of G/P
§16. The sheaf O_{X}(E) for infinite dimensional E

0 Introduction

The classical Bott–Borel–Weil Theorem [5] is the cornerstone of the geometric approach to representation theory. Analogs of this theorem have been studied in various contexts, including that of homogeneous spaces in characteristic p [14] and of homogeneous superspaces [23]. In these contexts the Bott–Borel–Weil Theorem does not carry over as a single theorem, and this has inspired important areas of investigation. Infinite dimensional group analogs of the Bott–Borel–Weil Theorem have also been studied. The loop group case was addressed in the 1980’s; see [18] and [24]. Direct limit Lie groups were first addressed recently by L. Natarajan, E. Rodríguez–Carrington and one of us [21]. Roughly speaking, the results of [21] extend the finite dimensional analytic Bott-Borel-Weil Theorem to direct limit Lie groups and direct limit unitary representations, in other words to the analytic category with representations on Hilbert spaces.

The classical Bott–Borel–Weil Theorem can be viewed as a result in algebraic geometry, see [6] and [7], so it is natural to try to find its analog for direct limits of linear algebraic groups. That turns out to be very interesting and, as in the characteristic p and the supergeometry contexts, one is led not to a single theorem but to the development of a new circle of ideas. To start, recall two forms^{1} of the classical Bott-Borel-Weil Theorem for a connected complex reductive linear algebraic group G:

(1) Let B be a Borel subgroup of G, L an irreducible (one dimensional) rational B–module, 𝕃 → G/B the associated homogeneous line bundle, and O_{G/B}(L) be the sheaf of regular local sections of 𝕃. Then at most one of the cohomology groups H^{q}(G/B; O_{G/B}(L)) is nonzero, and if H^{q_{0}}(G/B; O_{G/B}(L)) ≠ 0 then H^{q_{0}}(G/B; O_{G/B}(L)) is an irreducible G–module whose highest weight has an explicit expression in terms of the weight of L.

1A different purely algebraic approach to the Bott–Borel–Weil Theorem is due to B. Kostant [16]. Its analog for direct limit Lie algebras is beyond the scope of this paper.

(2) Let P be a parabolic subgroup of G with B ⊂ P and let π : G/B → G/P denote the canonical projection. Let E be an irreducible (finite dimensional) rational P –module, 𝔼 → G/P be the associated homogeneous vector bundle, and O_{G/P} (E) be the sheaf of regular local sections of 𝔼. Then O_{G/P} (E) is isomorphic to the direct image sheaf π_{∗}O_{G/B}(L) for some one dimensional B–module L, and H^{q}(G/P; O_{G/P} (E)) = H^{q}(G/B; O_{G/B}(L)) for any q. Therefore at most one cohomology group H^{q_{0}}(G/P; O_{G/P} (E)) is nonzero, and it is an irreducible G–module whose highest weight can be calculated explicitly.

Statement (1) admits a reasonably straightforward generalization to locally reductive ind– groups. As follows from Theorem 11.1(ii) below, if G is a locally reductive ind–group, B is a Borel subgroup, and L is any irreducible rational (and thus one dimensional) B–module, then the sheaf O_{G/B}(L) has at most one nonzero cohomology group H^{q_{0}}(G/B; O_{G/B}(L)). Furthermore, H^{q_{0}}(G/B; O_{G/B}(L)) is isomorphic to the dual of an irreducible rational G–module V, and therefore (in contrast to the classical case and to the analytic category case [21]) is a reducible G–module except in the very special situation of finite dimensional V. If q_{0} = 0, V has a highest weight which is the negative of the weight of L. For q_{0} > 0 the explicit description of V in terms of L is an interesting problem, and it is still open for sufficiently general ind–groups G. However, for a root–reductive^{2} ind–group G we prove in Proposition 14.1 that V is a highest weight module whose highest weight is calculated in terms of L using a Weyl group action which has many properties in common that of the classical Bott–Borel–Weil Theorem.

In this paper we develop a Bott–Borel–Weil theory for locally reductive ind–groups G, which in particular replaces (2). It involves a number of new constructions, and goes well beyond straightforward generalization. To indicate why this is needed, here are two observations on a parabolic subgroup P of G and an infinite dimensional irreducible rational P –module E. First, E need not be locally irreducible, i.e. need not be isomorphic to a direct limit of irreducible P_{n}–modules, and also E need not be a weight module. In particular, E need not have an extremal weight with respect to any Borel subgroup B ⊂ P. Second, there are two natural G–sheaves to consider: the sheaf O_{G/P} (E) of local sections of the G–bundle 𝔼 → G/P associated to E, and the sheaf O_{G/P} (E^{∗}) of local sections of the dual bundle (of uncountable rank) 𝔼^{∗} → G/P. It is not difficult to see that the “push down approach” of (2) applies only to O_{G/P} (E^{∗}) and then only under the additional condition that E be a highest weight P –module. Thus the “push down approach” does not lead to a complete description of the cohomology of O_{G/P} (E) or O_{G/P} (E^{∗}).

Our main results, collected in Theorem 11.1, concern the sheaf O_{G/P} (E^{∗}). When G is root– reductive, but E need not be locally irreducible, we prove that O_{G/P} (E^{∗}) has at most one nonzero cohomology group H^{q_{0}}(G/P; O_{G/P} (E^{∗})), and that this cohomology group is isomorphic to the dual of a rational G–module V. For q_{0} = 0, we show that, as in the finite dimensional case, V is parabolically generated by E itself. When E is a weight module this latter result turns out to imply that H^{0}(G/P; O_{G/P} (E^{∗})) ≠ 0 forces E to be locally irreducible; more precisely, E has to be a finite P –module (see Theorem 11.1(iv)). For q_{0} > 0, in contrast with the finite dimensional case, V need not be generated by an irreducible P –submodule. We prove that here V is generated by an irreducible submodule of a larger parabolic subgroup ^{w}P ⊃ P that depends on P and a certain element w of the Weyl group. The result is sharp in the sense that there are examples of irreducible P –modules E for which V does not admit an irreducible submodule for any parabolic subgroup of G properly contained in ^{w}P; see Example 13.1. This new infinite dimensional phenomenon is quite remarkable, for it provides a geometric construction of G–modules that are not parabolically generated, in particular of cuspidal weight modules [8]. According to Theorem 11.1(iv), cuspidal weight G–modules can only occur in higher cohomology groups. As a consequence, there is no “Demazure isomorphism” between the unique nonzero higher cohomology group of O_{G/P} (E^{∗}) and the zero^{th} cohomology group of O_{G/P} (E′^{∗}) for any other E′.

2Root–reductive ind–groups are the simplest ind–versions of reductive groups, see §4.

In connection with the above results, we establish an explicit criterion for the projectivity of the ind–variety G/P. Somewhat surprisingly, it turns out that G/P is rarely projective, even for G = GL(∞); see §15. Finally, we consider the sheaf O_{G/P} (E) and show by an example that it can have arbitrarily many nonzero cohomology groups. The problem of a systematic description of the cohomology for sheaves of type O_{G/P} (E) remains open.

Acknowledgments

We thank Robin Hartshorne, Ziv Ran, and David Ben–Zvi for encouraging discussions at the early stages of this work. In particular, David Ben–Zvi brought [17] to our attention. We thank also Yuri Manin, Vera Serganova and Gregg Zuckerman for constant support throughout the writing of the paper.

Notational Conventions

The ground field is ℂ, though all results extend easily to any algebraically closed field of characteristic zero. If V is a vector space, V^{∗} stands for its dual space. ℤ_{+} := {0, 1, 2, … }, ℤ_{−} = −ℤ_{+} , ℕ := ℤ_{+}\\{0\}, and ℂ^{×} is the multiplicative group \{a ∈ ℂ | a ≠ 0\}. Expressions such as \underrightarrow{lim} G_{n} and \underleftarrow{lim} H^{q}(X_{n}; …) denote the direct and inverse limit, respectively of a direct or inverse system indexed by ℤ_{+}. The notation H^{q}(X; F) always stands for the q-th cohomology group of a sheaf F on a topological space X. If π : X → Y is a morphism of algebraic varieties, π^{∗} and π_{∗} denote respectively the inverse and direct image functors of coherent sheaves. Ind–groups (in particular complex algebraic groups) are denoted by capital letters (e.g. K), and their Lie algebras are denoted by the corresponding lower case Gothic letter (e.g. 𝔨). Furthermore, as the ground field is fixed, we will write simply GL(n), SO(n), etc. instead of GL(n, ℂ), SO(n, ℂ), etc. U(𝔨) denotes the universal enveloping algebra of the Lie algebra 𝔨. The sign ⊂^{+} stands for semidirect sum of Lie

algebras: if 𝔨 = 𝔪⊂^{+} 𝔫, then 𝔪 is an ideal in 𝔨 and 𝔫 is a complementary subalgebra. The sign ×^{f} denotes ”finite” or ”weak” direct product of groups or of homogeneous spaces. If G^{t}, t ∈ T, is an infinite family of groups, we set

×^{f}_{t∈T} G^{t} := {×_{t∈T} g^{t} | g^{t} ∈ G^{t} for all t and g^{t} = 1 for almost all t ∈ T }.

Similarly, if G^{t}/H^{t} is a family of homogeneous spaces, ×^{f}_{t∈T} (G^{t}/H^{t}) is the image of ×^{f}_{t∈T} G^{t} in ×_{t∈T} (G^{t}/H^{t}). Finally, if 𝔨′ ⊂ 𝔨 as a pair of finite dimensional Lie algebras, and E_{𝔨} and E_{𝔨′} are respectively a finite dimensional irreducible 𝔨–module and a finite dimensional irreducible 𝔨′–module, we write E_{𝔨′} ≺ E_{𝔨} if there is an injection of 𝔨′–modules E_{𝔨′} → E_{𝔨}.

Part I. Ind–varieties and Ind–groups

1 Ind–varieties

This is a quick summary of the basic definitions on ind–varieties. See both [25] and [17] for more detailed summaries.

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

X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ … (1.1)

such that^{3} X = ⋃_{n} 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≥0} 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. An ind–variety is connected if it is connected as a topological space.

The (Zariski) tangent space T_{x}(X), to an ind–variety X = \underrightarrow{lim} X_{n} at a closed point x ∈ X is, by definition, the direct limit \underrightarrow{lim} T_{x}(X_{n}) where x ∈ X_{n} for n sufficiently large. Any ind–variety morphism φ : X → Y induces linear maps dϕ_{x} : T_{x}(X) → T_{ϕ(x)}(Y) for all closed points x ∈ X.

Example 1.2

(1) ℂ^{∞} = \underrightarrow{lim} ℂ^{n} = \{(a_{1}, a_{2}, a_{3}, …) | a_{n} ∈ ℂ, only finitely many a_{n} ≠ 0\} is an affine ind–variety.

(2) Every (complex) vector space V of countable dimension has a canonical structure of an affine ind–variety: any basis \{v_{1}, …, v_{n}, … \} identifies V = \underrightarrow{lim} Span\{v_{1}, …, v_{n}\} with ℂ^{∞} as sets and thus defines an ind–variety structure on V. Any other basis of V defines the same ind–variety structure because the identity map on V is an ind–variety isomorphism.

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

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

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

3In the following we will often write X = \underrightarrow{lim} X_{n}.

2 Projective and locally projective ind–varieties

An ind–variety X is locally projective if it admits an ind–variety filtration (1.1) such that all the X_{n} are projective varieties. An ind–variety X is projective^{4} if it can be embedded as a closed ind– subvariety into the projective ind–space ℙ(ℂ^{∞}). Any projective ind–variety is proper and locally projective, but the converse is not true. Below we introduce twisted projective ind–spaces which are the simplest examples of locally projective, generically not projective, ind–varieties.

Let Y be a proper Noetherian algebraic variety. A very ample invertible O_{Y} –module L_{Y} (i.e. a locally free sheaf L_{Y} of O_{Y} –modules of rank 1 which is generated by its global sections) determines a closed immersion of Y into the projective space ℙ := ℙ(H^{0}(Y; L_{Y})^{∗}), and L_{Y} is identified with the inverse image under this immersion of the standard sheaf O_{ℙ}(1) on ℙ. See [12, II, §5]. If X is a proper ind–variety, consider an invertible O_{X}–module L_{X} and an ind–variety filtration (1.1) such that all inverse images L_{X_{n}} of L_{X} are very ample and the restriction maps H^{0}(X_{n}; L_{X_{n}}) → H^{0}(X_{n−1}; L_{X_{n−1}}) are surjective. Then the system dual to the inverse system

… → H^{0}(X_{n}; L_{X_{n}}) → H^{0}(X_{n−1}; L_{X_{n−1}}) → … → H^{0}(X_{0}; L_{X_{0}}) → 0

is a direct system of injections, and L_{X} defines a closed immersion of X into ℙ(\underrightarrow{lim} H^{0}(X_{n}; L_{X_{n}})^{∗}).

We define X to be a twisted projective ind–space if it admits a filtration (1.1) such that X_{n} is a projective space for all n. To a twisted projective ind–space we attach its twisting sequence {c_{1}, c_{2}, … }, where c_{n} is the natural number that denotes the first Chern class of the inverse image of O_{X_{n+1}}(1) on X_{n}. Any sequence of natural numbers can be obtained as a twisting sequence.

Proposition 2.1 A twisted projective ind–space X is projective if and only if its twisting sequence stably equals the sequence {1, 1, … }, i.e. c_{n} = 1 for n sufficiently large.

Proof. If c_{n} = 1 for all n > n_{0}, then X_{n_{0}} ⊂ X_{n_{0}+1} ⊂ … is an ind–variety filtration for X and, for any n ≥ n_{0}, the invertible O_{X_{n}}–module O_{X_{n}}(1) with Chern class 1 is the inverse image of a well–defined O_{X}–module O_{X}(1). Then O_{X}(1) establishes an isomorphism between X and the projective ind–space ℙ(\underrightarrow{lim} H^{0}(X_{n}; O_{X_{n}}(1))^{∗}). Thus X itself is isomorphic to a projective ind–space. Conversely, let the sequence c_{1}, c_{2}, … corresponding to X admit a subsequence c_{n_{1}}, c_{n_{2}}, … with c_{n_{t}} ≥ 2 for all t. Assume that X is a closed ind–subvariety of a projective ind–space ℙ and consider the inverse images O_{ℙ}(1)_{|}X_{n} of O_{ℙ}(1) on X_{n}, where O_{ℙ}(1) := \underleftarrow{lim} O_{ℙ_{n}}(1). Denote the Chern class of O_{ℙ}(1)_{|}X_{n} by C_{n}. Then, for any s > k, C_{k}/C_{s} = c_{k+1}c_{k+2} … c_{s}. Therefore the positive integer C_{n_{0}} is divisible by all products c_{n_{0}+1}c_{n_{0}+2} … c_{n} for any n > n_{0}, which is an obvious contradiction. □

We remark that the ind–grassmannian Gr(k, V) of Example 1.2 (3) is projective, and that the classical Plücker embeddings induce a closed immersion P l^{k} : Gr(k, V) ↪ ℙ(⋀^{k} V) where ⋀^{k} denotes k–th exterior power.

4Tjurin’s notion of projectivity [26] is stronger, as it requires the existence of a finite codimensional projective embedding.

3 Ind–groups and direct limit Lie algebras

An algebraic ind–group, or, briefly, ind–group, is an ind–variety G with group structure such that the map

G × G → G, (g_{1}, g_{2}) ↦ g_{1}g_{2}^{−1}

is a morphism of ind–varieties. By definition, an ind–group homomorphism is a group homomorphism κ : G → K that is also an ind–variety morphism. An ind–subgroup K of G is a subgroup K ⊂ G that is an ind–subvariety.

We define a locally linear ind–group as an (affine) ind–variety G with an ind–variety filtration

G_{0} ⊂ G_{1} ⊂ G_{2} ⊂ … (3.1)

such that all the G_{n} are linear algebraic groups and all the inclusions are closed immersions that are group morphisms. Clearly, every locally linear ind–group is an affine ind–group but the converse is not true. An example of an affine ind–group which is not locally linear is provided by the automorphism group Aut(𝔸^{m}) of the m–dimensional affine space 𝔸^{m}; see [25]. We will only study connected locally linear ind–groups, and all ind–groups considered below are assumed to be connected and locally linear. By a slight abuse of language we refer to them simply as ind–groups.

Let G be a (connected) ind–group. In this paper we define a parabolic subgroup of G as an ind–subgroup P of G such that, for a suitable filtration (3.1), P_{n} := P ∩ G_{n} is a parabolic subgroup of G_{n} for each n. Similarly a Borel subgroup of G is an ind–subgroup B of G such that, for a suitable filtration (3.1), B_{n} := B ∩ G_{n} is a Borel subgroup of G_{n} for each n, and a Cartan subgroup of G is an ind–subgroup H of G such that, for a suitable filtration (3.1), H_{n} := H ∩ G_{n} is a Cartan subgroup of G_{n} for each n. An element u ∈ G is unipotent if for some (or equivalently, for any) ind–group filtration (3.1) u is unipotent in G_{n} whenever n is large enough so that u ∈ G_{n}. The unipotent radical U_{G} of G is defined to be the largest closed normal ind–subgroup of G such that every element u is unipotent. Finally, an ind–group G is locally reductive if we can choose the ind– group filtration (3.1) so that each G_{n} is a reductive linear algebraic group. Whenever G = \underrightarrow{lim} G_{n} is locally reductive we will assume that the G_{n} are also reductive.

Throughout the rest of the paper we will consider ind–groups G with a fixed filtration (3.1) of connected linear algebraic groups. We will assume without explicit mention that the parabolic, Borel and Cartan subgroups of G are aligned with respect to that filtration in the sense that, for all n, their intersections with G_{n} are parabolic, Borel or Cartan subgroups of G_{n}.

Proposition 3.2 Let G = \underrightarrow{lim} G_{n} be a locally reductive ind–group and let P = \underrightarrow{lim} P_{n} be a parabolic ind–subgroup.

(i) The unipotent radical U_{P} of P is well–defined. For each n, the intersection U_{P_{n}} := U_{P} ∩ G_{n} is the unipotent radical of P_{n}.

(ii) The Chevalley semidirect product decompositions P_{n} = U_{P_{n}}×^{|} P_{n}^{red} , into the unipotent radical and a complementary reductive subgroup, can be chosen so that P_{n}^{red} ⊂ P_{n+1}^{red} for all n. Then P = U_{P} ×^{|} P^{red} where P^{red} := \underrightarrow{lim} P_{n}^{red} .

Proof. The main point for part (i) is that the inclusion P_{n} ↪ P_{n+1} is a morphism of linear algebraic groups, so it sends unipotent elements to unipotent elements. Thus \underrightarrow{lim} U_{P_{n}} is a well– defined ind–subgroup of G. \underrightarrow{lim} U_{P_{n}} is closed and normal in P because, for each n, U_{P_{n}} is closed and normal in P_{n}. Finally, \underrightarrow{lim} U_{P_{n}} is the largest closed normal subgroup of P in which every element is unipotent, because the existence of a larger such subgroup would contradict the fact that U_{P_{n}} is the unipotent radical of P_{n} for every n. Therefore \underrightarrow{lim} U_{P_{n}} is the unipotent radical U_{P} of P.

A theorem of Mostow [19, Theorem 7.1] ensures that the maximal reductive subgroups of P_{n+1} are just the reductive subgroups R such that P_{n+1} = U_{P_{n+1}}×^{|} R, and that any two such groups R are conjugate by an element of U_{P_{n+1}} . See [13, VIII, Theorem 4.3] for a systematic development. Therefore, given a maximal reductive subgroup P_{n}^{red} in P_{n}, we can choose P_{n+1}^{red} to be any maximal reductive subgroup of P_{n+1} that contains it. This implies (ii), and the equality P = U_{P} ×^{|} P^{red} follows. □

The Lie algebra of an ind–group G = \underrightarrow{lim} G_{n} is the direct limit Lie algebra 𝔤 = \underrightarrow{lim} 𝔤_{n} for the direct system

𝔤_{0} ⊂ 𝔤_{1} ⊂ 𝔤_{2} ⊂ …, (3.3)

where 𝔤_{n} is the Lie algebra of G_{n} and the inclusions 𝔤_{n} ⊂ 𝔤_{n+1} are the differentials of the group immersions G_{n} ⊂ G_{n+1} . This Lie algebra is the tangent space T_{1_{G}}(G) with its natural Lie algebra structure [17]. An ind–group homomorphism κ : G → K induces a Lie algebra homomorphism dκ : 𝔤 → 𝔨 [17]. We shall consider direct limit Lie algebras more generally, sometimes without regard to ind–groups. These will always correspond to direct systems of injections of finite dimensional Lie algebras.

Let 𝔤 be a direct limit Lie algebra. In this paper we define a subalgebra 𝔭 ⊂ 𝔤 to be a parabolic subalgebra if, for a suitable direct system {𝔤_{n}}, 𝔭_{n} := 𝔭 ∩ 𝔤_{n} is a parabolic subalgebra of 𝔤_{n} for each n. Similarly, a subalgebra 𝔟 ⊂ 𝔤 (respectively 𝔥 ⊂ 𝔤) is a Borel (respectively Cartan) subalgebra of 𝔤 if, for a suitable direct system {𝔤_{n}}, each 𝔟_{n} := 𝔟 ∩ 𝔤_{n} (respectively 𝔥_{n} := 𝔥 ∩ 𝔤_{n}) is a Borel subalgebra (respectively Cartan subalgebra) of 𝔤_{n}. This definition of parabolic and Borel subalgebras is equivalent to that of [1], and is more general than that of [8]. In the rest of the paper we will automatically assume that 𝔤 is equipped with a fixed filtration (3.3) and that all parabolic, Borel or Cartan subalgebras we consider are such that their intersections with 𝔤_{n} are respectively parabolic, Borel or Cartan subalgebras of 𝔤_{n} for all n.

We define a direct limit Lie algebra 𝔤 to be locally reductive if we have an expression 𝔤 = \underrightarrow{lim} 𝔤_{n} where each of the 𝔤_{n} is reductive. Whenever we express a locally reductive Lie algebra 𝔤 as \underrightarrow{lim} 𝔤_{n}, it will be assumed that the 𝔤_{n} are reductive. The following Proposition is the Lie algebra analog of Proposition 3.2. Part (ii) of this Proposition is an adaptation of a result of Baranov [2, Lemma 4.3]. We leave the proof to the reader.

Proposition 3.4 Let 𝔤 = \underrightarrow{lim} 𝔤_{n} be a locally reductive direct limit Lie algebra and let 𝔭 be a parabolic subalgebra.

(i) Let 𝔲_{𝔭_{n}} denote the nilpotent radical of 𝔭_{n} := 𝔭 ∩ 𝔤_{n}. Then 𝔲_{𝔭} := \underrightarrow{lim} 𝔲_{𝔭_{n}} is a well–defined ideal in 𝔭. By definition, 𝔲_{𝔭} is the nilpotent radical of 𝔭.

(ii) One can choose semidirect sum decompositions 𝔭_{n} = 𝔲_{𝔭_{n}}⊂^{+} 𝔭_{n}^{red} , into the nilpotent radical of 𝔭_{n} and a complementary reductive subalgebra, such that each 𝔭_{n}^{red} ↪ 𝔭_{n+1}^{red} . Then 𝔭 = 𝔲_{𝔭}⊂^{+} 𝔭^{red} where 𝔭^{red} := \underrightarrow{lim} 𝔭_{n}^{red} .

(iii) Let 𝔤 be the Lie algebra of a reductive ind–group G, P is a parabolic ind–subgroup of G with Lie algebra 𝔭, and U_{P} be the unipotent radical of P. Then 𝔲_{𝔭} is the Lie algebra of U_{P}. Furthermore, if we choose 𝔭_{n}^{red} to be the Lie algebra of P_{n}^{red} of Proposition 3.2 (ii), then 𝔭^{red} is the Lie algebra of P^{red}. □

4 Root–reductive ind–groups and parabolic subgroups

Let 𝔤′ and 𝔤′′ be Lie algebras with root decomposition, so 𝔤′ = 𝔥′ ⊕ (⨁_{α′∈Δ′}(𝔤′)^{α′}) and 𝔤′′ = 𝔥′′ ⊕ (⨁_{α′′∈Δ′′}(𝔤′′)^{α′′}). Here 𝔥′ and 𝔥′′ are respectively Cartan subalgebras, and Δ′ and Δ′′ are the corresponding root systems of 𝔤′ and 𝔤′′. A Lie algebra homomorphism φ : 𝔤′ → 𝔤′′ is a root homomorphism, if φ(𝔥′) ⊂ 𝔥′′ and φ maps every root space (𝔤′)^{α′} into a root space (𝔤′′)^{α′′}, thus mapping Δ′ into Δ′′. To be precise one should write φ : (𝔤′, 𝔥′, Δ′) → (𝔤′′, 𝔥′′, Δ′′), but we often leave this to be understood by the reader. A root subalgebra of 𝔤′′ is the image of a root homomorphism. A locally reductive direct limit Lie algebra 𝔤 is root–reductive if it can be expressed as a direct limit 𝔤 = \underrightarrow{lim} 𝔤_{n} where all 𝔤_{n} are finite dimensional and reductive, 𝔥 = \underrightarrow{lim} 𝔥_{n} is a Cartan subalgebra, and each injection 𝔤_{n} ⊂ 𝔤_{n+1} carries 𝔥_{n} into 𝔥_{n+1} and is a root homomorphism. Finally, a locally reductive ind–group G is root–reductive if its Lie algebra 𝔤 is root–reductive.

Let 𝔤 = \underrightarrow{lim} 𝔤_{n} be a root–reductive direct limit Lie algebra, expressed as a direct limit of root injections (𝔤_{n}, 𝔥_{n}, Δ_{n}) ↪ (𝔤_{n+1}, 𝔥_{n+1}, Δ_{n+1}). Then 𝔤 admits a root decomposition with respect to the Cartan subalgebra 𝔥 = ⋃_{n} 𝔥_{n}. The root system of (𝔤, 𝔥) is Δ = ⋃_{n} Δ_{n}, and all root spaces 𝔤^{α} are one dimensional. It is easy to check (see [8]) that the direct limit W = \underrightarrow{lim} W_{n} of the Weyl groups W_{n} (considered as the groups generated by root reflections) of 𝔤_{n} is well–defined. In this paper W is by definition the Weyl group of 𝔤. Let furthermore 𝔟 ⊂ 𝔤 be a Borel subalgebra such that 𝔥 ⊂ 𝔟. It determines a decomposition Δ = Δ^{+} ⊔ Δ^{−} such that Δ^{−} = −Δ^{+}, the positive roots Δ^{+} being the roots of 𝔟. We say that a positive root α is 𝔟–simple if α cannot be decomposed as the sum of two positive roots. A Weyl group element w ∈ W is of finite length with respect to 𝔟 if w is a (finite) product of simple root reflections, w = σ_{α_{1}} ∘ … ∘ σ_{α_{k}} for some 𝔟–simple roots α_{1}, …, α_{k}. The length of the shortest such expression is by definition the length of w with respect to 𝔟.

Root–reductive direct limit Lie algebras were introduced and studied in [8]; also see [21, Section 7]. They are closely related to the classical simple direct limit Lie algebras 𝔞(∞), 𝔟(∞), 𝔠(∞) and 𝔡(∞) defined by letting 𝔤_{n} be the corresponding finite dimensional Lie algebra of rank n and by requiring that all φ_{n} be root injections. The isomorphism class of the resulting direct limit Lie algebra does not depend on the injections φ_{n} [8]. Moreover, every simple infinite dimensional root–reductive Lie algebra is isomorphic to one of the four classical simple direct limit Lie algebras^{5}.

We give now an explicit description of the classical simple ind–groups A(∞), B(∞), C(∞), and D(∞) whose Lie algebras are respectively 𝔞(∞), 𝔟(∞), 𝔠(∞), and 𝔡(∞).

A(∞). Here G = A(∞) = \underrightarrow{lim} A(n − 1) where G_{n} = A(n − 1) = SL(n) and the inclusion A(n − 1) ⊂ A(n) is given by g ↦ ( \begin{matrix}g & 0 \\ 0 & 1 \end{matrix} ). Fixing the Cartan subalgebra 𝔥 of all diagonal matrices in 𝔤 = 𝔞(∞), we have Δ = {ε_{i} − ε_{j} | i ≠ j} where ε_{i} ∈ 𝔥^{∗} is given by ε_{i}(diag{t_{1}, t_{2}, … }) = t_{i}. The Weyl group W consists of all permutations of {ε_{i}} which leave all but finitely many ε_{i} fixed.

5A further interesting fact is that 𝔟(∞) and 𝔡(∞) are isomorphic as Lie algebras, see [3] and [22]. However, no such isomorphism is a root isomorphism.

B(∞). Here G = B(∞) = \underrightarrow{lim} B(n), where B(n) = SO(2n+1) is the complex special orthogonal group corresponding to the non-degenerate symmetric bilinear form (u, v) = u_{1}v_{1} + ∑_{1}^{n}(u_{2i}v_{2i+1} + u_{2i+1}v_{2i}) on ℂ^{2n+1}, and where the inclusion B(n) ⊂ B(n + 1) is given by g ↦ ( \begin{matrix}g & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ). Fixing the Cartan subalgebra 𝔥 of all diagonal matrices in 𝔤 = 𝔟(∞), we have Δ = {±ε_{i}, ±ε_{i} ± ε_{j} | i ≠ j} where ε_{i} ∈ 𝔥^{∗} is given by ε_{i}(diag{0, t_{1}, −t_{1}, t_{2}, −t_{2}, … }) = t_{i}. The Weyl group W consists of all signed permutations of {ε_{i}} which leave all but finitely many ε_{i} fixed.

C(∞). Here G = C(∞) = \underrightarrow{lim} C(n), where C(n) = Sp(2n) is the complex symplectic group corresponding to the non-degenerate antisymmetric bilinear form ⟨u, v⟩ = ∑_{1}^{n}(u_{2i−1}v_{2i} − u_{2i}v_{2i−1}) on ℂ^{2n}, and where the inclusion C(n) ⊂ C(n + 1) is given by g ↦ ( \begin{matrix}g & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ). Fixing the Cartan subalgebra 𝔥 of all diagonal matrices in 𝔤 = 𝔠(∞), we have Δ = {±2ε_{i}, ±ε_{i} ± ε_{j} | i ≠ j} where ε_{i} ∈ 𝔥^{∗} is given by ε_{i}(diag{t_{1}, −t_{1}, t_{2}, −t_{2}, … }) = t_{i}. The Weyl group W consists of all signed permutations of {ε_{i}} which leave all but finitely many ε_{i} fixed.

D(∞). Here G = D(∞) = \underrightarrow{lim} D(n), where D(n) = SO(2n) is the complex special orthogonal group corresponding to the non-degenerate symmetric bilinear form (u, v) = ∑_{1}^{n}(u_{2i−1}v_{2i}+u_{2i}v_{2i−1}) on ℂ^{2n}, and where the inclusion D(n) ⊂ D(n + 1) is given by g ↦ ( \begin{matrix}g & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ). Fixing the Cartan subalgebra 𝔥 of all diagonal matrices in 𝔤 = 𝔡(∞), we have Δ = {±ε_{i} ± ε_{j} | i ≠ j} where ε_{i} ∈ 𝔥^{∗} is given by ε_{i}(diag{t_{1}, −t_{1}, t_{2}, −t_{2}, … }) = t_{i}. The Weyl group W consists of all signed permutations of {ε_{i}} which leave all but finitely many ε_{i} fixed.

If 𝔤 is a root–reductive direct limit Lie algebra and 𝔨 is a root subalgebra, we set Δ_{𝔨}^{ss} := Δ_{𝔨} ∩ (−Δ_{𝔨}) (where Δ_{𝔨} denotes the set of roots of 𝔨) and define 𝔨^{ss} to be the Lie subalgebra of 𝔨 generated by ⨁_{α∈Δ_{𝔨}^{ss}} 𝔨^{α}.

The Lie algebra part of the following Proposition 4.1 reformulates [8, Theorem 1]. It describes the relationship between an arbitrary root–reductive direct limit Lie algebra and the classical simple direct limit Lie algebras. The group level statements follow from the algebra level statements.

Proposition 4.1 Let G be a root–reductive ind–group and 𝔤 be its Lie algebra.

(i) 𝔤 = 𝔤^{ss}⊂^{+} 𝔞 for some abelian Lie subalgebra 𝔞 ⊂ 𝔤. Furthermore, 𝔥 = 𝔥^{ss} ⊕ 𝔞 where 𝔥^{ss} := 𝔥 ∩ 𝔤^{ss}.

(ii) G has a connected closed normal ind–subgroup G^{ss} with Lie algebra 𝔤^{ss} and a connected abelian ind–subgroup A with Lie algebra 𝔞, and (g, a) ↦ ga defines a homomorphism of the semidirect product G^{ss}×^{|} A onto G with discrete kernel Z. Furthermore H ≅ (H^{ss} × A)/Z where H^{ss} := H ∩ G^{ss} has Lie algebra 𝔥^{ss}.

(iii) 𝔤^{ss} ≅ ⨁_{t∈T} 𝔤^{t} where the 𝔤^{t} are classical simple direct limit algebras or simple finite di-

mensional Lie algebras. G^{ss} ≅ (×^{f}_{t∈T} G^{t})/Z_{1} where G^{t} is the connected ind–group of G with Lie algebra 𝔤^{t} and where Z_{1} is a discrete central subgroup of G^{ss}. □

Example 4.2 Set GL(∞) := \underrightarrow{lim} GL(n) where the inclusion GL(n) ↪ GL(n + 1) is given by

g ↦ ( \begin{matrix}g & 0 \\ 0 & 1 \end{matrix} ). Denote A := \{diag \{a, 1, 1, 1, … \} | a ∈ ℂ^{×}\} ≅ ℂ^{×}. Then GL(∞) ≅ SL(∞)×^{|} A under

g ↦ (gα^{−1}, α) where α := diag {det(g), 1, 1, 1, … }.

Throughout this paper, when considering a root–reductive Lie algebra 𝔤 we fix a Cartan subalgebra 𝔥 = ⋃_{n} 𝔥_{n} corresponding to a fixed system of root injections 𝔤_{n} ⊂ 𝔤_{n+1} . Without loss of generality we assume that any Borel or parabolic subalgebras of 𝔤 are chosen so that they contain 𝔥. Parabolic subgroups (including Borel subgroups) of a root–reductive ind–group G will thus contain the Cartan subgroup H = ⋃_{n} H_{n} with Lie algebra 𝔥. When we refer to A(∞), B(∞), C(∞), D(∞) or GL(∞) (or to their Lie algebras) we will furthermore assume that the G_{n} (or 𝔤_{n}), as well as the Cartan subgroup H (respectively, the Cartan subalgebra 𝔥) are chosen precisely as in our above explicit description.

We now define the parabolic subgroup ^{w}P needed in the statement of Theorem 11.1 below. If P is a proper parabolic subgroup (containing H) of a root–reductive ind–group G and 𝔭 is the Lie algebra of P, then 𝔥 + 𝔭^{ss} is a natural choice for 𝔭^{red}. Let w ∈ W be a Weyl group element. We define ^{w}𝔭 to be the parabolic subalgebra of 𝔤 generated by 𝔥 and the 𝔥–root spaces 𝔤^{α} for α ∈ Δ_{𝔭} ∪ w(Δ_{𝔭^{red}}). Then ^{w}P is the parabolic subgroup of G with Lie algebra ^{w}𝔭. The subgroup wP is not necessarily proper, as shown by the following example.

Example 4.3 Let G = GL(∞), let 𝔭 ⊂ 𝔤𝔩(∞) be the (maximal) parabolic subalgebra with roots {ε_{1} − ε_{i} | 2 ≤ i} ∪ {ε_{i} − ε_{j} | 2 ≤ i ≠ j}, and let P ⊂ G be the corresponding parabolic subgroup. If w = σ_{α} for α = ε_{1} − ε_{2}, then ^{w}P = G.

Next we describe the parabolic and Borel subalgebras of root–reductive direct limit Lie algebras 𝔤. In view of Proposition 4.1 it suffices to describe parabolic and Borel subalgebras for classical simple 𝔤. This gives also a description of the parabolic and Borel subgroups of root–reductive ind– groups G, for if P is a parabolic subgroup of G with Lie algebra 𝔭, then P = {g ∈ G | Ad(g)𝔭 = 𝔭}. The following statement, which of course is standard in the finite dimensional case, is a variation of Proposition 5 from [8].

Proposition 4.4 Suppose either that 𝔤 is a classical simple direct limit Lie algebra or that 𝔤 ≅ 𝔤𝔩(∞). Let 𝔭 ⊂ 𝔤 be a parabolic subalgebra. Then, for some index set S_{𝔭}, there is an isomorphism of Lie algebras

𝔭^{red} ≅ ⨁_{s∈S_{𝔭}} 𝔩^{s}, (4.5)

where each 𝔩^{s} is a subalgebra of 𝔭^{red} isomorphic to 𝔤𝔩(n), to 𝔤𝔩(∞), to a finite dimensional simple Lie algebra, or to a classical simple direct limit Lie algebra. Furthermore, if 𝔤 is classical simple S_{𝔭} can be chosen so that 𝔩^{s} is isomorphic to 𝔤𝔩(n) or 𝔤𝔩(∞) for any s ∈ S_{𝔭} except for at most one index s_{0} ∈ S_{𝔭} for which 𝔩^{s_{0}} is isomorphic to:

𝔞(n) or 𝔞(∞) for 𝔤 = 𝔞(∞); 𝔞(n), 𝔞(∞), 𝔟(n) or 𝔟(∞) for 𝔤 = 𝔟(∞);

𝔞(n), 𝔞(∞), 𝔠(n) or 𝔠(∞) for 𝔤 = 𝔠(∞); 𝔞(n), 𝔞(∞), 𝔡(n) or 𝔡(∞) for 𝔤 = 𝔡(∞). □

For a classical simple 𝔤, there is a natural choice for the set S_{𝔭}, and moreover S_{𝔭} is linearly ordered. Indeed, fix a Borel subalgebra 𝔟 of 𝔭. For every n the parabolic subalgebra 𝔭_{n} = 𝔤_{n} ∩ 𝔭 determines “marked” nodes in the Dynkin diagram of 𝔤_{n} which correspond to simple roots α such that both α and −α are roots of 𝔭_{n}. Let S_{𝔭_{n}} be the set whose elements are all the unmarked nodes together with all connected components of marked nodes. We fix an order on the nodes of the Dynkin diagram of 𝔤_{n} which is increasing from left to right. For 𝔤 ≠ 𝔡(∞) this order is unique. For 𝔤 = 𝔡(∞), if the two rightmost nodes are both marked or both unmarked, we set the upper one to precede the lower one, otherwise, we set the marked one to precede the unmarked one. This order on the Dynkin diagram induces an order on S_{𝔭_{n}}, and it is straightforward to check that the orders on S_{𝔭_{n}} are compatible for different n. Hence they determine a linear order on the union ⋃_{n} S_{𝔭_{n}}. In what follows, S_{𝔭} will be fixed as the union ⋃_{n} S_{𝔭_{n}}. Note that the order on S_{𝔭} depends only on 𝔭 and not on 𝔟. Furthermore, if s_{0} ∈ S_{𝔭} and 𝔩^{s_{0}} is not isomorphic to 𝔤𝔩(∞), 𝔤𝔩(n), 𝔞(∞) or 𝔞(n), then s_{0} is necessarily the unique maximal element of S_{𝔭}.

5 Diagonal ind–groups and beyond

The class of root–reductive ind–groups is part of the more general class of diagonal ind–groups. The corresponding class of Lie algebras has been studied quite extensively; see [2], [4], [1] and [21]. An essential difference between root–reductive direct limit Lie algebras and general diagonal direct limit Lie algebras is that the latter need not admit Cartan subalgebras that yield a root decomposition compatible with the direct limit. In this paper we do not develop a complete Bott–Borel–Weil theory for diagonal ind–groups.

Here is a diagonal ind–group that is not root–reductive. Consider the sequence of closed immersions

GL(2^{n}) ↪ GL(2^{n+1}) , g ↦ ( \begin{matrix}g & 0 \\ 0 & g \end{matrix} ),

and let GL(2^{∞}) denote the corresponding ind–group. Its Lie algebra 𝔤𝔩(2^{∞}) is a diagonal direct limit Lie algebra [1]. If H = \underrightarrow{lim} H_{n} where H_{n} denotes the diagonal matrices in GL(2^{n}), then H is a Cartan subgroup of GL(2^{∞}) and 𝔤𝔩(2^{∞}) has no root decomposition with respect to 𝔥.

Let Δ_{n} = {ε_{i,n} − ε_{j,n} | 1 ≤ i, j ≤ 2^{n}} be the root system of 𝔤𝔩(2^{n}). The Borel subalgebras of 𝔤𝔩(2^{∞}) containing 𝔥 are in bijective correspondence with the systems of triangular decompositions Δ_{n} = Δ_{n}^{+} ⊔ Δ_{n}^{−} satisfying the following compatibility condition: if ε_{i,n} − ε_{j,n} ∈ Δ_{n}^{+}, then ε_{i,n+1} − ε_{j,n+1} ∈ Δ_{n+1}^{+} and ε_{2^{n}+i,n+1} − ε_{2^{n}+j,n+1} ∈ Δ_{n+1}^{+}. One can give a similar description of all parabolic subalgebras containing 𝔥 in terms of compatible systems of partitions Δ_{n} = Δ_{n}^{+} ⊔ Δ_{n}^{0} ⊔ Δ_{n}^{−}.

Here is another interesting ind–group. Its Lie algebra was introduced in [1]. Fix k ∈ ℕ, k > 1. Let the κ_{n} : P GL(k^{2^{n}}) ↪ P GL(k^{2^{n+1}}) be the unique closed immersions of algebraic groups for which the diagrams

GL(k^{2^{n}}) ⟶^{Ad} GL(k^{2^{n+1}})
pr_{n} ↓  ↓ pr_{n+1}
P GL(k^{2^{n}}) ↪^{κ_{n}} P GL(k^{2^{n+1}})

are commutative, where pr_{n} and pr_{n+1} are the natural projections. We define the reductive ind– group P GL^{Ad}(k^{2^{∞}}) := \underrightarrow{lim} P GL(k^{2^{n}}). If H_{n} ⊂ P GL(k^{2^{n}}) denotes the subgroup of diagonal matrices, then κ_{n}(H_{n}) ⊂ H_{n+1} and H := \underrightarrow{lim} H_{n} is a Cartan subgroup of P GL^{Ad}(k^{2^{∞}}).

A Borel subgroup B ⊂ P GL^{Ad}(k^{2^{∞}}) is determined by the B_{n} := B ∩ P GL(k^{2^{n}}). Given B_{n}, we describe the Borel subgroups of P GL(k^{2^{n+1}}) that contain H_{n+1} and κ_{n}(B_{n}), providing recursive description of all Borel subgroups of P GL^{Ad}(k^{2^{∞}}) that contain H. Note first that specification of B_{n+1} is the same as specification of a pr_{n+1}^{−1}(H_{n+1})–invariant maximal flag in the natural representation space of GL(k^{2^{n+1}}), and that natural representation space is the adjoint representation space for GL(k^{2^{n}}). Hence, a pr_{n+1}^{−1}(H_{n+1})–invariant maximal flag in the natural representation of GL(k^{2^{n+1}}) is determined by a linear order on root basis of 𝔤𝔩(k^{2^{n}}), i.e. a basis

{x_{α}}_{α∈Δ_{n}} ∪ {h_{i}}_{1≤i≤k^{2^{n}}} (5.1)

consisting of root vectors x_{α} and of a basis {h_{i}}_{1≤i≤k^{2^{n}}} of 𝔥_{n}, where Δ_{n} denotes the root system of 𝔤𝔩(k^{2^{n}}). The Borel subgroup B_{n} determines a partition Δ_{n} = Δ_{n}^{+} ⊔ Δ_{n}^{−} and thus also the following partial order > on Δ_{n}: for α, β ∈ Δ_{n}^{+} , α > β whenever the 𝔟_{n}–height of α is greater then the 𝔟_{n}–height of β; α > β for any α ∈ Δ_{n}^{+} and any β ∈ Δ_{n}^{−}; and finally, for α, β ∈ Δ_{n}^{−}, α > β whenever −β > −α. Now, since Δ_{n} is naturally identified with the set {x_{α}}_{α∈Δ_{n}}, we can consider all extensions of this partial order on Δ_{n} to a linear order > on the root basis (5.1) such that x_{α} > h_{i} > x_{β} for all i whenever α ∈ Δ_{n}^{+} and β ∈ Δ_{n}^{−}. Any such extension determines a unique pr_{n+1}^{−1}(H_{n+1})–invariant maximal flag in the natural representation and thus a unique Borel subgroup of P GL(k^{2^{n+1}}). One can check that the Borel subgroups obtained in this way are precisely the Borel subgroups of P GL(k^{2^{n+1}}) that contain H_{n+1} and κ_{n}(B_{n}).

Part II. Representations

6 Rational and pro–rational G–modules

Let G = \underrightarrow{lim} G_{n} be an ind–group. We define a G–module to be a vector space V endowed with G_{n}–module structures φ^{n} : G_{n} × V → V (ℂ–linear in V) such that φ^{n+1}|_{G_{n}×V} = φ^{n} for all n. The maps φ^{n} define the structure map φ : G × V → V of the G–module V. We say that V is a rational G–module if in addition the dimension of V is countable and φ is a morphism of ind–varieties, where V has the canonical ind–variety structure of Example 1.2. An equivalent definition of a rational G–module: V is isomorphic to the limit of a direct system of injections of rational finite dimensional G_{n}–modules φ_{n} : G_{n} × V_{n} → V_{n}. Every rational G–module is locally finite, in other words, φ(G_{n} × ℂv) generates a finite dimensional for every v ∈ V and every n.

Any rational module over a reductive algebraic group is completely reducible. Thus, if G is locally reductive and V = \underrightarrow{lim} V_{n} is a rational G–module, then V is a completely reducible G_{n}– module for every n.

The category of rational G–modules is too restrictive for our purposes. For example the dual of a rational G-module is no longer rational (as dim V^{∗} is in general uncountable). We define a G–module U to be pro–rational if it is the dual of a rational G-module. This is equivalent to saying that U is isomorphic to the inverse (or projective) limit \underleftarrow{lim} U_{n} of a system

… ⟶^{ψ_{n+1}} U_{n} ⟶^{ψ_{n}} U_{n−1} ⟶^{ψ_{n−1}} … ⟶^{ψ_{1}} U_{0} ⟶ 0

of finite dimensional rational G_{n}–modules U_{n}, where ψ_{n} is a rational G_{n−1}–module surjection for each n. The G–module structure on \underleftarrow{lim} U_{n}: if u = (…, u_{k}, u_{k−1}, …, u_{0}) ∈ \underleftarrow{lim} U_{n}, so each u_{n} ∈ U_{n} and ψ_{n}(u_{n}) = u_{n−1} for n ≥ 1, and if g_{m} ∈ G_{m}, then

g_{m} · u := (…, g_{m} · u_{k}, g_{m} · u_{k−1}, …, g_{m} · u_{m}, ψ_{m}(g_{m} · u_{m}), …, ψ_{1} ∘ … ∘ ψ_{m}(g_{m} · u_{m}))

where k > m.

7 𝔤–modules

If G is an ind–group and V is a rational G–module with structure map φ : G × V → V, then V is a module for the Lie algebra 𝔤 of G with structure map dφ : 𝔤 × V → V. We say that a 𝔤–module V is rationally G–integrable (or, briefly, G–integrable) if it is obtained by this construction from a rational G–module structure on V. As G is assumed to be connected, that G–module structure is unique. Furthermore, a G–integrable 𝔤–module V necessarily is locally finite. By definition this means^{6} that V is locally finite as 𝔤_{n}–module for every n, in other words, dim U(𝔤_{n})·v < ∞ for every v ∈ V and every n. A straightforward verification shows that a countably dimensional 𝔤–module V is locally finite if and only if V is isomorphic to a direct limit \underrightarrow{lim} V_{n} of finite dimensional 𝔤_{n}– modules V_{n}. Unless the contrary is stated explicitly, in what follows we will assume automatically that an expression of a locally finite 𝔤–module as \underrightarrow{lim} V_{n} corresponds to a direct system of injections V_{n} ↪ V_{n+1}.

In the rest of this Section, G is a locally reductive ind–group, 𝔤 is its Lie algebra, and 𝔭 is a parabolic subalgebra of 𝔤. We study irreducible locally finite 𝔭–modules and parabolically generated irreducible 𝔤–modules.

Proposition 7.1 Let 𝔭 be a parabolic subalgebra of 𝔤. Let E be an irreducible locally finite 𝔭–module and let 𝔲_{𝔭} denote the nilpotent radical of 𝔭. Then 𝔲_{𝔭} · E = 0.

Proof. Suppose that ξ · e ≠ 0 for some ξ ∈ 𝔲_{𝔭} and e ∈ E. As E is irreducible there exists u ∈ U(𝔭) such that u · ξ · e = e. Let n be sufficiently large so that ξ, u ∈ U(𝔭_{n}). Then E′_{n} := 𝔲_{𝔭_{n}} · E_{n} is a proper 𝔭_{n}–submodule of E_{n} := U(𝔭_{n})·e such that e ∉ E′_{n}. But as ξ ·e ∈ E′_{n}, the equality u·ξ ·e = e is contradictory. Therefore ξ · e cannot be nonzero. □

Corollary 7.2 If P is a parabolic subgroup of G and E is an irreducible rational P –module, then the unipotent radical U_{P} of P acts trivially on E. □

6In many papers on Lie algebra representations, in particular in [15] and [8], the term “integrable” is a synonym for various versions of local finiteness. This is not acceptable in the present paper because such representations need not integrate from the Lie algebra to the ind–group. Therefore a distinction between G–integrable and locally finite 𝔤–modules is in order.

Let E be an irreducible 𝔭–module as above. We introduce the induced 𝔤–module

Ṽ(E) := U(𝔤) ⊗_{U(𝔭)} E.

If 𝔭 is a Borel subalgebra then dim E = 1, i.e. E = ℂ_{μ} for some μ ∈ 𝔥^{∗}, and Ṽ(ℂ_{μ}) is called a Verma module. In general, dim E = ∞, but nevertheless the following Proposition holds. It follows from the more general Theorem 2.1 in [1].

Proposition 7.3 Let E be an irreducible locally finite 𝔭–module. Then Ṽ(E) has a unique maximal proper 𝔤–submodule I_{E}. □

An important corollary of Proposition 7.3 is that Ṽ(E) has a unique irreducible quotient module V (E) := Ṽ(E)/I_{E} . We call V (E) the irreducible 𝔤–module parabolically generated by E. Any 𝔤– module V which is generated by a 𝔭–submodule isomorphic to E has the 𝔤–module V (E) as a quotient. Indeed, V admits an obvious surjection σ̃ : Ṽ(E) → V whose kernel is necessarily a 𝔤–submodule of I_{E} . Thus σ̃ induces a 𝔤–surjection s : V = (Ṽ(E)/ ker σ̃) → V (E) = Ṽ(E)/I_{E} .

In general, V (E) is not locally finite, and therefore is not rationally G–integrable. The problem of characterizing, for a fixed 𝔭, all irreducible locally finite 𝔭-modules E for which V (E) is G– integrable, is a generalization of the problem of computing all dominant integral weights for a finite dimensional reductive group. The following Proposition reduces this problem in an explicit way to the structure of E. The case where 𝔭 is a Borel subalgebra of a classical simple linear Lie algebra was studied in [1] and [20].

Proposition 7.4 Let E be an irreducible locally finite 𝔭–module. Then V (E) is G–integrable if and only if E is P –integrable and, for any expression E = \underrightarrow{lim} E_{n}, V (E_{n}) is a finite dimensional 𝔤_{n}–module for each n. When these conditions hold, each V (E_{n}) is a well–defined G_{n}–module and we have a canonical G–module isomorphism V (E) ≅ \underrightarrow{lim} V (E_{n}).

Proof. First, if E is P -integrable and V (E_{n}) is finite dimensional for each n, the standard theory of connected algebraic groups applied to G_{n} implies that V (E_{n}) is G_{n}–integrable for any n. Furthermore, \underrightarrow{lim} V (E_{n}) is a rational G–module which admits a P –module injection i : E ↪ \underrightarrow{lim} V (E_{n}) such that i(E) generates \underrightarrow{lim} V (E_{n}) as a G–module. This is sufficient to conclude that \underrightarrow{lim} V (E_{n}) = V (E). Indeed, one need only check that the 𝔤–surjection s : \underrightarrow{lim} V (E_{n}) → V (E) induced by i is an isomorphism. Assuming that ker s ≠ 0 we find an n such that ker s ∩ V (E_{n}) ≠ 0. Then, as V (E_{n}) is an irreducible 𝔤_{n}–module, V (E_{n}) ⊂ ker s. Therefore ker s ∩ i(E_{n}) ≠ 0, which contradicts the injectivity of i.

Conversely, if V (E) is G–integrable, E must be P –integrable. Indeed, if E = \underrightarrow{lim} E_{n}, then for each n E_{n} is a 𝔭_{n}–submodule of the finite dimensional G_{n}–submodule of V (E) generated by E_{n}. Therefore, again the theory of connected algebraic groups implies that E_{n} is necessarily P_{n}– integrable. Thus \underrightarrow{lim} E_{n} is a P –integrable 𝔭–module. To complete the proof we need to show also that V (E_{n}) is finite dimensional for each n (and is thus a G_{n}–module) whenever V (E) is G– integrable. This follows from a standard geometric version of Frobenius Reciprocity and we present this argument in Section 12 in the proof of Theorem 11.1(i). □

In the following, we say that an irreducible rational P –module E is dominant if V (E) is a locally finite 𝔤–module, and thus is a well–defined irreducible rational G–module. An integral weight λ ∈ 𝔥^{∗} is B–dominant for a Borel subgroup B ⊂ G if the one dimensional B–module ℂ_{λ} of weight λ is dominant.

We conclude this Section by recalling some basic definitions for weight modules. Let 𝔤 be root–reductive. A 𝔤–module V is a weight module if it has an 𝔥–module decomposition

V = ⨁_{μ∈𝔥^{∗}} V^{μ}, (7.5)

where 𝔥 is the fixed Cartan subalgebra of 𝔤 and V^{μ} := {v ∈ V | h · v = μ(h)v for any μ ∈ 𝔥^{∗}}. The support supp V of a weight module V is {μ ∈ 𝔥^{∗} | V^{μ} ≠ 0}. A weight 𝔤–module V is finite if the support of V is finite in the direction of every root of 𝔤, i.e. if for every μ ∈ supp V and every α ∈ Δ, the intersection {μ + kα | k ∈ ℤ_{+}} ∩ supp V is finite. See [8]. A finite 𝔤–module is locally finite. In the following we will consider finite 𝔤– and 𝔭–modules, the latter being defined as 𝔭–modules which are weight modules (i.e. which satisfy (7.5)) and which are finite as 𝔭^{red}–modules. Also, we define a rational G– or P –module to be finite if it is finite respectively as a weight 𝔤– or 𝔭–module.

8 Irreducible rational G–modules

Let G be a reductive ind–group. An irreducible rational G– (or P –) module V is called locally irreducible if V = \underrightarrow{lim} V_{n} for some direct system of irreducible rational (finite dimensional) G_{n}– (or P_{n}–) modules V_{n}. Our starting point in this Section is that an irreducible rational G–module V is not necessarily locally irreducible. Here are some examples.

Example 8.1 For every n fix a pair of non–isomorphic irreducible finite dimensional G_{n}–modules U_{n}, W_{n} such that U_{n}, W_{n} ≺ U_{n+1}, and U_{n}, W_{n} ≺ W_{n+1}, where here the sign ≺ indicates the existence of a G_{n}–module injection. Extend the diagonal injection U_{n} ↪ U_{n} ⊕ U_{n} to a G_{n}–module injection η_{n}^{U} : U_{n} ↪ U_{n+1} ⊕ W_{n+1}. Similarly, fix a G_{n}–module injection η_{n}^{W} : W_{n} ↪ U_{n+1} ⊕ W_{n+1}. Define η_{n} : U_{n} ⊕ W_{n} ↪ U_{n+1} ⊕ W_{n+1} by η_{n} := η_{n}^{U} ⊕ η_{n}^{W}. Let V_{n} := U_{n} ⊕ W_{n}. Then the G–module V = \underrightarrow{lim} V_{n} is an irreducible rational G–module that is not locally irreducible. It is irreducible as a consequence of Proposition 8.3 below; see Example 8.4(1). It is not locally irreducible because, for every nonzero v ∈ V and for sufficiently large n, the G_{n}–submodule of V generated by v is isomorphic to U_{n} ⊕ W_{n}, and thus is reducible.

Here is an explicit choice of the G_{n}–modules U_{n} and W_{n}. Let G = GL(∞). Let U_{n} and W_{n} be the respective irreducible GL(n)–modules with highest weights λ_{n} := ε_{1} − (n − 1)ε_{n} and μ_{n} := ε_{1} − nε_{n}. The standard branching rule [9] ensures that U_{n}, W_{n} ≺ U_{n+1}, and U_{n}, W_{n} ≺ W_{n+1}. For this particular choice of U_{n} and W_{n} the resulting module V is not a weight module. Other choices of U_{n} and W_{n} yield irreducible weight GL(∞)–modules that are irreducible but not locally irreducible.

Now we introduce an essential invariant of any locally finite 𝔤–module V. Represent V as the limit of a direct system of finite dimensional semisimple 𝔤_{n}–modules,

0 ⟶ V_{1} ⟶^{φ_{1}} … ⟶^{φ_{n−1}} V_{n} ⟶^{φ_{n}} V_{n+1} ⟶^{φ_{n+1}} …, (8.2)

where the φ_{i} need not be injective. Each V_{n} decomposes canonically as a direct sum of its isotypic components, V_{n} = ⨁_{i} V_{n}^{i}, and we fix decompositions V_{n}^{i} = ⨁_{k} V_{n}^{i,k} into simple 𝔤_{n}–modules. To each V_{n}^{i,k} we assign an abstract vector v_{n}^{i,k} and define a vector space V_{n}^{i} := ⨁_{k} ℂv_{n}^{i,k}. When the composition of φ_{n}|_{V_{n}^{i}} with the projection V_{n+1} → V_{n+1}^{j} is nonzero, we define a linear map

α_{i,j} : V_{n}^{i} → V_{n+1}^{j} by α_{i,j}(v_{n}^{i,k}) := ∑_{k′} v_{n+1}^{j,k′},

where k′ runs over the simple components of V_{n+1}^{j} onto which φ(V_{n}^{i,k}) projects nontrivially. The collection {V_{n}^{i}, α_{i,j}} is, by definition, a multiplicity diagram D_{V} of V. D_{V} is a commutative diagram of finite dimensional vector spaces.

A subdiagram D′ of a multiplicity diagram D_{V} is stably proper if there is no index n_{0} such that D′ contains the subdiagram of D_{V} consisting of all spaces V_{n}^{i} for n ≥ n_{0}. A subdiagram D′ of D_{V} is invariant, if whenever V′ ⊂ V_{n}^{i} is a vector space in D′ and 0 ≠ v′ ∈ V′, α_{i,j}(v′) belongs to a vector space of D′ for all possible pairs i, j. Finally, we call a multiplicity diagram D_{V} minimal (and call the direct system (8.2) minimal) if no stably proper invariant subdiagram D′ of D_{V} is a multiplicity diagram of V. Any countably dimensional locally finite 𝔤–module can be represented as the limit of a minimal direct system of injections (8.2).

The following is a straightforward but important Proposition.

Proposition 8.3 Assume that V is represented as the limit \underrightarrow{lim} V_{n} of a minimal direct system of injections (8.2). Then V is irreducible if and only if D_{V} admits no nonzero stably proper invariant subdiagram.

Proof. A nonzero invariant stably proper subdiagram D′ of D_{V} is immediately checked to give rise to a nonzero 𝔤–submodule V′ ⊂ V such that D′ = D_{V′}. As D_{V} is minimal, V′ is necessarily a proper submodule. Conversely, if D_{V} admits no nonzero stably proper invariant subdiagram, V is necessarily generated by each isotypic “vector”, i.e. by each v ∈ V such that v is in the image of some v′ ∈ V_{n}^{i}. This is sufficient to conclude that V is irreducible. □

Example 8.4

(1) If V = \underrightarrow{lim} V_{n}, V_{n} = U_{n} ⊕ W_{n} as in the beginning of this Section (in particular as in Example 8.1), then D_{V} has the form

V_{1}^{1} → V_{2}^{1} → … → V_{n}^{1} → …
↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘
V_{1}^{2} → V_{2}^{2} → … → V_{n}^{2} → …,

all spaces V_{n}^{i} being one dimensional and all maps being isomorphisms. Clearly, D_{V} is a minimal multiplicity diagram which has no stably proper invariant subdiagrams. Therefore V is irreducible.

(2) Let V_{n} := 𝔤𝔩(n), V_{n}^{1} ≅ 𝔞(n − 1), V_{n}^{2} ≅ ℂ, and let V_{n} ↪ V_{n+1} be the Lie algebra injection induced by the inclusion GL(n) ↪ GL(n + 1) of Example 4.2. Then V ≅ 𝔤𝔩(∞) and D_{V} has the form

V_{1}^{1} → V_{2}^{1} → … → V_{n}^{1} → …
↗ ↗ ↗ ↗
V_{1}^{2} → V_{2}^{2} → … → V_{n}^{2} → …,

where all spaces V_{n}^{i} are one dimensional and all maps are isomorphisms. D_{V} is a minimal multiplicity diagram of V. Its upper row is a stably proper invariant subdiagram which gives rise to the 𝔤𝔩(∞)–submodule 𝔞(∞) ⊂ 𝔤𝔩(∞).

Next we establish an important fact about the structure of parabolically generated irreducible locally finite 𝔤–modules which are weight modules.

Proposition 8.5 Let E be an irreducible dominant 𝔭–module which is a weight module. Assume furthermore that 𝔭 contains no entire simple component 𝔤^{t} of 𝔤^{ss} (see Proposition 4.1). Then

(i) E is a finite 𝔭–module and V (E) is a finite 𝔤–module;

(ii) both E and V (E) are locally irreducible.

Proof. Proposition 6 in [8] implies that, for any t′ ∈ T, V (E) is a finite 𝔥 + 𝔤^{t′}–module if and only if supp V (E) is finite in the direction of some root α ∈ Δ^{t′}. As V (E) is immediately seen to be finite in the direction of any root α ∈ Δ with 𝔤^{α} ⊂ 𝔲_{𝔭}, and as 𝔭 does not contain an entire simple component 𝔤^{t}, V (E) is necessarily a finite 𝔤–module. In particular, supp E is finite in the direction of every root of 𝔭^{red}, i.e. E is a finite 𝔭–module. Statement (i) is proved. Statement (ii) is a direct corollary of (i) and of Theorem 6(i) in [8]; the latter says that a finite module is locally irreducible. □

The following example shows that local irreducibility in Proposition 8.5 really requires that E be a weight module.

Example 8.6 Let G = GL(∞), let P be as in Example 4.3, and let V be the G–module from Example 8.1. Set E := {v ∈ V | g · v = v for all g ∈ U_{P} }. Then a direct verification shows that E is an irreducible, but not locally irreducible, P –module and V = V (E).

Concluding this Section, we note that a locally irreducible rational G–module does not have to be a weight G–module. Here are some examples.

Example 8.7 Assume that G is root–reductive and fix an irreducible finite dimensional G_{n}–module V_{n} for every n, such that V_{n+1} , considered as a G_{n}–module, contains at least two irreducible components V′_{n} and V′′_{n} isomorphic to V_{n} whose supports are disjoint as subsets of supp V_{n+1} . Define the G_{n}–injection V_{n} ↪ V_{n+1} to be the composition of the diagonal injection of V_{n} into V′_{n} ⊕ V′′_{n} and the injection of V′_{n} ⊕ V′′_{n} into V_{n+1}. Then V := \underrightarrow{lim} V_{n} is a locally irreducible rational G–module which is not a weight module.

For an explicit example, let G = A(∞) and let V_{n} be the irreducible G_{n}–module with highest weight λ_{n} := −ε_{1}−2ε_{2}−…−nε_{n}. Set λ′_{n+1} := λ_{n+1} and λ′′_{n+1} := −2ε_{1}−3ε_{2}−…−(n+1)ε_{n}−ε_{n+1}.

Both λ′_{n+1} and λ′′_{n+1} are extremal weights of V_{n+1}. The G_{n}–submodules of V_{n+1} generated by V_{n+1}^{λ′_{n+1}}

and V_{n+1}^{λ′′_{n+1}} are isomorphic to V_{n} and can be chosen as V′_{n} and V′′_{n}.

9 Combinatorics of locally irreducible locally finite 𝔭–modules

Here we introduce and study certain combinatorial invariants of locally irreducible locally finite 𝔭–modules. Those invariants appear naturally in the geometric study of these modules. Fix a finite dimensional reductive Lie algebra 𝔨, a Cartan subalgebra 𝔥_{𝔨} and a parabolic subalgebra 𝔭_{𝔨} with 𝔥_{𝔨} ⊂ 𝔭_{𝔨} ⊂ 𝔨. Denote the roots of 𝔨 by Δ_{𝔨}. Given a Borel subalgebra 𝔥_{𝔨} ⊂ 𝔟_{𝔨} ⊂ 𝔭_{𝔨}, we have a partition Δ_{𝔨} = Δ_{𝔨}^{+} ⊔ Δ_{𝔨}^{−}. As usual, ρ_{𝔟_{𝔨}} denotes the half–sum of the elements of Δ_{𝔨}^{+}. If E_{𝔨} is an irreducible finite dimensional 𝔭_{𝔨}–module, we call E_{𝔨} regular if λ + ρ_{𝔟_{𝔨}} is a regular weight of 𝔨, where λ is the 𝔟_{𝔨}–highest weight of E_{𝔨}. To any E_{𝔨} we assign an integer ℓ(E_{𝔨}), its length, as follows. If E_{𝔨} is regular then ℓ(E_{𝔨}) is the length of the unique Weyl group element w for which w(λ + ρ_{𝔟_{𝔨}}) is 𝔟_{𝔨}–dominant. If E_{𝔨} is not regular, we define ℓ(E_{𝔨}) inductively. If rk 𝔨 = 0, ℓ(E_{𝔨}) := 0. For rk 𝔨 > 0, we set ℓ(E_{𝔨}) := max{ℓ(E_{𝔨′}) | E_{𝔨′} ≺ E_{𝔨}}, where the maximum is taken over all modules E_{𝔨′} ≺ E_{𝔨} of all proper root subalgebras 𝔨′ of 𝔨.

Lemma 9.1 Let 𝔨 ≅ 𝔞(k −1) and suppose that 𝔰 ≅ 𝔞(s−1) is a root subalgebra of 𝔨, or let 𝔨 ≅ 𝔤𝔩(k) and suppose that 𝔰 ≅ 𝔤𝔩(s) is a root subalgebra of 𝔨. Set 𝔭_{𝔰} := 𝔭_{𝔨} ∩ 𝔰, and fix an irreducible finite dimensional 𝔭_{𝔨}–module E_{k} and an irreducible 𝔭_{𝔰}–module E_{s} ≺ E_{k}. Then

(i) ℓ(E_{s}) ≤ ℓ(E_{k});

(ii) if ℓ(E_{s}) = ℓ(E_{k}) and both E_{s} and E_{k} are regular, there exists a chain of proper root subalgebras 𝔰 ⊂ 𝔨_{s} ⊂ … ⊂ 𝔨_{m} ⊂ … ⊂ 𝔨_{k} = 𝔨 together with a chain of regular irreducible 𝔭_{𝔨} ∩ 𝔨_{m}–modules E_{m},

E_{s} ≺ … ≺ E_{m} ≺ … ≺ E_{k}.

Proof. (i) It suffices to prove that ℓ(E_{s}) ≤ ℓ(E_{k}) when both E_{k} and E_{s} are regular. If E_{k} is not regular, the inequality follows from the definition of ℓ(E_{k}). If E_{k} is regular but E_{s} is not regular, then ℓ(E_{s}) = ℓ(E_{s′}) for some regular E_{s′} ≺ E_{s}, and the inequality ℓ(E_{s′}) ≤ ℓ(E_{k}) gives ℓ(E_{s}) ≤ ℓ(E_{k}).

Assume Δ_{𝔨} = {ε_{i} − ε_{j} | 1 ≤ i ≠ j ≤ k} and Δ_{𝔭_{𝔨}} = ({ε_{i} − ε_{j} | 1 ≤ i < j ≤ k}) ∪ (⋃_{1≤q≤a}{−(ε_{i} − ε_{j}) | p_{q−1} + 1 ≤ i < j ≤ p_{q}}) for 1 ≤ p_{1} < p_{2} < … < p_{a} = k with p_{0} = 0. Then Δ_{𝔰} = {ε_{i} − ε_{j} | i ≠ j ∈ I_{s}} for some subset I_{s} = {r_{1}, …, r_{s}} of I_{k} := {1, …, k} where r_{1} < … < r_{s}. Fix a Borel subalgebra 𝔟_{𝔨} ⊂ 𝔭_{𝔨} such that Δ_{𝔨}^{+} = {ε_{i} − ε_{j} | 1 ≤ i < j ≤ k} and [p_{q−1} + 1, p_{q}] ∩ {r_{1}, …, r_{s}} = {p_{q−1} + 1, …, p_{q−1} + b_{q}} where b_{q} is the cardinality of [p_{q−1} + 1, p_{q}] ∩ {r_{1}, …, r_{s}}. Define the chain of sets I_{s} ⊂ … ⊂ I_{m} ⊂ … ⊂ I_{k} by I_{m+1} := I_{m} ∪ {i_{m}} where i_{m} is the smallest element of I_{k}\I_{m} for s ≤ m < k. This determines a chain of root subalgebras 𝔰 = 𝔨_{s} ⊂ … ⊂ 𝔨_{m} ⊂ … ⊂ 𝔨_{k} = 𝔨, 𝔨_{m} ≅ 𝔞(m − 1) (resp. 𝔨_{m} ≅ 𝔤𝔩(m)) with Δ_{𝔨_{m}} = {ε_{i} − ε_{j} | i ≠ j ∈ I_{m}}. Set Δ_{m} := Δ_{𝔨_{m}} and ρ_{m} := \frac{1}{2} ∑_{γ∈Δ_{𝔨}^{+}∩Δ_{m}} γ.

We define now a chain of irreducible 𝔭 ∩ 𝔨_{m}–modules E_{m},

E_{s} ≺ … ≺ E_{m} ≺ … ≺ E_{k},

by means of their 𝔟_{𝔨} ∩ 𝔨_{m}–highest weights λ_{m}. E_{s} and E_{k} are given. Expand λ_{k} and λ_{s} as λ_{k} = λ_{k}^{1}ε_{1} + … + λ_{k}^{k}ε_{k} and λ_{s} = λ_{s}^{r_{1}}ε_{r_{1}} + … + λ_{s}^{r_{s}}ε_{r_{s}}. Determine the integers s = i_{0} < … < i_{q} < … < i_{a} = k by the property that {1, 2, …, p_{q}} ⊂ I_{i_{q}} but {1, 2, …, p_{q}} ⊄ I_{i_{q}−1}, and set

λ_{i_{q}} := λ_{k}^{1}ε_{1} + … + λ_{k}^{p_{q}}ε_{p_{q}} + λ_{s}^{r_{q}}ε_{r_{q}} + … + λ_{s}^{r_{s}}ε_{r_{s}}, (9.2) where r_{q} is the smallest integer in I_{s} such that r_{q} > p_{q}. Let i_{q−1} < m < i_{q}. Then λ_{i_{q−1}} = λ_{k}^{1}ε_{1} + … + λ_{k}^{p_{q−1}}ε_{p_{q−1}} + λ_{s}^{p_{q−1}+1}ε_{p_{q−1}+1} + … + λ_{s}^{p_{q−1}+b_{q}}ε_{p_{q−1}+b_{q}} + λ_{s}^{r_{q}}ε_{r_{q}} + … + λ_{s}^{r_{s}}ε_{r_{s}} and λ_{i_{q}} = λ_{k}^{1}ε_{1} + … + λ_{k}^{p_{q−1}}ε_{p_{q−1}} + λ_{k}^{p_{q−1}+1}ε_{p_{q−1}+1} + … + λ_{k}^{p_{q−1}+(p_{q}−p_{q−1})}ε_{p_{q−1}+(p_{q}−p_{q−1})} + λ_{s}^{r_{q}}ε_{r_{q}} + … + λ_{s}^{r_{s}}ε_{r_{s}}. The standard branching rule gives

λ_{k}^{p_{q−1}+j} ≥ λ_{s}^{p_{q−1}+j} ≥ λ_{k}^{p_{q−1}+j+((p_{q}−p_{q−1})−b_{q})} (9.3)

for 1 ≤ j ≤ b_{q}. For 1 ≤ j ≤ b_{q} + (m − i_{q−1}), set

λ_{m}^{p_{q−1}+j} := \{ \begin{matrix}λ_{k}^{p_{q−1}+j} & if & j ≤ m − i_{q−1}, \\ min\{λ_{k}^{p_{q−1}+j}, λ_{s}^{p_{q−1}+j−(m−i_{q−1})}\} & if & j > m − i_{q−1}. \end{matrix} (9.4)

Put λ_{m} := λ_{k}^{1}ε_{1} + … + λ_{k}^{p_{q−1}}ε_{p_{q−1}} + λ_{m}^{p_{q−1}+1}ε_{p_{q−1}+1} + … + λ_{m}^{p_{q−1}+b_{q}+(m−i_{q−1})}ε_{p_{q−1}+b_{q}+(m−i_{q−1})} + λ_{s}^{r_{q}}ε_{r_{q}} + … + λ_{s}^{r_{s}}ε_{r_{s}}, and define E_{m} as the finite dimensional irreducible 𝔭_{𝔨} ∩ 𝔨_{m}–module with 𝔟_{𝔨} ∩ 𝔨_{m}–highest weight λ_{m}. Using (9.3) and (9.4), one verifies that E_{m} ≺ E_{m+1} for i_{q−1} ≤ m < i_{q}.

Note that the weights λ_{m} satisfy

(9.5) if α, β ∈ Δ_{𝔨}^{+} ∩ Δ_{m} and (λ_{m} + ρ_{m}, α) = (λ_{m} + ρ_{m}, β) = 0, then α + β ∉ Δ_{m}.

Indeed, if m = i_{q} for some 1 ≤ q ≤ a, then (9.5) is immediate because (λ_{m} + ρ_{m}, α) = 0 with α ∈ Δ_{𝔨}^{+}∩ Δ_{m} implies α = ε_{x} − ε_{y} for some 1 ≤ x ≤ p_{q} and p_{q} < y ≤ k. Assume now i_{q−1} < m < i_{q}. If α, β ∈ Δ_{𝔨}^{+} ∩ Δ_{m} and (λ_{m} + ρ_{m}, α) = (λ_{m} + ρ_{m}, β) = 0, one verifies (possibly after interchanging α and β) that α = ε_{x} − ε_{y} and β = ε_{y} − ε_{z}, where 1 ≤ x ≤ p_{q−1}, p_{q−1} < y ≤ p_{q} and p_{q} < z ≤ k. Then, using the definition of λ_{m} (i.e. formula (9.4)) one checks that, λ_{m}^{y} = λ_{k}^{y} gives (λ_{m} + ρ_{m}, α) = (λ_{k} + ρ_{𝔨}, α) = 0, i.e. a contradiction with the regularity of E_{k}. Similarly, λ_{m}^{y} = λ_{k}^{y−(m−i_{q−1})} gives (λ_{m} + ρ_{m}, β) = (λ_{s} + ρ_{𝔰}, β) = 0, which contradicts regularity of E_{s}. This establishes (9.5).

To complete the proof of (i), for each m we will construct an injection ι_{m} : R_{m} → R_{m+1}, where R_{m} := {α ∈ Δ_{𝔨}^{+} ∩ Δ_{m} | (λ_{m} + ρ_{m}, α) < 0}. Then ι := ι_{k−1} ∘ … ∘ ι_{s} : R_{s} → R_{k} will be an injection. Hence ℓ(E_{s}) = |R_{s}| ≤ |R_{k}| = ℓ(E_{k}) where | · | denotes cardinality.

Let I_{m+1} = {c_{1}, …, c_{m+1}} where c_{1} < … < c_{m+1} , and I_{m} = {c_{1}, …, c_{i′−1}, c_{i′+1}, …, c_{m+1}} where p_{q_{0}−1} + 1 ≤ c_{i′} ≤ p_{q_{0}}. In view of the choice of Borel subalgebra 𝔟_{𝔨} we have b_{j} = j for 1 ≤ j ≤ i′, and b_{i′} = i′ is the largest element of I_{m+1} ∩ [p_{q_{0}−1} + 1, p_{q_{0}}]. Set λ_{m+1} + ρ_{m+1} =: u′_{1}ε_{c_{1}} + … + u′_{m+1}ε_{c_{m+1}}, λ_{m} + ρ_{m} =: v_{1}ε_{c_{1}} + … + v_{i′−1}ε_{c_{i′−1}} + v_{i′+1}ε_{i′+1} + … + v_{m+1}ε_{c_{m+1}}, and u_{i} := u′_{i} − \frac{1}{2}. Then, from (9.2), (9.3) and (9.4),

\begin{matrix}u_{i} = v_{i} for 1 ≤ i ≤ p_{q−1}, \\ u_{p_{q−1}+1} ≥ v_{p_{q−1}+1} > … > u_{i′−1} ≥ v_{i′−1} > u_{i′}, \\ u_{i} = v_{i} − 1 for i′ < i ≤ m + 1. \end{matrix} (9.6)

Furthermore, the definition of I_{m} and the regularity of E_{s} and E_{k}, imply that v_{1}, …, v_{p_{q−1}} are distinct and so are v_{i′+1}, …, v_{m+1}.

Let α ∈ R_{m}. If α ∈ R_{m+1}, we set ι_{m}(α) := α. Assume that α = ε_{c_{x}} − ε_{c_{y}} ∉ R_{m+1}. Then v_{x} − v_{y} < 0 and u_{x} − u_{y} ≥ 0. Now (9.6) implies x < i′ and i′ < y. If p_{q−1} < x, we have u_{i′} < v_{x} < v_{y} = u_{y} + 1, and hence (λ_{m+1} + ρ_{m+1}, ε_{c_{i′}} − ε_{c_{y}}) = u_{i′} − u_{y} < 0. In this case we set ι_{m}(α) := ε_{c_{i′}} − ε_{c_{y}}. If x ≤ p_{q−1}, then v_{x} = u_{x} = u_{y} = v_{y} − 1. If there exists p_{q−1} + 1 ≤ z < i′ with v_{z} ≤ u_{y} ≤ u_{z}, we have (λ_{m+1} + ρ_{m+1}, ε_{c_{x}} − ε_{c_{z}}) = u_{x} − u_{z} ≤ 0. Notice that (9.5) rules out the equality u_{y} = u_{z}, and hence (λ_{m+1} + ρ_{m+1}, ε_{c_{x}} − ε_{c_{z}}) = u_{x} − u_{z} < 0. Furthermore, (λ_{m} + ρ_{m}, ε_{c_{x}} − ε_{c_{z}}) = v_{x} − v_{z} ≥ 0, and thus ε_{c_{x}} − ε_{c_{z}} ∉ R_{m}. In that case we set ι_{m}(α) := ε_{c_{x}} − ε_{c_{z}}. Finally, if there is no z as above, (λ_{m+1} + ρ_{m+1}, ε_{c_{x}} − ε_{c_{i′}}) + (λ_{m+1} + ρ_{m+1}, ε_{c_{i′}} − ε_{c_{y}}) = u_{x} − u_{y} = 0, and therefore at least one of the terms in the sum is non–positive. On the other hand, (9.5) again rules out the vanishing of either term, and thus one of them is positive and the other one — negative. In this last case we set ι_{m}(α) to be the root (ε_{c_{x}} − ε_{c_{i′}} or ε_{c_{i′}} − ε_{c_{y}}) corresponding to the negative term. The construction of ι_{m} ensures that it is an injection, and the proof of (i) is complete.

(ii) We claim that when ℓ(E_{s}) = ℓ(E_{k}) and both E_{s} and E_{k} are regular, each E_{m} (as constructed in the proof of (i)) is also regular. Indeed, let R′_{m} := {α ∈ Δ_{𝔨}^{+}∩Δ_{m} | (λ_{m} +ρ_{m}, α) ≤ 0}. Then there is an injection ι′_{m} : R′_{m} → R′_{m+1} whose construction is almost identical to the one above. Notice that R_{m} ⊂ R′_{m}, and R_{s} = R′_{s}, R_{k} = R′_{k}. The assumption ℓ(E_{s}) = ℓ(E_{k}) implies now that both ι : R_{s} → R_{k} and ι′ := ι′_{k−1} ∘ … ∘ ι′_{s} : R′_{s} → R′_{k} are bijections. Hence |R′_{m}| = ℓ(E_{s}) = ℓ(E_{k}) = |R_{m}| for every s ≤ m ≤ k, and thus R_{m} = R′_{m}. This means that E_{m} is regular. □

The following Proposition is crucial in the proof of Theorem 11.1. Note the connection with the notions of cohomological finiteness in [21, Section 4].

Proposition 9.7 Let 𝔤 = \underrightarrow{lim} 𝔤_{n} be root–reductive, 𝔭 = \underrightarrow{lim} 𝔭_{n} be a parabolic subalgebra of 𝔤, and E = \underrightarrow{lim} E_{n} be a locally irreducible locally finite 𝔭–module.

(i) Either lim ℓ(E_{n}) = ∞, or there exist q_{0} and m such that ℓ(E_{n}) = q_{0} for every n ≥ m.

(ii) Assume that ℓ(E_{n}) = q_{0} for n ≥ m and that E_{n} is regular for infinitely many n. Then there exist a Borel subalgebra 𝔟 ⊂ 𝔭, an integer m′ ≥ m, and an element w ∈ W of length q_{0} with respect to 𝔟, such that μ_{n} := w(λ_{n} + ρ_{n}) − ρ_{n} is a 𝔟_{n}–dominant weight for n ≥ m′, where 𝔟_{n} := 𝔟 ∩ 𝔤_{n} and ρ_{n} := ρ_{𝔟_{n}}. Furthermore, E_{n} is regular for every n ≥ m′.

Proof. A direct verification shows that it is enough to prove the Proposition under the assumption that 𝔤 is simple. More precisely, if 𝔤^{t} for t ∈ T are as in Proposition 4.1 (iii), then E_{n} = ⨂_{t∈T} E_{n}^{t}, E_{n}^{t} being irreducible finite dimensional 𝔭_{n}^{t} := 𝔭 ∩ 𝔤^{t}–modules. Using the observation that ℓ(E_{n}) = ∑_{t∈T} ℓ(E_{n}^{t}), one notes that the claims of the Proposition follow directly from the same claims for all 𝔭^{t}–modules E^{t} = \underrightarrow{lim} E_{n}^{t}. If 𝔤^{t} is finite dimensional there is nothing to prove, so in the rest of the proof we will assume that 𝔤 = 𝔞(∞), 𝔟(∞), 𝔠(∞), or 𝔡(∞).

We start by introducing notation. Recall decomposition (4.5) and the fact that S_{𝔭} is an ordered set. Consider the ordered set S̄_{𝔭} := S_{𝔭} ∪ {−∞, ∞}, where −∞ < s < ∞ for every s ∈ S_{𝔭}. Note that as an ordered set S̄_{𝔭} is isomorphic to a subset of ℝ ∪ {−∞, ∞}. In the following, s_{0} denotes the unique element of S_{𝔭} for which 𝔩^{s_{0}} ≅ 𝔤̸𝔩(p) or 𝔤𝔩(∞); if such an element does not exist, we set s_{0} := ∞ ∈ S̄_{𝔭} and 𝔩^{s_{0}} := 0. For s_{1} ≤ s_{2} ∈ S̄_{𝔭}, let Δ_{n}^{[s_{1},s_{2}]} denote the subset of Δ_{n} consisting of all integral linear combinations of simple roots of 𝔟_{n} whose nodes in the Dynkin diagram of 𝔤_{n} correspond to elements s ∈ S_{𝔭_{n}} with s_{1} ≤ s ≤ s_{2}. Put Δ^{[s_{1},s_{2}]} := ⋃_{n} Δ_{n}^{[s_{1},s_{2}]}. Let (𝔤^{[s_{1},s_{2}]})′ be the Lie subalgebra of 𝔤 generated by all root spaces 𝔤^{α} for α ∈ Δ^{[s_{1},s_{2}]} (see §4) and let 𝔤^{[s_{1},s_{2}]} := (𝔤^{[s_{1},s_{2}]})′ + (⨁_{s_{1}≤s≤s_{2}} 𝔩^{s}). When s_{1} = s_{2} = s we often write simply 𝔤^{s} instead of 𝔤^{[s_{1},s_{2}]}.

Set also 𝔭^{[s_{1},s_{2}]} := 𝔭∩𝔤^{[s_{1},s_{2}]}. Next, define the locally irreducible locally finite 𝔭^{[s_{1},s_{2}]}–module E^{[s_{1},s_{2}]} as follows. Fix a Borel subalgebra 𝔟′ of 𝔭, and let λ_{n} be the 𝔟′_{n}–highest weight of E_{n}. Let E_{n}^{[s_{1},s_{2}]} be the irreducible 𝔭_{n}^{[s_{1},s_{2}]}–module with 𝔟′_{n} ∩𝔭_{n}^{[s_{1},s_{2}]}–highest weight λ_{n}_{|𝔥_{n}^{[s_{1},s_{2}]}}, where 𝔥_{n}^{[s_{1},s_{2}]} := 𝔥∩𝔤_{n}^{[s_{1},s_{2}]}.

The modules E_{n}^{[s_{1},s_{2}]} form a direct system and we put E^{[s_{1},s_{2}]} := \underrightarrow{lim} E_{n}^{[s_{1},s_{2}]}. We define similarly the Lie algebras 𝔤^{(s_{1},s_{2})}, 𝔤^{(s_{1},s_{2}]} and 𝔤^{[s_{1},s_{2})}, their respective parabolic subalgebras 𝔭^{(s_{1},s_{2})}, 𝔭^{(s_{1},s_{2}]} and 𝔭^{[s_{1},s_{2})}, and the corresponding modules E^{(s_{1},s_{2})}, E^{(s_{1},s_{2}]}, and E^{[s_{1},s_{2})}. Throughout the proof, an upper index ^{[s_{1},s_{2}]}, ^{(s_{1},s_{2})}, ^{[s_{1},s_{2})} or ^{(s_{1},s_{2}]} always refers to a corresponding algebra, module or Weyl group. For example, 𝔥_{n}^{(s_{1},s_{2})} = 𝔥 ∩ 𝔤_{n}^{(s_{1},s_{2})}, W^{[s_{1},s_{2}]} is the Weyl group of 𝔤^{[s_{1},s_{2}]}, W^{s} is the Weyl group of 𝔤^{s} = 𝔤^{[s,s]}, etc. Finally, we set R_{n} := {α ∈ Δ_{n}^{+} | (λ_{n} + ρ_{n}, α) < 0}.

Here is the idea of the proof. Assume \underrightarrow{lim} ℓ(E_{n}) ≠ ∞. For 𝔤 = 𝔞(∞), (i) follows from Lemma 9.1(i). Applying Lemma 9.1(ii) to the 𝔭_{n}^{[s′,s′′]}–module E_{n}^{[s′,s′′]} for various s′ ≤ s′′, we are able locate the roots α ∈ R_{n} for n large enough. Namely, we show that there exist finitely many pairs s′_{i} ≤ s′′_{i} such that 𝔤^{(s′_{i},s′′_{i})} is finite dimensional and every α ∈ R_{n} belongs to some Δ^{[s′_{i},s′′_{i}]}. Therefore we can find finitely many finite dimensional subalgebras 𝔤^{i} ≅ 𝔞(p_{i}) (or 𝔤𝔩(p_{i})) of 𝔤 with 𝔤^{i} ⊃ 𝔤^{(s′_{i},s′′_{i})} so that R_{n} ⊂ ⋃ Δ_{𝔤^{i}}. Now we fix a Borel subalgebra 𝔟̄ of ⨁ 𝔤^{i} and choose 𝔟 so that every simple root of 𝔟̄ is a simple root of 𝔟. The desired element w exists and is of the claimed length because ⨁ 𝔤^{i} is finite dimensional. The regularity of all E_{n} now follows easily from Lemma 9.1(ii). For 𝔤 ≠ 𝔞(∞), we consider two cases depending on whether 𝔤^{s_{0}} is finite or infinite dimensional. If 𝔤^{s_{0}} is infinite dimensional, we first apply the Proposition to 𝔤^{(−∞,s_{0})} ≅ 𝔤𝔩(∞), 𝔤𝔩(p), 𝔞(∞), or 𝔞(p) to obtain w_{1} ∈ W. We then show that no sign changes are involved in the Weyl group elements that make λ_{n} + ρ_{n} dominant, and find an element w_{2} ∈ W which involves only reflections along roots of 𝔤 which are not roots of 𝔤^{(−∞,s_{0})} or 𝔤^{s_{0}}. The desired element w then equals w_{2} ∘ w_{1}. If 𝔤^{s_{0}} is finite dimensional, we first apply the Proposition to a natural subalgebra 𝔤̄ ⊂ 𝔤, 𝔤̄ ≅ 𝔞(∞) to obtain w_{1} ∈ W which does not involve sign changes. The element w_{1} maps the weights λ_{n} +ρ_{n} into weights ν_{n} +ρ_{n}, where ν_{n} are represented in coordinates by decreasing sequences of (not necessarily positive) integers or half–integers. We then find an element w_{2} which maps ν_{n} + ρ_{n} into dominant weights. The element w := w_{2} ∘ w_{1} ∈ W then satisfies the Proposition. The details are presented separately for 𝔤 = 𝔞(∞) and 𝔤 ≠ 𝔞(∞).

The case 𝔤 = 𝔞(∞). Statement (i) follows immediately from Lemma 9.1(i). We now prove (ii). Recall that Δ = {ε_{i} −ε_{j} | 1 ≤ i ≠ j} and Δ_{n} = {ε_{i} −ε_{j} | 1 ≤ i ≠ j ≤ n}. The parabolic subalgebra 𝔭 fixes a partition ℕ = ⊔_{s∈S_{𝔭}}I^{s} such that Δ^{[s_{1},s_{2}]} = {±(ε_{i}−ε_{j}) | i ∈ I^{s′_{1}}, j ∈ I^{s′_{2}}, where s_{1} ≤ s′_{1} ≤ s′_{2} ≤ s_{2}}. The Borel subalgebra 𝔟′ fixes a linear order <_{𝔟′} on ℕ: i <_{𝔟′} j if ε_{i} − ε_{j} ∈ Δ^{+}. Notice also that each of the Lie algebras 𝔤^{[s_{1},s_{2}]}, 𝔤^{(s_{1},s_{2}]}, 𝔤^{[s_{1},s_{2})} and 𝔤^{(s_{1},s_{2})} is isomorphic to 𝔤𝔩(∞), 𝔤𝔩(p), 𝔞(∞), or 𝔞(p) for some p. Fix an infinite sequence n_{1} < n_{2} < …, n_{i} ∈ ℕ, such that E_{n_{k}} is a regular module of length q_{0} for every k. Lemma 9.1(ii) implies the existence of root subalgebras 𝔤′_{n} ≅ 𝔞(n − 1) of 𝔤 and regular 𝔤′_{n}–modules E′_{n} such that 𝔤′_{n_{k}} = 𝔤_{n_{k}} and E′_{n_{k}} = E_{n_{k}} for n, n_{k} ≥ m′′ := n_{1}. Denote by λ′_{n} the 𝔟′_{n}–highest weight of E′_{n} and set λ′_{n} := (λ′_{n})^{1}ε_{1} + … + (λ′_{n})^{n}ε_{n}.

First we show that rank 𝔤^{(s′,s′′)} < q_{0} whenever (λ′_{m′′} + ρ_{m′′}, α) < 0 for α = ε_{i} − ε_{j} ∈ Δ_{m′′}^{+} with i ∈ I^{s′}, j ∈ I^{s′′}. For n ≥ m′′, let i_{n} be the <_{𝔟′}–maximal element of I^{s′} ∩ {1, …, n} and j_{n} be the <_{𝔟′}–minimal element of I^{s′′} ∩ {1, …, n}. We prove by induction on n that (λ′_{n} + ρ_{n}, α_{n}) < 0 for α_{n} := ε_{i_{n}} − ε_{j_{n}}. When n = m′′, (ρ_{m′′}, α) ≥ (ρ_{m′′}, α_{m′′}) and (λ′_{m′′}, α) ≥ (λ′_{m′′}, α_{m′′}), thus (λ′_{m′′} +ρ_{m′′}, α_{m′′}) ≤ (λ′_{m′′} +ρ_{m′′}, α) < 0. Assume (λ′_{n}+ρ_{n}, α_{n}) < 0. If n+1 ∉ I^{s} for any s′ < s < s′′, then (ρ_{n+1}, α_{n+1}) = (ρ_{n}, α_{n}), and by the branching rule (λ′_{n+1})^{i_{n+1}} ≤ (λ′_{n})^{i_{n}}, (λ′_{n+1})^{j_{n+1}} ≥ (λ′_{n})^{j_{n}}. Hence (λ′_{n+1}, α_{n+1}) ≤ (λ′_{n}, α_{n}), which implies (λ′_{n+1} +ρ_{n+1}, α_{n+1}) ≤ (λ′_{n} +ρ_{n}, α_{n}) < 0. If n+1 ∈ I^{s} for some s′ < s < s′′, then (ρ_{n+1}, α_{n+1}) = (ρ_{n}, α_{n}) + 1. Furthermore, (λ′_{n+1})^{i_{n+1}} = (λ′_{n})^{i_{n}}, (λ′_{n+1})^{j_{n+1}} = (λ′_{n})^{j_{n}}, and (λ′_{n+1}, α_{n+1}) = (λ′_{n}, α_{n}). Thus (λ′_{n+1}+ρ_{n+1}, α_{n+1}) = (λ′_{n}+ρ_{n}, α_{n})+1 ≤ 0. But since E′_{n} is regular, (λ′_{n+1} + ρ_{n+1}, α_{n+1}) ≠ 0 and then (λ′_{n+1} + ρ_{n+1}, α_{n+1}) < 0. The inequality (λ′_{n} + ρ_{n}, α_{n}) < 0 implies now rank 𝔤^{(s′,s′′)} < q_{0} as the height of α_{n} equals rank(𝔤′_{n})^{(s′,s′′)} + 1.

We show next by induction on q_{0}, that there exists a sequence s′_{1} ≤ s′′_{1} ≤ s′_{2} ≤ s′′_{2} ≤ … ≤ s′_{r} ≤ s′′_{r} ∈ S̄_{𝔭} such that, when n ≥ m′′, ℓ((E′_{n})^{[s′_{1},s′′_{1}]}) + … + ℓ((E′_{n})^{[s′_{r},s′′_{r}]}) = q_{0} and rk(𝔤′_{n})^{(s′_{i},s′′_{i})} ≤ q_{0} for every 1 ≤ i ≤ r. If q_{0} = 0 we set r := 0. Assume q_{0} > 0. Let, as above, (λ′_{m′′} + ρ_{m′′}, α) < 0 for α = ε_{i} −ε_{j} ∈ Δ_{m′′}^{+} with i ∈ I^{s′}, j ∈ I^{s′′} and s′ < s′′. Then ℓ((E′_{n})^{(−∞,s′]})+ℓ((E′_{n})^{[s′,s′′]})+ℓ((E′_{n})^{[s′′,∞)}) ≤ q_{0} for every n ≥ m′′. If ℓ((E′_{n})^{(−∞,s′]}) + ℓ((E′_{n})^{[s′,s′′]}) + ℓ((E′_{n})^{[s′′,∞)}) = q_{0}, we apply the induction assumption to the 𝔭^{(−∞,s′]}–module E^{(−∞,s′]} to get s′_{1}, s′′_{1}, …, s′_{r′−1}, s′′_{r′−1}, and to the 𝔭^{[s′′,∞)}–module E^{[s′′,∞)} to get s′_{r′+1}, s′′_{r′+1}, …, s′_{r}, s′′_{r}. By setting s′_{r} := s′, s′′_{r} := s′′, we obtain a sequence with the desired properties. If ℓ((E′_{n})^{(−∞,s′]}) + ℓ((E′_{n})^{[s′,s′′]}) + ℓ((E′_{n})^{[s′′,∞)}) < q_{0}, there exists β = ε_{i′} − ε_{j′} ∈ Δ_{m′′}^{+} where i′ ∈ I^{s̄′}, j′ ∈ I^{s̄′′} with (λ′_{m′′} + ρ_{m′′}, β) < 0 and ℓ((E′_{n})^{[s̄′,s̄′′]}) > ℓ((E′_{n})^{[s′,s′′]}). Then ℓ((E′_{n})^{(−∞,s̄′]}) + ℓ((E′_{n})^{[s̄′,s̄′′]}) + ℓ((E′_{n})^{[s̄′′,∞)}) ≤ q_{0} and we again consider two cases. For ℓ((E′_{n})^{(−∞,s̄′]}) + ℓ((E′_{n})^{[s̄′,s̄′′]}) + ℓ((E′_{n})^{[s̄′′,∞)}) = q_{0}, the desired sequence is constructed as above by replacing s′ and s′′ by s̄′ and s̄′′ respectively. If ℓ((E′_{n})^{(−∞,s̄′]}) + ℓ((E′_{n})^{[s̄′,s̄′′]}) + ℓ((E′_{n})^{[s̄′′,∞)}) < q_{0}, we replace in a similar manner s̄′ and s̄′′ by ¯ s̄′ and ¯ s̄′′ respectively. The process is finite because q_{0} ≥ ℓ((E′_{n})^{[¯s̄′,¯s̄′′]}) > l((E′_{n})^{[s̄′,s̄′′]}) > ℓ((E′_{n})^{[s′,s′′]}).

Fix a sequence s′_{1} ≤ s′′_{1} ≤ s′_{2} ≤ s′′_{2} ≤ … ≤ s′_{r} ≤ s′′_{r} ∈ S̄_{𝔭} as above. Let m′ = n_{k_{0}} ≥ m be such that 𝔤_{m′}^{(s′_{i},s′′_{i})} = 𝔤^{(s′_{i},s′′_{i})} for 1 ≤ i ≤ r, rk 𝔤_{m′}^{s′_{i}} = min{rk 𝔤^{s′_{i}}, 2q_{0}} and rk 𝔤_{m′}^{s′′_{i}} = min{rk 𝔤^{s′′_{i}}, 2q_{0}}. Choose a Borel subalgebra 𝔟 ⊂ 𝔭 with the property that whenever rk 𝔤^{s′_{i}} > 2q_{0} (respectively, rk 𝔤^{s′′_{i}} > 2q_{0}), there are q_{0} elements x_{s′_{i}}^{q_{0}} >_{𝔟} … >_{𝔟} x_{s′_{i}}^{1} ∈ I^{s′_{i}} (resp. y_{s′′_{i}}^{1} <_{𝔟} … <_{𝔟} y_{s′_{i}}^{q_{0}} ∈ I^{s′′_{i}}) such that x >_{𝔟} x_{s′_{i}}^{q_{0}}

(resp. y_{s′′_{i}}^{q_{0}} <_{𝔟} y) for any other x ∈ I^{s′_{i}} (resp. y ∈ I^{s′′_{i}}). Let λ_{n} = λ_{n}^{1}ε_{1} +…+λ_{n}^{n}ε_{n} be the 𝔟_{n}–highest weight of E_{n}.

Fix n ≥ m′. If α ∈ R_{m′}, then α = ε_{x} − ε_{y} for x ∈ I^{s_{x}}, y ∈ I^{s_{y}}. The construction of the sequence s′_{1}, s′′_{1}, …, s′_{r}, s′′_{r} and the definition of 𝔟 implies s′_{i} ≤ s_{x} < s_{y} ≤ s′′_{i} for some i, 1 ≤ i ≤ r. If s_{x} = s′_{i} , there are at most q_{0} elements of I^{s′_{i}} <_{𝔟}–greater than x, and if s_{y} = s′′_{i}, there are at most q_{0} elements of I^{s′′_{i}} <_{𝔟}–smaller than y. Furthermore, the choice of 𝔟 gives (ρ_{n}, α) = (ρ_{m′}, α) and the branching rule implies λ_{n}^{x} ≤ λ_{m′}^{x} and λ_{n}^{y} ≥ λ_{m′}^{y}. Hence (λ_{n}, α) = λ_{n}^{x} − λ_{n}^{y} ≤ λ_{m′}^{x} − λ_{m′}^{y} = (λ_{m′}, α) < 0, and thus α ∈ R_{n}. This establishes the inclusion R_{m′} ⊂ R_{n}. In particular R_{m′} = R_{n_{k}} whenever n_{k} ≥ m′. If, on the other hand, β ∈ Δ_{n}^{+}\R_{n}, fix k so that n_{k} ≥ n. If β = ε_{x} − ε_{y}, with x ∈ I^{s_{x}}, y ∈ I^{s_{y}}, assume that x is the j_{x} ^{th} largest element (with respect to <_{𝔟}) of I^{s_{x}} ∩ {1, …, n}, and y is the j_{y} th smallest element (with respect to <_{𝔟}) of I^{s_{y}} ∩ {1, …, n}. Let x′ be the j_{x} ^{th} largest element of I^{s_{x}} ∩ {1, …, n_{k}}, and let y′ be the j_{y} ^{th} smallest element of I^{s_{y}} ∩ {1, …, n_{k}}. Set β′ := ε_{x′} − ε_{y′}. The assumption β′ ∈ R_{n_{k}} would give β = β′, and hence β ∈ R_{n}, which contradicts the choice of β. Therefore (λ_{n_{k}} + ρ_{n_{k}}, β′) > 0. Notice that (ρ_{n}, β) = (ρ_{n_{k}}, β′), and, by the branching rule, λ_{n}^{x} ≥ λ_{n_{k}}^{x′}, λ_{n}^{y} ≤ λ_{n_{k}}^{y′}. Thus (λ_{n}, β) = λ_{n}^{x} − λ_{n}^{y} ≥ λ_{n_{k}}^{x′} − λ_{n_{k}}^{y′} = (λ_{n_{k}}, β′). Now it is clear that (λ_{n} + ρ_{n}, β) ≥ (λ_{n_{k}} + ρ_{n_{k}}, β′) > 0. This proves that E_{n} is regular. Hence |R_{n}| = |R_{m′}| which yields R_{n} = R_{m′}. The latter combined with the observation that every α ∈ R_{m′} has the same height in Δ_{n} for any n ≥ m′ implies also the existence of w ∈ W as required in (ii).

The case 𝔤 ≠ 𝔞(∞). The cases 𝔤 = 𝔟(∞), 𝔠(∞) and 𝔡(∞) are treated in an almost identical way. We present the details for 𝔤 = 𝔟(∞) only.

In the rest of the proof 𝔤 = 𝔟(∞). We establish (i) and (ii) simultaneously. If there are only finitely many regular modules among E_{n}, (i) follows from the definition of ℓ(E_{n}) (as E_{n} ≺ E_{n+1} and hence ℓ(E_{n}) ≤ ℓ(E_{n+1}) whenever E_{n+1} is not regular) and (ii) is an empty statement. Assume that infinitely many E_{n} are regular and lim ℓ(E_{n}) ≠ ∞. Then there exists q_{0} and an infinite sequence n_{1} < n_{2} < …, n_{k} ∈ ℕ, such that E_{n_{k}} is a regular module of length q_{0} for every k. Indeed, lim ℓ(E_{n}) ≠ ∞ implies the existence of q′_{0} and a sequence {n′_{k}} for which ℓ(E_{n′_{k}}) = q′_{0}. If there are infinitely many regular modules among E_{n′_{k}}, set q_{0} := q′_{0} and take {n_{k}} to be the subsequence of {n′_{k}} corresponding to regular modules E_{n_{k}}. If there are only finitely many regular modules among E_{n′_{k}}, then ℓ(E_{n}) ≤ q′_{0} for every n, as there is n′_{k} > n with E_{n′_{k}} not regular, and hence ℓ(E_{n}) ≤ ℓ(E_{n′_{k}}) = q′_{0} by the definition of ℓ(E_{n′_{k}}). In particular, ℓ(E_{n}) ≤ q′_{0} for every regular E_{n}, and thus, for some q_{0} ≤ q′_{0}, ℓ(E_{n}) = q_{0} for infinitely many regular E_{n}. Fix q_{0} and a sequence n_{1} < n_{2} < … so that E_{n_{k}} is a regular module of length q_{0} for every k.

Recall that Δ = {±ε_{i}, ±ε_{i} ± ε_{j} | 1 ≤ i ≠ j} and Δ_{n} = {±ε_{i}, ±ε_{i} ± ε_{j} | 1 ≤ i ≠ j ≤ n}. The Borel subalgebra 𝔟′ determines the sign function

sgn : ℕ → {±1}, sgn(i) = 1 if and only if ε_{i} ∈ Δ^{+}

and a linear order <_{𝔟′} on ℕ: i <_{𝔟′} j if sgn(i)ε_{i} − sgn(j)ε_{j} ∈ Δ^{+}. The parabolic subalgebra 𝔭 determines a partition ℕ = ⊔_{s∈S_{𝔭}}I^{s} such that Δ^{[s_{1},s_{2}]} = {±(sgn(i)ε_{i} − sgn(j)ε_{j}) | i ∈ I^{s′_{1}}, j ∈ I^{s′_{2}}, where s_{1} ≤ s′_{1} ≤ s′_{2} ≤ s_{2}} and Δ^{s_{0}} = {±ε_{i}, ±ε_{i} ± ε_{j} | i ≠ j ∈ I^{s_{0}}}.

Assume first that 𝔤^{s_{0}} is infinite dimensional and let rk 𝔤_{m_{0}}^{s_{0}} > q_{0} for some m_{0}. Then (λ_{n_{k}} + ρ_{n_{k}}, sgn(i)ε_{i}) > 0 for 1 ≤ i ≤ n_{k} and n_{k} ≥ m_{0},. Indeed, if i ∈ I^{s_{0}}, (λ_{n_{k}}, sgn(i)ε_{i}) ≥ 0 because λ_{n_{k}}_{|𝔥_{n_{k}}^{s_{0}}} is a 𝔟′_{n_{k}}^{s_{0}}–dominant weight, and (ρ_{n_{k}}, sgn(i)ε_{i}) > 0. If, on the other hand, i ∉ I^{s_{0}}, then sgn(i)ε_{i} ∈ Δ_{n_{k}} is of height greater than q_{0}, and (λ_{n_{k}} + ρ_{n_{k}}, sgn(i)ε_{i}) < 0 would imply ℓ(E_{n_{k}}) > q_{0}. Notice that ℓ(E_{n}^{(−∞,s_{0})}) ≤ ℓ(E_{n}), and hence the Proposition applied to 𝔭^{(−∞,s_{0})} ⊂ 𝔤^{(−∞,s_{0})} and the 𝔭^{(∞,s_{0})}–module E^{(−∞,s_{0})} yields integers m′_{1} ≥ m_{0} and q_{1}, an element w_{1} ∈ W^{(−∞,s_{0})}, and a Borel subalgebra 𝔟̃′ of 𝔭^{(−∞,s_{0})}. Let 𝔟′′ be a Borel subalgebra of 𝔭 for which 𝔟′′ ∩ 𝔭^{(−∞,s_{0})} = 𝔟̃′ and 𝔟′′ ∩ 𝔭^{s_{0}} = 𝔟′ ∩ 𝔭^{s_{0}}. Since W^{(−∞,s_{0})} ⊂ W, w_{1} has length q_{1} respect to 𝔟′′. Set ν_{n} := w(λ_{n} + ρ_{n}) − ρ_{n} and let ^{w_{1}}E_{n} be the irreducible ^{w_{1}}𝔭_{n}–module with 𝔟′′_{n}–highest weight ν_{n}. We claim that there exists a ^{w_{1}}𝔭–module of the form ^{w_{1}}E := \underrightarrow{lim}_{n≥m′_{1}}^{w_{1}}E_{n}. Indeed, the existence of ^{w_{1}}E follows from Theorem 11.1(iii) below for G = A(∞) and E^{(∞,s_{0})}, see the discussion after that Theorem. (As the proof of Theorem 11.1 for G = A(∞) does not depend on the case G = B(∞), C(∞) or D(∞), there is no contradiction in applying Theorem 11.1(iii) here.) Clearly ^{w_{1}}E_{n_{k}} is regular of length q_{0} − q_{1} for every n_{k} ≥ m′_{1}.

Notice that 𝔟′ may be fixed from the very beginning so that there are at least q_{0} − q_{1} <_{𝔟′∩𝔭^{s_{0}}}– smallest elements in I^{s_{0}}. Since 𝔟′′∩𝔭^{s_{0}} = 𝔟′∩𝔭^{s_{0}}, 𝔟′′ has the same property. Let y^{1} <_{𝔟′′} … <_{𝔟′′} y^{q_{0}−q_{1}} be the q_{0} − q_{1} <_{𝔟′′}-smallest elements of I^{s_{0}}. Observe that, for n_{k} ≥ m′_{1}, (ν_{n_{k}} + ρ_{𝔟′′_{n_{k}}}, α) < 0 with α ∈ Δ_{𝔟′′} implies that α = sgn(x)ε_{x} − sgn(y^{i})ε_{y^{i}} for some x ∉ I^{s_{0}} and 1 ≤ i ≤ q_{0}. Denote by l_{k,i} the number of indices x for which (ν_{n_{k}} +ρ_{𝔟′′_{n_{k}}}, sgn(x)ε_{x} −sgn(y^{i})ε_{y^{i}}) < 0. The definition of the length of w_{1}E_{n_{k}} implies ∑_{1≤i≤q_{0}−q_{1}} l_{k,i} = q_{0} − q_{1}, and dominance of ν_{n_{k}}_{|𝔥_{n_{k}}^{s_{0}}} gives l_{k,1} ≥ l_{k,2} ≥ … ≥ l_{k,q_{0}−q_{1}}. On the other hand, the branching rule implies l_{k′,i} ≥ l_{k,i} for k′ ≥ k. Combining these facts about l_{k_{i}}, we obtain the existence of l_{1}, …, l_{q_{0}−q_{1}} with l_{k,i} = l_{i} whenever n_{k} ≥ m′_{1}. It is now easy to see that there is a Borel subalgebra 𝔟 of ^{w_{1}}𝔭 for which there are l_{1} <_{𝔟}–largest elements of ℕ\I^{s_{0}} and 𝔟 ∩ 𝔤^{s_{0}} = 𝔟′′ ∩ 𝔤^{s_{0}}. Fix such a 𝔟 and denote by x^{1} >_{𝔟} … >_{𝔟} x^{l_{1}} the l_{1} <_{𝔟}–largest elements of ℕ\I^{s_{0}}. Let m′ = n_{k_{0}} ≥ m′_{1} be larger than x^{1}, …, x^{l_{1}}. For n_{k} ≥ m′ the element w_{n_{k}} ∈ W_{n_{k}} which makes ν_{n_{k}} + ρ_{𝔟_{n_{k}}} dominant permutes only x_{1}, …, x_{l_{1}}, y_{1}, …, y_{q_{0}−q_{1}}. Furthermore, the equality l_{k,i} = l_{i} implies that w_{n_{k}} does not depend on k. Set w_{2} := w_{n_{k}}. Finally, one observes that each E_{n} is regular, as otherwise (ν_{n} + ρ_{𝔟_{n}}, α) = 0 would imply (ν_{n_{k}} + ρ_{𝔟_{n_{k}}}, α) < 0 for n_{k} < n and (ν_{n_{k}} + ρ_{𝔟_{n_{k}}}, α) > 0 for n_{k} > n, i.e. ℓ(^{w_{1}}E_{n_{k}}) > ℓ(^{w_{1}}E_{n_{k_{0}}} ) for n_{k} > n_{0}. Hence μ_{n} = w(λ_{n} + ρ_{n}) − ρ_{n} is a regular 𝔟_{n}–dominant weight for w := w_{2} ∘ w_{1}. This completes the proof when 𝔤^{s_{0}} is infinite dimensional.

Assume that 𝔤^{s_{0}} is finite dimensional. Since the details in this case are similar to the details in the previous case, we will only outline the argument. Fix m_{0} with 𝔤_{m_{0}}^{s_{0}} = 𝔤^{s_{0}}. Set Δ̄ := {sgn(i)ε_{i} − sgn(j)ε_{j} | i ≠ j}. Let 𝔤̄ denote the root subalgebra of 𝔤 generated by the root spaces 𝔤^{α} for α ∈ Δ̄. Note that 𝔤̄ ≅ 𝔞(∞). If 𝔭̄ := 𝔭 ∩ 𝔤̄ and 𝔟̄′ := 𝔟′ ∩ 𝔤̄, then 𝔭̄ ⊂ 𝔤̄ is a parabolic subalgebra and 𝔟̄′ ⊂ 𝔭̄ is a Borel subalgebra. Let Ē_{n} be the irreducible 𝔭̄_{n}–module with 𝔟̄′_{n}– highest weight λ_{n}. For n ≥ m_{0}, λ_{n}_{|𝔥_{n}^{s_{0}}} = λ_{m_{0}}_{|𝔥_{m_{0}}^{s_{0}}} and thus Ē := \underrightarrow{lim} Ē_{n} is a well–defined 𝔭̄– module where the injections Ē_{n} ↪ Ē_{n+1} are the restrictions of the injections E_{n} ↪ E_{n+1}. Since ℓ(Ē_{n_{k}}) ≤ ℓ(E_{n_{k}}), the Proposition applied to the 𝔭̄–module Ē yields integers m_{1} and q_{1}, an element w_{1} ∈ W̄ ⊂ W, and a Borel subalgebra 𝔟̄′′ of 𝔭̄. Let 𝔟′′ be the Borel subalgebra of 𝔭 with roots Δ𝔟̄′′ ∪ {sgn(i)ε_{i}, sgn(i)ε_{i} + sgn(j)ε_{j} | i ≠ j}. Exactly as in the case when 𝔤^{s_{0}} is infinite dimensional, there exists a well–defined ^{w_{1}} 𝔭̄–module ^{w_{1}} Ē := \underrightarrow{lim}^{w_{1}} Ē_{n}. Set ν_{n} := w_{1}(λ_{n} + ρ_{n}) − ρ_{n}. The weights ν_{n} are 𝔟̄′′–dominant but not necessarily 𝔟′′–dominant: for n_{k} ≥ m_{1}, at most q_{0} − q_{1} of the integers (ν_{n} + ρ_{n}, sgn(i)ε_{i}) are negative. On the other hand, the branching rule for the modules ^{w_{1}} Ē_{n} implies that if l_{n} is the number of integers i for which (ν_{n} + ρ_{n}, sgn(i)ε_{i}) < 0, then {l_{n}}_{n≥m_{1}} is a non–decreasing bounded sequence. Fix m_{2} ≥ m_{1} such that l_{n} = l_{m_{2}} for n ≥ m_{2}.

As in the case where 𝔤^{s_{0}} is infinite dimensional, there exists a Borel subalgebra 𝔟 of 𝔭 such that the order <𝔟̄ admits q′ largest elements x^{1} >𝔟̄ … >𝔟̄ x^{q′}. Let m′ := n_{k_{0}} ≥ m_{2} be larger than x^{1}, …, x^{q′}. If 𝔤^{q′} is the Lie algebra generated by 𝔤^{α} for α ∈ {±ε_{x^{i}}, ±ε_{x^{i}} ± ε_{x^{j}} | 1 ≤ i ≠ j ≤ q′}, then, for n_{k} ≥ m′, the element w_{n_{k}} that makes ν_{n_{k}} + ρ_{n_{k}} dominant belongs to W^{q′}. We then notice that w_{n_{k}} does not depend on k, and set w_{2} := w_{n_{k}}. Furthermore, by comparing E_{n} with both E_{n_{k′}} and E_{n_{k′′}} for m′ ≤ n_{k′} ≤ n ≤ n_{k′′}, we conclude that E_{n} is regular for n ≥ m′. Finally, μ_{n} = w(λ_{n} + ρ_{n}) − ρ_{n} is a regular 𝔟_{n}–dominant weight for w := w_{2} ∘ w_{1}. This completes the proof. □

Part III. Cohomology

10 Homogeneous spaces and G–sheaves

Let P be a parabolic subgroup of a locally reductive ind–group G. For each n the map ψ_{n} : X_{n} = G_{n}/P_{n} ↪ G_{n+1}/P_{n+1} = X_{n+1} is a closed immersion of smooth proper varieties, and the union is the proper ind–variety

X = G/P := ⋃_{n} G_{n}/P_{n} = ⋃_{n} X_{n}.

The translation action τ : G × X → X is an ind–variety morphism. In other words, it endows X with the structure of a G–ind–variety.

Example 10.1

(1) Let V be a vector space of countably infinite dimension and let k ∈ ℕ. Choose an ordered basis in V and use it to identify V with the natural representation of G = A(∞). Let P be the stabilizer in G of the span of the first k vectors in the basis of V. Then P is a maximal parabolic subgroup of G and the G–ind–variety G/P is identified with the Grassmannian Gr(k, V) introduced in Example 1.2.

(2) Let G = GL(2^{∞}) and P = \underrightarrow{lim} P_{n}, where P_{n} is the parabolic subgroup of GL(2^{n}) containing the diagonal matrices and such that the roots of its Lie subalgebra 𝔭_{n} are

{ε_{i} − ε_{j} | 1 ≤ i < j ≤ 2^{n}} ∪ {ε_{2s+1} − ε_{2s} | 1 ≤ s ≤ 2^{n−2}}.

Then G_{n}/P_{n} is the flag space F l(1, 3, …, 2^{n}−1, ℂ^{2^{n}}) of maximal flags of odd dimensional subspaces in ℂ^{2^{n}}. The ind–variety G/P is the union ⋃_{n} G_{n}/P_{n} where the closed immersions σ_{n} : G_{n}/P_{n} ↪ G_{n+1}/P_{n+1} are described as follows. According to the construction of G, the natural representation ℂ^{2^{n+1}} of G_{n+1} has a canonical G_{n}–module decomposition as ℂ^{2^{n}} ⊕ ℂ^{2^{n}} . The map

σ_{n} : F l(1, 3, …, 2^{n} − 1, ℂ^{2^{n}}) ↪ F l(1, 3, …, 2^{n+1} − 1, ℂ^{2^{n}} ⊕ ℂ^{2^{n}})

sends a flag F_{1} ⊂ F_{3} ⊂ … ⊂ F_{2^{n}−1} in ℂ^{2^{n}} to the flag F_{1} ⊕ {0} ⊂ F_{3} ⊕ {0} ⊂ … ⊂ F_{2^{n}−1} ⊕ {0} ⊂ ℂ^{2^{n}} ⊕ F_{1} ⊂ ℂ^{2^{n}} ⊕ F_{3} ⊂ … ⊂ ℂ^{2^{n}} ⊕ F_{2^{n}−1} in ℂ^{2^{n}} ⊕ ℂ^{2^{n}} .

(3) Let G = P GL^{Ad}(k^{2^{∞}}) and B be any Borel ind–subgroup as described in §5. Then G/B = ⋃_{n} G_{n}/B_{n}, where for each n G_{n}/B_{n} is the space F l(1, 2, …, k^{2^{n}} − 1, ℂ^{k^{2^{n}}}) of maximal flags in ℂ^{k^{2^{n}}}. The G_{n}–equivariant closed immersion

θ_{n} : F l(1, 2, …, k^{2^{n}} − 1, ℂ^{k^{2^{n}}}) ↪ F l(1, 2, …, k^{2^{n+1}} − 1, ℂ^{k^{2^{n+1}}} )

is characterized by the property that the maximal flag in ℂ^{k^{2^{n}}} stabilized by the pre-image of B_{n} in GL(k^{2^{n}}) is sent to the pr_{n}^{−1}(H_{n+1})–invariant

maximal flag in ℂ^{k^{2^{n+1}}} whose stabilizer in GL(k^{2^{n+1}}) is pr_{n+1}^{−1}(B_{n+1}), see §5.

The structure sheaf O_{X} of X is the inverse limit sheaf \underleftarrow{lim} O_{X_{n}} of the structure sheaves O_{X_{n}} of X_{n}. If F is a sheaf of O_{X}–modules, or an O_{X}–module for short, we define F to be a G– homogeneous O_{X}–module (= G–linearized O_{X}–module), or simply a G–sheaf, if for every g ∈ G an O_{X}–module isomorphism φ_{g} is given, φ_{g} : g̃^{∗}(F) → F, so that φ_{g′} ∘ φ_{g′′} = φ_{g′∘g′′} for all g′, g′′ ∈ G. Here g̃ : X → X stands for the composition X ≃ g × X ⟶^{τ_{|g×X}} X. If F is G–sheaf, then its total cohomology H^{·}(X; F) := ⨁_{n≥0} H^{n}(X; F) is a graded G–module, i.e. for each n H^{n}(X, F) is endowed with a G–module structure. The structure map G × H^{n}(X; F) → H^{n}(X; F) is the map (g, f) ↦ φ_{g}^{H^{n}}(f), where φ_{g}^{H^{n}} : H^{n}(X; F) → H^{n}(X; F) is the linear automorphism of H^{n}(X; F) induced by φ_{g}.

If E = \underrightarrow{lim} E_{n} is a rational P –module, there are certain natural G–sheaves on X associated with E. Note first that, for each n, we have a closed G_{n}–equivariant immersion

𝔼_{n} := (G_{n} ×_{P_{n}} E_{n}) ↪ (G_{n+1} ×_{P_{n+1}} E_{n+1}) =: 𝔼_{n+1}

of the homogeneous vector bundles of finite rank associated respectively with E_{n} and E_{n+1}. These immersions determine a G–ind–variety 𝔼 := \underrightarrow{lim} 𝔼_{n}, the G–homogeneous vector bundle over X associated to E. We define O_{X}(E) to be the sheaf of local sections to 𝔼 → X. It is a locally free G–sheaf of O_{X}–modules, and rank O_{X}(E) = dim E. In particular, dim E can be infinite and, when P/U_{P} has no noncommutative finite dimensional factors, we have either dim E = 1 or dim E = ∞.

Another natural G–sheaf associated with E is O_{X}(E^{∗}). As noted earlier, E^{∗} is a pro–rational P –module, and there is a well–defined inverse system

… → O_{X_{n+1}}(E_{n+1}^{∗}) → O_{X_{n}}(E_{n}^{∗}) → … → 0,

O_{X_{n}}(E_{n}^{∗}) being the sheaf of local sections of 𝔼_{n}. The sheaf O_{X}(E^{∗}) is defined now as the inverse limit \underleftarrow{lim} O_{X_{n}}(E_{n}^{∗}) with the O_{X_{n}}(E_{n}^{∗}) viewed as sheaves of O_{X}–modules supported on X_{n}. O_{X}(E^{∗}) is a locally free G–sheaf of O_{X}–modules, and rank O_{X}(E^{∗}) = dim E^{∗}. Note that dim E^{∗} is uncountable whenever dim E = ∞.

The following Proposition is a key tool for calculating the cohomology groups of O_{X}(E^{∗}) and O_{X}(E). In order to formulate it we need a definition. Let

… ⟶^{ν_{n+1}} M_{n} ⟶^{ν_{n}} M_{n−1} ⟶^{ν_{n−1}} … ⟶^{ν_{2}} M_{1} ⟶ 0 (10.2)

be an inverse system of vector spaces. If m > n, ν_{n,m} denotes the map ν_{n+1} ∘ ⋯ ∘ ν_{m} from M_{m} to M_{n}. For each n let M̂_{n} := \underleftarrow{lim}_{m>n}M_{n}/ν_{n,m}(M_{m}), the completion of M_{n} with respect to its filtration by the subspaces ν_{n,m}(M_{m}), m > n. Then the value of the first right derived functor of \underleftarrow{lim} on the system (10.2) is the vector space \underleftarrow{lim}^{(1)}M_{n} := ∏_{n}(M̂_{n}/M_{n}).

Proposition 10.3 Let … ⟶^{ζ_{n+1}} F_{n} ⟶^{ζ_{n}} F_{n−1} ⟶^{ζ_{n−1}} … ⟶^{ζ_{2}} F_{1} → 0 be an inverse system of sheaves of locally free O_{X}–modules.

(i) For every q ≥ 0 there is a canonical exact sequence

(10.4) 0 → \underleftarrow{lim}^{(1)}H^{q−1}(X; F_{n}) → H^{q}(X; \underleftarrow{lim} F_{n}) → \underleftarrow{lim} H^{q}(X; F_{n}) → 0,

where (for q = 0) \underleftarrow{lim}^{(1)}H^{−1}(X; F_{n}) := 0.

(ii) If for some q > 0 the inverse system of vector spaces

… ⟶^{ζ_{n+1}^{q}} H^{q}(X; F_{n}) ⟶^{ζ_{n}^{q}} H^{q}(X; F_{n−1}) ⟶^{ζ_{n−1}^{q}} … ⟶ 0

satisfies the Mittag–Leffler condition, i.e. if for every n the filtration on H^{q}(X; F_{n}) by the subspaces ζ_{n+1,m}^{q}(H^{q}(X; F_{m})) is eventually constant, then there is a canonical isomorphism

H^{q}(X; \underleftarrow{lim} F_{n}) ≅ \underleftarrow{lim} H^{q}(X; F_{n}). (10.5)

(iii) If the F_{n}’s are G_{n}–sheaves and ζ_{n} are morphisms of G_{n−1}–sheaves, then (10.4) is an exact sequence of G–modules. In particular, if the Mittag–Leffler condition is satisfied, (10.5) is a G– module isomorphism.

Proof. The statements (i) and (ii) are adaptations of [11, Theorem 4.5] to our situation, and we will not reproduce the proofs here. See also [10, O_{III}, 13.3.1], which is the original source for Theorem 4.5 in [11].

To prove (iii), note that the canonical surjection

H^{q}(X; \underleftarrow{lim} F_{n}) → \underleftarrow{lim} H^{q}(X; F_{n}) (10.6)

is induced (by universality of \underleftarrow{lim} H^{q}(X; F_{n})) by the maps H^{q}(X; \underleftarrow{lim} F_{n}) ⟶ H^{q}(X; F_{n}), which in turn are induced by the canonical maps \underleftarrow{lim} F_{n} ⟶ F_{n}. As the latter are morphisms of G_{n}–sheaves, the former is a compatible system of maps of G_{n}–modules, and this implies (iii). (In this way the space \underleftarrow{lim}^{(1)}H^{q−1}(X; F_{n}) is equipped with the G–module structure defined by the surjection (10.6) by means of the exact sequence (10.4).) □

If E is a rational P –module then Proposition 10.3 applies to both sheaves O_{X}(E^{∗}) and O_{X}(E). Consider O_{X}(E^{∗}) first. We have O_{X}(E^{∗}) = \underleftarrow{lim} O_{X_{n}}(E_{n}^{∗}), and here the Mittag–Leffler condition is automatic because dim H^{q}(X; O_{X_{n}}(E_{n}^{∗})) = dim H^{q}(X_{n}; O_{X_{n}}(E_{n}^{∗})) < ∞ for all q and all n. To apply Proposition 10.3 to O_{X}(E) note that O_{X}(E) = \underleftarrow{lim} O_{X_{n}}(E), where O_{X_{n}}(E) is the sheaf of local sections of the G_{n}–homogeneous vector bundle G_{n} ×_{P_{n}} E → X_{n} associated to E. In this case the cohomology groups H^{q}(X_{n}; O_{X_{n}}(E)) are in general infinite dimensional, and therefore the Mittag–Leffler condition is not automatic. In fact we will see in Section 16 below that the Mittag–Leffler condition fails for some quite straightforward irreducible P –modules E.

11 Statement of main results on cohomology of O_{X}(E^{∗})

In this and the following Sections, G is a fixed locally reductive ind–group, P is a fixed parabolic subgroup of G, X = G/P, and E is an irreducible rational P -module. We assume further that E is expressed as \underrightarrow{lim} E_{n} for a minimal direct system of P_{n}–module injections E_{n} ↪ E_{n+1}. Throughout Sections 11 through 15 we write ζ_{n} for the canonical restriction map O_{X_{n}}(E_{n}^{∗}) → O_{X_{n−1}}(E_{n−1}^{∗}).

These are our main results on the cohomology of the sheaf O_{X}(E^{∗}).

Theorem 11.1 (i) For the group G and the P –module E specified above,

(a) H^{q}(X; O_{X}(E^{∗})) is a pro–rational G–module, and more precisely, for every integer q ≥ 0, there is a canonical G–module isomorphism H^{q}(X; O_{X}(E^{∗})) = \underleftarrow{lim} H^{q}(X_{n}; O_{X_{n}}(E_{n}^{∗}));

(b) H^{0}(X; O_{X}(E^{∗})) ≠ 0 if and only if E is dominant. In this case there is a canonical G–module isomorphism H^{0}(X; O_{X}(E^{∗})) = V (E)^{∗}.

(ii) Under the additional assumption that E is locally irreducible, H^{q_{0}}(X; O_{X}(E^{∗})) ≠ 0 for at most one integer q_{0} ≥ 0 . If q_{0} is such an integer then H^{q_{0}}(X; O_{X}(E^{∗})) = V^{∗} for some locally irreducible rational G–module V.

(iii) Suppose that G is root–reductive and that E is irreducible but not necessarily locally irreducible. Then H^{q_{0}}(X; O_{X}(E^{∗})) ≠ 0 for at most one q_{0}, and H^{q_{0}}(X; O_{X}(E^{∗})) ≠ 0 if and only if H^{q_{0}}(X_{n}; O_{X_{n}}(E_{n}^{∗})) ≠ 0 for large enough n. Furthermore, whenever H^{q_{0}}(X; O_{X}(E^{∗})) ≠ 0, there is a Borel subgroup B of P and an element w of W of length q_{0} with respect to 𝔟 such that H^{q_{0}}(O_{G/P} (E^{∗})) ≅ V (^{w}E)^{∗}, where ^{w}E is an irreducible ^{w}P –module unique up to isomorphism.

12 Proof and discussion of (i) and (ii)

Proof of (i). Statement (a) is a direct corollary of Proposition 10.3(ii) and (iii). To prove (b) note first that, precisely as in the classical case, Frobenius Reciprocity holds for H^{0}(X; O_{X}(E^{∗})): we have a canonical isomorphism

Hom_{G}(W, H^{0}(X; O_{X}(E^{∗}))) = Hom_{P} (W, E^{∗}) (12.1)

for any G–module W. Indeed, (12.1) follows immediately from (10.5) for F = O_{X}(E^{∗}) and q = 0, from the universality property of the inverse limit, and from Frobenius Reciprocity for H^{0}(X_{n}; O_{X_{n}}(E_{n}^{∗})) for all n. Now since the right hand of (12.1) is nonzero for W = V (E)^{∗}, H^{0}(X; O_{X}(E^{∗})) is nonzero whenever E is dominant. Therefore, by (10.5), H^{0}(X_{n}; O_{X_{n}}(E_{n}^{∗})) ≠ 0 for large enough n, and by the classical Borel–Weil Theorem we have H^{0}(X_{n}; O_{X_{n}}(E_{n}^{∗}))^{∗} = V (E_{n}). Thus, for a dominant E, \underrightarrow{lim} H^{0}(X_{n}; O_{X_{n}}(E_{n}^{∗}))^{∗} = \underrightarrow{lim} V (E_{n}). But then, as in the first part of the proof of Proposition 7.4, there is a G–isomorphism V (E) ≅ \underrightarrow{lim} V (E_{n}), and consequently H^{0}(X; O_{X}(E^{∗})) = V (E)^{∗}.

Conversely, let H^{0}(X; O_{X}(E^{∗})) = \underleftarrow{lim} H^{0}(X_{n}; O_{X_{n}}(E_{n}^{∗})) ≠ 0. Then \underrightarrow{lim} H^{0}(X_{n}; O_{X_{n}}(E_{n}^{∗}))^{∗} is a well–defined G–module which admits a P –module injection E ↪ \underrightarrow{lim} H^{0}(X_{n}; O_{X_{n}}(E_{n}^{∗}))^{∗} (whose dual corresponds via (12.1) to the identity map H^{0}(X; O_{X}(E^{∗})) → H^{0}(X; O_{X}(E^{∗}))) and is moreover generated over G by the image of E. Again, as in the proof of Proposition 7.4 this implies that \underrightarrow{lim} H^{0}(X_{n}; O_{X_{n}}(E_{n}^{∗}))^{∗} is canonically isomorphic to V (E) as a G–module. In this way, whenever H^{0}(X; O_{X}(E^{∗})) ≠ 0, E is dominant and H^{0}(X; O_{X}(E^{∗})) = V (E)^{∗}. □

The above argument shows in particular that V (E_{n}) is finite dimensional for each n whenever E is dominant. Therefore the proof Proposition 7.4 is now complete. Furthermore, Theorem 11.1(i)(b) and Proposition 8.5 have the following essential corollary.

Corollary 12.2 Let G, P and E be as in Theorem 11.1(i). Suppose that E is a weight P –module. If H^{0}(X; O_{X}(E^{∗})) ≠ 0 then E is a finite (and thus locally irreducible) P –module, and H^{0}(X; O_{X}(E^{∗})) is isomorphic to the dual of the rational locally irreducible finite G–module V (E). □

Of course, the requirement that E be a weight module is crucial in Corollary 12.2. For if G, P, E and V are as in Example 8.6, then H^{0}(X; O_{X}(E^{∗})) = V^{∗}, and V is an irreducible but not locally irreducible G–module which is not a weight module.

Proof of (ii). Since O_{X_{n}}(E_{n}^{∗}) is an irreducible G_{n}–homogeneous O_{X_{n}}-module, by the classical Bott–Borel–Weil Theorem H^{q_{0}}(X_{n}; O_{X_{n}}(E_{n}^{∗})) ≠ 0 for at most one integer q_{0}. Therefore \underleftarrow{lim} H^{q_{0}}(X_{n}; O_{X_{n}}(E_{n}^{∗})) ≠ 0 for at most one q_{0}, and this, combined with (i)(a), implies that H^{q_{0}}(X; O_{X}(E^{∗})) ≠ 0 for at most one q_{0}. If now H^{q_{0}}(X; O_{X}(E^{∗})) ≠ 0, V is defined simply as the limit of the direct system of irreducible finite dimensional G_{n}–modules

0 → H^{q_{0}}(X_{0}; O_{X_{0}}(E_{0}^{∗}))^{∗} → … → H^{q_{0}}(X_{n}; O_{X_{n}}(E_{n}^{∗}))^{∗} → …, (12.3) and is therefore a locally irreducible rational G–module. □

The problem left open by (ii) is that of an explicit description of the G–module V in terms of the P –module E. In the setting where G is root–reductive, (iii) is an essential step toward solving that problem. For general ind–groups the problem is open even in the case when P = B is a Borel group and E is a one dimensional B–module E; see Example 14.5 below.

13 Proof and discussion of (iii)

Proof of (iii) for locally irreducible E. The fact that H^{q_{0}}(X; O_{X}(E^{∗})) ≠ 0 for at most one q_{0} was established in the proof of (ii). Assume, more generally, that H^{q_{0}}(X_{n}; O_{X_{n}}(E_{n}^{∗})) ≠ 0 for some q_{0} and for large enough n. (If no such q_{0} exists, there is nothing to prove.) Then, for large enough n, all E_{n} are regular and ℓ(E_{n}) = q_{0}. Furthermore, Proposition 9.7 yields a Borel subalgebra 𝔟 of 𝔭 and an element w ∈ W of length q_{0} with respect to 𝔟. Let λ_{n} is the 𝔟_{n}–highest weight of E_{n} and μ_{n} := w(λ_{n} + ρ_{n}) − ρ_{n}. The Bott–Borel–Weil Theorem gives H^{q_{0}}(X_{n}; O_{X_{n}}(E_{n}^{∗})) ≅ V (ℂ_{μ_{n}})^{∗}, where ℂ_{μ_{n}} denotes the one dimensional 𝔟_{n}–module of weight μ_{n}. As in the proof of (ii), we define V as the limit of the direct system dual to the system of projections ζ_{n+1}^{q_{0}} : H^{q_{0}}(X_{n+1}; O_{X_{n+1}}(E_{n+1}^{∗})) → H^{q_{0}}(X_{n}; O_{X_{n}}(E_{n}^{∗})).

We claim that V ≠ 0, i.e. that ζ_{n+1}^{q_{0}} ≠ 0 for large enough n. We will prove this by induction on q_{0}. If q_{0} = 0, E is dominant by Proposition 7.4, and thus the statement follows from (i)(b). Let q_{0} > 0 and let w = σ_{α_{1}} ∘ … ∘ σ_{α_{q_{0}}} be a minimal decomposition of w as a product of 𝔟_{n+1}–simple reflections. Define P_{n+1}^{α_{q_{0}}} as the minimal proper parabolic subgroup of G_{n+1} such that B_{n+1} ⊂ P_{n+1}^{α_{q_{0}}} and ±α_{q_{0}} ∈ Δ_{𝔭_{n+1}^{αq_{0}}}. Then P_{n}^{α_{q_{0}}} := P_{n+1}^{α_{q_{0}}} ∩ G_{n} is a parabolic subgroup of G_{n}, and there is a commutative diagram

G_{n}/B_{n} ↪ G_{n+1}/B_{n+1}
p_{n}^{α_{q_{0}}} ↓  ↓ p_{n+1}^{α_{q_{0}}}
G_{n}/P_{n}^{α_{q_{0}}} ↪ G_{n+1}/P_{n+1}^{α_{q_{0}}}.

Notice first that O_{X_{n+1}}(E_{n+1}^{∗}) = (π_{n+1})_{∗}O_{G_{n+1}/B_{n+1}} (ℂ_{−λ_{n+1}}) and O_{X_{n}}(E_{n}^{∗}) = (π_{n})_{∗}O_{G_{n}/B_{n}}(ℂ_{−λ_{n}}), where π_{n+1} : G_{n+1}/B_{n+1} → X_{n+1} and π_{n} : G_{n}/B_{n} → X_{n} are the canonical projections. Demazure’s proof [6] of the Bott–Borel–Weil Theorem implies

\begin{aligned}H^{q_{0}}(X_{n+1}; O_{X_{n+1}}(E_{n+1}^{∗})) & = H^{q_{0}}(G_{n+1}/B_{n+1}; π_{n+1}^{∗}O_{X_{n+1}}(E_{n+1}^{∗})) \\ & = H^{q_{0}−1}(G_{n+1}/P_{n+1}^{α_{q_{0}}}; R^{1}(p_{n+1}^{α_{q_{0}}})_{∗}(π_{n+1}^{∗}O_{X_{n+1}}(E_{n+1}^{∗}))) \end{aligned}

and

H^{q_{0}}(X_{n}; O_{X_{n}}(E_{n}^{∗})) = H^{q_{0}}(G_{n}/B_{n}; π_{n}^{∗}O_{X_{n}}(E_{n}^{∗})) = H^{q_{0}−1}(G_{n}/P_{n}^{α_{q_{0}}} ; R^{1}(p_{n}^{α_{q_{0}}} )_{∗}(π_{n}^{∗}O_{X_{n}}(E_{n}^{∗}))).

Thus, for the induction step it suffices to check that ζ_{n+1} : O_{X_{n+1}}(E_{n+1}^{∗}) → O_{X_{n}}(E_{n}^{∗}) induces a nonzero G_{n}–morphism

ζ̃_{n+1}^{1} : R^{1}(p_{n+1}^{α_{q_{0}}})_{∗}(π_{n+1}^{∗}O_{X_{n+1}}(E_{n+1}^{∗})) → R^{1}(p_{n}^{α_{q_{0}}} )_{∗}(π_{n}^{∗}O_{X_{n}}(E_{n}^{∗})).

Indeed, the observation that ζ_{n+1}^{q_{0}} = (ζ̃_{n+1}^{1})^{q_{0}−1} will then allow us to conclude that ζ_{n+1}^{q_{0}} ≠ 0 as (ζ̃_{n+1}^{1})^{q_{0}−1} is nonzero by the induction assumption. But the fact that ζ̃_{n+1}^{1} ≠ 0 follows immediately from the long exact sequence

\begin{aligned}0 & → (p_{n+1}^{α_{q_{0}}})_{∗} ker ζ_{n+1} → (p_{n+1}^{α_{q_{0}}})_{∗}O_{X_{n+1}}(E_{n+1}^{∗}) → (p_{n}^{α_{q_{0}}} )_{∗}O_{X_{n}}(E_{n}^{∗}) \\ & → R^{1}(p_{n+1}^{α_{q_{0}}})_{∗} ker ζ_{n+1} → R^{1}(p_{n+1}^{α_{q_{0}}})_{∗}O_{X_{n+1}}(E_{n+1}^{∗}) → R^{1}(p_{n}^{α_{q_{0}}} )_{∗}O_{X_{n}}(E_{n}^{∗}) → 0. \end{aligned}

Therefore finally ζ_{n+1}^{q_{0}} ≠ 0, and thus also V ≠ 0.

It remains to show that V admits an irreducible ^{w}𝔭–submodule ^{w}E. Then V ≅ V (^{w}E) by the remark after Proposition 7.3, and the uniqueness of ^{w}E follows from the fact that ^{w}E is the subspace of 𝔲_{^{w}𝔭}–invariants in V (^{w}E). To define ^{w}E set ^{w}E := \underrightarrow{lim} U(^{w}𝔭_{n})·V (ℂ_{μ_{n}})^{μ_{n}}. As V (ℂ_{μ_{n}})^{μ_{n}} is the 𝔟_{n}–highest weight space of V (ℂ_{μ_{n}}), U(^{w}𝔭_{n}) · V (ℂ_{μ_{n}})^{μ_{n}} is an irreducible ^{w}𝔭_{n}–submodule of V (ℂ_{μ_{n}}). To show that ^{w}E is well–defined, we need to check only that (ζ_{n+1}^{q_{0}})^{∗}(U(^{w}𝔭_{n})·V (ℂ_{μ_{n}})^{μ_{n}}) ⊂ U(^{w}𝔭_{n+1}) · V (ℂ_{μ_{n+1}})^{μ_{n+1}}, i.e. that (ζ_{n+1}^{q_{0}})^{∗}(V (ℂ_{μ_{n}})^{μ_{n}}) ⊂ U(^{w}𝔭_{n+1}) · V (ℂ_{μ_{n+1}})^{μ_{n+1}}. Fix a large enough n and let κ_{1}, …, κ_{s} be all weights of V (ℂ_{μ_{n+1}})^{μ_{n+1}} onto which (ζ_{n+1}^{q_{0}})^{∗}(V (ℂ_{μ_{n}})^{μ_{n}}) projects non–trivially. (As V is not necessarily a weight module, s may be greater than one.) Then κ_{i|𝔥_{n}} = w(λ_{n} + ρ_{n}) − ρ_{n} for all i. But since μ_{n+1} = w(λ_{n+1} + ρ_{n+1}) − ρ_{n+1}, an immediate verification shows that all κ_{i} are of the form

μ_{n+1} + (ℤ_{+}–linear combination of 𝔟_{n+1}–negative roots of ^{w}𝔭_{n+1}).

Therefore, indeed (ζ_{n+1}^{q_{0}})^{∗}(V (ℂ_{μ_{n}})^{μ_{n}}) ⊂ U(^{w}𝔭_{n+1}) · V (ℂ_{μ_{n+1}})^{μ_{n+1}}, and the proof is complete. □

Note that (iii) has the following combinatorial consequence. The ^{w}𝔭_{n}–modules ^{w}E_{n} satisfy wE_{n} ≺ ^{w}E_{m} for large enough m > n. This fact can be checked directly in the spirit of Proposition 9.7, but the construction of the ^{w}P –module ^{w}E gives it as a trivial corollary. Furthermore, the modules ^{w}E_{n} are defined simply by the sequence of modules {E_{n}}, however in general the injections wE_{n} → ^{w}E_{n+1} (and hence the module ^{w}E) depend on the 𝔭–module E as a whole. In the special case when every infinite dimensional 𝔩^{s} in the decomposition (4.4) of every simple 𝔤^{t}, cf. Proposition 4.1, is isomorphic to 𝔤𝔩(∞), both E and ^{w}E are determined by the sequence {E_{n}} only, and are automatically weight modules. The reason is that the branching of an irreducible finite dimensional 𝔤𝔩(n + 1)–module into irreducible 𝔤𝔩(n)–modules is multiplicity–free.

The next example shows that ^{w}E may not admit an irreducible submodule of any parabolic subgroup which is properly contained in ^{w}P, in particular, of P. Furthermore, if ^{w}P = G then wE = V. As a consequence, even when E is a finite weight P –module, V need not be a finite G–module as Proposition 8.5 no longer applies. See Example 13.1(2) for a non–finite weight G– module V such that V^{∗} ≅ H^{2}(X; O_{X}(E^{∗})). Note also that in general E and ^{w}E are not determined by the weight sequence {λ_{n}}.

Example 13.1

(1) Let G = GL(∞), Let P be as in Example 4.3, and let B ⊂ P be the Borel subgroup of all upper triangular matrices in G. So {ε_{i} − ε_{j} | 1 ≤ i < j} is the set of roots of 𝔟. Define the P –module E as the direct limit of irreducible P_{n}–modules with B_{n}–highest weights λ_{n} := −ε_{1}+nε_{2} . The branching rule ensures that E is well–defined as the respective inclusions are unique (up to a multiplicative constant). Then H^{1}(X; O_{X}(E^{∗})) ≅ V^{∗}, where V is the direct limit of the irreducible G_{n}–modules with B_{n}–highest weights μ_{n} = (n−1)ε_{1}. The assertion of Theorem 11.1(iii) is satisfied with w = σ_{α} for α = ε_{1} − ε_{2} . Here ^{w}P = G; see Example 4.3 above. It is not difficult to check that V is a finite weight G–module that has no P –submodule that is irreducible for any proper parabolic subgroup P ⊂ G.

(2) Now let 𝔭 ⊂ 𝔤𝔩(∞) be the parabolic subalgebra with roots

{ε_{i} − ε_{j} | 3 ≤ i, 3 ≤ j, i ≠ j} ∪ {ε_{1} − ε_{i}, ε_{i} − ε_{2} | 3 ≤ i} ∪ {ε_{1} − ε_{2}},

let 𝔟 ⊂ 𝔭 be the Borel subalgebra with roots

\begin{aligned}\{ε_{i} − ε_{j} | 5 ≤ i, 5 ≤ j, i < j\} & ∪ \{ε_{1} − ε_{i}, ε_{3} − ε_{i}, ε_{i} − ε_{2}, ε_{i} − ε_{4} | 5 ≤ i\} \\ & ∪ \{ε_{1} − ε_{2}, ε_{1} − ε_{3}, ε_{1} − ε_{4}, ε_{3} − ε_{2}, ε_{3} − ε_{4}, ε_{4} − ε_{2}\}, \end{aligned}

and let P and B be the corresponding parabolic and Borel subgroups of G = GL(∞). Define the P –module E as the direct limit of the irreducible P_{n}–modules with B_{n}–highest weights λ_{n} := −ε_{1} + ε_{2} + nε_{3} − nε_{4}. Then H^{2}(X; O_{X}(E^{∗})) ≅ V^{∗}, where V is the direct limit of the irreducible G_{n}–modules with B_{n}–highest weights μ_{n} = (n − 1)ε_{1} + (1 − n)ε_{2}. Here the assertion of Theorem 11.1(iii) is satisfied with w = σ_{α} ∘ σ_{β} for α = ε_{1} − ε_{3} and β = ε_{2} − ε_{4}, and again ^{w}P = G. A direct verification shows that neither is E a finite P –module nor is V a finite G–module. Therefore V^{∗} cannot appear as the zero^{th} cohomology of O_{X}(E′^{∗}) for any rational P –module E′.

Proof of (iii) for general E. Note first that, as 𝔲_{𝔭} · E = 0 (Proposition 7.1), E is an irreducible 𝔭^{red}–module and is in particular completely reducible as 𝔭_{n}^{red}–module for all n. Let E_{n} = ⨁_{i=1}^{i_{n}} E_{n}^{i} be the decomposition of E_{n} into isotypic 𝔭_{n}^{red}–components. Fix furthermore a decomposition of each E_{n}^{i} as ⨁_{k} E_{n}^{i,k}, E_{n}^{i,k} being simple. For any sequence of simple 𝔭_{n}^{red}–components E_{n}^{i_{n},k_{n}}, let f_{m,n} : E_{n}^{i_{n},k_{n}} → E_{m}^{i_{m},k_{m}} denote the composition of the injection E_{n}^{i_{n},k_{n}} ↪ E_{m} followed by the projection E_{m} → E_{m}^{i_{m},k_{m}} determined by the fixed decomposition of E_{m}^{i_{m}}.

For any q, H^{q}(X_{n}; O_{X_{n}}(E_{n}^{∗})) = ⨁_{i} H^{q}(X_{n}; O_{X_{n}}((E_{n}^{i})^{∗})) = ⨁_{i,k} H^{q}(X_{n}; O_{X_{n}}((E_{n}^{i,k})^{∗})). If for no q H^{q}(X_{n}; O_{X_{n}}((E_{n})^{∗})) ≠ 0 for all large enough n, H^{q}(X; O_{X}((E)^{∗})) = 0 for all q, and there is nothing to prove. Assume therefore that there is q_{0} with H^{q_{0}}(X_{n}; O_{X_{n}}((E_{n})^{∗})) ≠ 0 for all large enough n. By the Bott–Borel–Weil Theorem this is the same as assuming that for large enough n some isotypic component E_{n}^{i_{0}} is regular and ℓ(E_{n}^{i_{0}}) = q_{0}.

We claim that then, for large enough n, every isotypic component E_{n}^{i} is regular and ℓ(E_{n}^{i}) = q_{0}. Assume not. We will construct a locally irreducible locally finite 𝔭-module E′ whose existence contradicts Proposition 9.7(ii). Let E_{n_{1}}^{i_{1},k_{1}} be regular of length q_{0}. Find E_{n_{2}}^{i_{2},k_{2}} with n_{2} > n_{1} which is either not regular or not of length q_{0} and such that the corresponding f_{n_{2},n_{1}} is nonzero. E_{n_{2}}^{i_{2},k_{2}} exists because of our assumption and the minimality of the system of injections E_{n} ↪ E_{n+1}. Let furthermore, E_{n_{3}}^{i_{3},k_{3}} be regular of length q_{0} with n_{3} > n_{2} and f_{n_{3},n_{2}} ≠ 0. E_{n_{3}}^{i_{3},k_{3}} exists also due to the minimality of the system of injections E_{n} ↪ E_{n+1}. Continuing in this way, we construct E′ := \underrightarrow{lim}_{j}E_{n_{j}}^{i_{j},k_{j}} ≠ 0, where E_{n_{j}}^{i_{j},k_{j}} is regular of length q_{0} for odd j and is not such for even j. The existence of E′ is contradicts Proposition 9.7(ii), therefore, for large enough n, each isotypic component E_{n}^{i} is regular and ℓ(E_{n}^{i}) = q_{0}. In particular, H^{q}(X; O_{X}(E^{∗})) = 0 for q ≠ q_{0}, i.e. H^{q_{0}}(X; O_{X}(E^{∗})) is the only possibly non–vanishing cohomology group of O(E^{∗}).

In a similar way we construct a Borel subgroup B ⊂ P and an element w ∈ W of length q_{0} with respect to 𝔟. Indeed, for a sequence of simple components E_{n}^{i_{n},k_{n}} such that \underrightarrow{lim} E_{n}^{i_{n},k_{n}} ≠ 0, Proposition 9.7(ii) yields a Borel subgroup B of P and an element w ∈ W of length q_{0} with respect to 𝔟. We claim that any sequence of simple components E_{m}^{i_{m},k_{m}} with \underrightarrow{lim} E_{m}^{i_{m},k_{m}} ≠ 0 yields the same Borel subgroup and the same element w ∈ W, as otherwise, a construction similar to the above would give a module whose existence is contradictory.

We are ready to complete the argument. As in the proof of (ii), define the rational G–module V as the limit of the direct system (12.3). Note that, as all isotypic components E_{n}^{i} are regular for large enough n, and, as the length of w with respect to 𝔟 is q_{0}, the decomposition H^{q_{0}}(X_{n}; O_{X_{n}}(E_{n}^{∗}))^{∗} = ⨁_{i} H^{q_{0}}(X_{n}; O_{X_{n}}((E_{n}^{i})^{∗}))^{∗} is a decomposition into 𝔤_{n}–isotypic components. Furthermore, H^{q_{0}}(X_{n}; O_{X_{n}}((E_{n}^{i,k})^{∗}))^{∗} are the simple 𝔤_{n}–components of H^{q_{0}}(X_{n}; O_{X_{n}}((E_{n}^{i})^{∗}))^{∗}. The crucial observation is that our argument for (iii) in the locally irreducible case implies that the multiplicity diagram D_{V} of V, defined by the decompositions ⨁_{i} H^{q_{0}}(X_{n}; O_{X_{n}}((E_{n}^{i})^{∗}))^{∗} = ⨁_{i,k} H^{q_{0}}(X_{n}; O_{X_{n}}((E_{n}^{i,k})^{∗}))^{∗}, is identical to the multiplicity diagram of E considered as a 𝔭^{red}– module. Therefore V ≠ 0. Moreover, by Proposition 8.3, D_{V} admits no stably proper invariant subdiagram. Consequently, the multiplicity subdiagram of any nonzero 𝔤–submodule V′ ⊂ V contains all spaces V_{n}^{i} for large enough n, i.e. V′ = V. This proves the irreducibility of V. It remains to show that V = V (^{w}E) for a unique irreducible ^{w}P –submodule ^{w}E of V. If w ∈ W_{n_{0}} for some n_{0}, the irreducible ^{W} P_{n}–module ^{w}E_{n}^{i} is well–defined for any E_{n}^{i} with n > n_{0}. Furthermore, as we showed when proving (iii) in the locally irreducible case, an injection E_{n}^{i} → E_{n+1}^{j} induces an injection ^{w}E_{n}^{i} → ^{w}E_{n+1}^{j}. Hence, for n ≥ n_{0}, there is a well defined G_{n}–module injection of wE_{n} := ⨁_{i}^{w}(E_{n}^{i}) into ^{w}E_{n+1} := ⨁_{i}^{w}(E_{n+1}^{i}), i.e. ^{w}E := \underrightarrow{lim}^{w}E_{n} is a well–defined ^{w}P –submodule of V. The irreducibility of V yields the irreducibility of ^{w}E, the uniqueness of ^{w}E, and the existence of a canonical isomorphism V (^{w}E) = V. □

14 The situation when E has a highest weight

The case when the P –module E has a highest weight with respect to some Borel subgroup B ⊂ P is a close analog of the classical Bott–Borel–Weil Theorem. We show below that, when E has a highest weight, statement (iii) of Theorem 11.1 can reformulated in very familiar terms without reference to the parabolic subgroup ^{w}P. Of course this applies to the case when E is finite dimensional, in particular to the case where P = B and dim E = 1.

Proposition 14.1 Suppose that G is root–reductive and E has highest weight λ with respect to a Borel subgroup B ⊂ P. Let Δ^{+} denote the roots of 𝔟. Then H^{q_{0}}(X; O_{X}(E^{∗})) ≠ 0 if and only if there exists w ∈ W of length q_{0} with respect to 𝔟, such that μ := w(λ) − (∑_{α∈Δ^{+},w(α)∉Δ^{+}} α) is a B–dominant integral weight. In this case

H^{q_{0}}(X; O_{X}(E^{∗})) = V (ℂ_{μ})^{∗}, H^{q}(X; O_{X}(E^{∗})) = 0 for q ≠ q_{0}, (14.2)

where ℂ_{μ} is the one dimensional B–module of weight μ.

Proof. Under the hypotheses of the Proposition, there is a canonical G–isomorphism

H^{·}(X; O_{X}(E^{∗})) ≅ H^{·}(G/B; O_{G/B}(ℂ_{−λ})). (14.3)

Indeed, the classical Bott–Borel–Weil Theorem yields a compatible family of G_{n}–isomorphisms

H^{·}(X_{n}; O_{X_{n}}(E_{n}^{∗})) ≅ H^{·}(G_{n}/B_{n}; O_{G_{n}/B_{n}}(ℂ_{−λ_{n}})) (14.4)

where λ_{n} := λ|_{𝔥_{n}}, and (14.3) follows from Theorem 11.1(i)(a) via (14.4).

Suppose that there exists w ∈ W of length q_{0} as above. Then μ_{n} := μ|_{𝔥_{n}} is a B_{n}–dominant weight for each n. The Bott–Borel–Weil Theorem, together with Proposition 7.4 and Theorem 11.1, gives

H^{q_{0}}(G/B; O_{G/B}(ℂ_{−λ})) = \underleftarrow{lim} H^{q_{0}}(G_{n}/B_{n}; O_{G_{n}/B_{n}}(ℂ_{−λ_{n}})) = \underleftarrow{lim} V (ℂ_{μ_{n}})^{∗} = V (ℂ_{μ})^{∗},

and

H^{q}(G/B; O_{G/B}(ℂ_{−λ})) = 0 for q ≠ q_{0}.

This combines with (14.3) to give (14.2).

Conversely, suppose H^{q_{0}}(G/B; O_{G/B}(ℂ_{−λ})) = \underleftarrow{lim} H^{q_{0}}(G_{n}/B_{n}; O_{G_{n}/B_{n}}(ℂ_{−λ_{n}})) ≠ 0. Then, by the Bott–Borel–Weil Theorem, the 𝔟_{n}–module ℂ_{λ_{n}} has length q_{0} for n sufficiently large. Apply Proposition 9.7 to E = ℂ_{λ} with 𝔭 = 𝔟 for the existence of w ∈ W as required. Now (14.2) holds by the same argument as above. □

The following examples illustrate the problem of describing the G–module structure on the only non–vanishing cohomology H^{q_{0}}(X; O_{X}(E^{∗})) for more general ind–groups G in the special case when P = B and E is one dimensional.

Example 14.5

(1) This example is similar to an example in [21]. Let G = GL(2^{∞}) and let P = B be the Borel subgroup of upper triangular matrices. Given an integer q_{0} ≥ 0, it is an interesting combinatorial problem to describe all systems of integral weights {λ_{n}} (λ_{n} being a weight of 𝔤_{n} = 𝔤𝔩(2^{n})) such that λ_{n} projects onto λ_{n−1}, and H^{q_{0}}(X_{n}; O_{X_{n}}(ℂ_{−λ_{n}})) ≠ 0 for large enough n. Another natural question is whether, given such a weight system, it always yields a nonzero cohomology H^{q_{0}}(X; O_{X}(E^{∗})) for E := \underrightarrow{lim} ℂ_{λ_{n}}, and whether H^{q_{0}}(X; O_{X}(E^{∗})) is always the dual of a 𝔟–highest weight module. Here is a straightforward way to produce weight systems {λ_{n}} as above for any q_{0} ≥ 0. Fix n_{0} ≥ 1 and an integral weight λ_{n_{0}} =: ∑_{1≤i≤2^{n_{0}}} λ^{i}ε_{i,n_{0}} such that H^{q_{0}}(X_{n_{0}}; O_{X_{n_{0}}} (ℂ_{−λ_{n_{0}}} )) ≠ 0. Let c be a non– positive integer, less or equal to min{λ^{1}, …, λ^{2^{n_{0}}} }. For n > n_{0}, define the weight λ_{n} recursively by

λ_{n} := λ_{n−1} − c ∑_{1≤i≤2^{n−1}} ε_{i,n} + c ∑_{2^{n−1}+1≤j≤2^{n}} ε_{j,n}.

A trivial verification shows that \{λ_{n}\} is a system of integral weights as desired. Set E := \underrightarrow{lim} ℂ_{λ_{n}}. An easy computation based on the Bott–Borel–Weil Theorem implies that H^{q}(X; O_{X}(E^{∗})) = 0 for q ≠ q_{0}, and

H^{q_{0}}(X; O_{X}(E^{∗})) = V (Ẽ)^{∗},

where the B–module Ẽ equals \underrightarrow{lim}_{n≥n_{0}}ℂ_{w(λ_{n}+ρ_{n})−ρ_{n}} (and w ∈ S_{2^{n_{0}}} permutes only the first 2^{n_{0}} coordinates of λ_{n} + ρ_{n}).

(2) Here is another interesting situation in which there are no nonzero higher cohomology groups.

Let G = P GL^{Ad}(k^{2^{∞}}), let P = B be any Borel subgroup as described in §5, and let L be an invertible O_{X}–module. We claim that in this situation all cohomology groups of L equal zero unless L ≅ O_{X} .

The inverse images of L on X_{n} determine a sequence of integral 𝔤𝔩(k^{2^{n}})–weights λ_{n}, and L ≅ O_{X} if and only if λ_{n} = 0 for all n. Assuming that the λ_{n} are not all zero, and assuming that H^{q_{0}}(X; L) ≠ 0 for some q_{0} ≥ 0, we show (following an idea of [1]) that, for n sufficiently large, λ_{n}(α_{n}) > λ_{n+1}(α_{n+1}), where α_{n} denotes the highest root of B_{n}. This computation is based on the explicit form of the immersions θ_{n} : X_{n} ↪ X_{n+1} ; see Example 10.1(2) above. The details are left to the reader. In this way we obtain an infinite strictly decreasing sequence of positive integers, i.e. a contradiction. Therefore λ_{n} = 0 for all n, so L ≅ O_{X} , H^{0}(X; L) = ℂ, and H^{q}(X; L) = 0 for q > 0.

There are uncountably many non–isomorphic G–homogeneous invertible sheaves of the form O_{X}(E^{∗}). For any such sheaf, the above result implies H^{q}(X; O_{X}(E^{∗})) = 0 for all q > 0, and that H^{0}(X; O_{X}(E^{∗})) ≠ 0 only when E is the trivial B–module. This latter fact is closely related to the discovery of Yu. Bahturin and G. Benkart, [1], that 𝔤 = 𝔭𝔤𝔩^{Ad}(k^{2^{∞}}) does not admit locally finite highest weight modules except the trivial one. Indeed, if H^{0}(X; O_{X}(E^{∗})) were nonzero for some non–trivial one dimensional B–module E, we would have had H^{0}(X; O_{X}(E^{∗})) = V^{∗} for some highest weight module V, V ≅ ℂ̸. Therefore the result of Bahturin and Benkart implies the part of our result concerning H^{0}(X; O_{X}(E^{∗})).

15 Projectivity of G/P

We are now ready to address the question of whether the ind–variety X = G/P is projective. In general, X is not projective as one can see from Example 14.5(2). Moreover, for G = P GL^{Ad}(k^{2^{∞}}) and for any proper parabolic subgroup P that contains a Borel subgroup as in §5, X is never projective. For the existence of a very ample invertible sheaf L_{X} on X would imply the existence of an invertible sheaf L ≅ O̸_{G/B} on G/B with H^{0}(G/B; L) ≠ 0.

On the other hand, the ind–variety X = G/P from Example 10.1(2) (here G = GL(2^{∞})) is projective. A straightforward verification shows that, in this case, for any fixed n, every very ample invertible O_{X_{n}}–module admits an extension to an invertible O_{X}–module whose restrictions to X_{m} are very ample for all m > n.

For a root–reductive ind–group G we will give now an explicit projectivity criterion for X =

G/P. Assuming that G is root–reductive, we have X ≅ ×^{f}_{t∈T} G^{t}/P^{t}, and the reader can check that X is projective if and only if G^{t}/P^{t} is projective for each t. Therefore it suffices to give a projectivity criterion for X under the assumption that G is classical simple.

Proposition 15.1 Let G be classical simple. Then X = G/P is projective if and only if as an ordered set S_{𝔭} (see §4) is isomorphic to a subset of ℤ for G = A(∞), and to a subset of ℤ_{−} otherwise.

Proof. Fix a Borel subgroup B ⊂ P. If S_{𝔭} is isomorphic to a subset of ℤ for G = A(∞), or to a subset of ℤ_{−} for G = B(∞), C(∞), D(∞), there exists a strictly decreasing function φ : S_{𝔭} → ℤ such that, if G ≠ A(∞), φ(s) ≥ 0 and φ(s_{0}) = 0 whenever s_{0} ∈ S_{𝔭}. Using φ, we will now determine a B–dominant integral weight λ as follows. First note that, just by the construction of S_{𝔭_{n}}, each simple root of B_{n} has a well–defined image in S_{𝔭_{n}}. Given α ∈ Δ^{+}, let α = c_{n}^{1}α_{n}^{1} + … + c_{n}^{k_{n}}α_{n}^{k_{n}}, c_{n}^{j} ≠ 0, be its decomposition into a linear combination of simple roots of B_{n}. (Here the images s_{n}^{1}, …, s_{n}^{k_{n}} ∈ S_{𝔭_{n}} of α_{n}^{1}, …, α_{n}^{k_{n}} are in decreasing order.) To define λ, we set

λ(α) := \{ \begin{matrix}φ(s_{n}^{1}) − φ(s_{n}^{k_{n}}) & if & c_{n}^{j} = 1 & for & i = j, …, k_{n}, \\ φ(s_{n}^{1}) + φ(s_{n}^{p_{n}}) & if & c_{n}^{j} = 1 & for & j = 1, …, k_{p} − 1, c_{n}^{k_{p}} = 2, \\ 2φ(s_{n}^{1}) & if & c_{1} = 2. \end{matrix}

A direct verification shows that λ(α) does not depend on n for large enough n. Therefore λ is a well–defined integral weight. Furthermore, the definition of λ implies that ℂ_{λ} is a well–defined dominant one dimensional P –module. Using the fact that S_{𝔭} is isomorphic to a subset of ℤ, for G = A(∞), and to a subset of ℤ_{−} otherwise, the reader will check that the restriction of the sheaf O_{X}(ℂ_{−λ}) to X_{n} is very ample for every n. Thus the sheaf O_{X}(ℂ_{λ}) provides a closed immersion of X into ℙ(V (ℂ_{λ})).

Conversely, let X be projective. Fix a closed immersion i : X ↪ ℙ(ℂ^{∞}). The restriction O_{ℙ(ℂ^{∞})}(1)_{|X_{n}} is a very ample invertible O_{X_{n}}–module. By choosing a G_{n}–linearization of O_{X}(1)_{|X_{n}} for each n (and by changing the signs of the corresponding weights) we obtain a system of integral weights {λ_{n}} of 𝔤_{n}. This system is not canonical, but the integers λ_{n}(α) for α ∈ Δ_{n} are canonical, i.e. depend only on the choice of the closed immersion i. The crucial observation now is that, if S_{𝔭} is not isomorphic to a subset of ℤ for G = A(∞), or to a subset of ℤ_{−} for G = B(∞), C(∞), D(∞), there always exists α ∈ Δ\Δ_{P} such that for every r there are α_{1}, α_{2}, …, α_{r} ∈ Δ\Δ_{P} with α = α_{1} + … + α_{r}. Fix n_{0} for which α ∈ Δ_{n_{0}}. Set r := λ_{n_{0}}(α) + 1 and let n ≥ n_{0} be such that α_{1}, …, α_{r} ∈ Δ_{n}. Then λ_{n_{0}}(α) = λ_{n}(α) = ∑_{i=1}^{r} λ_{n}(α_{i}) ≥ r = λ_{n_{0}}(α) + 1. This contradiction completes the proof. □

16 The Sheaf O_{X}(E) for infinite dimensional E

If E is finite dimensional, then so is E^{∗}, and O_{X}(E) = O_{X}((E^{∗})^{∗}). Therefore in this case the consideration of O_{X}(E) does not lead to anything new. The situation is very different for an infinite dimensional E. Here the Mittag–Leffler condition can fail, and O_{X}(E) has in general arbitrarily many non-vanishing cohomology groups. It is an interesting problem to study the structure of these G–modules.

In the following example \underleftarrow{lim}^{(1)}H^{0}(X_{n}; O_{X_{n}}(E_{n})) ≠ 0, and \underleftarrow{lim} H^{i}(X_{n}; O_{X_{n}}(E_{n})) ≠ 0 for all i in a prescribed finite interval. The interested reader can construct a similar example in which \underleftarrow{lim}^{(1)}H^{i}(X_{n}; O_{X_{n}}(E_{n})) ≠ 0 for any i, and therefore H^{j}(X; O_{X}(E)) ≠ 0 for all j > 0.

Example 16.1

(1) Let B ⊂ P ⊂ G be as in Example 13.1(1). We define the irreducible rational P –module E as follows. Let 1 < k ∈ ℤ_{+} and let E′ be the irreducible rational P –module with B–highest weight 2kε_{2} + (2k − 1)ε_{3} + … + (k + 1)ε_{k+1} . Then E′ := \underrightarrow{lim}_{n>k}E′_{n} where E′_{n} is the irreducible P_{n}–module with B_{n}–highest weight 2kε_{2} + (2k − 1)ε_{3} + … + (k + 1)ε_{k+1} . For n > k, (E′_{n})^{∗} is a dominant P_{n}–module. Furthermore, there is a canonical P_{n}–module surjection E′_{n+1} → E′_{n}, and thus a P_{n}–module injection (E′_{n})^{∗} ↪ (E′_{n+1})^{∗}. Set E_{n} := (E′_{n})^{∗} and E := \underrightarrow{lim}_{n>k}E_{n}. Consider O_{X}(E) := \underleftarrow{lim} O_{X_{n}}(E), ζ_{n} : O_{X_{n}}(E) → O_{X_{n−1}}(E) being the restriction maps. We claim that \underleftarrow{lim}^{(1)}H^{0}(X_{n}; O_{X_{n}}(E)) ≠ 0, and furthermore that H^{q}(X; O_{X}(E)) ≠ 0 for 1 ≤ q ≤ k. In particular, for F_{n} = O_{X_{n}}(E), the exact sequence (10.4) has three nonzero terms when q = 1.

Consider \underleftarrow{lim}^{(1)} H^{0}(X_{n}; O_{X_{n}}(E)). When n > k, H^{0}(X_{n}; O_{X_{n}}(E)) is the space E^{P_{n}} of P_{n}– invariants on E. Here dim E^{P_{n}} = ∞, for as a P_{n}–module E is isomorphic to a direct sum of finitely many non–trivial irreducible P_{n}–modules and infinitely many copies of the trivial one dimensional P_{n}–module. Furthermore, the natural injection E^{P_{n+1}} ⊂ E^{P_{n}} is immediately seen to be a strict inclusion. In the cohomology picture this injection is the restriction map ζ_{n+1}^{0} : H^{0}(X_{n+1}; O_{X_{n+1}}(E)) → H^{0}(X_{n}; O_{X_{n}}(E)). Therefore \underleftarrow{lim}^{(1)}H^{0}(X_{n}; O_{X_{n}}(E)) ≠ 0, the action of G on \underleftarrow{lim}^{(1)}H^{0}(X_{n}; O_{X_{n}}(E)) being trivial. Note also that H^{0}(X; O_{X}(E)) = \underleftarrow{lim} H^{0}(X_{n}; O_{X_{n}}(E)) = ⋂_{n} E^{P_{n}} = 0.

Now consider \underleftarrow{lim} H^{k}(X_{n}; O_{X_{n}}(E)). The P_{n}–module E decomposes as (E′_{n})^{∗} ⊕ Ẽ_{n} where Ẽ_{n} is a direct sum of irreducible P_{n}–modules not isomorphic to (E′_{n})^{∗}. Therefore we have a canonical G_{n}–module injection

H^{k}(X_{n}; O_{X_{n}}((E′_{n})^{∗})) = V (E′_{n})^{∗} ↪ H^{k}(X_{n}; O_{X_{n}}(E)).

Furthermore, the map

ζ_{n+1}^{k} : H^{k}(X_{n+1}; O_{X_{n+1}}(E)) → H^{k}(X_{n}; O_{X_{n}}(E))

induces a surjection

H^{k}(X_{n+1}; O_{X_{n+1}}((E′_{n+1})^{∗})) → H^{k}(X_{n}; O_{X_{n}}((E′_{n})^{∗})).

Thus there is a canonical G–module injection

V (E′)^{∗} = \underleftarrow{lim}_{n>k}V (E′_{n})^{∗} ↪ \underleftarrow{lim} H^{k}(X_{n}; O_{X_{n}}(E)),

and, in particular, H^{k}(X; O_{X}(E)) ≠ 0.

A similar argument shows that \underleftarrow{lim} H^{i}(X_{n}; O_{X_{n}}(E)) ≠ 0 for 1 ≤ i < k. In effect, for any such fixed i the P_{n}–module E decomposes as (E_{n}^{i})^{∗} ⊕ Ẽ_{n}^{i}, where E_{n}^{i} is the irreducible P_{n}–module with B_{n}–highest weight 2kε_{2} + (2k − 1)ε_{3} + … + (i + 1)ε_{2k−i+1} and where Ẽ_{n}^{i} is a direct sum of irreducible P_{n}–modules not isomorphic to (E_{n}^{i})^{∗}. Then, as above, one verifies that there is a G–module injection

V (\underrightarrow{lim}_{n>2k−i+1}E_{n})^{∗} ↪ \underleftarrow{lim} H^{i}(X_{n}; O_{X_{n}}(E)).

Consequently H^{i}(X; O_{X}(E)) ≠ 0 for 1 ≤ i ≤ k.

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  dimitrov@math.ucla.edu    penkov@math.ucr.edu       jawolf@math.berkeley.edu