Elements in Pointed Invariant Cones in Lie Algebras and Corresponding Affine Pairs

In this note, we study in a finite dimensional Lie algebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}$$\end{document} the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone Cx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_x$$\end{document}. Assuming that g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}$$\end{document} is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying [h,x]=x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[h,x]=x$$\end{document} for which Cx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_x$$\end{document} pointed. Given x, we show that such elements h can be constructed in such a way that adh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {\mathrm{ad}}\nolimits h$$\end{document} defines a 5-grading, and characterize the cases where we even get a 3-grading.


Introduction
Convexity properties of adjoint orbits O x = Inn(g)x in a finite dimensional real Lie algebra, where Inn(g) = e ad g is the group of inner automorphisms, play a role in many contexts. Most directly, they appear in the theory of invariant convex cones. For instance, if U : G → U(H) is a unitary representation and ∂U (x) denotes the infinitesimal generator of the unitary one-parameter group (U (exp t x)) t∈R , then the positive cone of U is a closed convex invariant cone in g which is pointed (contains no affine lines) if and only if ker(U ) is discrete. In the literature pointed generating invariant cones have been studied from the perspective of their interior: If W ⊆ g is pointed and generating, then its interior consists of elliptic elements (ad x is semisimple with imaginary spectrum) and W is determined by its intersection with a compactly embedded Cartan subalgebra [5,12]. Unfortunately, this theory provides not much information on the non-elliptic elements in the boundary of W . A notable exception is [6] which, for a simple hermitian Lie algebra, provides a classification of all nilpotent adjoint orbits in an invariant cone.
In the present paper, we address adjoint orbits in invariant cones. For more precise formulations, we introduce some notation. For x ∈ g, we write We call a closed convex subset C of a real linear space V pointed if it contains no non-trivial affine lines and generating if span C = V . We study the subsets g co := {x ∈ g : co(x) pointed} ⊇ g c := {x ∈ g : C x pointed} (1.2) and characterize the elements in this set in terms of explicitly available data. If g is a simple Lie algebra, then we have to distinguish three cases. If g is compact, then all sets co(x) are compact, hence pointed, so that g co = g. As every convex cone invariant under a compact group contains a fixed point in its interior, we have g c = {0} for compact Lie algebras. If g is non-compact, then g c = {0} is equivalent to g being hermitian, i.e., maximal compactly embedded subalgebras have non-trivial center. In this case g c = g co is a double cone (cf. Lemma 3.7, Kostant-Vinberg Theorem). If g is neither compact nor hermitian, then g c = g co = {0}.
For a direct sum g = g 1 ⊕ g 2 we have g co = g 1,co × g 2,co . This reduces for a reductive Lie algebra the determination of this set to the case of simple hermitian Lie algebras. However, the determination of the subset g c of g co is less obvious for reductive Lie algebras (Proposition 3.14).
This discussion shows that, among the reductive Lie algebras only the quasihermitian ones (all simple ideals are either compact or hermitian), play a role in our context. Beyond reductive Lie algebras, the natural context for our investigation is the class of admissible Lie algebras, i.e., those containing an Inn(g)-invariant pointed generating closed convex subset C (cf. [12,Def. VII.3.2]). Since every pointed invariant convex subset spans an admissible ideal, we shall assume throughout that g is admissible.
It is of vital importance for our arguments, that admissible Lie algebras permit a powerful structure theory. Their coarse structure is given by g = g(l, V , z, β) = z ⊕ V ⊕ l, where l is a reductive Lie algebra, V an l-module, z a vector space, and β : V × V → z an l-invariant skew-symmetric bilinear map; the Lie bracket on g is given by For their fine structure, we use the existence of a compactly embedded Cartan subalgebra t = z ⊕ t l and the corresponding root decomposition.
The key observation underlying our analysis of the sets g c and g co in Sect. 3, is the Reduction Theorem 3.2, asserting that every adjoint orbit in g co intersects the reductive subalgebra z + l. We therefore take in Sect. 3.2 a closer look at the reductive case, where we provide in Propositions 3.10 and 3.14 a complete description of the sets g c and g co . The central result in Sect. 3 is the Characterization Theorem 3.20 that characterizes elements x in g c and g co in terms of the closed convex hull co z (x) of the z-valued Hamiltonian function In particular, we show that co(x) is pointed if and only if co z (x) is pointed and that C x is pointed if, in addition, co z (x) generates a pointed cone whenever the l-component x l of x is nilpotent. We also discuss to which extent the pointedness of C x implies the existence of a pointed generating invariant cone W ⊆ g containing x; which is not always the case (Example 3.25).
In Sect. 4 we study affine pairs (x, h) for a pointed invariant cone W ⊆ g. These pairs are characterized by the relations (1.4) The interest in these pairs stems from their relevance in Algebraic Quantum Field Theory (AQFT), where they arise from unitary representations (U , H) of a corresponding Lie group G and their positive cones W = C U (see (1.1)). If U extends to an antiunitary representation of G {id G , τ G }, where τ G ∈ Aut(G) is an involution and the corresponding involution τ ∈ Aut(g) satisfies τ (h) = h and τ (x) = −x, then we can associate with h a so-called standard subspace and these subspaces encode localization data in QFT [2]. In this context is called a Borchers pair or a standard pair [8], and one would like to understand all those pairs arising from a given unitary representation (see [10] for more details). This leads naturally to the problem to describe and classify affine pairs. For an affine pair (x, h) the element x is nilpotent, hence in particular not elliptic if it is not central. As we know from [11], the most important affine pairs are those for which h in an Euler element of g, i.e., ad h is diagonalizable with possible eigenvalues {−1, 0, 1} and the Lie algebra g is generated by h and the cones C U ∩ g ±1 (h) (see [15,17] for related classification results). Considering representations with discrete kernel then leads to the situation, where the cone C U ⊆ g is pointed, so that g = g U Rh, and g U = C U − C U is an ideal containing the pointed generating invariant cone C U . This motivates our investigations in Sect. 4, where we start with a nilpotent element x ∈ W , W a pointed generating invariant cone, and then consider derivations D on g satisfying Dx = x and e RD W = W . Our first main result on affine pairs is the Existence Theorem 4.7, asserting that, for any nilpotent element x ∈ g co there exists such a derivation D with Spec(D) ⊆ 0, ± 1 2 , ±1 . The second main result characterizes the existence of Euler derivations with this property, i.e., where we even have Spec(D) ⊆ {0, ±1} (Theorem 4.17).

Notation:
• For a Lie algebra g, we write Inn(g) = e ad g for the group of inner automorphisms. For a Lie subalgebra h ⊆ g, we write Inn g (h) ⊆ Inn(g) for the subgroup generated by e ad h . "Invariance" of subsets of g always refers to the group Inn(g). • A subalgebra k ⊆ g is said to be compactly embedded if the closure of Inn g (k) in Aut(g) is compact. A compactly embedded Cartan subalgebra is a compactly embedded subalgebra which is also maximal abelian.
• A closed convex cone in a finite dimensional real vector space is simply called a cone. We write cone(S) = R + conv(S) for the closed convex cone generated by a subset S and for the dual cone.

Structure of Admissible Lie Algebras
In this section, we collect some relevant results on the structure of admissible Lie algebras.

Definition 2.1
Let C be a closed convex subset of the finite dimensional real vector space V . Then the recession cone of C is We

Remark 2.3
If C ⊆ g is an invariant pointed closed convex subset, then g C := span C g is an ideal in which C is also generating, so that g C is an admissible Lie algebra. [18]) Let l be a Lie algebra, V an l-module, z a vector space, and β : V × V → z an l-invariant skew-symmetric bilinear map. Then z × V × l is a Lie algebra with respect to the bracket
The following theorem describes the structure of non-reductive admissible Lie algebras.
Theorem 2.5 Any admissible Lie algebra g is of the form g(l, V , z, β), where (a) z = z(g), u = z + V is the maximal nilpotent ideal. (b) l is reductive and quasihermitian, i.e., all simple ideals are compact or hermitian. (c) l contains a compactly embedded Cartan subalgebra t l , and t := z + t l is a compactly embedded Cartan subalgebra of g.
There exists an element f ∈ z * such that (V , f • β) is a symplectic l-module of convex type, i.e., there exists an element x ∈ l, such that the Hamiltonian function is positive definite.
Proof In the context of Theorem 2.5(e), we see that w is admissible, then the following assertions hold: (a) Every abelian ideal of g is central.
Proof (a) Let a g be an abelian ideal and C ⊆ g be a pointed generating invariant closed convex subset. For x ∈ C we then have e ad a implies that a is abelian. In view of (a), a is central, so that a ⊆ V ∩ z = {0}. Definition 2.8 (a) Let t ⊆ g be a compactly embedded Cartan subalgebra, g C the complexification of g, z = x + iy → z * := −x + iy the corresponding involution, and t C the corresponding Cartan subalgebra of g C . For a linear functional α ∈ t * C , we define the root space g α C := {x ∈ g C : (∀y ∈ t C ) [y, x] = α(y)x} and write for the set of roots of g.
contains a unique element α ∨ with α(α ∨ ) = 2 which we call the coroot of α. We write s for the set of semisimple roots and call the roots in r := \ s the solvable roots.
(c) For each compact root α ∈ k , the linear mapping s α : t → t, x → x − α(x)α ∨ is a reflection in the hyperplane ker α. We write W k for the group generated by these reflections. It is called the Weyl group of the pair (k, t). According to [12, Prop. VII.2.10], this group is finite.

Definition 2.9 (a) A subset + ⊆
is called a positive system if there exists an element x 0 ∈ it with + = {α ∈ : α(x 0 ) > 0} and α(x 0 ) = 0 holds for all α ∈ . A positive system + is said to be adapted if for α ∈ k and β ∈ + p we have β(x 0 ) > α(x 0 ) for some x 0 defining + . In this case, we call + p := + ∩ p an adapted system of positive non-compact roots. (b) We associate with an adapted system + p of positive non-compact roots the convex cones and The structure theoretic concepts introduced above play a crucial role in the analysis of invariant convex subsets. In particular, [12,Thm. VII.3.8] asserts that the existence of a pointed generating Inn(g)-invariant closed convex cone W ⊆ g implies the existence of a compactly embedded Cartan subalgebra (cf. Theorem 2.5), and that, for every compactly embedded Cartan subalgebra t, there exists an adapted positive system + where p t : g → t denotes the projection with kernel [t, g]. Moreover, W is uniquely determined by W ∩ t, the cone C min is pointed. By [12, Thm. VIII.2.12] (cf. also [12, Thm. VIII.3.7]), for an adapted positive system + p and an admissible Lie algebra, the pointedness of C min implies that C min ⊆ C max . Note that is pointed if and only if C min,z is pointed because {α ∨ : α ∈ + p,s } is a finite subset contained in an open half space. If this condition is satisfied, then [12,Prop. VIII.3.7] shows that is a generating closed convex invariant cone with W max ∩ t = C max , and is a pointed, closed convex invariant cone with W min ∩ t = C min . In general, W min is not generating. The most extreme situation occurs if g is a compact Lie algebra. Then W min = {0} and W max = g.

Elements in Pointed Cones
In this section, we study elements x in an admissible Lie algebra g = g(l, V , z, β) for which co(x) is pointed. Splitting l into an ideal l 0 commuting with V and an ideal l 1 acting effectively on V , reduces this problem to the two cases, where either g = l is reductive (Sect. 3.2) or where the reductive subalgebra l acts faithfully on V (Sect. 3.3).

General Observations
Theorem 2.5 provides powerful structural information that is crucial to analyze the subsets g c and g co for non-reductive admissible Lie algebras. Throughout this section we write The kernel l 0 l of the representation of l on V has a complementary ideal l 1 , and g is a direct sum (3.1) Any x ∈ g decomposes accordingly as x = x 1 + x 0 with x 1 ∈ g 1 and x 0 ∈ l 0 , and implies that This reduces the description of this set to the two cases, where g is reductive or the representation of l on V is faithful.
Then the following assertions hold: To see that we also have V This rather simple formula will be a key tool throughout this paper. We conclude in particular that e ad y x ∈ z + l is equivalent to the vanishing of the V -component, i.e., to As Inn(g) = e ad V Inn g (l) , If this condition is satisfied, then (3.4), applied with x V = 0, shows that e ad V x ∩(z+l) is a single element. For x ∈ z + l, we thus obtain The following Reduction Theorem can be used to reduce many question concerning the sets co(x) and C x to elements in reductive Lie algebras.
follows from Lemma 3.1(b) and the assertion follows from (3.5).

Corollary 3.3
Let g ∼ = g(l, V , z, β) be an admissible Lie algebra. Then every adnilpotent element x ∈ g with co(x) pointed is conjugate under inner automorphisms to an element of z + s for s = [l, l]. Any ad-nilpotent element of z + l is contained in z + s. Proof By Theorem 3.2, we may assume that x = x z + x l ∈ z + l, i.e., that x V = 0.
is the Jordan decomposition of ad(x l ). Therefore, the nilpotency of this element implies ad(x 0 ) = 0, hence that x 0 = 0 because z(l) ∩ z(g) ⊆ l ∩ z = {0}. We conclude that x l = x s ∈ s and thus x ∈ z + s.
Proof We may assume that the nilpotent element x s ∈ s is non-zero, otherwise the assertion is trivial.
Lemma 3.5 If x is a nilpotent element of the pointed generating invariant cone W ⊆ g and x s = 0, then there exists a Lie subalgebra m ⊆ g, isomorphic to gl 2 (R), such that z(m) ⊆ z(g) and m ∩ W is pointed and generating.
Proof By Corollary 3.3, we may assume that x = x z + x s ∈ z(g) + s holds for a Levi complement s. We first choose an sl 2 (R)-subalgebra s x ⊆ s containing x s (cf. Proposition B.1). If x z = 0, then Further, Corollary 3.4 implies that m ∩ W contains x z and x s , hence is generating in m.
The following observation provides some information on the central part of lim(co(x)).

Lemma 3.6 For x
Proof For t → ∞, formula (3.4) leads to

Reductive Lie Algebras
If g is a simple real Lie algebra and k ⊆ g a maximal compactly embedded subalgebra, then the existence of a pointed generating invariant cone W implies the existence of a non-zero element z ∈ z(k), i.e., that g is hermitian. If this is the case, then is a minimal invariant cone, the dual cone with respect to the non-degenerate form κ(x, y) = − tr(ad x ad y) is a maximal invariant cone containing W min , and any other pointed generating invariant cone W either satisfies (a) C x is pointed.
(b) C x s and C x n are pointed and, if x s = 0, then x n ∈ C x s .
(a) ⇔ (c) follows from the Kostant-Vinberg Theorem ( [5, Thm. III.4.7], [19]). That (d) follows from (a) is clear. Suppose that co(x) is pointed. We may assume that x = 0, and observe that this implies that co(x) has interior points. Let k ⊆ g be maximal compactly embedded. Then K := Inn g (k) ⊆ Aut(g) is compact and co(x) K = co(x) ∩ z(k) contains an interior point z. Let W max be the maximal pointed generating invariant cone containing z.
We claim that x ∈ W max . The projection p z(k) : g → z(k) = z g (k) is the fixed point projection for the action of the compact group K . Therefore it preserves closed convex invariant subsets. This shows that p z(k) (co(x)) ⊆ co(x). As lim(co(−z)) = −C z , the subset p z(k) (co(x)) ⊆ z(k) = Rz must be contained in the half line [0, ∞)z. Therefore, The right-hand side is a closed convex invariant cone containing z. Therefore x ∈ W ⊆ W max implies that C x is pointed.

Corollary 3.9
If g is simple hermitian, then Proposition 3.10 (g co for reductive Lie algebras) Let g = g k + g 1 + · · · + g k be reductive, where g k is the maximal compact ideal and the ideals g j are simple noncompact. If g j is hermitian, we write W g j max for a maximal proper invariant cone in g j and otherwise we put W g j max := {0}. Then Proof This follows from g k,co = g k and Corollary 3.9, which entails g By the preceding proposition, the structure of g co is rather simple and adapted to the decomposition into simple ideals. The subset g c is slightly more complicated. From Corollary B.2, we obtain the following characterization of elements contained in a given invariant cone W in a reductive Lie algebra. Proposition 3.11 (Reduction to nilpotent and semisimple elements) Let g be a reductive Lie algebra and x ∈ g be contained in the invariant cone W ⊆ g. Write x = x 0 + x s + x n with x 0 ∈ z(g) and the Jordan decomposition x s + x n of the component of x in [g, g]. Then the following are equivalent: Corollary 3.12 Let g be reductive and x ∈ g. We write x n ∈ [g, g] for its nilpotent Jordan component and x s := x − x n for its ad-semisimple Jordan component. Then C x is pointed if and only if C x n + C x s is pointed.
Proof If C x is pointed, then Proposition 3.11 implies that x s , x n ∈ C x , so that C x n + C x s ⊆ C x has pointed closure. The converse follows from x = x n + x s ∈ C x n + C x s , which implies C x ⊆ C x n + C x s .

Lemma 3.13
Suppose that g is a reductive Lie algebra and x ∈ g is such that C x is pointed. Then the following are equivalent: Proof (a) ⇒ (b): We write g = z(g) ⊕ g 1 ⊕ · · · g n , where the g j are simple ideals and accordingly It, therefore, suffices to show that, if g is simple hermitian and 0 = x ∈ g is such that co(x) is pointed with 0 ∈ co(x), then x is nilpotent. If x is not nilpotent, then x s = 0. Proposition 3.8(b) shows that x n ∈ C x s , and thus The semisimplicity of x s implies that its orbit O x s is closed (Theorem of Borel-Harish-Chandra, [20, 1.3.5.5]), and since C x s is pointed, the orbit O x s is admissible in the sense of [12, Def. VII.3.14].
Here we use that g ∼ = g * as Inn(g)-modules, so that adjoint orbits correspond to coadjoint orbits under a linear isomorphism. Next we use [12,Prop. VIII.1.25] to see that, if p t : g → t is the projection onto a compactly embedded Cartan subalgebra t, then Proposition 3.14 (g c for reductive admissible Lie algebras) Let g = g k ⊕ g p be a reductive Lie algebra, where g k is the maximal compact ideal. We write elements x ∈ g accordingly as x = x k + x p . Then the following are equivalent: (a) C x is pointed.
(b) C x p is pointed and, if x p is nilpotent and x k = 0, then x k / ∈ [g k , g k ]. (c) co(x) is pointed, and if 0 ∈ co(x), then x k = 0 and x p is nilpotent.
where co(x k ) is compact. Therefore, co(x) is pointed if and only if co(x p ) is pointed, which by Proposition 3.8(d), applied to the simple ideals in g p , implies that C x p is pointed.
(a) ⇒ (b): If C x is pointed, then the argument above shows that C x p is pointed as well. If x p is nilpotent, then 0 ∈ co(x p ) by Lemma 3.13, so that (3.8) implies that C x k ⊆ C x is also pointed, and this further implies that, if x k = 0, then x k / ∈ [g k , g k ]. Here we use that the relative interior of C x k intersects z(g k ) because the projection p z : g k → z(g k ) is the fixed point projection for the compact group Inn(g k ).
(b) ⇒ (c): Suppose that C x p is pointed. Then and since co(x k ) is compact, lim(co(x)) ⊆ C x p is pointed. If 0 ∈ co(x), then 0 ∈ co(x k ) and 0 ∈ co(x p ). As C x p is pointed, Lemma 3.13 implies that x p is nilpotent.
(c) ⇒ (a): Suppose that co(x) is pointed and that, if 0 ∈ co(x), then x = x p is nilpotent. If 0 / ∈ co(x), then C x is pointed by Lemma A.2. If 0 ∈ co(x), then x = x p is nilpotent, so that co(x p ) = C x p = C x is pointed.

Proposition 3.15 (Extension of invariant cones)
Suppose that g is reductive and quasihermitian, i.e., a direct sum of a compact Lie algebra and hermitian simple ideals. If C x is pointed, then there exists a pointed generating invariant cone W ⊆ g containing x.
Proof Let g(x) = C x − C x g be the ideal generated by x. As g is reductive, g = g(x) ⊕ g 1 , where g 1 g is a complementary ideal. If g 1 itself contains a pointed generating invariant cone W 1 , then C x + W 1 is a pointed generating invariant cone in g. As g 1 also is quasihermitian, it contains no pointed generating invariant cone if and only if it is compact semisimple (Proposition 3.14). Then we consider a product B x × B 1 ⊆ g, where B x ⊆ C x is a compact base of the pointed cone C x , i.e., of the form f −1 (1) ∩ C x for f in the interior of the dual cone C x , and B 1 ⊆ g 1 is a compact invariant 0-neighborhood. Then W := cone(B x + B 1 ) is a pointed generating invariant cone in g containing C x = cone(B x ).
We shall see below that the preceding proposition does not extend to non-reductive admissible Lie algebras (Example 3.25).

The Characterization Theorem
We now turn to the non-reductive admissible Lie algebras g = g(l, V , z, β). With the reduction (3.1) in mind, we may assume that l acts faithfully on V . We start with a crucial lemma.
For a root α ∈ r with root space , we consider the closed convex cone Lemma 3. 16 For any pointed closed convex cone C z ⊆ z, is a pointed invariant closed convex cone in l.
Proof That W l is a closed convex invariant cone in l follows from the l-invariance of the bracket β : where + r ⊆ r is any positive system. We then find for x ∈ t: and thus This shows that iα(x) > 0 implies that C α ⊆ C x,z . As −iα(x)C −α = iα(x)C α by (3.9), the cone C x,z is generated by the cones C α for iα(x) > 0, α ∈ r .
Proposition 3.18 Let + r ⊆ r be a positive system and be a pointed closed convex cone in z. Then Proof In view of Lemma 3.16, it remains to show that W l is generating. First we observe that If x ∈ t l is such that iα(x) > 0 for all α ∈ + r , then C x,z = C min,z by Lemma 3.17. Therefore, and this cone has inner points. This implies that W l is generating because Inn(l)(W l ∩t l ) contains inner points. This in turn follows from the fact that the differential of the map is surjective if y ∈ t l is regular. If x ∈ t l is such that iα(x) < 0 for some α ∈ + r , then C α ⊆ −C min,z ⊆ −C z and the pointedness of C z thus shows that C x,z ⊆ C z . We conclude in particular that x / ∈ W l , so that W l ∩ t ⊆ (i + r ) , and thus (3.10) follows.

Remark 3.19
In the context of the preceding proposition, we assume, in addition, that + p is adapted and consider the closed convex cone Then the pointedness of C min,z implies the pointedness of C min and hence that C min ⊆ C max ( [12, Thm. VIII.2.12]). As the cone W is clearly contained in the ideal g W := g(l, V , z W , β). Further, implies that and this cone is generating in t W := t ∩ g W , which in turn implies that W is generating in g W . By construction, H (W ) ⊆ V is trivial by Lemma 2.7(b), so that W is pointed.
and put Then (a) co(x) is pointed if and only if co z (x) is pointed, i.e., g co = {x ∈ g : co z (x) pointed}. (

b) C x is pointed if and only if co z (x) is pointed and, if x l is nilpotent, then co z (x) is contained in a pointed cone.
We shall see in Example 3.25 below that the pointedness of C x does in general not imply the existence of a pointed generating invariant cone W containing x.
Proof (a) If co(x) is pointed, then we may assume that x = x z + x l by Theorem 3.2. Note that H z x only depends on the orbit e ad V x. Now co(x l ) = co(x)−x z is also pointed.
That the cone C x,z = C x l ,z = co z (x l ) is pointed follows from C x,z ⊆ z ∩ lim(co(x)) (Lemma 3.6).
Suppose, conversely, that co z (x) is pointed. As C x,z ⊆ lim(co z (x)), (3.11) by Lemma 3.6, the cone C x,z is pointed.
is an affine space. Our assumption implies that it is trivial, so that [x V , V x l ,0 ] = {0} and thus x V ∈ [x l , V ] by Lemma 3.1(b). Hence, O x ∩ (z ⊕ s) = ∅ follows from (3.5). We may, therefore, assume that x V = 0. Then x = x z + x l and co(x) = x z + co(x l ). It therefore suffices to show that co(x l ) is pointed if the cone C x l ,z is pointed.
By Lemma 3.16, is a pointed invariant cone in l. As x l ∈ W l , we obtain Therefore the linear subspace H (co(x l )) is an ideal of g contained in V , hence trivial (Lemma 2.7(b)). This shows that co(x l ) is pointed. (b) Suppose that C x is pointed. Then co(x) is pointed and thus co z (x) is pointed by (a). Suppose that x l is nilpotent. We have to show that co z (x) is contained in a pointed cone. From C x,z ⊆ lim(co(x)) ⊆ C x (3.14) (Lemma 3.6) it follows that C x,z is pointed. As we have seen in (a), we may assume that x V = 0, so that x = x z + x l . Further, the nilpotency of x l implies that it is contained in s (Corollary 3.3). Moreover, 0 ∈ co(x l ) by Lemma 3.13, so that C x ⊇ co(x) = x z + co(x l ) x z . We conclude that Conversely, suppose that co z (x) is pointed and further that x V = 0 by (a) and Theorem 3.2.
• If co z (x) is contained in a pointed cone D x ⊆ z, then co(x) ⊆ D x + V + co l (x l ).
In the proof of (a) we have seen that C x l ,z ⊆ lim(co z (x)) is pointed, and that x l is contained in the pointed invariant cone W l ⊆ l from (3.12). Therefore C x ⊆ D x + V + W l shows that H (C x ) g is an ideal contained in V , hence trivial (Lemma 2.7(b)).
• If co z (x) is not contained in a pointed cone, then we assume that x l is not nilpotent.
We show that C x is pointed by verifying that 0 / ∈ co(x) and applying Lemma A.2. As co(x) = x z + co(x l ), we have to show that −x z / ∈ co(x l ). We claim that 0 / ∈ co l (x l ). In fact, if x l = y s + y n is the Jordan decomposition, where y n ∈ [l, l] is nilpotent and y s = x l − y n , then y s , y n ∈ W l (Proposition 3.11). This shows that As y s = 0 by assumption and W l is pointed by (a), 0 / ∈ co l (y s ) by Lemma 3.13, and this implies that 0 / ∈ co l (x l ). Finally we observe that and by the preceding argument, this convex set intersects z trivially. Therefore, −x z / ∈ co(x l ).

Remark 3.21 (a) From (3.13) we derive in particular that the pointedness of co(x) implies the pointedness of co
which is pointed if and only if C x,z is pointed. If this is the case, then co z (x) is contained in a pointed cone if and only if which means that the intersection either is empty or x z = 0 (cf. Lemma A.3).

Remark 3.22
Let f ∈ z * , considered as a linear functional on g = g(l, Then f is fixed under the coadjoint action of Inn g (l), so that its coadjoint orbit is For elements x = x z + x l ∈ z + l, it follows that Proof We shall obtain this is a special case of Theorem 3.20. The Jacobi algebra is admissible of the form g(l, V , z, β) with is the Hamiltonian function on (V , ω), corresponding to x ∈ g.
and the action of l on V is given by With ε j (x 1 , x 2 ) = i x j , this means that For y := (−1, 0) ∈ l, we obtain with iε 1 (y) = 1: We likewise obtain C ε 2 = [0, ∞)e 2 , and Therefore + r := {ε 1 , ε 2 , ε 1 + ε 2 } is an adapted positive system for which g is an ideal contained in V . As β : V × V → z is non-degenerate, this ideal is trivial and therefore W min is pointed. Now [12,Lemma VIII.3.22] implies that g is admissible.
We claim that there exists no pointed generating invariant cone W ⊆ g containing x. Suppose that W is such a cone. Then there exists an adapted positive system + r with C min,z ⊆ W [12, Thm. VII.3.8]. As C x,z ⊆ C x ⊆ W (Lemma 3.6), we must have ε 2 , −ε 1 ∈ + r . If ε 1 + ε 2 is positive, then C ε 1 ⊆ C ε 1 +ε 2 ⊆ C min,z and − C ε 1 = C −ε 1 ⊆ C min,z contradict the pointedness of C min,z . If ε 1 + ε 2 is negative, then −C ε 2 ⊆ C −ε 1 −ε 2 ⊆ C min,z and C ε 2 ⊆ C min,z contradict the pointedness of C min,z . Hence there exists no pointed generating invariant cone W containing x.
In the preceding example it was important that dim z > 1. We have the following positive result for the Jacobi-Lie algebra, where z = R.

Proposition 3.26
If g = hsp 2n (R) = g(sp 2n (R), R 2n , R, ω) and x ∈ g is such that C x is pointed, then x is contained in a pointed generating invariant cone W ⊆ g.
Proof In view of Corollary 3.23, we may assume that the Hamiltonian function H x is bounded from below. We write . If x l is nilpotent, then even H x ≥ 0, so that x ∈ W := {y ∈ g : H y ≥ 0}, and W is a pointed generating invariant cone in g.
We may therefore assume that x l is not nilpotent. By the Reduction Theorem 3.2, we may further assume that x V = 0. If H x ≥ 0, then x ∈ W ; so we assume that As x ∈ z + V + W l for W l := W ∩ l (the cone of non-negative quadratic forms), the invariance of the set on the right implies C x ⊆ z + V + W l . We conclude that W ∩ −C x ⊆ z + V is a pointed invariant cone. As e ad V x = x + [V , x] for x ∈ z + V , it follows that W ∩ −C x ⊆ z. We thus obtain If C x,z = {0}, then x l = 0 and x = x z ∈ −W . So we may also assume that Now [12,Prop. V.1.7] implies that the invariant cone W + C x ⊆ g is closed and pointed. It is generating because W is generating. Definition 4.1 (Affine pair) Let g be a finite dimensional real Lie algebra and W ⊆ g a pointed invariant cone. We call (x, h) ∈ g × g an affine pair for the cone W if (4.1) For an affine pair, the subalgebra Rh + Rx is isomorphic to the non-abelian 2dimensional Lie algebra aff(R); hence the name. As this Lie algebra is solvable, ad x is nilpotent [4,Prop. 5.4.14].

Invariance of W Under One-Parameter Groups of Outer Automorphisms
On g = g(l, V , z, β) we consider the canonical derivation D can , defined by The derivation 2D can corresponds to the Z-grading of g, defined by g 0 = l, g 1 = V and g 2 = z.
In the Existence Theorem 4.7 below, the one-parameter group e RD with D ∈ D can + ad g leaves an invariant cone W in g invariant if and only if e RD can does. The following proposition characterizes the cones W for which this is the case.

Proposition 4.2
For a pointed generating invariant cone W ⊆ g = g(l, V , z, β) and a compactly embedded Cartan subalgebra t ⊆ g, the following are equivalent: If W satisfies these conditions, then W ∩ (z + l) = W z + W l for W z := W ∩ z and W l := W ∩ l, and the Reduction Theorem 3.2 implies that , showing that W is uniquely determined by the two cones W z and W l .
Then Theorem 3.2 implies the existence of ϕ ∈ Inn(g) with y := ϕ(x) ∈ z + l. Then y z ∈ W and y l ∈ W by (a). Therefore, e t D can y = e t y z + y l ∈ W for t ∈ R. Now e t D can x = e t D can ϕ −1 (y) = e t D can ϕ −1 e −t D can (e t D can y) ∈ Inn(g)W = W .
This shows that e t D can W ⊆ W for every t ∈ R. As e RD can is a group, this implies (a).
we have a 2-dimensional compactly embedded Cartan subalgebra Up so sign, there is a unique positive system + (which is adapted). Then C min = C min,z ⊕ C min,s is a quarter plane and C max = R ⊕ C max,s = R ⊕ C min,s is a half plane. Any pointed generating closed convex cone W t ⊆ t with C min ⊆ W t ⊆ C max is of the form W t = W ∩ t for a pointed generating invariant cone W ⊆ g because the Weyl group W k is trivial [12,Thm. VIII.3.21]. In particular, we may have p z (W t ) = z(g) ⊆ W t . Therefore, we do not always have p z (W ) ⊆ W .

Remark 4.4
Although the conditions in Proposition 4.2 are not always satisfied, this is the case for many naturally constructed cones.
Let g = g(l, V , z, β) be an admissible Lie algebra and D ∈ der(g). We assume that the representation of l on V is faithful. The cone W , constructed from a pointed cone C z ⊆ z in Remark 3.19 is generated by C z + W l and satisfies p z (W ) ⊆ C z ⊆ W and More generally, any derivation D ∈ der(g) with l ⊆ ker D and e RD C z = C z satisfies e RD W = W because W = e ad V (W ∩ (z + l)) = Inn(g)(W ∩ (z + l)) follows from the Reduction Theorem 3.2.

Extending Nilpotent Elements to Affine Pairs
Let W ⊆ g be a pointed generating invariant cone. In this section, we consider a nilpotent element x ∈ W and ask for the existence of a derivation D ∈ der(g) with Dx = x and e RD W = W .
Note that the latter condition implies that W is an invariant cone in the extended Lie algebra g D := g RD and (x, D) is an affine pair for W .
This problem is trivial for semisimple Lie algebras: Then D := ad h is a derivation with Dx = x and e RD W = W . (b) If g = z(g)⊕[g, g] is reductive and x = x z +x s with 0 = x z ∈ z(g) and x s ∈ [g, g], then it cannot be reproduced with inner derivations. For any derivation D on g there exists an endomorphism D z of z(g) and an element h ∈ [g, g] with If W x is a pointed generating invariant cone, then the nilpotency of x s implies that R + x s ⊆ O x s (Corollary B.2), so that x z , x s ∈ W . Putting D z := id z(g) , we then have Dx = x and at least e RD (W z + W s ) ⊆ W for W z := W ∩ z(g) and W s := W ∩ [g, g].
As the maximal cone W max ⊆ g contains z(g), it is invariant under e RD for any D ∈ der(g).
The following lemma provides crucial information that we shall need below to explore the existence of Euler derivations on g(l, V , z, β), i.e., a diagonalizable derivation with Spec(D) ⊆ {0, ±1}.
is generating, hence intersects the Cartan subalgebra t l . Therefore (V , ω) also is a symplectic module of convex type for the reductive subalgebra l := t l + s. The edge l 1 of the generating invariant cone is the kernel of the representation of l on V , hence an ideal. We write with a complementary ideal l 2 . Then where the pointed cone W V ,s 2 = W V ,l 2 ∩ s 2 is also generating because it contains W min,s 2 . Therefore, (V , ω) is a symplectic s 2 -module of convex type. As V is a semisimple s-module, where V eff,s is a symplectic s 2 -module of convex type because W V ,s 2 = W V eff ,s is pointed and generating [13, Prop. II.5].
(b) Let h ∈ s be an Euler element for which s ±1 (h) generate s. We decompose s into a direct sum n j=1 s j of simple ideals, which are hermitian of tube type because they possess Euler elements [9,Prop. 3.11(b)]. By [17,Thm. 2.14], we can decompose V eff,s into a direct sum n j=1 V j of s-submodules such that s k acts trivially on V j for k = j. In particular, each (V j , ω) is a symplectic s j -module of convex type. We decompose h as h = n j=1 h j , with h j ∈ s j . Then each h j is an Euler element in s j . Hence [17,Lem. 3.4] implies that, for each j, the operator 2 ad h j defines an antisymplectic involution on V j , and thus 2 ad h defines an antisymplectic involution on V eff,s . Theorem 4.7 (Existence Theorem) Let g = g(l, V , z, β) be an admissible Lie algebra and x = x z + x l ∈ z + l be an ad-nilpotent element for which co(x) is pointed. Then there exists a derivation D ∈ D can + ad g with Dx = x and Any invariant cone W generated by W l := W ∩ l and a central cone W z ⊆ z satisfies e RD W = W .
Recall that, by Corollary 3.3, any ad-nilpotent element x with C x pointed is conjugate to an element of z + s.
As D can h s = 0, the derivation D is diagonalizable because both summands commute and are diagonalizable. The eigenvalues on s are contained in 0, ± 1 2 , ±1 by (4.4) and z ⊆ ker (D − 1). Let denote the h s -eigenspaces in V (Lemma 4.6). Then the corresponding eigenvalues of D on V are 1 2 , 1 and 0. This completes the proof of the first assertion. If the invariant cone W ⊆ g is generated by W l and a central cone W z , then the invariance of both cones under e RD can implies that e RD can W = W , hence that e RD W = W follows from D ∈ D can + ad g.

Remark 4.8
If the ideal s of s generated by a nilpotent element x s ∈ s contains only simple summands of tube type, then their restricted root systems are of type (C r ) and never of type (BC r ) [3, pp. 587-588]. Therefore, [6, Lemma IV.7] actually provides an Euler element h s of s , and hence also of s.

Euler Derivations
Definition 4. 9 We call D ∈ der(g) an Euler derivation if D is diagonalizable with In this section we ask, for a nilpotent element x for which C x is pointed for an Euler derivation D satisfying Dx = x. Recall that the latter relation implies that x is nilpotent (cf. Definition 4.1).  Hence there exists an element h ∈ s := [l, l] and endomorphisms D z ∈ End(z) and

Remark 4.10
It follows in particular that D preserves all ideals of l.
Write h = h 0 + h 1 with h 0 ∈ z(l) and h 1 ∈ s. These two summands commute and h 1 is an Euler element in s. As elements of z(l) acts with purely imaginary spectrum on V , we further obtain h 0 = 0, so that h ∈ s.
Let g = g(l, V , z, β) be admissible. For h ∈ l, we recall the subspaces Here is an example of a rather small admissible Lie algebra that already displays the complexity of the situation we encounter in Theorem 4.17.

Proposition 4.19
Let g = g(s, V , z, β) be an admissible Lie algebra, where s is a direct sum of hermitian simple Lie algebras of tube type and z = [V , V ]. Let x ∈ g be a nilpotent element with C x pointed. Then there exists an Euler derivation D ∈ der(g) such that Dx = x and g 0 (D) = [g 1 (D), g −1 (D)]. (4.6) Proof By Corollary 3.3, we may assume that x = x z + x s ∈ z + s. Then x s is nilpotent with C x s pointed (cf. Corollary 3.4). Hence, there exists an Euler element h ∈ s such that [h, x s ] = x s and such that s is generated by s ±1 (h) [6,Lemma IV.7]. Now the discussion in Remark 4.16(b) shows that D := D can + ad h is an Euler derivation satisfying (4.6).
Next we observe that C +lim(C) ⊆ C implies R + C +lim(C) ⊆ R + C. Therefore, S is an additive subsemigroup of V in which R + C is a semigroup ideal, i.e., S + R + C ⊆ R + C. Since 0 / ∈ R + C, it follows that This implies that cone(C) ⊆ S is pointed. (b) ⇒ (c): That the intersection of x + C and −C is contained in {0} happens in two cases. If the intersection is empty, then 0 ∈ −C shows that 0 is not contained in x + C. If the intersection is non-empty, then it is {0} and thus x ∈ −C, which in turn implies x = 0.
(c) ⇒ (a): If 0 / ∈ x + C, then Lemma A.2 implies that cone(x + C) is pointed. If x = 0, then cone(x + C) = cone(C) = C is trivially pointed. Proof Since the Jordan decomposition and the adjoint orbit of x adapts to the decomposition of g into simple ideals, we may w.l.o.g. assume that g is simple.

B: Tools Concerning Lie Algebras
Let q = l u ⊆ g denote the Jacobson-Morozov parabolic associated with the nilpotent element x n [6]. Then x s ∈ ker(ad x n ) ⊆ q implies that x s ∈ q. As x s is semisimple, it is conjugate under the group of inner automorphisms of q to an element of l. 1 By the Jacobson-Morozov Theorem [1, Ch. VIII, §11, Prop. 2], l contains a semisimple element h with [h, x n ] = 2x n and h ∈ [x n , g]. In terms of this element, we have q = n≥0 g n (h) and l = ker(ad h). We further find a nilpotent element y ∈ g −2 (h) such that [x n , y] = h, so that the Lie algebra generated by x and y is isomorphic to sl 2 (R). Replacing x by a suitable conjugate, we have seen above that we may assume that x s ∈ g 0 (h). We consider the Lie algebra m := span{x s , h, x n , y}. Corollary B.2 Let x be an element of the semisimple real Lie algebra g and x = x s + x n its Jordan decomposition, where x s is semisimple and x n = 0 is nilpotent. Then the adjoint orbit O x of x contains all elements of the form x s + t x n , t > 0. In particular, x s ∈ co(x) and x n ∈ lim(co(x)).
Proof With Proposition B.1 we find an element h ∈ g with [h, x s ] = 0 and [h, x n ] = 2x n . Then the assertion follows from e t ad h x = x s + e 2t x n for t ∈ R.