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Stability with respect to actions of real reductive Lie groups

We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of non-compact real reductive Lie groups on topological spaces that admit functions similar to the Kempf-Ness function. The point of this construction is that one can characterize stability, semi-stability and polystability of a point by numerical criteria, that is in terms of a function called maximal weight. We apply this setting to the actions of a real non-compact reductive Lie group G on a real compact submanifold M of a Kaehler manifold Z and to the action of G on measures of M.


Introduction
Stability theory in Kähler geometry has been intensively studied by many authors and from several points of view, see e.g. [15,17,18,20,21,30,31,36,40,41]. This paper is inspired by the works of I. Mundet i Riera [38] and A. Teleman [42] where a systematical presentation of the stability theory in the non-algebraic Kählerian geometry of complex reductive Lie groups is given, and by the recent paper [8] where the first author jointly with A. Ghigi develops a geometrical invariant theory on topological spaces, without assuming the existence of a symplectic structure. In particular, they apply the main results to the action of U C on measures on a compact Kähler manifold Z, where U is a compact connected Lie group acting in Hamiltonian fashion on Z. This was also motivated by an application to upper bounds for the first eigenvalue of the Laplacian on functions [2,3,4,10,28].
In this paper we identify an abstract setting to develop the geometrical invariant theory for actions of real reductive Lie groups. More precisely, given a Hausdorff topological space M with a continuous action of a non-compact real reductive Lie group G = K exp(p) and a set of functions formally similar to the classical Kempf-Ness function we define an analogue of the gradient map F : M −→ p and the usual concepts of stability.
The gradient map has been intensively studied in [22,23,24,26] and many other papers. The main idea is to investigate a class of actions of real reductive Lie groups on complex spaces and on real submanifolds using momentum map techniques. This means that we consider a Kähler manifold (Z, ω) acted on by a complex reductive Lie group U C of holomorphic maps. The Kähler form ω is U -invariant, where U is a compact form of U C , and there exists a momentum map µ : Z −→ u * . We recall that a momentum map µ is U -equivariant and for any ξ ∈ u, the gradient of the function µ ξ (x) = µ(x)(ξ) is given by J(ξ Z ), where ξ Z (p) = d dt | t=0 exp(tξ)p is the vector field corresponding to ξ ∈ u and J is the complex structure of Z (see [19,34] for more details about momentum map). Since U is compact we may identify u ∼ = u * by means of an Ad(U )-invariant scalar product on u. Hence we may think the momentum map as a u-valued map, i.e., µ : Z −→ u.
Let G ⊂ U C be compatible. Then G is closed and the Cartan decomposition U C = U exp(iu) induces a Cartan decomposition G = K exp(p), where K = G ∩ U and p = g ∩ iu. Identifying iu ∼ = u the inclusion p ֒→ iu induces a K-equivariant map µ p : Z −→ p. Finally if M is a G-stable real submanifold of Z, we may restrict µ p to M and so considering µ p : M −→ p. The map µ p : M −→ p is called gradient map. In Section 7 we extend the construction given in [37] for the gradient map, defining a Kempf-Ness function of (M, G, K).
The G-action on M induces in a natural way a continuous action on measures of M , that we denote by P(M ), with respect to the weak- * topology. In Section 8 we prove there exists a Kempf-Ness function for (P(M ), G, K) and the map is the analogue of the gradient map in this setting. These are our basic examples and the main motivations to develop a geometrical invariant theory for actions of real reductive Lie groups.
Stability and semi-stability are checked using the position of the G-orbit with respect to the vanishing locus of the gradient map. The main point of our construction is that one can characterize stability, semi-stability and polystability of a point by numerical criteria, that is in terms of a function called maximal weight, which is defined on the Tits boundary of the symmetric space of non-compact type G/K. Roughly speaking we extend criteria for stability, semi-stability and polystability due to Teleman [42], Mundet I Riera [37,38], Kapovich, Leeb and Milson [29], Biliotti and Ghigi [8] and probably many others, for a large class of actions of complex reductive Lie groups, to actions of non-compact real reductive Lie groups. Our criterion for polystability is weaker than those proved by Mundet i Riera [38] and by the first author and Ghigi in [8] for complex reductive Lie gropus. However if G = K C = K exp(ik) is complex reductive then condition (P 3) in Section 3, i.e., d 2 Ψ(x, exp(tv)) = 0 if and only if exp(Rv) ⊂ G x , does not imply exp(Cv) ⊂ G x as required by (P 3) in [8, p. 6]. This condition is crucial in Mundet's proof [38] and in the proof given in [8] for polystability. Indeed, thanks to the K-equivariance of F, if exp(Cv) ⊂ G x , then F(x) ∈ k v = {u ∈ k : [u, v] = 0} and thus a sort of a reduction principle applied.
In the abstract setting introduced in this paper, the above condition is equivalent to the following: if exp(Rv) ⊂ G x then F(x) ∈ p v = {u ∈ p : [u, v] = 0}. This does not hold for a general gradient map F since it is only K-equivariant. On the other hand this condition holds for the gradient map [24] and the gradient map defined by the Kempf-Ness function with respect to the G action on measure (Proposition 45). The authors believe that the polystability criterion due to Mundet [38] holds under the above condition. We leave this problem for future investigation.
What is satisfactory of Theorem 33 is that the reductivity of the stabilizer is obtained as a consequence of conditions involving only the maximal weight and the set on which the maximal weight is zero. We also prove a version of the Hilbert-Munford criterion and the openness of the set of stable points. Finally we completely characterize stable, semi-stable and polystable measures on real projective spaces.
The paper is organized as follows.
In Section 2 we review basic facts on real reductive Lie groups and Tits boundary of a Hadamard manifold.
In Section 3 we define the abstract setting and the general gradient map with respect to a Kempf-Ness function of (M , G, K).
In Section 4 we define the maximal weight on the Tits boundary of X = G/K. Since the Kempf-Ness function is K-invariant, for any x ∈ M the Kempf-Ness function descends to a function ψ x : X −→ R which is geodesically convex. If ψ x is Lipchitz then Lemma 9 defines what is called maximal weight on the Tits boundary of X. We also point out that the maximal weight is G-equivariant.
In Section 5 we define stable, semi-stable and polystable points giving a numerical criterion for an element x ∈ M to be stable (Theorem 23). We give a version of the Hilbert-Munford criterion (Corollary 28) and we prove the openness of the set of stable points (Corollary 29).
In Section 6 we give numerical criteria for semi-stability (Theorem 35) and polystability (Theorem 33) and a Hilbert-Munford criterion for semi-stable points (Corollary 36).
In Section 7 we discuss the basic example, i.e., the classical gradient map, and in Section 8 we apply our setting on the G action on measures. Using the Morse-Bott theory of the gradient map on M we compute rather explicitly the maximal weight. Moreover, assuming the convexity theorem holds for abelian subgroup A = exp(a) where a ⊂ p, see [6,26] for more details, if 0 ∈ E(µ p ), where E(µ p ) is the convex hull of the image of the gradient map µ p , then any smooth measures is semi-stable (Proposition 55). The condition 0 ∈ E(µ p ) is always satisfied up to shifting the gradient map with respect to some Ad(K)-fixed point of p. We also prove that the set of semi-stable points is dense. If 0 lies in the interior of E(µ p ) then any smooth measures is stable and the set of stable points is open and dense. This condition is always satisfied if M is an adjoint orbit of K in p and K acts irreducibly on p.
In Section 9 we completely describe stable, semi-stable and polystable measures on real projective spaces. Acknowledgements. The authors are very grateful to Alessandro Ghigi for all the intersting discussions and to Alberto Raffero for his comments. The second author wishes also to thank Parma's Mathematics Department for the wonderful hospitality during her stay as post-doc student.

Tits boundary of G/K
Let G be a non-compact real reductive Lie group and denote by g its Lie algebra. Recall that such G has a finite number of connected components and its algebra splits as g = [g, g] ⊕ z(g), where [g, g] is semisimple and z(g) is the center of g. Further, maximal compact subgroups of G always exist and meet every connected components, and any two of them are conjugate under an element of the identity component G o of G. Assume that there exists a Cartan involution θ : G −→ G with fixed points set K and let us denote also by θ : g −→ g its differential. Then g = k ⊕ p and the map f : K × p → G, f (g, v) = g exp v is a diffeomorphism. This means that G = K exp(p) and G/K is simply connected. Since θ | k = Id and θ |p = −Id, we have [k, k] ⊂ k, [k, p] ⊂ p and [p, p] ⊂ k. Therefore if a ⊂ p is a Lie subalgebra, then it must be abelian. Moreover, two maximal abelian subalgebras contained in p are conjugate with respect to the identity component K o . We refer the reader to [9,27,32] for more details on real reductive Lie groups. Set Observe that G acts isometrically on X from the left by: To simplify the notation, we will often write gx instead of L g (x). The choice of an Ad(K)invariant scalar product on p induces a G-invariant Riemannian metric on X. It is well known that X endowed with this metric is a symmetric space of non-compact type and thus a Hadamard manifold [13,27]. The Riemannian exponential map arises by the exponential map of Lie groups. Hence a geodesic on X is given by g exp(tv)K, where g ∈ G and v ∈ p. In the sequel we denote by γ v (t) = exp(tv)K. Since X is a Hadamard manifold there is a natural notion of boundary at infinity ∂ ∞ X which can be described using geodesic.
Set o := K ∈ X. Mapping v to the tangent vectorγ v (0) yields an isomorphism p ∼ = T o X. Since any geodesic ray in X is equivalent to a unique ray starting from o, the map where S(p) is the unit sphere in p, is a bijection. The sphere topology is the topology on ∂ ∞ X such that e is a homeomorphism. (For more details on the Tits boundary see for example [9, §I.2] and [13].) Since G acts by isometries on X, if γ is a unit speed geodesic in X, then for each g ∈ G also gγ is. Further, since γ ∼ γ ′ implies gγ ∼ gγ ′ , we get a G-action on the Tits boundary ∂ ∞ X by: which also induces by (1) a G-action on S(p) given by: Observe that this last one is discontinuous with respect to the sphere topology on S(p).

Definition 2.
Let H ⊂ G be a closed subgroup. Set L := H ∩ K andp := h ∩ p. According to [24,25], we say that H is compatible if H = L exp(p).
If H is a compatible subgroup of G, then it follows that it is a real reductive subgroup of G, the Cartan involution of G induces a Cartan involution of H, L is a maximal compact subgroup of H and finally h = l ⊕p. Note that H has finitely many connected components. Moreover, there are totally geodesic inclusions X ′ := H/L ֒→ X and ∂ ∞ X ′ ⊂ ∂ ∞ X.

Kempf-Ness functions
Let M be a Hausdorff topological space and let G be a non-compact real reductive group which acts continuously on M . Observe that with these assumptions we can write G = K exp(p), where K is a maximal compact subgroup of G. Starting with these data we consider a function Ψ : M × G → R, subject to five conditions. The first four are the following ones: (P 1) For any x ∈ M the function Ψ(x, ·) is smooth on G. (P 2) The function Ψ(x, ·) is left-invariant with respect to K, i.e.: Ψ(x, kg) = Ψ(x, g). (P 3) For any x ∈ M , and any v ∈ p and t ∈ R: (P 4) For any x ∈ M , and any g, h ∈ G: This equation is called the cocycle condition.
As in the previous section, let X = G/K. If Ψ is a function satisfying (P 1)-(P 4), then by (P 2) the function g → Ψ(x, g −1 ) descends to a function on X: and the cocycle condition (P 4) can be rewritten in terms of ψ x as: which is also equivalent to the following identity between two functions and a constant: where L g denotes the action of G on X (see previous section).
In order to state our fifth condition, let ·, · : p * × p → R be the duality pairing. For x ∈ M define F(x) ∈ p * by requiring that: The following is the fifth and last condition imposed on the function Ψ: We call F the gradient map of (M , G, K, Ψ). As immediate consequence of the definition of F we have the following result.
Proof. It is an easy application of the cocycle condition and the left-invariance with respect to K of Ψ(x, ·). Indeed, The following definition summarizes the above discussion.
Definition 6. Let G be a non-compact real reductive Lie group, K a maximal compact subgroup of G and M a topological space with a continuous G-action. A Kempf-Ness function for (M , G, K) is a function that satisfies conditions (P 1)-(P 5).

Remark 7.
Taking g = h = e in the cocycle condition (P 4) we have Ψ(x, e) = 0. Hence Ψ(x, k) = 0 for every k ∈ K, since Ψ(x, ·) is K-invariant on the second factor. Moreover, for any x ∈ M and for any g, h ∈ G x we have: which implies that Ψ(x, ·) : G x −→ R is a homomorphism.

Maximal weights
Let X = G/K and let u : X → R be a smooth function. We say that u is geodesically convex on X if u(γ(t)) is a convex function for any geodesic γ(t) in X. The following lemma is proven in greater generality by Kapovich, Leeb and Millson in [29, §3.1] (see also [8, §2.3]).
Lemma 9. Let u : X → R be a smooth geodesically convex function on X. Assume that u is globally Lipschitz continuous. Then the function u ∞ : ∂ ∞ X → R given by: is well-defined. Moreover u is an exhaustion if and only if u ∞ > 0 on ∂ ∞ X.
Recall that a continuous function f : X → R is an exhaustion if for any c ∈ R the set f −1 ((−∞, c]) is compact, condition which is equivalent for f to be bounded below and proper.
As in [8], the following result holds.
Lemma 11. The function ψ x is geodesically convex on X. More precisely, if v ∈ p and α(t) = g exp(tv)K is a geodesic in X, then ψ x • α is either strictly convex or affine. The latter case occurs if and only if g exp(Rv)g −1 ⊂ G x . In the case g = e, the function ψ x • α is linear if exp(Rv) ⊂ G x and strictly convex otherwise.
Due to Lemma 11, in order to apply Lemma 9 to ψ x , we need only to add this last assumption: (P 6) For any x ∈ M , the function ψ x : X → R is globally Lipschitz on X.
When property (P 6) holds, for any x ∈ M the function λ x := (ψ x ) ∞ given by: is well-defined and finite. We call λ x maximal weight. Moreover for any x ∈ M , any g ∈ G and any p ∈ ∂ ∞ X we have (see [8,Lemma 2.28] for a proof): It is also well-defined and finite the function: Since we set the sphere topology on ∂ ∞ X, i.e., the topology on ∂ ∞ X such that e : S(p) → ∂ ∞ X is an homeomorphism (see Section 2), by [8,Lemma 4.9], λ is lower semicontinuous and for v ∈ S(p) it follows:

Stability
Let (M , G, K) be as above and let Ψ be a Kempf-Ness function. In particular, according to Definition 6 we assume that Ψ satisfies conditions (P 1)-(P 5).
Remark 17. The four conditions above are G-invariant in the sense that if a point x satisfies one of them, then every point in the orbit of x satisfy the same condition. This follows directly from the definition for polystability, semi-stability and unstability, while for stability it is enough to recall that g gx = Ad(g)(g x ).
The following result establishes a relation between the Kempf-Ness function and polystable points.
The following conditions are equivalent: Proof. Let v ∈ p. Using the cocycle condition (P 4), one gets: Since for any k ∈ K, Ψ(x, kg) = Ψ(x, g), then F(gx) = 0 if and only if g is a critical point of Therefore d 2 dt 2 Ψ(x, exp(tv)) = 0 for 0 ≤ t ≤ 1. It follows from (P 3) that exp(tv)x = x for any t ∈ R and thus G x is compatible.
Next we give a numerical criteria for an element x ∈ M to be stable. We begin with the following lemma.
Lemma 21. If a ⊂ g is a subalgebra which is conjugate to a subalgebra of k, then a ∩ p = {0}.
Proof. It is enough to show that Ad(g)(k) ∩ p = {0} for any g ∈ G. Let X ∈ Ad(g)(k) ∩ p. By the Cartan decomposition G = K exp(p), it follows Γ = exp(RX) is a closed abelian subgroup of G isomorphic to R. On the other hand X = Ad(g)(Y ) for some Y ∈ k which implies Γ = Ad(g)(exp(RY )) is a torus. Hence X = 0.
Consider the function: The following Lemma is proven in [42, Lemma 2.10].

Lemma 22.
Let V be a subspace of p. For a point x ∈ M the following conditions are equivalent: a) The map Ψ(x, exp(ξ)) is linearly proper on V , i.e. there exist positive constants C 1 and C 2 such that: Proof. Let first x ∈ M be stable. Then F(gx) = 0 for some g ∈ G and by Proposition 18, g is a critical point of Ψ(x, ·). Set y = gx. We start by proving Λ(y, ξ) > 0 for any ξ ∈ p − {0}. By (P 3) the function f (t) = Ψ(y, exp(tξ)) is a convex function. Hence: Assume Λ(y, ξ) = 0. By assumption f is a convex function satisfying lim t→+∞ f ′ (t) = 0 and f ′ (0) = 0. Hence f ′ (t) = 0 for t ≥ 0 and so d 2 dt 2 Ψ(x, exp(tξ)) = 0 for any t ≥ 0. By (P 3) it follows that exp(Rξ) ⊂ G y , so ξ ∈ g y ∩ p. Since x is stable, g y = Ad(g)(g x ) is conjugate to a subalgebra of k, thus Lemma 21 implies that ξ = 0.
We claim that there exists γ o > 0 such that Ψ(x, exp(ξ)) > M 2 for ξ ∈ p with ||ξ|| ≥ γ o . Indeed, otherwise there exist sequences ξ n ∈ S(p) and t n ∈ R with t n → +∞ such that Ψ(x, exp(t n ξ n )) ≤ M 2 . We may assume ξ n → ξ o . Since Ψ(x, exp(tξ o )) ≥ M for t > t(ξ o ) and keeping in mind that the function and for any ξ ∈ U . Now, there existsñ ∈ N such that ξ n ∈ U and t n > t(ξ o ) for n ≥ñ. Since the function t → Ψ(x, exp(tξ)) increases, it means Ψ(x, exp(t n ξ n )) > M 2 for n ≥ñ which is a contradiction. Now, keeping in mind that ψ x • exp(ξ) = Ψ(x, exp(−ξ)), we have proved that the function ψ x • exp has a minimum and so a critical point. Since exp : p −→ G/K is a diffeomorphism, it follows that ψ x has a critical point. By Proposition 18 the point x is polystable.
Remark 24. One may prove that the condition Λ(x, ξ) > 0 for any ξ ∈ p − {0} is equivalent to ψ x being an exhaustion.
Proof. Let g ∈ G be such that F(gx) = 0 and set y = gx. By Proposition 20 the stabilizer of y, i.e. G y , is compatible and so has only finitely many connected components. Moreover G 0 y is compact since g y ⊂ k. It follows that G y and G x = g −1 G y g are both compact.
If M ′ is a G-invariant subspace of M , the restriction of Ψ to G×M ′ is a Kempf-Ness function for (M ′ , G, K). The functions Λ and F for (M ′ , G, K) are simply the restrictions of those for M . If G ′ ⊂ G is a compatible subgroup of G, i.e., G ′ = K ′ exp(p ′ ), then K ′ ⊂ K, p ′ ⊂ p and X ′ := G ′ /K ′ ֒→ X is a totally geodesic inclusion. If Ψ is a Kempf-Ness function for (G, K, M ), then Ψ K ′ := Ψ| M ×G ′ is a Kempf-Ness function for (G ′ , K ′ , M ). The related functions are A subalgebra contained in p must be abelian since [p, p] ⊂ k. The following Corollary is analogous to the stability part in the Hilbert-Mumford criterion.
where a is a subalgebra of g contained in p.
Proof. By Theorem 23 it is enough to prove that we have Λ(x, ξ) > 0 for any ξ ∈ p − {0} if and only if for any abelian group A = exp(a), where a is a subalgebra of g contained in p we have Λ A (x, ξ) > 0 for any ξ ∈ a − {0}. The necessary condition is trivial, being Λ A (x, ξ) the restriction of Λ(x, ξ) to a. For the sufficient, observe that for any ξ ∈ S(p) we can set a = Rξ and conclusion follows since with this choice we have Λ(x, ξ) = Λ A (x, ξ).
We conclude this section with the following interesting result. Proof. The proof of [8, Lemma 3.9] works also for Λ proving it is lower semicontinuos. The openness of the stable points can be proved as in [8, Corollary 3.10].

Polystability and semi-stability
The aim of this section is to characterize polystability and semi-stability of x ∈ M in terms of the maximal weight λ x . Throughout this section we assume that the Kempf-Ness function of (M , G, K) satisfies not only (P 1)-(P 5) but also (P 6). Further, for semi-stability we also assume that M is compact. This will be enough for the case of measures on a compact manifold.
Let us denote by M ps the set of polystable points, i.e. according to Definition 16: It follows by an easy argument that if x ∈ M is polystable then Gx ∩ F −1 (0) contains exactly one K-orbit. Indeed, let y ∈ Gx be such that F(y) = 0. We shall prove that Ky = Gy ∩ F −1 (0). Assume that gy ∈ F −1 (0). Set g = k exp(v). By the K-equivariance of F it follows F(exp(v)y) = 0. As in the proof of Proposition 20, we get Rv ∈ g y and so Gy ∩ F −1 (0) = Ky. Hence we have proven the following result. In this section we give a numerical criteria for an element x ∈ M to be a polystable point. Let x ∈ M . We define Z(x) = {p ∈ ∂ ∞ X : λ x (p) = 0}. We start with the following result.
Proof. By Proposition 20 the stabilizer G x is compatible. Hence g x = k x ⊕ q with k x ⊂ k and q ⊂ p. Further, observe that for ξ ∈ p, since Ψ(x, exp(tξ)) is a convex function, we get: To conclude, we shall prove that v ∈ S(q) if and only if λ x (e(−v)) = 0. Let first v ∈ S(q). By Remark 7 the function: x, exp(tv)) = 0. By property (P 3) we get Rv ∈ q concluding the proof.
Note that the inclusion G x /K x ֒→ X is totally geodesic. We claim that a vice-versa of Proposition 31 holds as well. We start with the following Lemma.
By properties (P 3) we have v ∈ g x . Let g ′ = [q, q] ⊕ q. Observe that g ′ is a subalgebra of g due to the fact that q is a Lie triple system of p (see e.g. [27]). Let G ′ denote the connected subgroup of G with lie algebra g ′ . Hence In general, for any g ∈ G we can consider the totally geodesic submanifold defined by X ′′ = gX ′ . Since by (13) it follows Z(gx) = g(Z(x)), we have: and we are done.
Corollary 34. Let x ∈ M be a polystable point. Then there exist g ∈ G and an abelian subalgebra a ⊂ p such that a ⊂ g gx and gx is G a polystable, where G a = {g ∈ G : Ag(g)(ξ) = ξ for all ξ ∈ a}. Moreover if we denote by G ′ ss the semisimple part of G a , then gx is stable with respect to G ′ ss .
Proof. Let x ∈ M be a polystable point and let g ∈ G be such that F(gx) = 0. Set y = gx. By Proposition 31, G y is compatible, and thus g y = k y ⊕ p y ⊂ k ⊕ p, and Z(y) = e(S(p y )). Let a be a maximal abelian subalgebra of p y and let G a be the centralizer of a in G. G a is a compatible subgroup of G (see [32]) and by (26) it follows F ′ (y) = 0 and so y is polystable with respect to G a . Let G ′ ss be the semisimple part of G a . By 26 it follows that y is G ′ ss polystable and so (g ′ ss ) y is compatible. We claim (g ′ ss ) y ∩ p = {0}. Indeed, if v ∈ (g ′ ss ) y ∩ p, then v ∈ p y and [v, a] = 0. Since v / ∈ a and a is a maximal abelian subalgebra of p y we get a contradiction. Since (g ′ ss ) y is compatible it follows (g ′ ss ) y ⊂ k and so (G ′ ss ) y is compact. Therefore y is G ′ ss stable concluding the proof. The following result is a Hilbert-Mumford criterion for semi-stability. The proof is totally similar to Corollary 28's one.

Corollary 36. A point x ∈ M is G semi-stable if and only if it is A semi-stable for any abelian group A = exp(a)
, where a is a subalgebra of g contained in p.
We conclude this section with the following corollaries. Corollary 38. If x ∈ M is semistable then any y ∈ Gx also is.
Proof. Let g α ∈ G be a net such that g α x → y and let v ∈ ∂ ∞ X. By the G-equivariance of the maximal weight (14) and the semicontinuity of λ, we get:

The integral of the gradient map
Let U be a compact connected Lie group and denote by u its Lie algebra and by U C its complexification. Let (Z, ω) be a Kähler manifold on which U C acts holomorphically. Assume that U acts in a Hamiltonian fashion with momentum map µ : Z −→ u * . Consider a closed connected subgroup G of U C compatible with respect to the Cartan decomposition of U C , i.e. G = K exp(p), for K = U ∩ G and p = g ∩ iu (see [24,25]) [24,25]. The inclusion ip ֒→ u induces by restriction a K-equivariant map µ ip : Z −→ (ip) * . There is a Ad (U C )-invariant and non-degenerate bilinear form B : u C × u C −→ R which is positive definite on iu, negative definite on u and such that B(u, iu) = 0 (see [5, p. 585]). Therefore B is Ad (U C )-invariant, non-degenerate and its restriction to g satisfies the following conditions: B is Ad(G)-invariant, B(k, p) = 0, B restricted to k is negative definite and B restricted to p is positive definite. Using ·, · , we identify u ∼ = u * . For z ∈ Z, let µ p (z) ∈ p denote −i times the component of µ(z) in the direction of ip. In other words we require that µ p (z), β = − µ(z), iβ , for any β ∈ p. Then, we view µ p as a map: which is called the G-gradient map or restricted momentum map associated to µ. We also set:

By definition, it follows that gradµ
Throughout this section we fix a G-stable subset M ⊂ Z and we consider the gradient map µ p : M −→ p restricted on M . Further, we denote by β M = d dt t=0 exp(tβ)x. Observe that if M is a manifold, then β M is the gradient of µ β p restricted to M with respect to the induced Riemannian structure on M .
As Mundet pointed out in [39], the existence of the Kempf-Ness function for an action of a complex reductive group on a Kähler manifold given in [37] also holds for the setting introduced in [23,24,26].
Theorem 39. There exists a Kempf-Ness function for (M, G, K) satisfying the conditions (P 1) − (P 5). Furthermore, if M is a G-stable compact submanifold of Z, then (P 6) holds as well.
Proof. Fix x ∈ M . Let π p : g −→ p be the linear projection induced by the decomposition g = k ⊕ p and identify T e G with g in the usual way. For g ∈ G and v ∈ T g G, one has dR g −1 (v) ∈ g. Thus, we can define a 1-form σ on G by setting: Observe that σ g ∈ T g G * and σ ∈ Λ 1 (G). When we need to stress the dependence on x we will write σ x . We claim that σ is closed. In order to prove it, fix g ∈ G, v, w ∈ T g G and let ξ, η ∈ g be such that dR g (ξ) = v and dR g (η) = w. Further, let also X, Y ∈ X(G) be the fundamental vector fields corresponding to ξ and η under the action of left multiplication. In other words X is the right-invariant vector field such that X(e) = v, i.e. for h ∈ G, For a left action the map that sends a vector in g to its fundamental vector field is an antiisomorphism of Lie algebras. Thus [X, Y ] is the fundamental vector field corresponding to −[ξ, η]. Hence: We can assume by linearity that ξ, η ∈ k ∪ p.
Thus recalling that: for ξ, η ∈ k the claim is proven.
In order to prove (P 4 ), let g ∈ G and x ∈ M . We claim that R * g σ x = σ gx . Indeed if v ∈ T h G and w = dR h −1 (v), then: . Thus the claim is proven. Therefore Ψ gx − R * g Ψ x = c is a constant. Evaluating at h = e we get: and thus: as desired. Property (P 2) follows by the cocycle condition together with the fact that for any Finally, if M is a compact G-stable submanifold of Z, then the norm square of the gradient map restricted to M is bounded. Hence ψ x is Lipschitz since its differential is bounded and thus (P 6) holds.
If f : M → N is a measurable map between measurable spaces and ν is a measure on M , the image measure f * ν is the measure on Y such that f * ν(A) := ν(f −1 (A)). Observe that it satisfies the change of variables formula If G acts on M , then we have an action on the probability measures on M as follows: Let U be a compact connected Lie group and U C its complexification. As in section 7 we assume that G = K exp(p) is a compatible subgroup of U C and M is a G-stable compact subset of a Kähler manifold (Z, ω). One can prove in a totally similar way as in the proof of [8, Lemma 5.5 p. 18] that the action (43) is continuous with respect to the weak topology.
Lemma 44. Let X be a vector field on Z such that its flow {ϕ t } preserves M . If ν ∈ M (M ) and X vanishes ν-almost everywhere, then ϕ t * ν = ν for any t. Hence, if v ∈ g and v M (x) = 0 for every x outside a set of ν-measure zero, then exp(Rv) ⊂ G ν .
Proof. Set N := {p ∈ M : X(p) = 0}. Then ν(N ) = 0 and for any t ∈ R and any x ∈ N , ϕ t (x) = x. In particular both N and M − N are ϕ t -invariant. If A ⊂ M is measurable, then Proposition 45. Let M , G, K and µ p be as in § 7 and let Ψ M be the Kempf-Ness function of (M, G, K). The function: is a Kempf-Ness function for (P(M ), G, K) satisfying conditions (P 1 ) − (P 5 ). If in addition M is compact then Ψ also satisfies condition (P 6). Moreover, if we denote by X = G/K, then and if F denotes the gradient map, then: For a sake of completeness we sketch the proof which is totally similar to that of Proposition 5.12 in [8].
Proof. Since Ψ M is left-invariant with respect to K, the same holds for Ψ P .
Again by (P 3) this implies that v M = 0 ν-almost everywhere. By Lemma 44 it follows that exp(Rv) ⊂ G ν . We have proven that Ψ P satisfies (P 1) − (P 3). The cocycle condition for Ψ P follows immediately from the cocycle condition for Ψ M . Fix ν ∈ P(M ). It is immediate to verify that the function ψ P associated to Ψ P as in (3) is the one given by (47). Therefore it is clearly continuous on P(M ). Finally, it is easy to check that Ψ P ν is Lipschitz whenever M is a compact manifold.
From now on we assume that M is a compact G-stable submanifold of Z. We shall compute the maximal weight using the geometry of the gradient map. We begin recalling the following slice theorem proved in [23,24]. Fix v ∈ p. The gradient flow of a function f ∈ C ∞ (M ) is usually defined as the flow of the vector field − grad f . Let {ϕ t } denote the gradient flow of µ v . Since grad µ v = β M , we have ϕ t (x) = exp(tv)x. Then the function µ v p is a Morse-Bott function [23,24,25]. If we denote by c 1 > · · · > c r the critical values of µ v , then the corresponding level sets of µ v p , C i := (µ v ) −1 (c i ) are submanifolds which are the components of Crit(µ v ). By Theorem 49 it follows that for any x ∈ M the limit: exists. Let us denote by W i the unstable manifold of the critical component C i for the gradient flow of µ v : Then: and for any i the map: is a smooth fibration with fibres diffeomorphic to R l i where l i is the index (of negativity) of the critical submanifold C i .
Proposition 52. Let ν be a polystable measure which is not stable. Hence there exist an abelian subalgebra a ⊂ g ν such that ν is supported on M a = {x ∈ M : ξ M (x) = 0 for any ξ ∈ a}.
Let a ′ ⊂ q be a maximal abelian subalgebra of q. Then a = Ad(g)(a ′ ) is an abelian subalgebra of g ν and S(a) ⊂ Z(ν). Let u ∈ a. Then exp(tu) ∈ G ν and thus: Let A ⊂ M be a measurable subset. Then ν(A) = lim n →−∞ ν(exp(nu)(A)) = ν(α(A)), where α is the gradient flow of µ u p . Hence ν is supported on the critical submanifolds of µ u p for any u ∈ a. Hence ν is supported on M a . Now, we explicitly compute the maximal weights.
Theorem 53. With the notation above we have We give a sketch of the proof, which follows essentially that of [8,Th. 5.23].
Proof. By definition of λ ν and by differentiating under the integral sign we get Applying the dominated convergence theorem, since Since for x ∈ W i , α(x) ∈ C i and so µ v p (α(x)) = c i , we finally obtain: Let E(µ p ) denote the convex hull of µ p (M ) ⊂ p, i.e. a K-invariant convex body in p. Let a ⊂ p be a abelian subalgebra and let π : p −→ a be the orthogonal projection onto a. Then µ a = π • µ p is the gradient map associated to A = exp(a). Denote by P = µ a (M ). It is wellknown that P is a finite union of polytopes [26]. In the sequel we always assume that P is a polytope, hypothesis which holds e.g. if G = U C and M is a complex connected submanifold by the Atiyah-Guillemin-Sternberg convexity theorem [1,16] or when M is an irreducible semialgebraic subset of a Hodge manifold Z [6,26,33]. We point out that the convex bodies E(µ p ) and P are strongly related [7]. Observe that under such hypothesis for any v ∈ p, a local maxima of µ v p is a global maxima. Then the Morse-Bott decomposition of M with respect to µ v p , i.e., M = r i=0 W i , has a unique open and dense unstable manifold W u r and the others unstable manifolds are proper submanifolds. Therefore, if ν is a smooth measure of M then W u r has full measure and so λ ν (e(−v)) = c r = max x∈M µ v p . Summing up we have proved the following result.
Corollary 54. If ν is a smooth measure on M , then for any v ∈ p: Since ν is a probability measures, it follows that F(ν) ∈ E(µ p ). Indeed, F(ν) is the barycenter of the gradient map µ p with respect to ν and so it lies in E(µ p ). If 0 / ∈ E(µ p ), then there exists v ∈ E(µ p ) realizing the minimum distance of E(µ p ) to the origin. Moreover v is a K fixed point due to the fact that E(µ p ) is K-invariant. Hence up to shifting the gradient map we may assume that 0 ∈ E(µ p ). Under this assumption we get the following result.
Proposition 55. If 0 ∈ E(µ p ) then any smooth measure on M is semi-stable.
Proof. Let v ∈ p. By the above corollary, we have λ ν (e(−v)) = max x∈M µ v p . Since 0 ∈ E(µ p ), it follows that λ ν (e(−v)) = max x∈M µ v p ≥ 0. By Theorem 35 ν is semi-stable. Proof. By the above Proposition any smooth measure is semi-stable. Since smooth measures are dense, then the set of semi-stable measures is dense. If 0 belongs to the interior of the E(µ p ), then for any v ∈ p the function µ v p change sign and so it has a strictly positive maxima. By Corollary 54 λ ν (e(−v)) > 0 and by Theorem 23 we get that it is stable. Since by Corollary 29 the set of the stable points is also open, it means P s (M ) is open and dense.
For the case (b), namely for ξ = (λ 1 , Finally, when ξ = (λ 1 , λ 1 , −2λ 1 ), Crit(µ ξ ) = P(V 1 ) ∪ {p 3 }, Denote by Li ⊂ R 3 a linear subspace of R 3 of dimension 2 and let p ∈ P 2 (R). Then, ν is stable iff for any choice of Li and p: ν is semistable iff for any choice of Li and p: and ν is polystable iff either it is stable or it is one of the following: i.e. it is supported by some P(Li) and by a point p or by three points (see the proof of Prop. 60 below for details).
We conclude with the following proposition which states necessary and sufficients conditions for stability and polystability in general dimension.
The measure ν is polystable iff there exists a splitting R n+1 = Li 1 ⊕ · · · ⊕ Li r such that ν is supported on P(Li 1 ) ∪ · · · ∪ P(Li r ). Moreover where δ P(Li j ) is a stable measure of P(Li j ) with respect to SL(Li j ).
Vice-versa, assume ν = r j dim(Li j ) n+1 δ P(Li j ) with respect to a splitting: where δ P(Li j ) is a stable measure of P(Li j ) with respect to SL(Li j ). Let g ∈ SL(n + 1, R) such that if we denote by V j = gLi j for j = 1, . . . , r, then: is an orthogonal splitting. By the above computation we get F(gν) = r j=1 dim V j n+1 F j (δ P(V j ) ), where δ P(V j ) = gδ P(Li j ) for j = 1, . . . , r. By the above discussion since δ P(Li j ) is stable with respect to SL(Li j ), then gδ P(V j ) is stable with respect to SL(V j ). Hence there exists g j ∈ SL(V j ) such that F j (g j δ P(V j ) ) = 0. Let h = g 1 × · · · × g r ∈ SL(V 1 ) × · · · × SL(V r ) ⊂ SL(n + 1, R). Then F(hgν) = r j=1 dim V j n + 1 F j (g j δ P(V j ) ) = 0, concluding the proof.