Journal of Lie Theory Volume 5 (1995) 41–79 C ◯1995 Heldermann Verlag

Invariant orders in simply connected Lie groups

Victor M. Gichev

Communicated by E. B. Vinberg

In this paper we give a criterion for an invariant cone in a real Lie algebra to generate a bi-invariant order in the corresponding simply connected Lie group.

Introduction

A convex, closed, pointed, generating cone C in a real Lie algebra which is invariant under all inner automorphisms will be called an invariant cone. Let S be the closed semigroup generated by exp(C). If S ∩ S^{−1} = {e}, where e is the identity of the group, then the invariant order is defined by g ≽ h ⇔ gh^{−1} ∈ S.

The assumption that a Lie algebra admits an invariant cone is rather restrictive. Among simple real Lie algebras, only hermitian ones have this property ([11],[10]). This class recently was described by K.-H. Neeb ([7]). A Lie algebra admitting an invariant cone is a semidirect product of a reductive Lie algebra which also admits an invariant cone and a two-step nilpotent ideal, and all possibilities for the representation which determines the product are known now. A reductive Lie algebra admits an invariant cone if and only if it is not compact semisimple and each it’s simple ideal is either compact or hermitian.

In the original paper [11], E.B. Vinberg proved that each simply connected group corresponding to a simple hermitian Lie algebra admits an invariant order. Among others, he also formulated the problem of a description of invariant cones which generates an invariant order (these cones are called global).

For simple algebras, this problem was solved by G.I. Ol’shanskiǐ who discovered an obstruction to the globality: an invariant cone is global if and only if the dual cone contains some special element l. This l is nonzero only for nontubular simple hermitian algebras, so in tubular ones any invariant cone is global.

In the solvable case, each invariant cone is global ([5], [1]).

Various results for the general case were obtained by K.-H. Neeb. In particular, he proved that in a Lie algebra which has no simple nontubular ideals each invariant cone is global. Furthermore, an invariant cone is global if it’s dual cone intersects the relative interior of the cone C_{l} generated by elements l in

ISSN 0940–2268 / $2.50 ^{_{C} ◯} Heldermann Verlag all nontubular simple ideals. It follows also from his results that in semisimple algebras the condition for the dual cone to intersect C_{l} is a necessary one.

Let J be a semisimple ideal in a Lie algebra G with nontubular simple summands only, L_{J} be the linear span of these elements l in all summands, C be an invariant cone in G, and π_{J} be the projection to J ^{∗} dual to the identical embedding. By the main result of this paper, the globality of C is equivalent to the condition

π_{J} C^{∗} ∩ L_{J} ≠ {0}

for all such ideals J. Since (C ∩ J)^{∗} = clos π_{J} C^{∗}, this condition is close to the evident necessary one that C ∩J is global in J but involves the degree of contact between C and J.

A cone in a Lie algebra, by left shifts, naturally defines a left invariant cone field on the corresponding group; if the cone is invariant then the cone field is bi-invariant. A smooth curve is timelike if it’s tangent vector at any point belongs to the cone at this point. It is possible to prove that the semigroup S above is the closure of the set of endpoints of timelike curves starting at e. Hence the existence of a closed timelike curve implies that the cone is not global. Another possibility for an invariant cone to be nonglobal is the existence of a sequence of timelike curves starting at e such that their lengths tend to the infinity but endpoints tend to e. In the setting of left invariant cone fields this effect occurs if the cone has a sufficiently high degree of contact with a Heisenberg subalgebra (Theorem 4.1). For bi-invariant cone fields, one may consider two different Heisenberg subgroups together with their actions, the first by left shifts and the second by right ones. Thus a global invariant cone cannot have a high degree of contact with a sum of two Heisenberg subalgebras. A construction of special closed curves in Heisenberg groups gives estimates for the degree of contact; gathered with some technical lemmas on the geometry of cones and a choice of two Heisenberg subalgebras in nontubular hermitian algebras this proves the necessity of the globality criterion.

If the interior of the invariant cone intersects a sum of two Heisenberg subalgebras then the mentioned construction gives an example of a closed timelike curve. This solves the problem of Ol’shanskiǐ formulated in [8] for su(2, 1). In simple simply connected groups, such a curve exists if and only if the cone is not global, but even in the algebra su(2, 1) ⊕ ℝ there exists an invariant cone which is not global but admits no closed timelike curves.

The proof of the sufficiency is more complicated technically but more standard. It uses the techniques of increasing functions, in fact, a modification of the construction of the paper [11]. An essential tool is also Theorem 5.1 which deals with the delicate situation of an invariant cone whose image in a factor algebra is included to some global invariant one. The globality problem is reduced to the reductive case by the consideration of simplectic algebras sp(2n, ℝ).

The material of the paper is organized as follows.

Section 1 contains basic definitions and notations as well as the statement of Main Theorem and the description of a geometrical procedure which allows to check the globality; there are also two examples.

In Section 2 the preparatory material on simple hermitian algebras is given in a form convenient for our purpose. The results of this section are known.

Mainly, we need a reformulation of the Ol’shanskiǐ criterion l ∈ C^{∗} (Proposition 2.1). A construction of an increasing function which, in fact, coincides with the function of Vinberg ([11]) shows that it is naturally connected with some finite dimensional representation.

In Section 3 it is shown that estimates of lengths of closed curves of a special kind can give estimates of the degree of contact (Proposition 3.1).

Section 4 is devoted to the consideration of Heisenberg groups. According to Theorem 4.1 proved in this section, the degree of contact of a global cone with a Heisenberg algebra at a point of it’s center cannot exceed 2. The necessity of the criterion of the globality is also proved in this section.

The main result of Section 5 is Theorem 5.1 mentioned above. It is proved that an invariant cone is global if it’s projection to a factor by an ideal is included to some global cone and the projection of the dual cone to the dual space of this ideal includes some cone dual to a global one. The additional hypothesis is that this ideal admits a comlementary subalgebra.

In Section 6, nonreductive algebras are considered; it is proved in Theorem 6.3 that the globality of an invariant cone in a nonreductive algebra is equivalent to the globality of it’s intersection with some reductive subalgebra.

The results formulated in Section 1 are proved in Section 7.

The paper is rather selfcontained. Special results which are given without proofs concern the structure of Lie algebras admitting an invariant cone (see [3], [6], [2]; the last article could be a preliminary part to this paper).

The main result of the paper could be understood as follows: the Heisenberg group is the unique obstruction to the existence of an bi-invariant order which is not an obstruction to the existence of an invariant cone. It seems to be faithful that this obstruction (with general nilpotent groups instead of the Heisenberg group) is an essential one in a more general situation of homogeneous spaces.

1. Statement of the main theorem

The identity in a group is denoted by e, in an associative algebra by 1; ℝ^{+} = [0, ∞). Vector spaces everywhere in this paper are supposed to be finite dimensional and, if the contrary is not supposed in an explicit form, real. For a vector space V , V^{∗} denotes the dual space. We consider only convex cones. A cone C ⊆ V is called pointed if it is closed and C ∩ (−C) = {0}.

The interior of a set S is denoted by Int(S), it’s boundary by ∂S; RelInt(S) denotes the interior of a set S ⊆ V in it’s linear span.

A cone C is called generating if Int(C) ≠ ∅. A closed cone C is pointed (generating) if and only if the dual cone

C^{∗} = {λ ∈ V^{∗} : λ(x) ≥ 0 for all x ∈ C}

is generating (pointed).

Let M be a smooth manifold (”smooth” everywhere in this paper means ”infinitely differentiable”). A cone field on M is a subset ℭ of the total space of the tangent bundle T M such that ℭ(m) = ℭ ∩ T_{m}M is a cone in the tangent space T_{m}M for all m ∈ M. A cone field on a domain in an Euclidean space E will be called continuous if the projections of cones to the projective space P E depends on the point continuously with respect to the Hausdorff distance between sets; the Hausdorff distance may correspond to any Riemannian metric in P E. Clearly, diffeomorphisms keep this property, thus the definition of the continuity naturally extends to smooth manifolds.

A smooth curve γ : [a, b] → G is called timelike if

(1.1)  γ′(t) ∈ ℭ(γ(t)) for all t ∈ [a, b]

For a piecewise smooth curve, one has to replace γ′(t) in (1.1) with both one-side derivatives. By T(m) will be denoted the set of all timelike piecewise smooth curves in M which start at m.

The future F(m) of the point m is the closure of the set of endpoints of curves in T(m). The past P(m) of m is the future of m with respect to the cone field −ℭ.

A cone field ℭ is called global if the tangent cone to the set F(m) coincides with ℭ(m) for each x ∈ M. If the cone field ℭ is global then the dual cone field ℭ^{∗} in the cotangent bundle T^{∗}M defined by ℭ^{∗}(m) = ℭ(m)^{∗} will be called exact.

A smooth function f is called increasing at a point m if d_{g}f ∈ ℭ(m)^{∗}. If d_{g}f ∈ Int(ℭ(m)^{∗}) then f will be called strictly increasing at m. An increasing (strictly increasing) function on M is a function which is increasing (strictly increasing) at each point. Set

(1.2)  ℭ^{⋆}(m) = {d_{m} f : f is an increasing function on M}

Clearly, ℭ^{⋆}(m) ⊆ ℭ^{∗}(m).

The length of a vector x in an euclidean space E is denoted by |x|; for the distance between a point and a set or between two sets we shall use the notation dist. The length of a curve γ is denoted by l(γ).

For a subalgebra X ⊆ G, let Ad(X) be the subgroup of GL(G) generated by the set {e^{ad(x)} : x ∈ X}. If clos Ad(X) is compact, X is called compactly embedded.

A pointed generating Ad(G)-invariant cone in a real Lie algebra G will be called an invariant cone. Each cone C ⊆ G defines the unique left invariant cone field ℭ on a Lie group G with the tangent Lie algebra G such that ℭ(e) = C. If C is an invariant cone then ℭ is also right invariant. More generally, a cone in a tangent (cotangent) space at a point p of a homogeneous space defines an invariant cone field ℭ if and only if it is invariant under the isotropy group. Such a cone will be called global (exact) if it generates the global (exact) invariant cone field. A cone dual to an invariant cone in a Lie algebra will be called an exact invariant cone.

A simple real Lie algebra is called hermitian if the center of it’s maximal compactly embedded subalgebra is nontrivial. Simple hermitian Lie algebras are exactly Lie algebras of groups of holomorphic automorphisms of irreducible hermitian symmetric spaces. A simple hermitian Lie algebra G is called tubular if the corresponding symmetric space is equivalent to a tube domain, i.e. a domain in ℂ^{n} of the type ℝ^{n} + iC, where iC is a pointed generating cone in ℝ^{n}. Otherwise, G is nontubular. An equivalent algebraic definition is: G is nontubular if and only if some special element l is nonzero (for the exact definition of l, see (2.10)).

Let G be a real Lie algebra, J_{1}, …, J_{n} be all it’s simple hermitian nontubular ideals, l_{k} ≠ 0 be the element defined by (2.10) for J_{k}, \overline{n} = \{1, …, n\}. For a set I ⊆ \overline{n}, let J_{I} be the sum of J_{k}, k ∈ I, L_{I} be the linear subspace generated by l_{k}, k ∈ I, π_{I} be the projection G^{∗} → J_{I} dual to the embedding J_{I} → G, where J_{I}^{∗} is identified with J_{I} by the Killing form. If I = \overline{n}, set π_{I} = π, J_{I} = J, and L_{I} = L. Note that l_{k} are linearly independent.

Theorem (Main Theorem). An invariant cone C in a real Lie algebra G is global if and only if

(GlC)  π_{I} C^{∗} ∩ L_{I} ≠ {0}

for each subset I ⊆ \overline{n}. Moreover, the globality of C is equivalent to each of the following conditions:

(1) the cone πC^{∗} includes some exact cone in J;

(2) π_{I} C^{∗} includes some exact cone in J_{I} for any I ⊆ \overline{n}.

The globality criterion (GlC) may be checked by the intersection of the cone πC^{∗} with a rather small subspace of G. Let V_{r} be the two-dimensional subspace generated by l_{r} and m_{r}, where l_{r} and m_{r} are defined by (2.10) for the nontubular ideal J_{r}, V = V_{1} + ⋯ + V_{n}. Note that V is invariant under all projections π_{I}, I ⊆ \overline{n}. Set C′ = πC^{∗} ∩ V.

The checking procedure deals with a set I ⊆ \overline{n} and the cone C_{I} = π_{I} C′ (if I = ∅, put L_{I} = C_{I} = \{0\}). Initially, I = \overline{n}. Each step consists of two operations:

(1) find the least subset I′ ⊆ I such that C_{I} ∩ L_{I} ⊆ L_{I′};

(2) replace I by I \ I′.

The procedure finishes if either I′ = ∅ or I = ∅.

Proposition 1.1. The invariant cone C is global if and only if the procedure above finishes with I = ∅.

This section is concluded by two illustrating examples. We use the notation (2.10).

Among simple nontubular hermitian algebras, su(2, 1) has the least dimension. This is a real form of sl(3, ℂ) whose root system is A_{2} . Recall that A_{2} consists of vertices of the regular hexagon. Let α be a compact root, β the simple noncompact one. Then 3ik = α + 2β, the automorphism ν transposes α and β, and (2.10) may be rewritten as follows

il = \frac{1}{3}(β − α), im = \frac{1}{3}(2α + β), ip = α + β

Note that ip is the second positive noncompact root.

The essential part of possible geometrical effects obstructing to the globality occurs in algebras su(2, 1)⊕ℝ and su(2, 1)⊕su(2, 1). By a result of Hilgert and Hofmann, any closed convex generating Weyl invariant cone c in a Cartan subalgebra C of a reductive Lie algebra G such that c_{min} ⊆ c ⊆ c_{max} is the intersection of the unique invariant cone C in G with C. So we give examples of cones c. For su(2, 1), c_{min} is generated by −iβ and p, c_{max} by l and m. The Weyl group of su(2, 1) contains only one nontrivial element, namely, the reflection ρ with respect to the line orthogonal to α. Since k is a fixed point of it and ρα = −α, any invariant cone in su(2, 1) is uniquely determined by the condition that the vector 3k + itα with t ∈ [1, 3] belongs to it’s boundary. The space V coincides with C.

Example 1.1. Let G = su(2, 1) ⊕ ℝ , C be a Cartan subalgebra of G. Then C = C′ ⊕ ℝ , where C′ is a Cartan subalgebra of su(2, 1). Let f be a convex function on (−3, 3) such that

(1) f is even;

(2) f(t) = 0 for t ∈ [−1, 1];

(3) lim_{t→3} f(t) = +∞.

There exists the unique convex homogeneous of degree 1 function F on the interior of c_{max} in C′ which coincides with f(t) on the interval k − itα, t ∈ (−3, 3). Set

c = {(x, s) : x ∈ Int(c_{max} ), s ≥ F (x)} ∪ {(0, s) : s ≥ 0}

Then c is a closed convex generating Weyl invariant cone in C. It’s projection to C′ is equal to Int(c_{max} ) ∪{0}, so (GlC) is not satisfied for the corresponding invariant cone C. Hence the predual cone is not global while it’s intersection with su(2, 1) is global because the closure of the projection is c_{max} . Note that the degree of contact of the predual cone with C′ at the boundary of c_{min} depends on the growth of f as t → 3 but always exceeds 1 (for the exact definition of the degree of contact, see Section 3).

Example 1.2. Set G = G_{1} ⊕ G_{2} , where G_{1} and G_{2} are isomorphic to su(2, 1). Lets denote objects (vectors, cones, subspaces) in an ideal G_{r} introduced above for su(2, 1), r = 1, 2, by the same letters with the index r.

For t ∈ [1, 3], let c_{t} be the closed Weyl invariant cone in C generated by

(1.3)  p_{1}, l_{1} + k_{2}, k_{1} + l_{2}, 3 k_{2} +it α_{2}

and C_{t} be the corresponding invariant cone in G. Then

c_{min,1} ⊕ c_{min,2} ⊆ c_{t} ⊆ c_{max,1} ⊕ c_{max,2}

The projection of c_{t} to G_{r} contains l_{r}. Let λ be the linear functional on C = C_{1} ⊕ C_{2} such that λ(l_{1}) = λ(l_{2}) = 0, λ(m_{1}) = λ(m_{2}) = 1. If t < 3 then λ is strictly positive on all generating elements (1.3), so c_{t} doesn’t contain any nonzero element of the type a l_{1} +b l_{2} with a, b ∈ ℝ . Hence C_{t} is not exact for t < 3 by (GlC). If t = 3 then l_{2} ∈ c_{t} and the projection of c_{t} to G_{1} contains l_{1} . Thus C_{3} is exact by Proposition 1.1.

2. Simple hermitian Lie algebras

Let G be a real simple Lie algebra, κ be a Cartan involution which defines the Cartan decomposition

G = K ⊕ M

where κξ = ξ for ξ ∈ K and κξ = −ξ if ξ ∈ M, K is a maximal compactly embedded subalgebra. Then

K̃ = K ⊕ iM

is a compact real form of the complexification G^{C} = G ⊕ iG of G. The group Ad(K) is irreducible in M.

If τ is the complex conjugation in G^{C} with respect to K̃ in G^{C} then

(2.1)  μ = κτ = τκ

is the complex conjugation with respect to G.

The algebra G is called hermitian if Z(K) ≠ 0. If this is the case then the center of K coincides with the centralizer Z(K), dim Z(K) = 1 and M admits an Ad(K)-invariant complex structure

(2.2)  J = ad(k)

for some k ∈ Z(K). This k is unique up to a sign. The automorphism κ admits an expression by ad(k)

(2.3)  κ = e^{π ad(k)}

A complex Lie algebra G^{C} is simple because G admits no complex structure (otherwise, its maximal compactly embedded subalgebra would be simple).

Any Cartan subalgebra C of K is a Cartan subalgebra of G. Let Δ denotes the root system for C. Since C is compactly embedded, Δ ⊂ (i C)^{∗}.

Let B be any base of the root system Δ. Then there exists a Chevalley base 𝔅 for G^{C}

(2.4)  𝔅 = {h_{α} : α ∈ B} ∪ {x_{α} : α ∈ Δ}

where x_{α} is the eigenvector of C corresponding to a root α (so [h, x_{α}] = α(h)x_{α} for all α ∈ Δ and h ∈ C),

h_{α} = [x_{α}, x_{−α}], α(h_{α}) = 2 for all α ∈ Δ

and structure constants of G^{C} for the base 𝔅 are integer; in particular, they are real. Thus {h_{α}, x_{α}, x_{−α}} is a sl_{2} -triple.

We may also assume that

(2.5)  τx_{α} = −x_{−α}

The union of vectors h_{α} is a root system Δ^{∨} dual to Δ.

Let q be an invariant symmetric bilinear form in G^{C} which is positive definite in iC (so q is a scalar multiplier of the Killing form), | | be the corresponding length. Then for all α ∈ Δ and h ∈ iC α(h) = 2q(h_{α}, h)/q(h_{α}, h_{α}). Thus we may assume that Δ is embedded into iC in such a way that

(2.6)  α = \frac{2}{q(h_{α}, h_{α})}h_{α}, h_{α} = \frac{2}{q(α, α)}α

It will be convenient to normalize q by the condition

(2.7)  max\{ |α| : α ∈ Δ\} = \sqrt{2}

If α is a root of the maximal length, we shall say that α is a long one; others will be called short; short roots are of length 1. The normalization (2.7) and (2.6) imply that

(2.8)  h_{α} = α, |α| = \sqrt{2}, and h_{α} = 2α, |α| = 1

Let W be the Weyl group of Δ, V be a Weyl chamber in iC which contains ik. There exists the unique base B of Δ such that

(2.9)  V = {x ∈ iC : α(x) ≥ 0 for all α ∈ B}

It will be supposed that B in (2.4) satisfies (2.9).

Since V is a fundamental domain for W, there exists the unique δ ∈ W such that δV = −V . This is an involution because δ^{2}V = V . Let ν be an automorphism of G^{C} which is equal to −δ in C; then ν is an involution in iC, νV = V , and ν is either outer or identical in iC. Set

(2.10)  l = k + δk, m = νk, p = k + m

Clearly, l⊥p.

Let W_{k} and W_{m} denote subgroups of W generated by reflections in W which fix points ik and im respectively, W_{km} = W_{k} ∩ W_{m}, W_{p} be the group generated by W_{km} and ν.

Lemma 2.1. The set of fixed points in iC of a group W_{x} is equal to iℝx, x = k, m, p. The group W_{km} fixes points of the subspace generated by k and m, and only these points. Proof. Note that W_{k} includes the Weyl group of the reductive Lie algebra K which could be naturally identified with the Weyl group of the compact semisimple algebra K′ = [K, K] corresponding to its Cartan subalgebra K′ ∩ C. The last group has no fixed points in K′ ∩ C. Therefore, Z(K) = ℝk is exactly the set of fixed points of W_{k} in C.

For x = m, the assertion is clear because m = νk = −δk and δ ∈ W. If x = p then an obvious case is k = m. Since iℝk is the set of W_{k} -fixed points in iC, iℝ^{+}k is an extreme ray of V . The same is true for m. The cone V is simplicial, so the subspace generated by ik and im is the intersection of those boundary hyperplanes of V which contains ik and im; since all reflections with respect to these hyperplanes are contained in W_{km}, the set of W_{km} -fixed points is exactly this subspace. It remains to note that if k ≠ m then iℝp is the set of ν -fixed points in the plane generated by ik and im because ν transposes them.

Remark 2.1. In fact, W_{k} is the Weyl group of the reductive Lie algebra K because any reflection in C with respect to a hyperplane orthogonal to a root α ∈ Δ_{0} could be realized by the inner automorphism exp(\frac{π}{2} ad(x_{α} − x_{−α})).

The following corollary was already used in the proof of the lemma; the proof is clear.

Corollary 2.1. iℝ^{+}k and iℝ^{+}m are extreme rays of V .

By the corollary, ik is proportional to a fundamental weight of the base B and the dual base B^{∨} of Δ^{∨}. Since eigenvalues of ad(k) are 0, ±i,

(2.11)  α(ik) ∈ {0, 1, −1}

for all α ∈ Δ. Therefore, ik is a fundamental weight of Δ^{∨}. Moreover, ik is a microweight of Δ^{∨} ((2.11) is a definition).

Set

Δ_{j} = {α ∈ Δ : α(ik) = j}

Then Δ = Δ_{−1} ∪ Δ_{0} ∪ Δ_{1} . Roots in Δ_{0} will be called compact, in Δ_{−1} ∪ Δ_{1} noncompact, in Δ_{1} — positive noncompact. Note that there is only one root in B which is not orthogonal to ik; this is the simple noncompact root.

By (2.1), (2.3), and (2.5),

(2.12)  μx_{α} = −x_{−α}, α ∈ Δ_{0} and μx_{α} = x_{−α}, α ∈ Δ_{1} ∪ Δ_{−1}

So, there are natural bases for K and M in the decomposition G = K ⊕ M

𝔅_{K} = iB^{∨} ∪ {x_{α} − x_{−α} : α ∈ Δ0^{+}} ∪ {i (x_{α} + x_{−α}) : α ∈ Δ0^{+}}

(2.13)  𝔅_{M} = {x_{α} + x_{−α} : α ∈ Δ_{1}} ∪ {i (x_{α} − x_{−α}) : α ∈ Δ_{1}}

where Δ0^{+} is a subset of positive roots for some ordering (so Δ0^{+} ∪ (−Δ0^{+}) = Δ_{0}

and Δ0^{+} ∩ (−Δ0^{+}) = ∅).

Since m is conjugated to k in G^{C} , ad(m) has the same eigenvalues as ad(k). Put

Δ_{r,s} = {α ∈ Δ : α(ik) = r, α(im) = s}

and denote by Gr,s^{C} the corresponding eigenspace

Gr,s^{C} = {x ∈ G^{C} : ad(ik)x = rx, ad(im)x = sx}

Then G0,0^{C} is the complex linear span of C and {x_{α} : α ∈ Δ_{0,0}}, and other Gr,s^{C} are generated by x_{α}, α ∈ Δ_{r,s}.

Since ik and im belong to the same Weyl chamber V , α(ik) and α(im) cannot have opposite signs for all α ∈ Δ. Therefore, Δ_{−1,1} = Δ_{1,−1} = ∅. Thus we obtain an A_{2} -type gradation

G^{C} = G−1,−1^{C} ⊕ G−1,0^{C} ⊕ G0,−1^{C} ⊕ G0,0^{C} ⊕ G1,0^{C} ⊕ G0,1^{C} ⊕ G1,1^{C}

For the element p defined by (2.10), ad(ip) has the eigenvalue r + s in the space Gr,s^{C} . So we have a gradation of G^{C}

G^{C} = G−2^{C} ⊕ G−1^{C} ⊕ G0^{C} ⊕ G1^{C} ⊕ G2^{C}

where Gr+s^{C} = Gr,s^{C} + Gs,r^{C} .

Our aim is to receive the same gradation for G. To do it, we’ll find an element in M conjugated to ip in G^{C} .

Lemma 2.2. Let Q(Δ) be the lattice in iC generated by Δ, λ ∈ Q(Δ), λ ≠ 0. Then

(2.14)  λ = ∑_{α∈E} n_{α}α

for some positive integer n_{α} and E ⊂ Δ such that q(λ, α) > 0, q(α, β) ≥ 0 for all α, β ∈ E.

Proof. Consider the class of representations of λ of the type (2.14) with nonnegative n_{α} and q(λ, α). This class is not empty because these conditions are satisfied for the unique representation by the base corresponding to a Weyl chamber which contains λ. Suppose that for the data n_{α}, α ∈ E the sum of n_{α} attains the minimal value over this class. Then q(α, β) ≥ 0 for all α, β ∈ E since otherwise α + β is a root and it is possible to decrease the sum. So α ∈ E, q(λ, α) = 0 implies n_{α} = 0, and α could be excluded from E.

Lemma 2.3. Let Δ̃ be the subsystem of long roots in Δ. Then ik − iδk ∈ Q(Δ̃).

Proof. First, note that for any weight λ of a root system Δ (this means that q(λ, α^{∨}) is integer for all α ∈ Δ) and w ∈ W, λ − wλ ∈ Q(Δ) (for reflections in W, this is a simple calculation; for any w ∈ W, this is true because W is generated by reflections).

So, the assertion is clear if roots in Δ are of equal length. We may exclude from the consideration G_{2} and F_{4} since they have no hermitian real forms. Thus we have to consider B_{n} and C_{n}.

Let W̃ be the Weyl group of Δ̃. If Δ = C_{n} then δ = −1 ∈ W̃ since Δ̃ = nA_{1} . For Δ = B_{n}, δ = −1 but δ ∉ W̃ for odd n; being a microweight of Δ^{∨} = C_{n}, ik is a short root ε of B_{n}, so ik − iδk = 2ε could be represented as a sum of two long roots in any B_{2} -subsystem which contains ε.

Lemma 2.4. There exists a subset E ⊆ Δ_{1,1} consisting of long roots such that

(2.15)  ip = ∑_{α∈E} h_{α}

and the set E is a maximal in Δ_{1} subset of pairwise orthogonal roots.

Proof. By Lemma 2.3, Lemma 2.2, and the definition of p, there exists a representation of ip of the type (2.14), and roots in E may be supposed to be long. For each ξ ∈ Δ,

2 ≥ q(ip, ξ) = ∑_{α∈E} n_{α}q(α, ξ)

If ξ ∈ E then q(ξ, ξ) = 2, so n_{ξ} = 1, q(ip, ξ) = 2, and q(α, ξ) = 0 for α ≠ ξ, α ∈ E. Therefore, roots in E are pairwise orthogonal and E ⊆ Δ_{1,1} . By (2.11), α = h_{α} for α ∈ E. Thus (2.15) is proved. The maximality of E is a consequence of (2.15) and the obvious inequality q(ip, α) ≥ 1 for α ∈ Δ_{1} .

Lemma 2.5. The linear span of all x_{α}, x_{−α}, h_{α}, α ∈ E, is a subalgebra of G^{C} isomorphic to the direct sum of sl_{2} -subalgebras corresponding to roots in E.

Proof. For all α ∈ E, x_{α} and x_{−α}, are annihilated by all summands in (2.15) which are not equal to h_{α}; if α, β ∈ E, α ≠ β, then ±α ± β are not roots because α, β are orthogonal and long.

Corollary 2.2. The element ip can be embedded into a sl_{2} -triple in G^{C} .

Proof. Put

(2.16)  h = ip, e = ∑_{α∈E} x_{α}, f = ∑_{α∈E} x_{−α}

Then [e, f] = h, [h, e] = 2e, [h, f] = −2f by the lemma and (2.15).

In sl(2, ℂ), h and e+f are conjugated by an inner automorphism which may be written explicitly. Set

(2.17)  θ = e^{\frac{1}{4} π ad(e−f)}

Then a calculation with 2-matrices shows that

(2.18)  θh = e + f, θe = −\frac{1}{2}(h − e + f), θf = −\frac{1}{2}(h + e − f)

By Lemma 2.3 and the same calculation,

(2.19)  θx_{α} = −\frac{1}{2}(h_{α} − x_{α} + x_{−α}), θx_{−α} = −\frac{1}{2}(h_{α} + x_{α} − x_{−α})

for all α ∈ E. Thus

(2.20)  iθx_{α}, iθx_{−α} ∈ G for all α ∈ E

by (2.13).

Lemma 2.6. For θ defined by (2.17), θE ⊂ M. The real linear span A of θE is a maximal abelian subspace of M; for B = (E^{⊥} ∩ C), A + B is a Cartan subalgebra of G.

Proof. By Lemma 2.3 and (2.18), θh_{α} = x_{α} + x_{−α} ∈ M. This implies the first assertion.

The set of θ-fixed points in C includes E^{⊥}; therefore,

(2.21)  θC = iA ⊕ B

So Z(A + B) = θ(C + iC) ∩ G = A + B and A + B is a Cartan subalgebra of G.

The remainder is the equality Z(A) ∩ M = A which is a consequence of the following lemma.

Lemma 2.7. Let F be a set of θ-fixed points in Z(A). Then F ⊆ K and Z(A) = A ⊕ F. Proof. If β ∈ E and α⊥E then β + α cannot be a root because roots in E are long. Therefore, [e, x_{α}] = [f, x_{α}] = 0 and θx_{α} = x_{α} for all α ∈ E^{⊥}.

The centralizer of E in G^{C} is generated by C and root vectors x_{α}, α ∈ E^{⊥} ∩ Δ. So Z(A) = Z(θE) = A ⊕ F by (2.21).

By Lemma 2.4, noncompact roots cannot be orthogonal to E. Thus F ⊆ K.

Corollary 2.3. A simple hermitian Lie algebra G admits a gradation

(2.22)  G = G_{−2} ⊕ G_{−1} ⊕ G_{0} ⊕ G_{1} ⊕ G_{2}

Proof. This is the gradation by eigenvalues of ad(θh) (note that θh ∈ M by (2.15) and Lemma 2.6).

Let L = K ∩ G_{0} . Clearly, B ⊆ L. Lets denote by P_{X} the q -orthogonal projection in G onto the linear subspace X ⊂ G. Then

P_{M} = \frac{1}{2}(1 − κ), P_{K} = \frac{1}{2}(1 + κ)

Set H^{+} = G_{1} ⊕ G_{2} , H^{−} = G_{−1} ⊕ G_{−2} , and

P = G_{0} ⊕ H^{+}, H = H^{+} ⊕ H^{−}, H_{0} = H ∩ K

Lemma 2.8. In the decomposition (2.22), (1) κG_{j} = G_{−j}, j = 0, ±1, ±2; in particular, κH^{+} = H^{−};

(2) l = 2P_{L}k;

(3) l ∈ Z(L) ∩ L;

(4) P_{K}H = P_{K}H^{+} = P_{K}H^{−} = H_{0} ;

(5) K = L ⊕ H_{0} , and this decomposition is orthogonal;

(6) G = K + P;

(7) if G is nontubular then [G_{1}, G_{1}] = G_{2} and [G_{−1}, G_{−1}] = G_{−2} . Proof. By Lemma 2.6, θh ∈ M, so κθh = −θh. This implies (1).

Since l ⊥ Δ_{1,1} ⊆ E, l ∈ B ⊆ L. By (2.18), h = −θe − θf. Hence

(2.23)  p = iθ(e + f)

and

(2.24)  k = \frac{1}{2}(l + p) = \frac{1}{2}(l + iθe + iθf)

So (2) is true because iθe ∈ G_{2}, iθf ∈ G_{−2} by (2.20), and L ⊥ G_{2} ⊕ G_{−2} since the decomposition (2.22) is orthogonal.

It follows from (2) that l is a fixed point of Ad(L). Since l ∈ L, this proves (3).

Note that κH = H by (1), so P_{K}H = H ∩ K = H_{0} ; P_{K}H^{+} = P_{K}H^{−} because κH^{+} = H^{−}. Clearly, H ⊥ L, so H_{0} ⊥ L. This implies that H_{0}∩L = {0} since −q is positive definite in K. By (1), G_{0} is κ-invariant, so P_{K}G_{0} = L; therefore, K = P_{K}(G_{0} + H) = L + H_{0} . Thus (4) and (5) are proved.

By (4) and (5), K + H^{+} ⊃ H^{−}. This proves (6).

To prove (7), set G′_{2} = [G_{1}, G_{1}], G′_{−2} = [G_{−1}, G_{−1}], G′_{0} = [G_{1}, G_{−1}] and

G′ = G′_{−2} ⊕ G_{−1} ⊕ G′_{0} ⊕ G_{1} ⊕ G′_{2}

Then [G_{0}, G′_{k}] = G′_{k}, k = 0, ±2. Since [G_{2}, G_{−1}] ⊆ G_{1} , [G_{2}, G′_{−2}] ⊆ G′_{0} ; similarly, [G′_{2}, G_{−2}] ⊆ G′_{0} . Clearly, [G_{±1}, G′] ⊂ G′. Thus G′ is an ideal in G and this ideal is proper if (7) is not true.

In the theorem below, we summarize known properties of invariant cones in simple hermitian Lie algebras in the form convenient for our purpose. We follow papers of Vinberg [11] and Ol’shanskiǐ [8], [9]; see also [10].

Recall that the Weyl group of C in G may be identified with the group W_{k} of Lemma 2.1 (see Remark 2.1).

Theorem 2.1. Let C be a compactly embedded Cartan subalgebra of G.

(1) Any invariant cone C in G is uniquely determined by C ∩ C = c, and c is a W_{k} -invariant generating cone in C.

(2) There exist unique up to a sign maximal and minimal invariant cones C_{max} and C_{min} . The cone C_{min} may be chosen in such a way that k ∈ C_{min} . Then elements x ∈ C_{max} ∩K are distinguished by the property J ad(x) ≥ 0 in M ( for C_{max} ⊇ C_{min} ; J is defined by (2.2)).

(3) Let c_{max} = C_{max} ∩ C, c_{min} = C_{min} ∩ C. Then each W_{k} -invariant cone

c ⊂ C such that

c_{min} ⊆ c ⊆ c_{max}

is the intersection of some invariant cone with C.

(4) The cone C_{min} is generated by k; C_{max} is the cone dual to C_{min} with respect to the invariant form −q.

(5) The cone ic_{min} is the minimal cone which contains all positive noncompact roots; c_{max} = {x ∈ C : q(α, ix) ≥ 0 for all α ∈ Δ_{1}}.

In other words, ic_{max} is the union of Weyl chambers which contain ik. Note that, by (2.10), (5), and (2.15),

(2.25)  l, m ∈ c_{max}, p ∈ c_{min}

Ol’shanskiǐ ([8]) discovered a criterion of the globality of an invariant cone for simple simply connected Lie groups. For a cone C, let C^{∗} be the cone dual to C with respect to −q.

Theorem 2.2. An invariant cone C ⊇ C_{min} in a simple hermitian Lie algebra G is global if and only if l ∈ C^{∗}.

If l = 0 then a simple hermitian Lie algebra G is called tubular. Otherwise, it is called nontubular. By the theorem, in a tubular Lie algebra any invariant cone is global.

In fact, the sufficiency of the condition l ∈ C^{∗} was proved by Vinberg in the original paper [11] (he formulated the result for C_{min} ). Ol’shanskiǐ observed this and proved the necessity.

Since Vinberg defined the element l by another way, we shall outline a modified version of his proof.

By (2.11), ik is a weight of Δ; since ik ∈ V , this is a highest weight of a finite dimensional representation ρ of G^{C} . Weights of ρ lie in the W -invariant convex polytope M generated by ik. If α ∈ Δ_{0} then ik belongs to the convex hull of W_{k}(ik +α) = ik +W_{k}α. Since ik is a vertex of M, ik +α is not a weight of ρ. Clearly, the same is true for α ∈ Δ_{1} . So, if w denotes the corresponding highest vector then

(2.26)  ρ(x_{α})w = 0 for all α ∈ Δ_{0} ∪ Δ_{1}

The dual representation ρ^{∗} has the highest weight im = iνk. Therefore,

ρ^{∗}(x_{α})ω = 0 for all α ∈ Δ_{0,0} ∪ Δ_{−1,0} ∪ Δ_{0,1} ∪ Δ_{1}

where ω is a highest vector for ρ^{∗}. Note that P + iP is the complex linear span of θx_{α}, α ∈ Δ_{0,0} ∪ Δ_{0,1} ∪ Δ_{1} , and θC. Hence

(2.27)  ρ^{∗}(θ^{−1}x)ω = 0 for all x ∈ P′ = [P, P]

Let G^{C} be the simply connected complex Lie group with the Lie algebra G^{C} . Subgroups of G^{C} with tangent algebras G, K, P will be denoted by G, K, P. Note that K is compact.

The representation of G^{C} corresponding to ρ will be denoted by the same letter. Set

(2.28)  Θ = e^{ρ( \frac{π}{4} (e−f))} and f(g) = ω(Θρ(g)w)

Lemma 2.9. For the function f, the following conclusions hold:

(1) there exist functions a, b on G such that f(xgy) = a(x)b(y)f(g) for all x ∈ P, y ∈ K, g ∈ G;

(2) f(g) ≠ 0 for all g ∈ G.

Proof. By (2.26), w is a common eigenvector of ρ(η), η ∈ K; analogously, ω is a common eigenvector of ρ^{∗}(θ^{−1}ξ), ξ ∈ P, by (2.27).

So (1) may be verified by a calculation (note that ρ^{∗}(θ^{−1}ξ) = Θ^{−1}ρ^{∗}(ξ)Θ and ρ^{∗}(g) = ρ(g^{−1})^{∗}).

The function f cannot be identically zero on G^{C} since ρ is irreducible and on G because f is holomorphic. Thus f(g) ≠ 0 for all g ∈ G by (1) and the following lemma.

Lemma 2.10. G = KP. Proof. Let X be the homogeneous space G/P. Consider the action of the group K in X. By Lemma 2.8, (6), the orbit of the class P in X is open. Since K is compact, this is closed, so the action of K in X is transitive which proves the lemma.

Let f be as in Lemma 2.9. We may assume that f(e) = 1. Since f has no zeroes in G, the definition

(2.29)  F (g) = ℑ log f(g), F (e) = 0

of the function F on the universal covering group G̃ is correct.

Lemma 2.11. If l ∈ C^{∗} then F is an C -increasing function in G̃. Moreover,

(2.30)  F (xgy) = F (x) + F (g) + F (y)

for all x ∈ K̃, g ∈ G̃, y ∈ P̃, where K̃ and P̃ are subgroups in the universal covering group G̃ corresponding to subalgebras K and P. Furthermore, after the multiplication on a suitable constant and the identification of G and G^{∗} by means of the invariant form q,

(2.31)  d_{e}F = \frac{1}{2}l + iθe

Proof. First of all, note that (2.30) is a consequence of Lemma 2.9 and (2.29). Lemma 2.10 and (2.30) imply that the function F is increasing if d_{e}F ∈ C^{∗}. Thus the lemma will be proved if we’ll prove (2.31) and the inclusion iθe ∈ C_{min} .

It follows from (2.28) that d_{e}f(ζ) = ω(Θρ(ζ)w) for all ζ ∈ G^{C} . Set φ = d_{e}F. By (2.28) and (2.26), φ has the zero restriction to the semisimple part K′ = [K, K] of K; by (2.28) and (2.27), φ = 0 on P′. So restrictions of φ to K and P are scalar multipliers of functionals q(k, ·) and q(ℑ(iθm), ·) respectively, where ℑ is the imaginary part with respect to the algebra G in G^{C} . Since θl = l,

ℑ(iθm) = ℜ(θm) = \frac{1}{2}ℜθ(p − l) = −\frac{1}{2}l

according to (2.23) and Lemma 2.6. Combining this with Lemma 2.8, (2), we receive the equality k + ξ = \frac{1}{2}l + η, where ξ ∈ K^{⊥} = M, η ∈ P^{⊥} = H^{+} . It follows from Lemma 2.8, (6), that this equality determines φ up to a scalar multiplier. Since iθ(e − f) ∈ M and iθe ∈ G_{2} , by (2.24),

k + \frac{i}{2}θ(e − f) = \frac{1}{2}l + iθe

This proves (2.31).

Since ad^{2}(e)e = 0, ad^{2}(e)f = 2e, by (2.23),

lim_{t→∞} \frac{1}{t^{2}} e^{t ad(iθe)}p = 2iθe

hence iθe ∈ C_{min} by (2.25) and (2.20).

The proof of the globality could be concluded by an application of Theorem 2.1 and Corollary 5.6 (cf. [11]).

If G is tubular then l = 0. Hence l ∈ C^{∗}. So one may assume that G is nontubular proving the necessity. It will be done in Section 4 for a more general setting; in this subsection, some conditions equivalent to l ∈ C^{∗} are given.

Let V be the inverse image of the group W_{km} in the normalizer of Ad(C) in Ad(G) under the homomorphism of restriction. This is a compact group isomorphic to a semidirect product of the torus Ad(C) and the finite group W_{km}. The averaging operator over V will be denoted by A_{V} . As in Proposition 1.1, the two-dimensional subspace of G generated by k and m is denoted by V.

Proposition 2.1. Let G be a nontubular simple hermitian Lie algebra, C be an invariant cone in G, C ⊆ C_{max} . Then A_{V} = P_{V}, where P_{V} is the q - orthogonal projection to V. Moreover,

(2.32)  A_{V} (C^{∗} ∩ p^{⊥}) = C^{∗} ∩ ℝ^{+}l

and the condition l ∈ C^{∗} is equivalent to each of the following ones: (1) p ∉ Int(C);

(2) Int(C) ∩H = ∅;
(3) p^{⊥} ∩ C^{∗} ≠ {0};

Proof. Since C is the set of fixed points of Ad(C) and V is the set of fixed points of W_{km} in C by Lemma 2.1, A_{V} = P_{V} because the averaging operator over a compact group of orthogonal transformations is the orthogonal projection to the set of fixed points of this group. The group V fixes p, hence p^{⊥} is V - invariant. Therefore, A_{V} (C^{∗} ∩ p^{⊥}) = A_{V} C^{∗} ∩ p^{⊥} and (2.32) folows from the equality ℝl = p^{⊥} ∩ V and the inclusion C ⊆ C_{max} .

It follows from the equality A_{V} = P_{V} that the dual to C ∩ V cone in V is equal to C^{∗}∩V. Note also that RelInt(C ∩ V) = Int(C) ∩V since k ∈ Int(C) ∩V. The cone C ∩ V is an angle in the plane V which contains p by (2.25). Thus

(2.33)  p ∈ Int(C) if and only if l ∉ C^{∗}

and the condition l ∈ C^{∗} is equivalent to (1) as well as to (3). Since p ∈ H by (2.23), (2) implies (1); the converse is true because l⊥H (recall that l ∈ G_{0} ).

3. A necessary condition for the globality

In this section, we show that the degree of contact of a cone with some special subspaces could be an obstruction to the globality (Proposition 3.4). We start with preparatory lemmas on the geometry of cones.

Let η be an increasing function defined on some interval in ℝ with left endpoint 0, lim_{ε→0} η(ε) = 0, L be a linear subspace of an Euclidean space E. We shall say that a cone C has the degree of contact with L at the point x ∈ C greater than η if

dist(C, x + y) = o(η(|y|)) as y → 0 in L

The degree of contact of C with L at x is greater or equal to η if there exist a constant Q > 0 and a neighborhood U of zero in L such that

Qη(|y|) ≥ dist(C, x + y) for all y ∈ U

Suppose that x ∉ L; then the degree of contact is equal to η if it is greater or equal and the inverse inequality holds with some another constant. If x ∈ L then one has to replace L in this definition to any subspace L′ ⊂ L complementary to ℝx in L; clearly, the definition doesn’t depend on the choice of L′.

If η(ε) = ε^{a} then a will be called the degree of contact and denoted by cont(C, L, x); cont(C, L, x) > a (cont(C, L, x) ≥ a) will mean that the degree of contact is greater than (greater or equal to) ε^{a}.

Lemma 2.12. Suppose that cont(C, L, x) > 1. Then (1) either x ∈ Int(C) or x ∈ ∂C and (x + L) ∩ Int(C) = ∅;

(2) the orthogonal projection of C to ℝx + L includes a neighborhood of x in ℝx + L;

(3) if ξ ∈ C^{∗} and (ξ, x) = 0 then ξ ∈ L^{⊥}.

Proof. If x ∈ ∂C then there exists a hyperplane H in E such that x, 0 ∈ H, H ∩ Int(C) = ∅. Suppose that (x + L) ∩ Int(C) ≠ ∅ and consider the ray R in x + L with the origin x which is orthogonal to H ∩ (x + L) and separated from C by H. Then dist(x + y, C) > K|y| for y ∈ R and some K > 0 which is impossible because cont(C, L, x) > 1. The same construction for the projection of C to ℝx + L shows that x cannot be a boundary point of it. To prove (3), note that u ∈ L and (ξ, u) ≤ 0 imply

0 ≥ (ξ, x + tu) ≥ −|ξ| dist(x + tu, C) = o(t) as t → +0

Lemma 3.1. Let f ≥ 0 be a convex function on a neighborhood of the origin of an euclidean space E, Gr be it’s graph in ℝ × E endowed with the product Euclidean distance. Then there exist a neighborhood V ⊂ E of zero and a constant K > 0 such that f(x) ≤ K dist(x, Gr) for all x ∈ V .

Proof. We may assume that f^{−1}(0) is compact since it is possible to add to f a convex function vanishing in some neighborhood of the origin and strictly positive outside it. Then, for sufficiently small ε > 0, the set

V_{ε} = {x ∈ U : f(x) ≤ ε}

is compact. Let ε be such that both V_{ε} and V_{2ε} have this property. Set δ = inf{|x − y| : x ∈ V_{ε}, y ∉ V_{2ε}}. Then, for any x, y ∈ V_{ε}, x ≠ y,

(3.1)  \frac{|f(x) − f(y)|}{|x − y|} ≤ \frac{ε}{δ}

(to prove it, consider the straight line passing through x and y and the restriction of f to it).

Let d(x) = dist(x, Gr). For any x ∈ V_{ε}, there exists the unique x^{∗} ∈ U such that |x − x^{∗}|^{2} + f^{2}(x^{∗}) = d^{2}(x). For this x^{∗},

f(x) ≥ d(x) ≥ f(x^{∗}) and d(x) ≥ |x − x^{∗}|

and the equality in any of these inequalities holds if and only if f(x) = d(x) = f(x^{∗}) = 0, x = x^{∗}. By (3.1),

f(x) − f(x^{∗}) ≤ \frac{ε}{δ}|x − x^{∗}| ≤ \frac{ε}{δ}d(x)

Thus the assertion of the lemma holds for V = V_{ε} and K = \frac{ε}{δ} + 1.

For x ∈ C, v ∈ E, y ∈ E, set
Φ_{x,v}(y) = inf{t > 0 : x + y + tv ∈ C}

and Φ_{x,v}(y) = +∞ if x + y + tv ∉ C for all t > 0. For a subspace L ⊆ E and x ∈ E, set B(L, x, r) = {x + y : y ∈ L, |y| ≤ r}; B_{r} = {y ∈ E : |y| ≤ r}..

Lemma 3.2. Let C be a generating closed cone in E, x ∈ C, x ≠ 0, and C has the degree of contact with L at x greater than η(ε) = O(ε). Suppose that v ∈ E and x + v ∈ Int(C). Then there exists a function φ on some interval (0, ξ), ξ > 0, such that φ(ε) = o(η(ε)) as ε → 0 and

(3.2)  φ(ε)v + B(L, x, ε) ⊂ Int(C)

for all ε ∈ (0, ξ). Proof. The assertion is clear if x ∈ Int(C), so we may assume that x ∈ ∂C. Let H be a tangent hyperplane to C at x. It follows from the assumption of the lemma that cont(C, L, x) > 1, hence H ⊇ L + ℝx by Lemma 3.1, (3). Let δ be such that x + v + B_{δ} ⊂ Int(C). Then Φ_{x,v}(y) < 1 for all y ∈ B(H, x, δ), ∂C ∩ (x + y + [0, 1]v) is a single point, and ∂C ∩ (B(H, x, δ) + [0, 1]v) may be considered as the graph of the convex function Φ_{x,v}(y) on B(H, x, δ). It follows from Lemma 3.2 that

(3.3)  Φ_{x,v}(y) ≤ K dist(x + y, C) = o(η(|y|))

for all y ∈ B(L, x, δ) and some K > 0. Note also that

(3.4)  x + y + tv ∈ Int(C) if Φ_{x,v}(y) < t < 1

Let ψ be any positive increasing function on (0, δ) such that ψ(ε) = o(η(ε)). Set

φ(ε) = ψ(ε) + sup{Φ_{x,v}(y) : y ∈ B(L, x, ε)}

By (3.3) and (3.4), φ satisfies the lemma for some ξ, 0 < ξ ≤ δ.

Let G be a Lie group endowed with a left invariant Riemannian metric. For a subspace L of it’s Lie algebra G, any x ∈ G, and ε > o, let Γ(L, x, ε) be the set of all piecewise smooth curves γ : [a, b] → G such that, for both one-side derivatives and all t ∈ [a, b],

γ′(t) ∈ d_{e}λ_{γ(t)}(B(L, x, ε))

where λ_{g}(h) = gh is the left shift. It’s subset of closed curves will be denoted by Γ_{cl}(L, x, ε). Let l(γ) be the length of a curve γ. Set

Λ(L, x, ε) = inf{l(γ) : γ ∈ Γ_{cl}(L, x, ε)}

and Λ(L, x, ε) = +∞ if Γ_{cl}(L, x, ε) = ∅.

Proposition 3.1. Suppose that C is pointed and generating cone in G, x ∈ C, and

(3.5)  \frac{Q}{ε} < Λ(L, x, ε) < +∞

for some Q > 0 and all sufficiently small ε > 0. If there exists v ∈ G such that

(3.6)  [v, x] = 0, [v, L] ⊆ L, x + v ∈ Int(C)

and the degree of contact of C with L at x is greater than η(ε) = Λ(L, x, ε)^{−1} then C is not global. Proof. If x ∈ Int(C) then C ⊃ B(L, x, ε) for small ε and the assertion is true since Λ(L, x, ε) < +∞ by (3.5). So we may assume that x ∈ ∂C. Another inequality in (3.5) implies that cont(C, W, x) > 1. By Lemma 3.3, there exists a function φ(ε) = o(η(ε)) satisfying (3.2). For γ ∈ Γ_{cl}(L, x, ε), γ : [0, L] → G, set

(3.7)  γ̃(t) = γ(t) exp(φ(2ε)tv), 0 ≤ t ≤ L

Then

d_{γ̃(t)}λ_{γ̃(t)}^{−1}(γ̃′(t)) = Ad(exp(−φ(2ε)v))d_{γ(t)}λ_{γ(t)}^{−1}(γ′(t)) + φ(2ε)v

By (3.6), Ad(exp(τv))x = x and Ad(exp(τv))L ⊆ L for all τ ∈ ℝ . So there exists μ > 0 such that Ad(exp(τv))B(L, x, ε) ⊆ B(L, x, 2ε) if 0 ≤ τ ≤ μ. Since d_{γ(t)}λ_{γ(t)}^{−1}(γ′(t)) ∈ B(L, x, ε), by the choice of φ, γ̃ is timelike if

(3.8)  φ(2ε)L ≤ μ

and ε is sufficiently small. Note that

(3.9)  |γ̃′(t) − d_{e}λ_{γ̃(t)}(x)| ≤ 2ε + φ(2ε)|v|

Hence |L|x| − l(γ̃)| ≤ L(2ε + φ(2ε)|v|). Analogously, |L|x| − l(γ)| ≤ Lε. Since l(γ) may be arbitrary close to Λ(L, x, ε), for all sufficiently small ε > 0 there exists a curve γ ∈ Γ_{cl}(L, x, ε) such that (3.8) is true.

We may choose now a sequence {ε_{n}}, ε_{n} → 0 as n → ∞, and a sequence γ̃_{n} : [0, L_{n}] → G of timelike curves such that γ̃_{n}(L_{n}) = exp(φ(2 ε_{n}) L_{n}v) → e as n → ∞. Since ε_{n} → 0, by (3.7) and (3.9),

lim_{n→∞} γ̃_{n}(L_{n}−t) = exp(−tx)

for all sufficiently small t > 0. Therefore, the tangent to F(e) cone contains −x. Thus C is not global.

Remark 3.1. In general, C admits no timelike closed curves. This is true if L is a normal closed connected subgroup of G with the tangent Lie subalgebra L, the projection of C to G/L is global, and C ∩ L is global in L.

This is not clear if the converse to the Proposition 3.1 is true in the setting of left invariant cone fields. It follows from the results of this paper that for simply connected Lie groups and bi-invariant cone fields the converse is true.

We conclude this section with two geometrical lemmas which will be used in the proof of the necessity part of the main theorem.

Lemma 3.3. Let U, V be subspaces of the Euclidean space E, U∩V = {0}, C be a pointed generating cone, u, v ∈ C, cont(C, U, u) ≥ a ≥ 1 and cont(C, V, v) ≥ a. Then cont(C, U + V, u + v) ≥ a. If cont(C, U, u) > a and cont(C, V, v) > a then cont(C, U + V, u + v) > a.

Proof. This is a consequence of inequalities

|x + y| ≥ K(|x| + |y|) for some K > 0

which is true for all x ∈ U, y ∈ V since U ∩ V = {0},

dist(u + v + x + y, C) ≤ dist(u + x, C) + dist(v + y, C)

and t^{a} + s^{a} ≤ (t + s)^{a} for a ≥ 1, t, s ≥ 0.

Lemma 3.4. Let W, L be subspaces of E, W ⊂ L, C be a pointed generating cone in E, x ∈ W, and the degree of contact of the cone C ∩L with W at x in L is greater or equal to η, where 0 < η(ε) = o(ε) near zero. If P_{L}C^{∗} ∩ W^{⊥} = {0} then the degree of contact of C with W at x is greater than η.

Proof. Suppose that the degree of contact of C with W at x is not greater than η. Then there exists a sequence {x_{n}} in W and δ > 0 such that

(3.10) dist(x + x_{n}, C) ≥ δ dist(x + x_{n}, C ∩ L) > 0, x_{n} → 0 as n → ∞

For u ∈ W + x, let u^{∗} (u^{⋆}) be the nearest to u point in C (C ∩ L). Then

(3.11)  u^{∗} − u ∈ C^{∗}, u^{∗} − u⊥u^{∗}, and u^{⋆} − u ∈ L

In the triangle with vertices u, u^{∗}, u^{⋆}, the angle at the vertex u^{∗} cannot be acute since there is no points which are closer to u than u^{∗} in the interval [u^{∗}, u^{⋆}]. Hence

(3.12)  |u^{⋆} − u| cos α ≥ |u^{∗} − u|

where α is the angle at the vertex u. Set

y_{n} = (x + x_{n})^{∗} − (x + x_{n})

Then (3.12) and (3.10) imply that |P_{L}y_{n}| ≥ δ|y_{n}|. Thus the normalized sequence |y_{n}|^{−1}y_{n} has a limit point y with the nonzero projection to L. By (3.11), y ∈ C^{∗} and y⊥x, so P_{L}y ∈ (C ∩ L)^{∗} and P_{L}y ∈ W^{⊥}; the last inclusion (which gives a contradiction) follows from Lemma 3.1, (3), since cont(C ∩ L, W, x) > 1 by the assumption of the lemma.

4. Heisenberg group as an obstruction to the globality

A Lie group will be called a Heisenberg group if it is simply connected, twostep nilpotent, and has no abelian direct factors. At the level of Lie algebras, it means that the center of the algebra coincides with it’s commutator algebra. We shall receive estimates for lengths of closed curves of the type considered in Proposition 3.1.

A cone field on a manifold M is called controllable if the future F(m) of any point in M coincides with M. For simply connected nilpotent Lie groups, a criterion of controllability for a left invariant cone field defined by a generating cone C in it’s Lie algebra N is as follows (see [3])

(4.1)  [N, N] ∩ Int(C) ≠ ∅

The results of this section could be considered as a more precise version of (4.1) for Heisenberg groups; certainly, similar estimates for lengths of timelike closed curves could be done for all nilpotent groups.

Let H be a Heisenberg algebra, i.e. the Lie algebra of a Heisenberg group, Z be it’s center; by the definition above, [H, H] = Z. If W is any complementary to Z subspace of H then the multiplication in H is uniquely determined by the bilinear skew-symmetric form [ , ] : W × W → Z. Conversly, any bilinear skewsymmetric form ω : W × W → Z defines the Heisenberg algebra by the setting [H, Z] = 0 and [u, v] = ω(u, v) for all u, v ∈ W if the linear span of ω(W, W) is equal to Z. For a linear space W, there exists a free Heisenberg algebra H_{W} = W ⊕ Z_{W} with the center Z_{W} = W ∧ W and ω(u, v) = u ∧ v. Clearly, every Heisenberg algebra is isomorphic to a factor algebra of a free Heisenberg algebra by a subspace of it’s center.

Lemma 4.1. In a free Heisenberg algebra H_{W}, there exists an Euclidean norm | | such that Z_{W} ⊥ W and every z ∈ Z_{W} admits a representation

(4.2)  z = u_{1} ∧ v_{1} + ⋯ + u_{n} ∧ v_{n}

where 2n ≤ dim W, u_{k}, v_{k} ∈ W, k = 1, …, n, and

(4.3)  |z|^{2} = |u_{1}|^{4} + |v_{1}|^{4} + ⋯ + |u_{n}|^{4} + |v_{n}|^{4}

Proof. Let | | be any Euclidean norm in W and ( , ) be the corresponding scalar product. Elements of W ∧ W could be identified with skew-symmetric bilinear forms on W by

(4.4)  (u ∧ v)(ξ, η) = (ξ, u)(η, v) − (ξ, v)(η, u) for all ξ, η, u, v ∈ W

So, there is the natural isomorphism z → A_{z} between Z_{W} and the space of all skew-symmetric linear operators in W defined by the formula (A_{z}ξ, η) = z(ξ, η). Lets extend | | to H_{W} by the setting |z|^{2} = − Tr A_{z}^{2} . Then |z|^{2} = λ_{1}^{2} + ⋯ + λ_{n}^{2} , where λ_{k} > 0 and ±iλ_{1}, …, ±iλ_{n} are nonzero eigenvalues of A_{z} (so 2n ≤ dim W). For every k = 1, …, n, there exists a pair of vectors u_{k}, v_{k} ∈ W such that

A_{z}u_{k} = λ_{k}v_{k}, A_{z}v_{k} = −λ_{k}u_{k}, |u_{k}| = |v_{k}| = \sqrt{λ_{k}}, u_{k} ⊥ v_{k}

and u_{k}, v_{k}⊥u_{j}, v_{j} if k ≠ j. Since A_{u∧v}ξ = (ξ, u)v − (ξ, v)u by (4.4),

A_{u_{k}∧v_{k}}u_{k} = λ_{k}v_{k}, A_{u_{k}∧v_{k}}v_{k} = −λ_{k}u_{k}, k = 1, …, n

This data satisfies the assertion of the lemma because A_{z} = ∑_{k=1}^{n} A_{u_{k}∧v_{k}}.

Corollary 4.1. In any Heisenberg algebra H = W ⊕ Z, where W is a complementary to the center Z of H subspace, and every Euclidean norm | | in it, there exists a constant K > 0 such that each z ∈ Z admits a representation of the type

(4.5)  z = [u_{1}, v_{1}] + ⋯ + [u_{n}, v_{n}]

where u_{k}, v_{k} ∈ W, k = 1, …, n, 2n ≤ dim W, and

(4.6)  max{|u_{k}|^{2}, |v_{k}|^{2} : k = 1, …, n} ≤ K|z|

Proof. Clear.

The Heisenberg group H may be identified with H endowed with a multiplication given by the Campbell-Hausdorff formula

(4.7)  xy = x + y + \frac{1}{2}[x, y]

Then the mapping exp is identical, x^{−1} = −x, and the multiplicative commutator coincides with the Lie product

(4.8)  {x, y} = xyx^{−1}y^{−1} = [x, y]

Let | | be an Euclidean norm in H, z ∈ Z, r > 0. We keep the notation of Proposition 3.1 with G = H.

Lemma 4.2. There exists a constant K > 0 such that for any z ∈ Z \ {0} and some x_{1}, …, x_{4n} ∈ B(W, z, \sqrt{K|z|})

(4.9)  z^{−1} = x_{1} … x_{4n}

where 2n ≤ dim W. Proof. Let u_{k}, v_{k} be as in Corollary 4.1. For k = 1, …, n, set

x_{4k−3} = z + v_{k}, x_{4k−2} = z + u_{k}, x_{4k−1} = z − v_{k}, x_{4k} = z − u_{k}

Then x_{4k−3} x_{4k−2} x_{4k−1} x_{4k} = {v_{k}, u_{k}} = −[u_{k}, v_{k}] by (4.7) and (4.8). Since the group multiplication in Z is the addition, the assertion follows from Corollary 4.1 (with the same K).

Lemma 4.3. For x ∈ H, lets consider the parametrized interval \overline{x} : [0, 1] → H, \overline{x}(t) = tx, as a curve in H. If x ∈ B(W, z, r) for z ∈ Z and r > 0 then \overline{x} ∈ Γ(W, z, r) and l(\overline{x}) = |x| . Proof. Since exp is identical, ℝx with the natural parametization is an oneparametrical group. So the left invariant Riemannian and the Euclidean metrics coincide on it, l(\overline{x}) = |x| , and the restriction of the left invariant vector field on H with the vector x at the indentity to ℝx is the constant vector field x.

Suppose that γ_{1}, γ_{2} ∈ Γ(W, z, r), γ_{k} : [0, L_{k}] → H, γ_{k}(0) = 0, k = 1, 2. Then the curve γ_{1} · γ_{2} : [0, L_{1} + L_{2}] → H which coincides with γ_{1} on [0, L_{1}] and with γ_{1}(L_{1}) γ_{2}(t − L_{1}) on [L_{1}, L_{1} + L_{2}] also belongs to Γ(W, z, r). Note that

(4.10)  l(γ_{1} · γ_{2}) = l(γ_{1}) + l(γ_{2})

because the metric is left invariant.

Proposition 4.1. For any left invariant Riemannian metric on a Heisenberg group H and a subspace W complementary to the center Z of it’s Lie algebra H, there exists Q > 0 such that for every ζ ∈ Z \ {0} and ε ∈ (0, |ζ|)

(4.11)  Λ(W, ζ, ε) ≤ \frac{Q|ζ|^{2}}{ε^{2}}

Proof. Let z ∈ Z \ {0} and x_{1} … x_{4n} be as in Lemma 4.4. Then, by Lemma 4.5,

γ = \overline{z} · \overline{x}_{1} · . . . · \overline{x}_{4n} ∈ Γ_{cl}(W, z, \sqrt{K|z|})

and (4.10) together with (4.6) give the estimate

(4.12)  l(γ) = |z| + |x_{1}| + ⋯ + |x_{4n}| ≤ (4n + 1)|z| + 4n\sqrt{K|z|}

Note that the change t → rt of the parameter defines one-to-one correspondence between Γ_{cl}(W, rζ, rε) and Γ_{cl}(W, ζ, ε) for any r > 0. Set z = rζ, where r = Kε^{−2}|ζ|. Then rε = \sqrt{Kr|ζ|}, so (4.12) implies that

Λ(W, ζ, ε) ≤ (4n + 1)K \frac{|ζ|^{2}}{ε^{2}} + 4nK \frac{|ζ|}{ε} ≤ \frac{Q|ζ|^{2}}{ε^{2}}

for Q = (8n + 1)K.

Theorem 4.1. Let C be a pointed generating cone in a Lie algebra F, H ⊆ F be a Heizenberg subalgebra, H = W ⊕ Z, where Z is the center of H and W is a complementary to Z subspace of H. Suppose that z ∈ C ∩ Z, z ≠ 0, and set

N = {x ∈ F : [x, W] ⊆ W, [x, z] = 0}

If (z + N) ∩ Int(C) ≠ ∅ and cont(C, W, z) > 2 then C is not global.

Proof. Combine Proposition 3.1 and Proposition 4.1.

We are ready now to prove the necessity of (GlC) and this is convenient to do it here because the proof of the sufficiency uses other methods.

The simply connected Lie group with the Lie algebra G is denoted by G, it’s connected subgroup corresponding to a subalgebra is denoted by the same Latin letter with the same indices as the subalgebra. We keep the notation of Section 2, the Main Theorem, and Proposition 1.1 up to the end of this section. For any subspace or vector in a simple hermitian nontubular Lie algebra which was introduced in Section 2 the sum of these subspaces or vectors for all simple summands of an ideal J_{I} will be denoted by the same letter with the index I. We need also some preparatory lemmas.

Lemma 4.4. Let G be a simple hermitian nontubular Lie algebra, C be any invariant cone in G which includes c_{min} , W = G_{1} + G_{−1} , where p is defined by (2.10). Then cont(C, W, p) ≥ 2.

Proof. By (2.23), Lemma 3.3, and the equality κθe = θf, this is sufficient to prove that cont(C, G_{1}, iθe) ≥ 2. For any x ∈ C

(4.13)  t^{−2}e^{ad(tiθe)}x = t^{−2}x + t^{−1} ad(iθe)x + \frac{1}{2} ad^{2}(iθe)x ∈ C

If x = k + u, u ∈ G_{−1} then, by (2.22), (2.23), and (2.24),

t^{−2}e^{ad(tiθe)}x = t^{−2}x + t^{−1} ad(iθe)u + t^{−1} ad(iθe)k + iθe

Replacing e with f and u ∈ G_{−1} with v ∈ G_{1} and summarizing these two formulas we obtain

t^{−2}(2k + u + v) + t^{−1}(ad(iθe)u + ad(iθf)v) + p ∈ C

for all sufficiently small t > 0. Note that ad(iθe)G_{−1} = G_{1} and ad(iθe) is invertible on G_{−1} because

ad(iθh)y = [ad(iθe), ad(iθf)]y = ad(iθf) ad(iθe)y

for y ∈ G_{−1} . An analogous assertion is true for f and G_{1} . Thus, for any compact neighborhood U of zero in W, dist(C, e + tU) = O(t^{2}).

Lemma 4.5. For any simple hermitian nontubular Lie algebra G with the gradation (2.22), l, p, H^{+} , H^{−}, and c_{min} defined in Section 2, the following conclusions hold:

(1) ad(l) keeps the gradation (2.22);

(2) p + l ∈ Int(C_{min} );

(3) H^{+} and H^{−} are Heizenberg subalgebras of G;

(4) ad(l)G_{±2} = {0}.

Proof. The first assertion is true because l ∈ G_{0} by Lemma 2.8. To prove the second, note that p + l = 2k by (2.10) and that the averaging procedure over Ad(K) shows that ℝk ∩ Int(C) ≠ ∅ for any invariant cone C in G. The third is a consequence of Lemma 2.8, (7). The forth is true because l is orthogonal to all roots in Δ_{1,1} and Δ_{−1,−1} by the definition of these sets and is a fixed point of the automorphism θ.

Lemma 4.6. If I is a simple noncompact ideal of a Lie algebra with an invariant cone C then C ∩ I ≠ {0}; moreover, C ∩ I is an invariant cone in I.

Proof. Note that in (4.13) x ∈ C need not be an element of I. Thus C ∩ I ≠ {0}. Since the linear span of C ∩ I is an ideal in I, C ∩ I is generating; clearly, this cone is pointed and Ad(I)-invariant.

Proposition 4.2. Suppose that for some nontubular semisimple ideal J_{I} of a real Lie algebra G with an invariant cone C π_{I} C^{∗} ∩ L_{I} = {0}. Then C is not global.

Proof. Set W̃ = W + ℝp where W is as in Lemma 4.5. By Lemma 4.6 and (2.25), p ∈ C. So, by Lemma 4.5 and Lemma 3.3, for any nontubular ideal J_{I} of G cont(C, W̃_{I}, p_{I}) ≥ 2. Our aim is to prove that

(4.14)  cont(C, W̃_{I}, p_{I}) > 2

Suppose (4.14) is not true. Then, by Lemma 3.4 with L = J_{I}, W = W̃_{I}, there exists x ∈ C^{∗} such that π_{I} x ≠ 0 and π_{I} x⊥W̃_{I}. Let V_{I} be the group generated by groups V of Proposition 2.1 for all simple summands of J_{I}. The averaging procedure A_{V_{I}} over this group which clearly commutes with π_{I}, by Proposition 2.1, shows that

0 ≠ π_{I} A_{V_{I}} x ∈ L_{I} = W_{I}^{⊥} ∩ V_{I}

Since A_{V_{I}} x ∈ C^{∗}, this contradicts with the assumption of the proposition. Thus (4.14) is proved.

Let N = Z(J_{I}). Set F = H_{I}^{−} × P_{I} ×N (the direct product of groups). There is the unique action of F on G such that H_{I}^{−} and N act by lefts shifts and P_{I} acts by right ones. Since G = H_{I}^{−} ⊕P_{I} ⊕N, the orbit of e is open and the stabilizer is discrete. The restriction of the bi-invariant cone field corresponding C to this orbit is an invariant cone field on it. The universal covering space of the orbit coincides with the universal covering group F̃ of F. This is sufficient to prove that the left invariant cone field on F̃ defined by the cone C is not global.

By Lemma 4.5, (3), H_{I} = H_{I}^{+} ⊕ H_{I}^{−} is a Heizenberg subalgebra of F. By the same lemma, (1), [l_{I}, W_{I}] ⊆ W_{I}, where the Lie product is taken in F. By (2.23), p_{I} belongs to the center of H_{I}; by (2.23) and Lemma 4.5, (4), [l_{I}, p_{I}] = 0. Since W_{I} ⊂ W̃_{I}, (4.14) implies that cont(C, W_{I}, p_{I}) > 2. Thus it remains to prove that

(p_{I} + l_{I} +N) ∩ Int(C) ≠ ∅

to satisfy the assumption of the Theorem 4.1.

If (p_{I} + l_{I} +N)∩Int(C) = ∅ then p_{I} + l_{I} is a boundary point of the projection of the cone C to J_{I} along N. This is a contradiction with Lemma 4.5, (2), since C includes the sum of cones C_{min} for all simple summands by Lemma 4.6.

5. A sufficient condition for the globality

Let G_{1} be a closed connected normal subgroup of a Lie group G, G_{2} = G/G_{1} , ι : G_{1} → G and π : G → G_{2} be the corresponding homomorphisms of Lie algebras.

Theorem 5.1. Suppose that G_{1} admits a complementary as a vector space subalgebra in G. If C is an invariant cone in G such that πC is included to some global invariant cone in G_{2} and ι^{∗}C^{∗} includes some exact invariant cone in G_{1}^{∗} then C is global.

Note that neither ι^{∗}C^{∗} is closed nor clos(ι^{∗}C^{∗}) is pointed in general. We need some lemmas to prove this theorem.

We shall say that a smooth function f on a Lie group G has a left bounded differential if the set {d_{e}(f ∘ λ_{g}) : g ∈ G}, where λ_{g}(h) = gh is a left shift, is bounded in G^{∗}.

Lemma 5.1. For any global invariant cone C ⊂ G there exists an increasing strictly increasing at e smooth function f on G with the left bounded differential.

Proof. Let χ be the characteristic function of the set F(e), φ be a nonnegative nonzero smooth function with a compact support, U be a bounded neighborhood of zero in G, ≽ denotes the invariant order on G generated by C, and ∗ denotes the convolution. For ξ ∈ G, let L_{ξ} be the left invariant vector field on G such that L_{ξ}(e) = ξ. Clearly, f has the left bounded differential if and only if

(5.1)  sup{|L_{ξ}f(g)| : g ∈ G, ξ ∈ U} < ∞

Set f = χ ∗ φ. Then L_{ξ}f = χ ∗ L_{ξ}φ and (5.1) implies that the differential of f is left bounded. Since χ has the property that χ(g) ≥ χ(h) if g ≽ h, f is increasing. If supp φ = exp(V), where V is a sufficiently small neighborhood of zero in G, then the convolution is close the abelian one, hence f is strictly increasing at e if supp φ is a ball in a suitable Euclidean metric.

Lemma 5.2. Let L, M be linear spaces, α : L → M a linear mapping, and C be a closed cone in L. The cone αC is closed if and only if there exist compact neighborhoods of zero Q in M and K in L such that

α(C ∩ K) ⊇ (αC) ∩ Q

Proof. If x ∈ clos(αC) \ αC then the same is true for any point in the ray ℝ^{+}x. Hence αC is closed if and only if (αC) ∩ Q is compact. Standard compactness arguments show that this condition is equivalent to the following one: for any compact convex neighborhood K′ of zero in L and all sufficiently great integer n > 0, α(C ∩ (nK′)) ⊇ αC ∩ Q.

Lemma 5.3. Let C be a pointed generating cone in a linear space L = M⊕N, μ be the projection to M in L along N. Suppose that μC is closed and C ∩ N is generating in N. Then for any bounded subset Q ⊂ μC and every compact Q′ ⊂ RelInt(C ∩ N) there exists t > 0 such that Q + tQ′ ⊂ C. Moreover, for this t, Q ∩ RelInt(μC) +tQ′ ⊂ Int(C).

Proof. It follows from Lemma 5.2 that, for some compact K ⊂ C, μK ⊇ Q. Let ν = 1 − μ. Then νK ⊂ N and Q ⊆ K − νK. For some sufficiently great t > 0, tQ′ − νK ⊂ RelInt(C ∩ N) since νK is compact; therefore, Q + tQ′ ⊆ K − νK + tQ′ ⊂ C. By a suitable choice of K, we may assume that for any x ∈ RelInt(μC) ∩Q there exists y ∈ K ∩ Int(C) with μy = x; for this y, x + tQ′ ⊂ y + tQ′ − νK ⊂ Int(C).

Lemma 5.4. Let ℭ be a continuous pointed cone field on a smooth manifold M. Suppose that ℭ^{⋆}(m) ∩ Int(ℭ(m)^{∗}) ≠ ∅ for all m ∈ M, where ℭ^{⋆} is defined by (1.2). Then the cone field ℭ is global.

Proof. The assumption of the lemma could be reformulated as follows: for each m ∈ M there exists an increasing function on M which is strictly increasing at m. Let f be such a function for m ∈ M. We may assume that f(m) = 0. Set F_{ε} = {x ∈ M : f(x) ≥ ε}.

Choose a local Euclidean coordinate system near m. Let

λ ∈ Int(ℭ^{∗}(m))

Then λ considered as a function on the chosen neighborhood of m is strictly increasing in some smaller neighborhood because ℭ is continuous. Hence, for any λ_{1}, …, λ_{n} ∈ Int(ℭ^{∗}(m)), it is possible to find a neighborhood U of m, and ε > 0 such that

F(m) ⊆ F_{ε} ∪ {x ∈ U : λ_{k}(x) ≥ 0, k = 1, …, n}

and the tangent to F(m) cone cannot be greater than ℭ(m) by the Hahn-Banach theorem.

Corollary 5.1. If C is an invariant cone in a Lie algebra G such that C^{⋆} is generating then C is global.

Corollary 5.2. If ι : G_{1} → G_{2} is an isomorphic embedding of a Lie algebra G_{1} to G_{2} such that ιC_{1} ⊂ C_{2} , where C_{k} is an invariant cone in G_{k}, k = 1, 2, and C_{2}^{⋆} is generating then C_{1} is global.

Corollary 5.3. An invariant cone C is global if and only if C^{⋆}∩Int(C^{∗}) ≠ ∅

Proof. The first corollary is a consequence of the inclusion C^{⋆} ⊆ C^{∗}, the second is true because the operation dual to the intersection is the projection which clearly keeps the property of a cone to be generating, the third follows from Lemma 5.1.

Proof of Theorem 5.1. Let C_{k} ⊂ G_{k}, k = 1, 2, be global invariant cones such that ι^{∗}C^{∗} ⊇ C∗_{1} , C_{2} ⊇ πC. We may assume that G is equipped by a scalar product in such a way that V = G_{1}^{⊥} is a subalgebra of G. Let V be an open neighborhood of zero in V such that the mapping (g, v) → g exp(v) is a diffeomorphism between the inverse image E of exp(πV) under the homomorphism G → G/G_{1} and G_{1} × V . Note that it is possible to replace V by a smaller neighborhood of zero if it would be necessary. Since the cone C_{2} is global, we may assume that V has the property that any C_{2} -timelike curve starting at e which leave V cannot come back (the construction of Lemma 5.4 implies the existence of such V ). Hence any C -timelike curve in G which leaves E cannot come back and it is sufficient to prove the globality of the bi-invariant cone field ℭ corresponding to C on the set E.

Since C_{2} is pointed, C^{∗} ∩ V is generating in V. Choose φ ∈ RelInt(C_{2}^{∗}) and put φ̃(g exp(v)) = φ(v). Then φ̃ is a C_{2} -strictly increasing near e function on V . Let ρ_{g}(h) = hg be a right shift. Replacing V by a smaller neighborhood, we may assume that

Q′ = clos{d_{e}(φ̃ ∘ ρ_{exp(−v)}) : v ∈ V }

is a compact subset of RelInt(C_{2}^{∗}) ⊂ C^{∗} ∩ V. Then φ̃ is a C -increasing function on E since φ̃ ∘ λ_{g} = φ̃ for all g ∈ G_{1} .

Let η be a C_{1} -increasing strictly increasing at e smooth function on G_{1} with the left bounded differential. Set η̃(g exp(v)) = η(g). Then

η̃ ∘ ρ_{exp(u)}(g exp(v)) = η(g)

for u and v sufficiently close to zero in V because V is a subalgebra. Put

Q = clos{d_{e}(η ∘ λ_{g} ∘ ρ_{exp(−v)}) : g ∈ G_{1}, v ∈ V }

Since functions η ∘ λ_{g} ∘ ρ_{exp(−v)} are constant in exp(V) near e, Q ⊂ V^{⊥} = G_{1} , so Q ⊂ C_{1}^{∗}.

Let C′ = (ι^{∗})^{−1}(C_{1}^{∗}) ∩ C^{∗}; note that C′ ∩ V = C^{∗} ∩ V since ker ι^{∗} = V. By Lemma 5.3, Q + tQ′ ⊂ C′ ⊆ C^{∗} for some t > 0. The cone field ℭ is left and right invariant, so f = η̃ + tφ̃ is an increasing function on E. By the same lemma and the choice of η and φ, f is strictly increasing at e, hence near e in V ; replacing once more V by a smaller neighborhood, we may assume that this is true in V . Thus for any p = g exp(v) ∈ E the function f ∘ λ_{g} is increasing on E and strictly increasing at p. By Lemma 5.4, ℭ is global in E.

6. Globality of invariant cones in nonreductive algebras

This section contains the reduction of the problem to the reductive case. We use the known material on the structure of Lie algebras admitting an invariant cone formulated in Theorem 6.1. The reduction is based on the consideration of special cases of algebras sp(2n, ℝ) and their semidirect products with Heizenberg algebras.

Recall that the simplectic Lie algebra sp(2n, ℝ) is a hermitian tubular one and that an invariant cone in it is unique up to a sign. A homomorphism ρ of two algebras G_{1}, G_{2} with invariant cones C_{1}, C_{2} will be called positive if ρ(C_{1}) ⊆ C_{2} .

Theorem 6.1. Let G be a Lie algebra which admits an invariant cone C. Then

(1) G = R_{0} ⊕W⊕Z, where Z is the center of G, R_{0} is a reductive subalgebra of G, N = W⊕Z is the maximal nilpotent ideal, [W, W] ⊆ Z, and W is the unique Ad(R_{0})-invariant subspace of N complementary to Z which contains no nontrivial fixed points;

(2) if W ≠ {0} then C ∩ Z ≠ {0} and, for any w ∈ W, ad^{2}(w)C ⊆ C ∩ Z; (3) if λ ∈ Z^{∗} is strictly positive on C ∩ Z then the skew-symmetric bilinear form ω_{λ}(u, v) = λ([u, v]) on W is nondegenerate; moreover, ω_{λ}(ad(h)w, w) > 0 for every h ∈ Int(C) and w ∈ W, w ≠ 0;

(4) the representation of G in W ≅ N/Z with the form ω_{λ} induced by ad is a positive simplectic representation with the kernel I_{s} ⊕ N, where I_{s} is the maximal semisimple ideal of G;

(5) for any simple noncompact ideal J of G and the natural projection π to J along it’s centralizer Z(J), clos πC and C ∩ J are invariant cones in J;

(6) for any simple compact ideal J of G and the natural projection π to the centralizer Z(G) along J, πC = C ∩ Z(J); moreover, J ∩ C = {0};

(7) any h ∈ Int(C) is compact (i.e. ad(x) has purely imaginary eigenvalues and is semisimple).

The statement of the theorem is close to [2]; these results are known (see [6] or [3]). We keep the notation of Theorem 6.1 with an addition

(6.1)  R = R_{0} ⊕Z

up to the end of this section. The subalgebra R defined by (6.1) will be called the main reductive subalgebra of G.

Lemma 6.1. Let π be the projection onto R along W, h ∈ Int(C). Then (1) there exists the unique u ∈ W such that e^{ad(u)}h ∈ RelInt(C ∩ R);

(2) πh ∈ RelInt(C ∩ R).

Proof. Let h = r_{0} + w + z be the decomposition of h ∈ Int(C) as in Theorem 6.1, (1). Then, for any u ∈ W,

e^{ad(u)}h = r_{0} + (w + [u, r_{0}]) + (z + [u, w] + \frac{1}{2}[u, [u, h]])

Since ad(h) is nondegenerate on the factor space W/Z by Theorem 6.1, (3), ad(r_{0}) is nondegenerate on W, hence there exists the unique u ∈ W such that w = −[u, r_{0}]. So h is conjugated with h′ = r_{0} + z − \frac{1}{2}[u, [u, h]]; clearly, h′ ∈ Int(C) ∩R = RelInt(C ∩ R). By Theorem 6.1, (2), [u, [u, h]] ∈ C ∩ Z, hence

πh = h′ + \frac{1}{2}[u, [u, h]] ∈ RelInt(C ∩ R)

Let U be a real linear space with a nondegenerate skew-symmetric bilinear form ω. The Lie algebra sp(U, ω) of the Lie group Sp(U, ω) of linear transformations keeping ω could be realized as the Lie algebra Q_{2}(U) of quadratic forms on U with the Poisson brackets as a Lie product. Recall that ω defines the Poisson brackets of functions f and h by {f, h}(w) = ω(Jd_{w}f, Jd_{w}h), where J : U^{∗} → U is defined by ω(Jξ, x) = −ξ(x) for all x ∈ U. The natural isomorphism between Q_{2}(U) and sp(U, ω) corresponds to q the linear vector field Jdq.

Fix u ∈ U and consider the isotropy subgroup St_{u} of u. This group keeps the hyperplane u^{⊥} which contains u; it’s Lie algebra st_{u} is the centralizer of the form ω(u, x)^{2} . Choose v ∈ U, v ∉ u^{⊥}, such that ω(u, v) = 1 and set W = u^{⊥} ∩ v^{⊥}. Then ω is nondegenerate in W. The space Q_{2}(U) restricted to W + v may be identified with the space P_{2}(W) of polynomials of degree ≤ 2 which is also a Lie algebra with respect to the Poisson brackets in W. Set p̃(w + v) = p(w). Then, for any p ∈ P_{2}(W), p̃ admits the unique extension q ∈ Q_{2}(U) from W + v to U such that d_{u}q = 0. This construction defines an isomorphic embedding ι : P_{2}(W) → Q_{2}(U) with the image st_{u}. This construction may be realized up to an isomorphism in suitable coordinates as follows:

\{f, h\} = ∑_{k=1}^{n+1} (\frac{∂f}{∂x_{k}} \frac{∂h}{∂y_{k}} − \frac{∂h}{∂x_{k}} \frac{∂f}{∂y_{k}} )

and ι corresponds to the polynomial p_{2}(w) + p_{1}(w) + p_{0} the quadratic form p_{2}(w) + x_{n+1} p_{1}(w) + p_{0} x_{n+1}^{2} which doesn’t depend on y_{n+1} .

Each of these algebras admits an invariant cone — the cone of nonnegative functions. In Q_{2}(U), this is the unique up to a sign invariant cone, in P_{2}(W) it is not unique.

The natural decomposition

P_{2}(W) = Q_{2}(W) ⊕ Q_{1}(W) ⊕ Q_{0}(W)

where Q_{k}(W) is the space of homogenious polynomials of degree k, k = 0, 1, 2, is the decomposition of Theorem 6.1, (1), for P_{2}(W).

The algebra P_{2}(W) is the basic example of a nonreductive Lie algebra with an invariant cone. The subspace P_{1}(W) of polynomials of degree ≤ 1 is the Heisenberg algebra with the center P_{0}(W) consisting of constant functions.

Let Cq^{+} and Cp^{+} be cones of nonnegative functions in Q_{2}(U) and P_{2}(W) respectively.

The multiplication in N could be defined by the Campbell-Hausdorff formula (4.7). Let α be the representation of the simply connected group R_{0} in W with the tangent representation ad. Since G is the semidirect product of N and R_{0} via ad, a calculation leads to the following multiplication low in the simply connected group G = R_{0} × W × Z

(6.2) (r_{1}, w_{1}, z_{1})(r_{2}, w_{2}, z_{2}) = (r_{1} r_{2}, w_{1} +α(r_{1}) w_{2}, z_{1} + z_{2} +\frac{1}{2}[w_{1}, α(r_{1}) w_{2}])

Set g_{0} = (e, w_{0}, 0), g_{k} = (r_{k}, 0, z_{k}), k = 1, 2, g = (r, w, z). Then g_{0}^{−1} = (e, − w_{0}, 0). Hence

(6.3)  g_{0}^{−1}g g_{0} = (r, (1 − α(r)) w_{0} +w, z + \frac{1}{2}[w_{0}, w] − \frac{1}{2}[w_{0} +w, α(r) w_{0}])

Put β(r) = (1 − α(r))^{−1} ; then

(6.4)  α(r)β(r) = β(r) − 1 = −β(r^{−1})

Lemma 6.2. Let g_{k}, k = 0, 1, 2 be as above. Suppose that 1−α(r) is invertible. Then

(6.5)  g_{0} g_{1} g_{0}^{−1} g_{2} ∼(r_{1} r_{2}, 0, z_{1} + z_{2} +\frac{1}{2}[(β(r_{2}) − β(r_{1} r_{2}))u, u])

where ∼ means that left and right parts of (6.5) are conjugated by an element of G of the type (e, ξ, 0).

Proof. Put w_{0} = −β(r)w in (6.3). Then

(6.6)  g_{0}^{−1}g g_{0} = (r, 0, z − \frac{1}{2}[α(r) w_{0}, w_{0}]) = (r, 0, z − \frac{1}{2}[β(r)w, w])

The setting w = 0 and u = (1 − α(r)) w_{0} in (6.3) gives the equality

(6.7)  g_{0}^{−1}g g_{0} = (r, u, z + \frac{1}{2}[β(r)u, u])

By (6.2), (6.3), and (6.7),

g_{0} g_{1} g_{0}^{−1} g_{2} = (r_{1} r_{2}, u, z_{1} + z_{2} +\frac{1}{2}[β(r_{2})u, u])

Thus (6.6) implies (6.5).

Lemma 6.3. Suppose that 1 − α(r) is invertible. Then for any w ∈ W and x ∈ R_{0} such that (x + I_{s} ⊕ N) ∩ C ≠ ∅

(6.8)  −[β(r) ad(x)α(r)β(r)w, w] ∈ C ∩ Z

Proof. By the definition of α, [α(r)u, α(r)v] = [u, v] for all u, v ∈ W. Hence [α(r)u, v] = [u, α(r^{−1})v] and [(1 − α(r))u, v] = [u, (1 − α(r^{−1}))v] which implies that [β(r)u, v] = [u, β(r^{−1})v]. By (6.4),

[β(r) ad(x)α(r)β(r)w, w] = −[ad(x)β(r^{−1})w, β(r^{−1})w] = − ad^{2}(u)x

where u = β(r^{−1})w ∈ W. If ξ ∈ I_{s} ⊕ N then ad^{2}(u)ξ = 0. Thus Theorem 6.1, (2), and the assumption of the lemma implies (6.8).

For a linear operator X, let σ(X) denotes the set of eigenvalues of X, ρ(X) = max{|λ| : λ ∈ σ(X)} be the spectral radius of X. We identify sp(U, ω) and Q_{2}(U) by

q_{X}(u) = \frac{1}{2}ω(Xu, u), X ∈ sp(U, ω)

Lemma 6.4. Let U be a real linear space equipped by a skew-symmetric bilinear nondegenerate form ω and an Euclidean norm | |. Then there exists a neighborhood Q of zero in U and a norm ∥ ∥ in sp(U, ω) such that for every

X ∈ Q ∩ Int(Cq^{+} ) and u ∈ U

(6.9)  ω(β(e^{X})u, u) ≥ \frac{|u|^{2}}{∥X∥}

where β(t) = (1 − t)^{−1} .

Proof. We may assume that ω(u, v) = (Ju, v), where ( , ) is a scalar product in U such that |u|^{2} = (u, u), J : U → U, J^{2} = −1, J^{∗} = −J, and X^{∗}J +JX = 0 for all X ∈ sp(U, ω). If X ∈ Cq^{+} then JX is a nonnegative symmetric linear operator in U. Note that X ∈ Int(Cq^{+} ) implies ((JX)^{−1})^{∗} = −JX^{−1} ∈ Int(Cq^{+} ). Hence

(6.10)  (−JX^{−1}u, u) ≥ \frac{|u|^{2}}{ρ(JX)}

for all u ∈ U. Since JX is symmetric, ρ(JX) coincides with the operator norm of JX. The equality β(e^{t}) = −t^{−1} + \frac{1}{2} +o(1) implies that, for a suitable choice of a sufficiently small bounded neibourhood Q and a norm ∥ ∥ as a scalar multiplier of the operator norm, (6.9) is a consequence of (6.10).

Lemma 6.5. Let π be the natural homomorphism G → G′ = G/(I_{s} ⊕N). Then the cone clos π(C) is global in G′.

Proof. Theorem 6.1, (4), defines an embedding of πG to Q_{2}(W) such that π Int(C) ⊆ Int(Cq^{+} ). Recall that Q_{2}(W) is tubular. Hence, by Corollary 5.8, (Cq^{+})^{⋆} ≠ {0}; since Q_{2}(W) is simple, (Cq^{+})^{⋆} ≠ {0} is generating. Thus Corollary 5.7 implies the globality of clos π(C).

We need once more a known result; for a proof, see [6], Chapter 6.

Theorem 6.2. Let C be an invariant cone in a Lie algebra G, S be the semigroup topologically generated by exp(C). If C is global then there exists a neighborhood V of zero in G such that exp(C ∩ V) = S ∩ exp(V).

Lemma 6.6. Let π be as in Lemma 6.5. There exists a neighborhood U of zero in R_{0} such that r_{1} = exp(ξ_{1}), r_{2} = exp(ξ_{2}), ξ_{1}, ξ_{2} ∈ U, and π ξ_{1}, π ξ_{2} ∈ Int(πC) implies

(6.11)  [(β(r_{2}) − β(r_{1} r_{2}))w, w]) ∈ C ∩ Z

for all w ∈ W.

Proof. It follows from the matrix equality

(1 − BA)^{−1} − (1 − A)^{−1} = (1 − A)^{−1}(B − 1)A(1 − BA)^{−1}

that

\frac{d}{dt}(1 − e^{tX}A)_{t=0}^{−1} = (1 − A)^{−1}XA(1 − A)^{−1}

Hence for any x ∈ R_{0} and r ∈ R_{0} such that 1 − α(r) is invertible in W,

\frac{d}{dt}β(exp(tx)r)_{t=0} = β(r) ad(x)α(r)β(r)

Put x = ξ_{1} and r = r_{2} . Since π ξ_{k} ∈ πC, k = 1, 2, by Lemma 6.3,

−\frac{d}{dt}[β(exp(t ξ_{1}) r_{2})w, w] ∈ C ∩ Z

for all w ∈ W. Note also that π ξ_{k} ∈ π Int(C), k = 1, 2, implies invertibility of 1 − α(r_{k}) by Theorem 6.1, (3), and Lemma 6.4 if U is sufficiently small.

Set r_{t} = exp(t ξ_{1}) r_{2} . The integration by t and the same consideration show that (6.11) is true if r_{t} = exp(ξ_{t}) with ξ_{t} ∈ U such that π ξ_{t} ∈ Int(πC) for all t ∈ [0, 1].

By Lemma 6.5, the cone clos(πC) is global. Let V be as in Theorem 6.2 for the algebra G′ in this lemma and chose a convex neighborhood U of zero in R_{0} with the property exp(πU) exp(πU) ⊆ exp(V). It follows from Theorem 6.2 that exp(Int(πC) ∩V) = Int(exp(πC ∩ V)). Therefore,

exp(Int(πC) ∩U) exp(Int(πC) ∩U) ⊆ exp(Int(πC) ∩V)

Thus ξ_{t} has the desired property if ξ_{1}, ξ_{2} ∈ U, π ξ_{1}, π ξ_{2} ∈ Int(πC), and U is sufficiently small.

Lets denote by S∘_{R} the semigroup in R algebraically generated by the set exp(RelInt(C ∩ R)), S_{R} = clos(S∘_{R}),

(6.12)  S^{∘} = {g_{0} sg_{0}^{−1} : s ∈ S∘_{R}, g_{0} = (e, w_{0}, 0), w_{0} ∈ W}

We keep the notation of the Lemma 6.5. The semigroup topologically generated by exp(πC) in the simply connected group G′ corresponding to the Lie algebra G′ will be denoted by S′. Let p denotes the homomorphism of simply connected groups with the tangent homorphism π.

Lemma 6.7. There exists a neighborhood V of zero in G′ such that

(6.13)  S_{0} = S^{∘} ∪ p^{−1}(S′ \ exp(V))

is a semigroup in G.

Proof. Set g_{k} = (r_{k}, 0, z_{k}), k = 1, 2, and suppose that g_{1}, g_{2} ∈ S∘_{R}. Since S_{0} is invariant under all inner automorphisms and p(S_{0}) ⊆ S′, this is sufficient to find U such that

(6.14)  g_{0} g_{1} g_{0}^{−1}g_{2} ∈ S_{0}

if p(g_{1}), p(g_{2}) ∈ U and g_{0} is as in (6.12).

Note that S_{R} ⊃ C ∩ Z since exp is identical in N. Let U be as in Lemma 6.6. Then, by (6.5) and (6.11),

g_{0} g_{1} g_{0}^{−1} g_{2} ∼(r_{1} r_{2}, 0, z_{1} + z_{2} +\frac{1}{2}[(β(r_{2}) − β(r_{1} r_{2}))u, u]) ∈ g_{1} g_{2} exp(C ∩ Z)

if r_{1} r_{2} ∈ exp(U). Note that g_{2} exp(C ∩ Z) ⊂ S∘_{R} since g_{2} ∈ exp(Int(C) ∩U) by the choice of U and Theorem 6.8, (1). Hence g_{1} g_{2} exp(C ∩ Z) ⊂ S∘_{R}. It remains to set V = πU to satisfy (6.14).

Theorem 6.3. Let G be a nonreductive Lie algebra with an invariant cone C, R be the main reductive subalgebra of G. Then C is global in G if and only if C ∩ R is global in R.

Proof. If C is global in G then C ∩ R is global in R by Lemma 6.1 and Corollary 5.2.

Suppose that C ∩ R is global. Let S be the closed semigroup in G generated by exp(C), S̃ = clos(S_{0}), where S_{0} is defined by (6.13). By Lemma 6.1, (1), S̃ ⊇ exp(Int(C)), so Lemma 6.7 implies that S̃ ⊇ S. Clearly, S ⊃ S^{∘}, hence

(6.15)  S̃ ∩ p^{−1}(exp(V)) = S ∩ p^{−1}(exp(V))

where p and V are as in Lemma 6.7. Furthermore, by Theorem 6.2 there exists a neighborhood U of zero in R such that

(6.16)  exp(U) ∩ S_{R} = exp(C ∩ U)

We may assume that U is convex, open in R, exp is diffeomorphic on U, and πU ⊆ V , where π = d_{e}p. Moreover, the existence of an increasing strictly increasing at e function (Lemma 5.3) implies the existence of a neighborhood U′ of zero in R such that

(6.17)  g_{1}, g_{2} ∈ S_{R}, g_{1} g_{2} ∈ exp(U′) ⟹ g_{1}, g_{2} ∈ exp(U) ∩ S_{R}

We may also assume that U′ has the same properties as U. Set

Ũ = {(r, w, z) : (r, 0, z) ∈ exp(U′), w ∈ W}

Let g = (r, w, z) ∈ S^{∘} ∩ exp(Ũ). The neighborhood V in Lemma 6.7 could be supposed to be a diffeomorphic image of a sufficiently small neighborhood of zero in G′ by exp. By Theorem 6.1, (3), we may assume that 1 − α(r) is invertible for all r ∈ Int(exp(C ∩ U′)). Then, by (6.6) and the definition of S^{∘},

(6.18)  g ∼ h = (r, 0, z − \frac{1}{2}[β(r)w, w]) ∈ S∘_{R}

and there exists the unique u ∈ W such that g = (e, u, 0)^{−1}h(e, u, 0). It follows from Lemma 6.4 and Theorem 6.1, (3), that λ([β(r)w, w]) ≥ 0 for any strictly positive on C ∩ Z linear functional λ ∈ Z^{∗} if V is sufficiently small. Hence

ζ = \frac{1}{2}[β(r)w, w] ∈ C ∩ Z

Recall that exp is identical on N. Thus

(6.19)  exp(ζ) ∈ S_{R}

and

(6.20)  (r, 0, z) = h exp(ζ) ∈ S∘_{R} ∩ exp(U′)

By (6.20), (6.19), (6.18), and (6.17), h ∈ exp(U) ∩ S∘_{R}. Hence h ∈ exp(C ∩ R) by (6.16). Therefore,

(6.21)  g ∈ exp(Int(C))

Lets fix λ ∈ Z^{∗} strictly positive on C ∩ Z. It follows from Lemma 6.4 that

(6.22)  |w|^{2} ≤ Kλ(ζ)

for some K > 0. Hence (6.21) together with (6.22) imply that S^{∘} ∩ Ũ can be included into a neighborhood Q of e in G which is a diffeomorphic image of a neighborhood of zero in G if U′ is sufficiently small. By (6.21),

Q ∩ clos(S^{∘}) ⊆ exp(C)

Thus (6.15) implies the globality of C.

7. Proof of the Main Theorem

In this final section we deal with the remainder of the proof of the Main Theorem. Recall that the necessity of (GlC) was proved in Section 4. The sufficiency was partially reduced to the reductive case in Section 6; for the complete reduction, we need two more lemmas. We keep the notation of the Main Theorem and Theorem 6.1. So I_{s} is the maximal semisimple ideal of G. Let I = I_{s} ⊕ Z, π be the projection in G to I dual to the embedding. The dual space R^{∗} to the main reductive subalgebra R is identified with R by an invariant bilinear form which is positive definite on it’s maximal compactly embedded subalgebra, hence on Z. Then the projection π restricted to R is an orthogonal one.

Lemma 7.1. Suppose that πC^{∗} includes some exact invariant cone. Then C is global.

Proof. By Lemma 6.5, the projection of the cone C to G′ = G/(I_{s} ⊕ N) is included to some global invariant cone in G′. Lemma 6.1, (2), implies that the same is true for the cone C ∩R. Note that (I_{s} ⊕N)∩R = I. Thus Theorem 5.1 and the assumption of the lemma implies the globality of C∩R. By Theorem 6.3, C is global in G.

Clearly, a convex cone is exact if and only if it is closed and the dual cone is global. Thus the following lemma shows that if (GlC) implies the globality for reductive algebras then the assumption of Lemma 7.1 is satisfied.

Lemma 7.2. Let G = I⊕J, where I, J are ideals of G, π be the projection to I along J, C be an invariant cone in G. Suppose that I is reductive and has no compact simple ideals. Let Λ ⊂ πC be a finite set consisting of compact nonzero elements which could be included into some pointed cone. Then πC includes the closed Ad(I)-invariant cone generated by Λ; moreover, this cone is pointed.

Proof. Suppose first that I is semisimple. It follows from Theorem 6.1, (5), that clos πC is a pointed generating cone which includes the sum of minimal cones and is included to the sum of maximal ones. Therefore, for all x ∈ Λ ρ(ad(x)) > 0 (the spectral radius ρ is defined in Section 6). Hence the set Ad(I)Λ could be separated from zero by a neighborhood U which may be choosen in such a way that πC \ U is convex since clos πC is pointed. Thus the closure K of the convex hull of Ad(G)Λ doesn’t contain zero.

Let Λ′ be a finite subset of C such that πΛ′ = Λ, C′ be the closed Ad(J)-invariant cone generated by Λ′, Q be a compact convex neiborhood of zero in J such that P = Q ⊕ K ⊃ Λ′. Then clos(ℝ^{+}P) is a pointed cone with the trivial intersection with J and C′ ⊆ clos(ℝ^{+}P), so C′ ∩ J = {0}. Thus πC′ is pointed and the closed Ad(I)-invariant cone generating by Λ coincides with πC′.

For a reductive I, let Λ_{0} be the intersection of Λ with the center of I, Λ_{1} = Λ \ Λ_{0} . By the assumption of the lemma, Λ_{0} is included into some pointed cone in the center. By the consideration above, the Ad(I)-invariant closed cone generated by the projection of Λ_{1} to the semisimple part of I is pointed. Hence Λ_{1} is included to the sum of the two cones above, Ad(I)-invariant cone generated by Λ is pointed, and the closed convex hull of the orbit of Λ_{1} can be separated from zero. It remains to repeat once more the construction of the previous paragraph.

In the last lemma we consider a special case of a reductive algebra R. Let R = S ⊕ A, where S is semisimple and A abelian. We identify R and R^{∗} as above. Let σ be the orhogonal projection R → S.

Lemma 7.3. Suppose that each simple ideal of S is nontubular. Let L be the same as in (GlC) for S. If dim A ≤ 1 and σC ∩ L has nonempty interior in L then C is exact.

Proof. Let l ∈ σC ∩ L. Then l = s_{1}l_{1} + ⋯ + s_{n}l_{n}, where l_{k} are defined by (2.10) for the simple summand J_{k}. Since σC ∩ L has a nonempty interior in L, by a choice of a sign of l_{k}, we may assume that s_{k} > 0, k = 1, …, n. Let f_{k} be the function (2.29) for J_{k}, f = 2(s_{1}f_{1} + ⋯ + s_{n}f_{n}). It follows from the construction of f that for all x ∈ S, u ∈ K, and v ∈ P′, where K, P′ are products of subgroups K, P in Lemma 2.10 for all simple summands,

(7.1)  f(uxv) = f(x) + f(u) + f(v) and d_{e}f = l + ξ

where ξ ∈ C_{min} , C_{min} is the sum of minimal cones in all simple summands. Since any invariant cone includes the minimal one, Theorem 6.1, (5), (7), and Lemma 7.2 imply that ξ ∈ σC. Set P = P′ × Z. There exists l_{0} ∈ C such that σl_{0} = l. Put l_{1} = l_{0} − l. Then l_{1} ∈ A and could be considered as a homomorphism R → ℝ .

If l_{1} = 0 then l ∈ C. By Theorem 6.1, (5), Ad(R)-invariant closed cone generated by l is generating in S. Hence C ∩ S is exact by Corollary 5.7. This implies that the projection of the predual cone to S is included to the global cone (C ∩ S)^{∗}. Since the projection of C to A clearly includes some exact cone, the assumption of the Theorem 5.1 is satisfied for the predual cone, and C is exact.

Suppose that l_{1} ≠ 0. The function F (x) = f(x)+l_{1}(x) satisfies (7.1) for all u ∈ K, v ∈ P, and d_{e}F = l_{0}+ξ. By Lemma 2.11, F is an increasing function on R. By the definition of l, l_{0}, l_{1} , d_{e}F has nonzero projections to each simple summand of S. By Theorem 6.1, (5), the intersection of the Ad(G)-invariant cone C_{0} generated by d_{e}F with S is generating in S. Since codim S = 1, C_{0} is generating in R. Therefore, the cone generated by the orbit of d_{e}F has a nonempty interior in R (although it need not be closed). By Corollary 5.6, C is exact.

Proof of Main Theorem. The necessity of (GlC) was already proved (Proposition 4.2). If (GlC) is true for G then for any I ⊆ \overline{n} there exists l_{I} ∈ π_{I} C ∩ L_{I}. Set Λ = \{l_{I} : I ⊆ \overline{n}\}. By Lemma 7.2, there exists a closed invariant cone in I = I_{s} + Z which includes Λ. For this cone (GlC) is satisfied. Hence, by Lemma 7.1, this is sufficient to prove that (GlC) implies the globality in I. Thus we may assume that G is reductive proving the necessity.

If J is a compact simple ideal of G then the averaging procedure over the corresponding compact normal subgroup and Theorem 6.1, (6), show that the projection of C^{⋆} to the centralizer Z(J) coincides with C^{⋆} ∩ Z(J). So C is global in G if and only if C ∩ Z(J) is global in Z(J) and it is possible to assume that G has no simple compact ideals.

Suppose that G is semisimple and each simple ideal of G is noncompact tubular. Then Theorem 6.1, (5), Theorem 5.1, and induction arguments show that each invariant cone in G is global.

Since in abelian algebras any pointed generating cone is exact, Theorem 5.1 implies that the same is true for reductive algebras with the semisimple part which contains no nontubular ideals.

We prove the sufficiency of (GlC) by the induction on the number of simple nontubular summands in G; it’s base is provided by the consideration above.

Recall that J_{1}, …, J_{n} denote all nontubular ideals of G. We keep the notation of Main Theorem.

Suppose that (GlC) is satisfied for G. Let π be the projection to the semisimple part of G and l ∈ πC ∩ L\overline{n}, l ≠ 0, \overline{n} = \{1, …, n\}, I be the set of indices k such that l has the nonzero projection to J_{k}, K = \overline{n} \ I. Note that, by Theorem 6.1, (5), l generates a nontrivial invariant cone in J_{I}, hence π_{I} C has the nonempty interior in J_{I}.

Choose l_{0} ∈ C such that π_{I} l_{0} = l. Then the assumption of Lemma 7.3 is satisfied for the cone C ∩R_{0} in the linear span R_{0} of J_{I} and l_{0} . Hence C ∩R_{0} is exact. This means that the projection of the predual cone to R_{0} is included to a global one. Let π_{1} be the orthogonal projection to R_{1} = R_{0}^{⊥}. By Theorem 5.1, this is sufficient to prove that C_{1} = π_{1} C includes some exact cone in R_{1} .

Note that the element l defined by (2.10) is compact. Using the averaging over the maximal compact subgroup of a complementary semisimple summand and Theorem 6.1, (7), one may find a compact inverse image for each l_{I} ∈ L_{I}, I ⊂ \overline{n}. So there exists a finite set Λ ⊂ C_{1} consisting of compact elements such that for any J ⊂ K π_{J} Λ ∩ L_{J} ≠ {0}. Since it is possible to add to Λ any finite number of points with compact projections to R_{1} , we may also assume that the convex closed generated by Λ Ad(R_{1})-invariant cone in R_{1} is generating and has trivial intersection with the center of R_{1} if R_{1} is not abelian. By Lemma 7.2, this cone is pointed. Since (GlC) is satisfied for it, this cone is exact by the induction hypothesis.

The nontrivial part of assertions (1) and (2) of the theorem is that π_{I} C includes some closed invariant cone satisfying (GlC) for any I ⊆ \overline{n}. This is a consequence of Lemma 7.2 and could be proved as above.

Proof of Proposition 1.1. Note that the procedure analogous to the procedure of the proposition for global invariant cones in G finishes with I = ∅ if (GlC) is true since π_{I} C^{∗} ∩ L_{I} ≠ ∅ in each step. Conversly, if π_{K} C^{∗} ∩ K_{I} = ∅ for some K ⊆ \overline{n} then K ⊆ I in each step because K ∩ I′ = ∅, hence the procedure finishes with I′ = ∅.

The averaging over Ad(C) for a compactly embedded Cartan subalgebra C and, after that, over groups W_{km} for each simple summand as in the proof of Proposition 2.1 show that the projection of C to the sum of any family of spaces V_{k} coincides with the intersection with this sum. Therefore, π_{I} C = π_{I} C′ and the procedure of the proposition must finish with the same result as for invariant cones.

References

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Omsk State University prosp. Mira 55a 644077, Omsk, Russia gichev@univer.omsk.su

Received December 20, 1994 and in final form May 16, 1995