A Hilbert–Mumford criterion for polystability for actions of real reductive Lie groups

Abstract

We study a Hilbert–Mumford criterion for polystablility associated with an action of a real reductive Lie group G on a real submanifold X of a Kähler manifold Z. Suppose the action of a compact Lie group with Lie algebra $\mathfrak {u}$ extends holomorphically to an action of the complexified group $U^{\mathbb {C}}$ and that the U-action on Z is Hamiltonian. If $G\subset U^{\mathbb {C}}$ is compatible, there is a corresponding gradient map $\mu _\mathfrak {p}: X\rightarrow \mathfrak {p}$, where $\mathfrak {g}= \mathfrak {k}\oplus \mathfrak {p}$ is a Cartan decomposition of the Lie algebra of G. Under some mild restrictions on the G-action on X,  we characterize which G-orbits in X intersect $\mu _\mathfrak {p}^{-1}(0)$ in terms of the maximal weight functions, which we viewed as a collection of maps defined on the boundary at infinity ($\partial _\infty G/K$) of the symmetric space G/K. We also establish the Hilbert–Mumford criterion for polystability of the action of G on measures.

1 Introduction

The classical Hilbert–Mumford criterion in Geometric Invariant Theory (in projective algebraic geometry) is an explicit numerical criterion for finding the stability of a point in terms of an invariant known as maximal weight function [26]. This criterion has been extended to the non-algebraic Kählerian settings using the theory of Kähler quotients and a version of maximal weight function [10, 22, 27, 28, 30]. For this setting, a Kähler manifold $(Z,\omega )$ with a holomorphic action of a complex reductive Lie group $U^{\mathbb {C}}$, where $U^{\mathbb {C}}$ is the complexification of a compact Lie group U with Lie algebra $\mathfrak {u}$ is considered. Assume $\omega$ is U-invariant and that there is a U-equivariant momentum map $\mu : Z \rightarrow \mathfrak {u}^*.$ By definition, for any $\xi \in \mathfrak {u}$ and $z\in Z,$ $d\mu ^\xi = i_{\xi _Z}\omega ,$ where $\mu ^\xi (z):= \mu (z)( \xi )$ and $\xi _Z$ denotes the fundamental vector field induced on Z by the action of U,  i.e.,

$$\begin{aligned} \xi _Z(z):= \frac{d}{dt}\bigg \vert _{t=0} \exp (t\xi )z \end{aligned}$$

(see, for example, [23] for more details on the momentum map).

We aim to investigate a class of actions of real reductive Lie groups on real submanifolds of Z using gradient map techniques. This setting was recently introduced in [16,17,18]. More precisely, a subgroup G of $U^{\mathbb {C}}$ is compatible if G is closed and the map $K\times \mathfrak {p} \rightarrow G,$ $(k,\beta ) \mapsto k\exp (\beta )$ is a diffeomorpism where $K:= G\cap U$ and $\mathfrak {p}:= \mathfrak {g}\cap \text {i}\mathfrak {u};$ $\mathfrak {g}$ is the Lie algebra of G. The Lie algebra $\mathfrak {u}^{\mathbb {C}}$ of $U^{\mathbb {C}}$ is the direct sum $\mathfrak {u}\oplus \text {i}\mathfrak {u}.$ It follows that G is compatible with the Cartan decomposition of $U^{\mathbb {C}}= U\exp (\text {i}\mathfrak {u})$, K is a maximal compact subgroup of G with Lie algebra $\mathfrak {k}$ and that $\mathfrak {g} = \mathfrak {k}\oplus \mathfrak {p}$. The inclusion $\text {i}\mathfrak {p}\hookrightarrow \mathfrak {u}$ induces by restriction, a K-equivariant map $\mu _{\text {i}\mathfrak {p}}: Z \rightarrow (\text {i}\mathfrak {p})^*.$ One can choose and fix an $\textrm{Ad}(U^{\mathbb {C}})$-invariant inner product B of Euclidean type on the Lie algebra $\mathfrak {u}^{\mathbb {C}}$, see [10, Section 3.2], [25, Definition 3.2.4] and also [20, Section 2.1] for the analog in the algebraic GIT. Such an inner product will automatically induce a well-defined inner product on any maximal compact subgroup $U'$ of $U^{\mathbb {C}}$.

Let $\langle \cdot , \cdot \rangle$ denote the real part B. Then $\langle \cdot , \cdot \rangle$ is positive definite on $\text {i}\mathfrak {u}$, negative definite on $\mathfrak {u}$, $\langle \mathfrak {u}, \text {i} \mathfrak {u}\rangle =0$ and finally the multiplication by $\text {i}$ satisfies $\langle \text {i} \cdot ,\text {i} \cdot \rangle =-\langle \cdot , \cdot \rangle$. We use $-\langle \cdot , \cdot \rangle$ to identify ${\mathfrak {u}}^*$ with ${\mathfrak {u}}$ and we think the momentum map $\mu$ as a ${\mathfrak {u}}$-valued map. Hence we replace consideration of $\mu _{\text {i}\mathfrak {p}}$ by that of $\mu _\mathfrak {p}:Z \longrightarrow \mathfrak {p}$, where

$$\begin{aligned} \mu _\mathfrak {p}^\beta (x):=\langle \mu _\mathfrak {p}(x),\beta \rangle :=\langle \text {i}\mu (x),\beta \rangle =-\langle \mu (x),-\text {i}\beta \rangle =\mu ^{-\text {i} \beta }(x). \end{aligned}$$

The map $\mu _\mathfrak {p}:Z \longrightarrow \mathfrak {p}$ is called the G-gradient map associated with $\mu$. It is K-equivariant and grad$\, \mu _\mathfrak {p}^\beta = \beta _Z$ for any $\beta \in \mathfrak {p}$. Here the grad is computed with respect to the Riemannian metric induced by the Kähler structure. For a G-stable locally closed real submanifold X of Z,  we also denote the restriction $\mu _\mathfrak {p}$ to X by$\mu _\mathfrak {p}: X\rightarrow \mathfrak {p}$. We have $\textrm{grad}\, \mu _\mathfrak {p}=\beta _X$ for any $\beta \in \mathfrak {p}$, where the gradient is now computed with respect to the induced Riemannian metric on X.

Different notions of stability of points in X can be identified by taking into account the position of their G-orbits with respect to $\mu _\mathfrak {p}^{-1}(0).$ A point $x\in X$ is polystable if it’s G-orbit intersects the level set $\mu _\mathfrak {p}^{-1}(0)$ (i.e., $G\cdot x \cap \mu _\mathfrak {p}^{-1}(0) \ne \emptyset$). As pointed out in the introduction of [28] (see also [5]), a set of polystable points plays a critical role in the construction of a good quotient of X by the action of G. The aim of this article is to answer the first part of question 1.1 in [28] for actions of real Lie groups on real submanifolds of a Kähler manifold, generalizing [28]. Following [28], we require a mild technical restriction to be satisfied; namely, the fundamental vector field induced by the action grows at most linearly with respect to the distance function from a given base point. More precisely, we require the following assumption.

Assumption 1.1

X is connected, and there exists a point $x_0\in X$ and a constant $C > 0$ such that for any $x\in X$ and any $\beta \in \mathfrak {p},$

$$\begin{aligned} \parallel \beta _X(x)\parallel \le C\parallel \beta \parallel (1 + d_X(x_0, x)), \end{aligned}$$

(1)

where $d_X$ denotes the geodesic distance between points of X with respect to the induced Riemannian metric on X.

If X is compact or if X is a vector space and the G-action on X is linear then this condition is satisfied. Under this assumption, we construct the maximal weight function

$$\begin{aligned} \lambda _x: \partial _\infty (G/K) \rightarrow \mathbb {R}\cup \{\infty \} \end{aligned}$$

for any $x\in X$. It is well-known that G acts on $\partial _\infty (G/K)$ and the G-action on $\partial _\infty (G/K)$ is continuous with respect to the sphere topology [7]. The same idea given in [28] proves that the maximal weight functions are G-equivariant. If $g\in G,$ $p\in \partial _\infty (G/K)$ and $x\in X,$ then $\lambda _{gx} (p)=\lambda _x (g^{-1} p)$. We then prove that a point $x\in X$ is polystable if and only if $\lambda _x \ge 0$ and for any $p\in \partial _\infty (G/K)$ such that $\lambda _x(p) = 0$ there exists $p'\in \partial _\infty (G/K)$ such that p and $p'$ are connected in the sense of Definition 5.1 below. In the classical case of a group action on a Kähler manifold this characterization is due to Mundet i Riera [28].

The idea of viewing the maximal weights as defining functions on the boundary $\partial _\infty M$ appeared in [22]. They also give a characterization of polystability which they refer to as nice semistability [22, Definition 3.13]. Finally, we prove the polystability criterion for the G-action on measures. Polystable measures are interested in an application to upper bounds for the first eigenvalue of the Laplacian of functions, see, for instance, [3, Section 1.17], [8, 19] and the introduction to [1].

2 Compatible subgroups, parabolic subgroups, and gradient maps

Let U be a compact Lie group and let $U^{\mathbb {C}}$ be the corresponding complex linear algebraic group [11]. The group $U^{\mathbb {C}}$ is reductive and is the universal complexification of U in the sense of [21]. On the Lie algebra level, we have the Cartan decomposition $\mathfrak {u}^{\mathbb {C}}=\mathfrak {u}\oplus \text {i} \mathfrak {u}$ with a corresponding Cartan involution $\theta :\mathfrak {u}^{\mathbb {C}}\longrightarrow \mathfrak {u}^{\mathbb {C}}$ given by $\xi +\text {i}\nu \mapsto \xi -\text {i}\nu$. We also denote by $\theta$ the corresponding involution on $U^{\mathbb {C}}$. The real analytic map $F:U\times \text {i}\mathfrak {u}\longrightarrow U^{\mathbb {C}}$, $(u,\xi ) \mapsto u\exp (\xi )$ is a diffeomorphism. We refer to the composition $U^{\mathbb {C}}=U\exp (\text {i}\mathfrak {u})$ as the Cartan decomposition of $U^{\mathbb {C}}$.

Let $G\subset U^{\mathbb {C}}$ be a closed real subgroup of $U^{\mathbb {C}}$. We say that G is compatible with the Cartan decomposition of $U^{\mathbb {C}}$ if $F (K \times \mathfrak {p}) = G$ where $K:=G\cap U$ and $\mathfrak {p}:= \mathfrak {g}\cap \text {i}\mathfrak {u}$. The restriction of F to $K\times \mathfrak {p}$ is then a diffeomorphism onto G. It follows that K is a maximal compact subgroup of G and that $\mathfrak {g}= \mathfrak {k}\oplus \mathfrak {p}$. Since K is a retraction of G, it follows that G has only finitely many connected components and $G=KG^o$, where $G^o$ denotes the connected component of G containing e.

Lemma 2.1

( [2, Lemma 7])

• a) If $G\subset U^{\mathbb {C}}$ is a compatible subgroup, and $H\subset G$ is closed and $\theta$-invariant, then H is compatible if and only if H has only finitely many connected components.

• b) If $G\subset U^{\mathbb {C}}$ is a connected compatible subgroup, then $G_{ss}$ is compatible.

• c) If $G\subset U^{\mathbb {C}}$ is a compatible subgroup and $E\subset \mathfrak {p}$ is any subset, then $G^E = \{g \in G: {\text {Ad}}(g) (\beta ) = \beta , \; \forall \; \beta \in E\}$ is compatible. Indeed, $G^E=K^E\exp (\mathfrak {p}^E)$, where $K^E=K\cap G^E$ and $\mathfrak {p}^E=\{x\in \mathfrak {p}:\, [x,E]=0\}$. If $E=\{\beta \}$ then we simply write $K^\beta$, $\mathfrak {p}^\beta$ and $G^\beta$.

If $\beta \in \mathfrak {p}$ we define,

$$\begin{aligned} G^{\beta +}&:=\{g \in G : \lim _{t\rightarrow - \infty } \exp (t\beta )\, g \exp (-t\beta ) \text{ exists } \},\\ R^{\beta +}&:=\{g \in G : \lim _{t\rightarrow - \infty } \exp (t\beta )\, g \exp (-t\beta ) =e \}. \end{aligned}$$

Then $G^{\beta +}$ is a parabolic subgroup of G with unipotent radical $R^{\beta +}$. $G^{\beta +}$ is the semi-direct product of $G^\beta$ and $R^{\beta +}$.

Proposition 2.2

For any $\beta \in \mathfrak {p}$, we have $G=KG^{\beta +}$.

Proof

If G is connected, the result is well-known, see for instance [2, Lemma 9] and [17, Lemma 4.1]. Since $G=KG^o$, it follows that $G=KG^o=K(G^{o})^{\beta +}=KG^{\beta +},$ concluding the proof. $\square$

Let $(Z,\omega )$ be a Kähler manifold endowed with a holomorphic action $U^{\mathbb {C}}\times Z \longrightarrow Z$. We assume that $\omega$ is U-invariant and there exists a U-equivariant momentum map $\mu :Z \longrightarrow \mathfrak {u}^*$. We fix an $\textrm{Ad}(U^{\mathbb {C}})$ inner product B on $\mathfrak {u}^{\mathbb {C}}$ and we denote by $\langle \cdot , \cdot \rangle$ it’s real part. Then $\langle \cdot , \cdot \rangle$ is positive definite on $\text {i}\mathfrak {u}$, negative definite on $\mathfrak {u}$, $\langle \mathfrak {u}, \text {i} \mathfrak {u}\rangle =0$ and finally the multiplication by $\text {i}$ satisfies $\langle \text {i} \cdot ,\text {i} \cdot \rangle =-\langle \cdot , \cdot \rangle$. We may think of the momentum map as a $\mathfrak {u}$-valued map using $-\langle \cdot , \cdot \rangle$.

Let G be a closed and compatible subgroup of $U^{\mathbb {C}}$. The G-gradient map associated with $\mu$ is given by $\mu _\mathfrak {p}:Z \longrightarrow \mathfrak {p}$ where

$$\begin{aligned} \mu _\mathfrak {p}^\beta (x):=\langle \mu _\mathfrak {p}(x),\beta \rangle :=\langle \text {i}\mu (x),\beta \rangle =-\langle \mu (x),-\text {i}\beta \rangle =\mu ^{-\text {i} \beta }(x), \end{aligned}$$

for any $\beta \in \mathfrak {p}$. For the rest of the paper, we fix a G-invariant locally closed submanifold X of Z. We also denote the restriction of $\mu _\mathfrak {p}$ to X by $\mu _\mathfrak {p}$. The map $\mu _\mathfrak {p}:X \longrightarrow \mathfrak {p}$ is K-equivariant and $\textrm{grad}\, \mu _\mathfrak {p}^\beta =\beta _X,$ for any $\beta \in \mathfrak {p}$, where the gradient is computed with respect to the induced Riemannian structure.

Let $\beta \in \mathfrak {p}$ and let $X^\beta =\{z\in X:\, \beta _X(z)=0\}.$ If $A=\exp (\mathbb {R}\beta )$ we have a Slice Theorem at every point of X [17, Theorem 3.1]. In particular, $X^\beta$ is a smooth, possibly disconnected, submanifold of X. Since $\textrm{grad}\, \mu _\mathfrak {p}^\beta =\beta _X$ it follows that $X^\beta$ is the set of critical points of $\mu _\mathfrak {p}^\beta$ that we denote by $\textrm{Crit}\, \mu _\mathfrak {p}^\beta$. Moreover, $\mu _\mathfrak {p}^\beta :X \longrightarrow \mathbb {R}$ is a Morse-Bott function, see for instance [2, Corollary 2.3].

Assume that X is compact. The Slice Theorem implies that the limit $\lim _{t\rightarrow +\infty } \exp (t\beta )x$ exists and it lies in $X^\beta$ for any $x\in X$.

Let $C_1,\ldots ,C_k$ be the connected components of $X^\beta$. Let

$$\begin{aligned} W_i^\beta :=\{x\in X:\, \lim _{t\rightarrow +\infty } \exp (t\beta )x \in C_i\}, \end{aligned}$$

for $i=1,\ldots ,k$. One of the fundamental theorems of Morse theory is the following, see for instance [9].

Theorem 2.3

$W_i^\beta$ is an immersed submanifold, which is called the unstable manifold corresponding to $C_i$, and

$$\begin{aligned} \varphi _\infty : W_i^\beta \longrightarrow C_i, \qquad x \mapsto \exp (t\beta )x, \end{aligned}$$

is smooth. Moreover, $X=\bigsqcup W_i^\beta$ (disjoint union).

3 Symmetric spaces

Let $G\subset U^{\mathbb {C}}$ be a closed compatible subgroup. Then $G = K\exp (\mathfrak {p}),$ where $K:= G\cap U$ is a maximal compact subgroup of G and $\mathfrak {p}:= \mathfrak {g}\cap \text {i}\mathfrak {u};$ $\mathfrak {g}$ is the Lie algebra of G. Let $M = G/K$ and let $\langle \cdot , \cdot \rangle$ be the real part of the fixed $\textrm{Ad}(U^{\mathbb {C}})$-invariant inner product B of Euclidean type on the Lie algebra $\mathfrak {u}^{\mathbb {C}}$. Then $\langle \cdot , \cdot \rangle$ is positive definite on $\text {i}\mathfrak {u}$, negative definite on $\mathfrak {u}$, $\langle \mathfrak {u}, \text {i} \mathfrak {u}\rangle =0$ and finally the multiplication by $\text {i}$ satisfies $\langle \text {i} \cdot ,\text {i} \cdot \rangle =-\langle \cdot , \cdot \rangle$. Since $\mathfrak {k}\subset \mathfrak {u}$, respectively $\mathfrak {p}\subset \text {i} \mathfrak {u}$, the formula $\langle \xi _1 + \beta _1, \xi _2 + \beta _2\rangle := -\langle \xi _1, \xi _2\rangle + \langle \beta _1, \beta _2\rangle$ where $\xi _1,\xi _2\in \mathfrak {k}$ and $\beta _1,\beta _2\in \mathfrak {p}$, defines an $\textrm{Ad}(K)$-invariant inner product on $\mathfrak {g}$ and so it induces a G-invariant Riemannian metric of nonpositive curvature on M. Moreover, M is a symmetric space of non-compact type [7]. Let $\pi : G \rightarrow M$ be the projection onto the right cosets of G. G acts isometrically on M from left by

$$\begin{aligned} L_g: M \rightarrow M; \quad L_g(hK):= ghK, \quad g,h \in G. \end{aligned}$$

A geodesic $\gamma$ in M is given by $\gamma = g\exp (t\beta )K,$ where $g\in G$ and $\beta \in \mathfrak {p}.$ For $\beta \in \mathfrak {p}$, we set $\gamma ^\beta (t) = \exp (t\beta )K$ and $o:= K \in M.$

Since M is a Hadamard manifold there is a natural notion of a boundary at infinity which can be described using geodesics. We refer the reader to [7, 14] for more details. Two unit speed geodesics $\gamma , \gamma ': \mathbb {R}\rightarrow M$ are equivalent, denoted by $\gamma \sim \gamma '$, if $\sup _{t>0}d(\gamma (t), \gamma '(t)) < +\infty .$

Definition 3.1

The Tits boundary of M denoted by $\partial _\infty M$ is the set of equivalence classes of unit speed geodesics in M.

The map that sends $\beta \in \mathfrak {p}$ to the tangent vector $\dot{\gamma }^\beta (0)$ produces an isomorphism $\mathfrak {p}\cong T_oM.$ Since any geodesic ray in M is equivalent to a unique ray starting from o,  the map

$$\begin{aligned} e:&S(\mathfrak {p}) \rightarrow \partial _\infty M;\\&e(\beta ) := [\gamma ^\beta ] \end{aligned}$$

where $S(\mathfrak {p}):= \{\beta \in \mathfrak {p}: \parallel \beta \parallel = 1\}$ is the unit sphere in $\mathfrak {p},$ is a bijection. The sphere topology is the topology on $\partial _\infty M$ such that e is a homomorphism. Since G acts by isometries on M,  then for every unit speed geodesic $\gamma ,$ $g\gamma$ is also a unit speed geodesic for any $g\in G.$ Moreover, if $\gamma \sim \gamma '$ then $g\gamma \sim g\gamma '.$ There is a G-action on $\partial _\infty M$ given by:

$$\begin{aligned} g\cdot [\gamma ] = [g\cdot \gamma ] \end{aligned}$$

and this action also induces a G-action on $S(\mathfrak {p})$ given by

$$\begin{aligned} g\cdot \beta := e^{-1}(g\cdot e(\beta )) = e^{-1}[g\cdot \gamma ^\beta ]. \end{aligned}$$

This action is continuous with respect to the sphere topology on $\partial _\infty M$. The K-action on $\partial _\infty M$ induces the adjoint action of K on $S(\mathfrak {p})$, see for instance [7].

Let H be a compatible subgroup of G,  i.e $H:= L \exp (\mathfrak {q}),$ where $L:= H \cap K$ and $\mathfrak {q}= \mathfrak {h}\cap \mathfrak {p}$, where $\mathfrak {h}$ is the Lie algebra of H. It follows that H 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 $\mathfrak {h}= \mathfrak {l}\oplus \mathfrak {q}.$ The inclusion $M':= H/L \hookrightarrow M=G/K$ is totally geodesic and induces an inclusion $\partial _\infty M' \hookrightarrow \partial _\infty M.$

3.1 The Kempf–Ness function

Given G a real reductive group which acts smoothly on Z;  $G = K\text {exp}(\mathfrak {p}),$ where K is a maximal compact subgroup of G. Let X be a G-invariant locally closed submanifold of Z. As Mundet pointed out in [29], there exists a function $\Phi : X \times G \rightarrow \mathbb {R},$ such that

$$\begin{aligned} \langle \mu _\mathfrak {p}(x), \xi \rangle = \dfrac{\textrm{d} }{\textrm{dt}}\bigg \vert _{t=0} \Phi (x, \exp (t\xi )), \qquad \xi \in \mathfrak {p}, \end{aligned}$$

and satisfying the following conditions:

• a) For any $x\in X,$ the function $\Phi (x,.)$ is smooth on G.

• b) The function $\Phi (x,.)$ is left-invariant with respect to K,  i.e., $\Phi (x, kg) = \Phi (x, g).$

• c) For any $x\in X,$ $v\in \mathfrak {p}$ and $t\in \mathbb {R};$

  $$\begin{aligned} \frac{d^2}{dt^2}\Phi (x, \exp (tv)) \ge 0. \end{aligned}$$

  Moreover:

  $$\begin{aligned} \frac{d^2}{dt^2}\Phi (x, \exp (tv)) = 0 \end{aligned}$$

  if and only if $\exp (\mathbb {R}v)\subset G_x.$

• d) For any $x\in X,$ and any $g, h \in G;$

  $$\begin{aligned} \Phi (x, hg) = \Phi (x, g) + \Phi (gx, h). \end{aligned}$$

  This equation is called the cocycle condition. Finally, using the cocycle condition, we have

  $$\begin{aligned} \dfrac{\textrm{d} }{\textrm{dt}}\Phi (x,\exp (t\beta ))=\langle \mu _\mathfrak {p}(\exp (t\xi )x),\beta \rangle . \end{aligned}$$

The function $\Phi : X \times G \rightarrow \mathbb {R}$ is called the Kempf-Ness function for (X, G, K). It is just the restriction of the classical Kempf-Ness function $Z\times U^{\mathbb {C}}\longrightarrow \mathbb {R}$ considered in [29, 30] to $X\times G$ [4]. Moreover, if $H\subset G$ is compatible and $Y\subset X$ is a H-stable submanifold of X, then the restriction $\Phi _{\vert {Y\times H}}$ is the Kempf-Ness function of the H-gradient map on Y.

Let $x\in X$. By property b),  i.e. $\Phi (x, kg) = \Phi (x, g)$, the function $\Phi _x: G \rightarrow \mathbb {R}$ given by $\Phi _x(g):= \Phi (x, g^{-1})$ descends to a function on M which we denote by the same symbol. That is

$$\begin{aligned} \Phi _x: M \longrightarrow \mathbb {R}; \qquad \Phi _x (gK):=\Phi (x,g^{-1}). \end{aligned}$$

The cocycle condition d) can be rewritten as

$$\begin{aligned} \Phi _x(ghK) = \Phi _{g^{-1}x}(hK) + \Phi _x(gK), \end{aligned}$$

(2)

and it is equivalent to $L_g^{*}\Phi _x = \Phi _{g^{-1}x} + \Phi _{g^{-1}x}(gK),$ where $L_g$ denotes the action of G on X given above.

Note that

$$\begin{aligned} -(d\Phi _x)_o(\dot{\gamma }^\beta (0)) = \dfrac{\textrm{d} }{\textrm{dt}}\bigg \vert _{t=0} \Phi _x(\exp (-t\beta )K) = \dfrac{\textrm{d} }{\textrm{dt}}\bigg \vert _{t=0} \Phi (x, \exp (t\beta )) = \langle \mu _\mathfrak {p}(x), \beta \rangle . \end{aligned}$$

Lemma 3.1

Let $x\in X$ and let $\Phi _x: M \rightarrow \mathbb {R}.$ Suppose $\gamma (t) = g\exp (t\beta )K$ for $\beta \in \mathfrak {p}$ is a geodesic in M,  then $\Phi _x \circ \gamma$ is convex and so,

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{d}{dt}(\Phi _x\circ \gamma ) = \lim _{t\rightarrow \infty }\frac{\Phi _x\circ \gamma }{t} \end{aligned}$$

.

Proof

That $\Phi _x$ is a convex function on M follows from [3, Lemma 2.19]. Let $f(t) = (\Phi _x\circ \gamma )(t).$ Since f is convex,

$$\begin{aligned} \frac{f(s)}{s}\le f'(s) \le \frac{f(t) - f(s)}{t-s} \quad 0< s < t. \end{aligned}$$

Furthermore, the two quantities are increasing in s,  while the third in t. Hence,

$$\begin{aligned} \lim _{s\rightarrow \infty }\frac{f(s)}{s}\le \lim _{s \rightarrow \infty } f'(s) \le \lim _{t\rightarrow \infty } \frac{f(t) - f(s)}{t-s}. \end{aligned}$$

Since the last limit is the same with the first limit, we have $\lim _{t \rightarrow \infty } f'(t) = \lim _{t\rightarrow \infty }\frac{f(t)}{t}$ and the result follows. $\square$

4 Stability and maximal weight function

Let U be a compact Lie group and $U^{\mathbb {C}}$ its complexification. Let $(Z,\omega )$ be a Kähler manifold. In this paper, we assume that the complex reductive Lie group $U^{\mathbb {C}}$ acts holomorphically on Z. The Kähler form $\omega$ is U-invariant and the U-action on Z is Hamiltonian and so there exists a momentum map $\mu : Z \rightarrow \mathfrak {u}.$ Let $G\subset U^{\mathbb {C}}$ be a closed compatible subgroup. Then $G = K\exp (\mathfrak {p}),$ where $K:= G\cap U$ is a maximal compact subgroup of G and $\mathfrak {p}:= \mathfrak {g}\cap \text {i}\mathfrak {u};$ $\mathfrak {g}$ is the Lie algebra of G. Suppose $X\subset Z$ is a G-stable locally closed connected real submanifold of Z with the gradient map $\mu _\mathfrak {p}: X\rightarrow \mathfrak {p}.$ From now on, X satisfies Assumption 1.1.

We recall that by $G_x$ and $K_x$, we denote the stabilizer subgroup of $x\in X$ with respect to the G-action and the K-action respectively, and by $\mathfrak {g}_x$ and $\mathfrak {k}_x$ their respective Lie algebras.

Definition 4.1

Let $x\in X.$ Then:

• a) x is stable if $G\cdot x \cap \mu _\mathfrak {p}^{-1}(0) \ne \emptyset$ and $\mathfrak {g}_x$ is conjugate to a Lie subalgebra of $\mathfrak {k}.$

• b) x is polystable if $G\cdot x \cap \mu _\mathfrak {p}^{-1}(0) \ne \emptyset .$

• c) x is semistable if $\overline{G\cdot x} \cap \mu _\mathfrak {p}^{-1}(0) \ne \emptyset .$

We denote by $X^s_{\mu _\mathfrak {p}}$, $X^{ss}_{\mu _\mathfrak {p}}$, $X^{ps}_{\mu _\mathfrak {p}}$ the set of stable, respectively semistable, polystable, points. It follows directly from the definitions above that the conditions are G-invariant in the sense that if a point satisfies one of the conditions, then every point in its orbit satisfies the same condition, and for stability, recall that $\mathfrak {g}_{gx} = \text {Ad}(g)(\mathfrak {g}_x).$

The following well-known result establishes a relation between the Kempf-Ness function and the polystability condition. A proof is given in [5].

Proposition 4.1

Let $x\in X$ and let $g\in G$. The following conditions are equivalent:

• a) $\mu _\mathfrak {p}(gx)=0$.

• b) g is a critical point of $\Phi (x,\cdot )$.

• c) $g^{-1}K$ is a critical point of $\Phi _x$.

Proposition 4.2

Let $x\in X$.

• If x is polystable, then $G_x$ is compatible.

• If x is stable, then $G_x$ is compact.

4.1 4.0.1. maximal weight function

In this section, we introduce the maximal weight function associated with an element $x\in X.$

For any $t\in \mathbb {R},$ define $\lambda (x,\beta ,t) = \langle \mu _\mathfrak {p}(\exp (t\beta )x), \beta \rangle .$

$$\begin{aligned} \lambda (x,\beta ,t) = \langle \mu _\mathfrak {p}(\exp (t\beta )x), \beta \rangle = \frac{d}{dt}\Phi (x, \exp (t\beta )), \end{aligned}$$

(3)

where $\Phi : X\times G\rightarrow \mathbb {R}$ is the Kempf-Ness function. By the properties of the Kempf-Ness function,

$$\begin{aligned} \frac{d}{dt}\lambda (x,\beta ,t) = \frac{d^2}{dt^2}\Phi (x, \exp (t\beta )) \ge 0. \end{aligned}$$

This means that $\lambda (x,\beta ,t)$ is a non decreasing function as a function of t.

The maximal weight of $x\in X$ in the direction of $\beta \in \mathfrak {p}$ is defined in [5] as the numerical value

$$\begin{aligned} \lambda (x,\beta ) = \lim _{t\rightarrow \infty }\lambda (x,\beta ,t)=\lim _{t\rightarrow +\infty } \langle \mu _\mathfrak {p}(\exp (t\beta )x), \beta \rangle \in \mathbb {R}\cup \{\infty \}. \end{aligned}$$

Note that

$$\begin{aligned} \frac{d}{dt}\lambda (x,\beta ,t) = \parallel \beta _X(\exp (t\beta ) x)\parallel ^2, \end{aligned}$$

and so

$$\begin{aligned} \lambda (x,\beta ,t) = \langle \mu _\mathfrak {p}(x), \beta \rangle + \int _0^t\parallel \beta _X(\exp (s\beta ) x)\parallel ^2 \textrm{ds}. \end{aligned}$$

(4)

Lemma 4.3

Let $\beta ,\beta '\in \mathfrak {p}$. If $\beta \in \mathfrak {g}_x$ and $[\beta ,\beta ']=0$, then

$$\begin{aligned} \lim _{t\rightarrow +\infty } \frac{d}{dt}\Phi (x, \exp (t(\beta +\beta '))=\lim _{t\rightarrow +\infty } \frac{d}{dt}\Phi (x, \exp (t\beta ))+\frac{d}{dt}\Phi (x, \exp (t\beta ')), \end{aligned}$$

Proof

By the cocycle condition, keeping in mind that $[\beta ,\beta ']=0$, we have

$$\begin{aligned} \Phi (x, \exp (t(\beta +\beta '))= & {} \Phi (x,\exp (t\beta '))+\Phi (\exp (t\beta )x,\exp (t\beta '))\\= & {} \Phi (x,\exp (t\beta '))+\Phi (x,\exp (t\beta ')), \end{aligned}$$

and so the result follows. $\square$

Let $\gamma :[0,+\infty ) \longrightarrow G/K$ be a geodesic ray and let $\Phi _x$ be the Kempf-Ness function at x. We define

$$\begin{aligned} \lambda _x (\gamma )=\lim _{t \rightarrow +\infty } \frac{\Phi _x (\gamma (t))}{t}. \end{aligned}$$

The results proved in [28, sections 3.2 and 3.3] hold in our setting. Therefore, we have the following result.

Proposition 4.4

The function $\lambda _x: \partial _\infty M \rightarrow \mathbb {R}\cup \{+\infty \}$ defined by

$$\begin{aligned} \lambda _x ([\gamma ])=\lambda _x (\gamma ) \end{aligned}$$

is well-defined and G-equivariant, i.e., $\lambda _{gx}(p)=\lambda _x (g^{-1}p)$ for any $g\in G$ and any $p\in \partial _\infty (G/K)$.

We conclude this section by recalling the results that will be needed in the following sections.

By Lemma 3.1 for any $\beta \in S(\mathfrak {p})$, keeping in mind formula (3), we have

$$\begin{aligned} \lambda _x(e(\beta ))= & {} \lim _{t\rightarrow \infty }\frac{d}{dt}\Phi _x(\exp (t\beta )K) = \lim _{t\rightarrow \infty }\frac{d}{dt}\Phi (x, \exp (-t\beta ))\nonumber \\= & {} \lim _{t\rightarrow +\infty } \langle \mu _\mathfrak {p}(\exp (t\beta )x,-\beta \rangle . \end{aligned}$$

(5)

Lemma 4.5

Let $\beta \in \mathfrak {p}-\{0\}$ and let $v=\frac{\beta }{\parallel \beta \parallel }$. Then

$$\begin{aligned} \lambda _x (e(v))= \frac{1}{\parallel \beta \parallel } \lim _{t\rightarrow +\infty } \frac{d}{dt}\Phi (x, \exp (-t\beta )). \end{aligned}$$

Proof

$\Phi _x (\exp (tv))=\Phi _x (\exp \big (\frac{t}{\parallel \beta \parallel } \beta \big ))$. Then

$$\begin{aligned} \lambda _x (e(\beta ))=\frac{1}{\parallel \beta \parallel } \lim _{t\rightarrow +\infty } \frac{d}{dt}\Phi (x, \exp (-t\beta )). \end{aligned}$$

$\square$

A proof of the following lemma is given in [5, Lemma 3.5, p. 92], see also [30].

Lemma 4.6

Let V be a subspace of $\mathfrak {p}$. The following are equivalent for a point $x\in X$:

• a) the map $\Phi (x,\cdot )$ is linearly properly on V, i.e., there exist positive constants $C_1$ and $C_2$ such that

  $$\begin{aligned} \parallel v \parallel \le C_1 \Phi (x,\exp (v))+C_2,\, \forall v\in V. \end{aligned}$$

• b) $\lambda _x, (e(\beta ))>0$ for every $\beta \in S(V)$.

The following theorem is well-known and it gives a numerical criterion for stable points in terms of maximal weights. A proof is given in [5, Theorem 3.7].

Theorem 4.7

Let $x\in X$. Then x is stable if and only if $\lambda _x > 0$ on $\partial _\infty M.$

5 Polystability

Definition 5.1

We say that $p, q \in \partial _\infty M$ are connected if there exists a geodesic $\alpha$ in X such that $p = \alpha (\infty )$ and $q = \alpha (-\infty ).$

For any $x\in X,$ as in [28], see also [3], let $Z(x):= \{p\in \partial _\infty M: \lambda _x(p) = 0\}.$

Lemma 5.1

Let $x\in X$ be such that $\mu _\mathfrak {p}(x) = 0,$ then $\mathfrak {g}_x = \mathfrak {k}_x \oplus \mathfrak {p}_x$ and $Z(x) = e(S(\mathfrak {p}_x)) = \partial _\infty G_x/K_x.$

Proof

By Proposition 4.2 if $\mu _\mathfrak {p}(x) = 0,$ $G_x$ is a compatible subgroup of G. Hence, $\mathfrak {g}_x = \mathfrak {k}_x \oplus \mathfrak {p}_x.$ To prove the second assertion, let $\beta \in S(\mathfrak {p}).$ Suppose $e(\beta ) \in Z(x).$ This means that $\lambda _x(e(\beta )) = 0,$ then the convex function $f(t):= \Phi _x(\exp (t\beta )K)$ satisfies

$$\begin{aligned} f'(\infty ) = \lim _{t\rightarrow \infty }\frac{d}{dt}\Phi _x(\exp (t\beta )K) = \lambda _x(e(\beta )) = 0 \end{aligned}$$

and

$$\begin{aligned} f'(0) = \frac{d}{dt}{\bigg \vert }_{t= 0}\Phi _x(\exp (t\beta )K) = \langle \mu _\mathfrak {p}(x), -\beta \rangle = 0. \end{aligned}$$

These imply that f is constant for all $t> 0,$ and by the condition (c) of Kempf-Ness function, $\exp (\mathbb {R}\beta )\subset G_x.$ Since $G_x$ is compatible, $\beta \in S(\mathfrak {p}_x).$ Conversely, if $\beta \in S(\mathfrak {p}_x),$ then f is linear. Moreover, $f'(0) = 0.$ Therefore, $f \equiv 0$ and $e(\beta ) \in Z(x).$ $\square$

Let $x\in X$ and $\beta \in S(\mathfrak {p}).$ Since $\mathfrak {p}\subset i\mathfrak {u},$ then $i\beta \in \mathfrak {u}.$ We define the torus $T_\beta$ given as

$$\begin{aligned} T_\beta := \overline{\{\exp (ti\beta ): t\in \mathbb {R}\}} \subseteq U^o, \end{aligned}$$

where $U^o$ denotes the connected component of U containing the identity.

Lemma 5.2

Let $g\in G.$ Then $\dim T_\beta = \dim T_{g\cdot \beta }.$

Proof

It is well-known that $G^{\beta +}$ fixes $e(\beta )$, see for instance [14, Proposition 2.17.3, p.102]. Then for any $g\in G$, keeping in mind Proposition 2.2, write $g = kh,$ where $k\in K$ and $h\in G^{\beta +}$. Hence $g\cdot \beta =kh\cdot \beta =k\cdot \beta =\textrm{Ad}(k)(\beta )$ and so

$$\begin{aligned} T_{g\cdot \beta }&= \overline{\{\exp (it\textrm{Ad}(k)\beta ) : t\in \mathbb {R}\}}\\&= \overline{\{k\exp (it\beta )k^{-1} : t\in \mathbb {R}\}}\\&= kT_\beta k^{-1}. \end{aligned}$$

Therefore $\dim T_\beta = \dim T_{g\cdot \beta }.$ $\square$

Lemma 5.3

Let $x\in X$ and $p, p'\in Z(x)$ be connected. Then there exists $g\in G$ and $\xi \in S(\mathfrak {p})$ such that $\xi \in \mathfrak {p}_y$, where $y = gx.$

Proof

Since $p, p'\in Z(x)$ are connected, then there exists geodesic $\alpha \in M$ such that $\alpha (+\infty ) = p\in Z(x)$ and $\alpha (-\infty ) = p' \in Z(x).$ Assume $\alpha (t) = g\exp (t\xi )K,$ $g\in G.$ Then $p = g\cdot e(\xi )$ and $p' = g\cdot e(-\xi ).$ By the G-invariant property of the maximal weigh we get

$$\begin{aligned} \lambda _{g^{-1}x}(e(\xi )) = \lambda _x(g\cdot e(\xi )) = \lambda _x(p) = 0 \end{aligned}$$

and

$$\begin{aligned} \lambda _{g^{-1}x}(e(-\xi )) = \lambda _x(g\cdot e(-\xi )) = \lambda _x(p') = 0. \end{aligned}$$

Let $y = g^{-1}x.$ This means that the convex function $t \mapsto \Phi _y(\exp (t\xi )K)$ has zero derivatives at both $+\infty$ and $-\infty ,$ and so, it is constant and by property (c) of Kempf-Ness function, $\exp (\mathbb {R}\xi ) \subset G_y$, $\xi \in \mathfrak {p}_y.$ $\square$

Let $X^\beta := \{z\in X: \beta _X(z) = 0\}.$ $G^\beta$ preserves $X^\beta$ [5, Prop. 2.9] and $X^\beta$ is the disjoint union of closed submanifold of X [17]. The following result is proved in [5, Proposition 2.10,p.92]

Proposition 5.4

The restriction $(\mu _\mathfrak {p})_{\vert _{X^\beta }}$ takes value on $\mathfrak {p}^\beta$ and so it coincides with the $G^\beta$-gradient map $(\mu _{\mathfrak {p}^\beta })_{\vert _{X^\beta }}$.

Corollary 5.1

If $x\in X^\beta$ is $G^\beta$-polystable, then x is G-polystable.

Theorem 5.5

A point $x\in X$ is polystable if and only if $\lambda _x \ge 0$ and for any $p\in Z(x)$ there exists $p'\in Z(x)$ such that p and $p'$ are connected.

Proof

Let $x\in X.$ If $Z(x) = \emptyset$, $\lambda _x > 0$ and by Theorem 4.7, x is stable and hence polystable. Suppose $Z(x) \ne \emptyset .$ Let $p\in Z(x).$ Let $\beta \in S(\mathfrak {p})$ such that $p = e(\beta ).$ Suppose $p\in Z(x)$ is chosen such that the of the torus $T_\beta$ satisfies

$$\begin{aligned} \dim T_\beta = \max _{\eta \in e^{-1}(Z(x))} \dim T_\eta . \end{aligned}$$

By assumption there is a geodesic $\alpha \in M$ such that $\alpha (+\infty ) = p\in Z(x)$ and $\alpha (-\infty ) = p' \in Z(x).$ Assume $\alpha (t) = g\exp (t\xi )K,$ $g\in G.$ Then $p = g\cdot e(\xi )$ and $p' = g\cdot e(-\xi ).$ By Lemma 5.3, $\xi \in \mathfrak {p}_y$ where $y = g^{-1}x.$ Moreover, since $e(\beta ) = p = g\cdot e(\xi ),$ using Lemma 5.2,

$$\begin{aligned} \dim T_\xi = \dim T_\beta = \max _{\eta \in e^{-1}(Z(x))} \dim T_\eta . \end{aligned}$$

Let ${\mathfrak {t}}_\xi$ be the Lie algebra of $T_\xi$. Then ${\mathfrak {a}}=i {\mathfrak {t}}_\xi \cap \mathfrak {p}^\xi$ is an Abelian subalgebra contained $\mathfrak {p}^\xi$ different from zero since $\beta \in \mathfrak {a}$. Since $T_\xi =\overline{\exp (i\mathbb {R}\xi )}$ fixes y it follows that $\mathfrak {a}\subseteq \mathfrak {g}_y$

Let Y be the connected component of $X^{\mathfrak {a}}$ containing y. By Lemma 2.1, $(G^\mathfrak {a})^o=(K^\mathfrak {a})^o \exp (\mathfrak {p}^\mathfrak {a})$ is compatible and preserves Y. By Proposition 5.4 we get $(\mu _\mathfrak {p})|_{Y}=\mu _{\mathfrak {p}^\mathfrak {a}}$. Hence, if y is $(G^\mathfrak {a})^o$-polystable, then it is G-polystable. We split $\mathfrak {p}^\mathfrak {a}= \text {span}(\mathfrak {a}) \oplus \mathfrak {p}^{'}$, where $\mathfrak {p}^{'}$ is the orthogonal of $\mathfrak {a}$ and so it is a $K^\mathfrak {a}$-invariant splitting..

Claim: $\lambda _y(e(\beta ')) > 0$ for all $\beta ' \in S(\mathfrak {p}').$ Indeed, we prove this claim by contradiction. Suppose there exists $\beta ' \in S(\mathfrak {p}')$ such that $\lambda _y(e(\beta ')) = 0.$ Hence $[\xi , \beta '] = 0$ by the choice of $\xi$ and $\beta ',$ and they are linearly independent. Let $a > 0.$ Since $[\xi ,\beta ']=0$ and $\xi \in \mathfrak {g}_y$, by Lemma 4.3 it follows that

$$\begin{aligned} \lim _{t\rightarrow +\infty } \Phi (y,\exp (t(\xi +a\beta '))=\lim _{t\rightarrow +\infty } \Phi (y,\exp (t(\xi ))+a \lim _{t\rightarrow +\infty } \Phi (y,\exp (t\beta ')). \end{aligned}$$

Since $\lambda _y(e(\xi )) = \lambda _y(e(\beta ')) = 0,$ it follows by Proposition 4.4 that

$$\begin{aligned} \lim _{t\rightarrow +\infty } \Phi (y,\exp (t(\xi +a\beta '))=0. \end{aligned}$$

Applying Lemma 4.5, we have

$$\begin{aligned} \lambda _y(e(\frac{\xi + a\beta '}{\parallel \xi + a\beta '\parallel }))= 0, \end{aligned}$$

and so the vector

$$\begin{aligned} \frac{\xi + a\beta '}{\parallel \xi + a\beta '\parallel } \end{aligned}$$

belongs to $e^{-1}(Z(y)).$

We claim that for some $a>0$, $\dim T_{\xi +a\beta '}>\dim T_\xi$.

Let $T'= \overline{\exp (\mathbb {R}i\xi + \mathbb {R}i\beta ')}\subseteq (U^\xi )^o$ and $T_{\beta '}=\overline{\exp (\mathbb {R}i \beta ')}$. Let $U'\subseteq ( U^\xi )^o$ be a compact connected subgroup such that the morphism

$$\begin{aligned} T_\xi \times U' \rightarrow (U^\xi )^o, (a,b) \mapsto (ab) \end{aligned}$$

is surjective with a finite center. Since $\beta ' \notin \mathfrak {a}$, it follows that $i\beta \notin {\mathfrak {t}}_\xi$. Hence, $T_{\beta '}\subseteq U'$ and the morphism

$$\begin{aligned} f: T_\xi \times T_{\beta '} \rightarrow T', \quad f(a,b) = ab \end{aligned}$$

is a finite covering. Let $\{e_1, \ldots , e_n\},$ respectively $\{e^{'}_1, \ldots , e^{'}_m\}$, be a basis of the lattice $\ker \exp \subset \mathfrak {t}_\xi$, respectively $\ker \exp \subset \mathfrak {t}_{\beta '}.$ If $i\xi = X_1e_1 + \cdots + X_ne_n$ and $i\beta ' = Y_1e'_1 + \cdots + Y_me'_m,$ then $i(\xi + a\beta ' )= X_1e_1 + \cdots + X_ne_n + aY_1e'_1 + \cdots + aY_me'_m.$ Denote by $T'_{\xi +a \beta }$ the closure of $\exp (\mathbb {R}(i(\xi +a\beta '))$. Since f is a covering, $\dim T_{\xi + a\beta '} = \dim T'_{\xi + a\beta '}.$ Hence,

$$\begin{aligned} \dim T_{\xi + a\beta '} = \dim _\mathbb {Q}(\mathbb {Q}X_1 + \cdots + \mathbb {Q}X_n + \mathbb {Q}aY_1 + \cdots + \mathbb {Q}aY_m), \end{aligned}$$

see for instance [12, $p.\, 61$]. Since $\beta ' \ne 0,$ $Y_j \ne 0$ for some j. Choose a such that $aY_j \notin \mathbb {Q}X_1 + \cdots + \mathbb {Q}X_n.$ Then $\dim T_{\frac{\xi + a\beta '}{\parallel \xi + a \beta \parallel }} > \dim T_\xi$ which is a contradiction. Therefore, $\lambda _y > 0$ on $e(S(\mathfrak {p}'))$. By Lemma 4.6, $\Phi (y,\cdot )$ is linearly proper on $\mathfrak {p}'$. This implies that $\Phi (y,\cdot )$ is bounded from below on $\mathfrak {p}'$ and

$$\begin{aligned} m=\textrm{inf}_{\alpha \in \mathfrak {p}'} \Phi (y,\exp (\alpha )), \end{aligned}$$

is achieved. We claim that

$$\begin{aligned} m=\textrm{inf}_{\alpha \in \mathfrak {p}^\mathfrak {a}} \Phi (y,\exp (\alpha )). \end{aligned}$$

Indeed, let $v\in \mathfrak {p}^\mathfrak {a}$. Then $v=v_1 + v_2$, where $v_1 \in \mathfrak {a}$ and $v_2\in \mathfrak {p}'$. By the cocycle condition, keeping in mind that $[v_1,v_2]=0$ and $v_1 \in \mathfrak {g}_y$, we get

$$\begin{aligned} \Phi (y,\exp (v))=\Phi (y,\exp (v_1))+\Phi (y,\exp (v_2)). \end{aligned}$$

We claim that $\Phi (y,\exp (v_1))=0$. Indeed, since $\mathfrak {a}\subset \mathfrak {g}_y$, If $w \in S(\mathfrak {a})$ then by formula (5) we get

$$\begin{aligned} \lambda _y (e(w))=\langle \mu _{\mathfrak {p}^\mathfrak {a}} (y), -w \rangle \ge 0, \end{aligned}$$

for any $w\in S(\mathfrak {a})$. This implies $\lambda _y (e(-w))=-\lambda _y (e(w))$ and so $\lambda _y (e(w))=0$ for any $w\in \mathfrak {a}$.

Let $w\in \mathfrak {a}-\{0\}$ and let $s: \mathbb {R}\longrightarrow \mathbb {R}$ be the function $s(t)=\Phi (y,\exp (tw))$. Since $\exp (tw)y=y$ for any $t\in \mathbb {R}$, it follows that s(t) is a linear function. Therefore, $s(t)=b t$ for some $b\in \mathbb {R}$. On the other hand

$$\begin{aligned} 0=\lambda _y \left( e \left( \frac{w}{\parallel w \parallel } \right) \right) =\lim _{t\rightarrow +\infty }\frac{1}{\parallel w \parallel }\frac{d}{dt}\Phi (y, \exp (tw))=b. \end{aligned}$$

This proves

$$\begin{aligned} \textrm{inf}_{\alpha \in \mathfrak {p}'} \Phi (y,\exp (\alpha ))=\textrm{inf}_{\alpha \in \mathfrak {p}^\mathfrak {a}} \Phi (y,\exp (\alpha )), \end{aligned}$$

and so $\Phi _y: (G^\mathfrak {a})^o /(K^\mathfrak {a})^o \longrightarrow \mathbb {R}$ has a minumum and so a critical point. By Proposition 4.1, it follows that y is $(G^\mathfrak {a})^o$ polystable and by Corollary 5.1, y is G-polystable.

Suppose x is polystable. There exists $g\in G$ such that $\mu _\mathfrak {p}(gx) = 0.$ Let $y = gx$ and fix $\beta \in \mathfrak {p}.$ Since the Kempf-Ness function is convex along geodesics,

$$\begin{aligned} \lambda _y(e(\beta )) = \lim _{t\rightarrow \infty }\frac{d}{dt}\Phi _y(\exp (t\beta )K)) \ge \frac{d}{dt}|_{t= 0}\Phi _y(\exp (-t\beta )) = \langle \mu _\mathfrak {p}(y), -\beta \rangle = 0. \end{aligned}$$

This shows that $\lambda _y \ge 0$ on $\partial _\infty M.$ By the G-equivariance of the maximal weight it follows that $\lambda _x \ge 0.$ By Lemma 5.1, $G_y$ is compatible with Lie algebra $\mathfrak {g}_y = \mathfrak {k}_y \oplus \mathfrak {p}_y$ and $Z(y) = e(S(\mathfrak {p}_y)).$ Suppose there exist $p = e(\beta ) \in Z(y).$ Then $e(-\beta )\in Z(y)$ also. Furthermore, $e(\beta )$ and $e(-\beta )$ are connected by the geodesic $[\exp (t\beta )].$ This means that the condition of the Theorem holds for Z(y). Now, for $p\in Z(x),$ $g\cdot p\in Z(y).$ Let $q\in Z(y)$ be connected to $g\cdot p$ by a geodesic $\alpha .$ Then the geodesic $g^{-1}\circ \alpha$ connects p to $g^{-1}\cdot q\in Z(x).$ This concludes the proof of the theorem. $\square$

Corollary 5.2

A point $x\in X$ is polystable if and only if there exist $\beta \in S(\mathfrak {p}),$ $y\in G\cdot x$ and $g\in (G^\beta )^o$ such that $\lambda _x(e(\beta )) = 0$ and $\mu _\mathfrak {p}(gy) = 0.$

6 Measure

Let N be a compact manifold. We denote by $\mathscr {M}(N)$ the vector space of finite signed Borel measures on N. These measures are automatically Radon [15, Thm. 7.8, p. 217]. Denote by $\textrm{C}(N)$ the space of real continuous function on N. It is a Banach space with the $\textrm{sup}$–norm. By the Riesz Representation Theorem [15, p.223] $\mathscr {M} (N)$ is the topological dual of $\textrm{C}(N)$. The induced norm on $\mathscr {M}(N)$ is the following one:

$$\begin{aligned} ||\nu ||:= \sup \Bigl \{ \int _N f d\nu : f\in \textrm{C}(M), \textrm{sup}_M |f| \le 1 \Bigr \}. \end{aligned}$$

(6)

We endow $\mathscr {M}(N)$ with the weak-$*$ topology as dual of $\textrm{C}(N)$. Usually, this is simply called the weak topology on measures. We use the symbol $\nu _\alpha \rightharpoonup \nu$ to denote the weak convergence of the net $\{\nu _\alpha \}$ to the measure $\nu$. Denote by $\mathscr {P}(N) \subset \mathscr {M}(N)$ the set of Borel probability measures on N. We claim that $\mathscr {P}(N)$ is a compact convex subset of $\mathscr {M}(N)$. Indeed the cone of positive measures is closed and $\mathscr {P}(N)$ is the intersection of this cone with the closed affine hyperplane $\{\nu \in \mathscr {M}(N): \nu (N) = 1\}$. Hence $\mathscr {P}(N)$ is closed. For a positive measure $|\nu |=\nu$, so $\mathscr {P}(N)$ is contained in the closed unit ball in $\mathscr {M}(N)$, which is compact in the weak topology by the Banach-Alaoglu Theorem [13, p. 425]. Since $\textrm{C}(N)$ is separable, the weak topology on $\mathscr {P}(N)$ is metrizable [13, p. 426].

If $f: X \longrightarrow Y$ is a measurable map between measurable spaces and $\nu$ is a measure on X, the image measure $f_{\star } \nu$ is defined by $f_{\star } \nu (A): = \nu (f^{-1} (A))$. It satisfies the change of variables formula

$$\begin{aligned} \int _Y u (y) \mathrm d ( f_{\star } \nu )(y) = \int _X u(f(x)) \mathrm d \nu (x). \end{aligned}$$

(7)

Lemma 6.1

[3, Lemma 5.5] Let N be a compact manifold. If G is a Lie group acting continuously on N, the map

$$\begin{aligned} G\times \mathscr {P}(N) \longrightarrow \mathscr {P}(N), \qquad (g, \nu ) \mapsto g_{\star }\nu , \end{aligned}$$

(8)

defines a continuous action of G on $\mathscr {P}(N)$ provided with the weak topology.

Let $(Z,\omega )$ be a compact connected Kähler manifold. Let U be a compact Lie group and $U^{\mathbb {C}}$ its complexification. As before, we assume that $U^{\mathbb {C}}$ acts holomorphically on Z, and the Kähler form is U-invariant. It is also assumed that there exists a momentum map $\mu :Z \longrightarrow \mathfrak {u}$. If $G\subset U^{\mathbb {C}}$ is closed and compatible we denote by $\mu _\mathfrak {p}:Z \longrightarrow \mathfrak {p}$ the associated G-gradient map. Finally, If X is a compact connect G-invariant submanifold of Z then $\mu _\mathfrak {p}:X \longrightarrow \mathfrak {p}$ is a $\textrm{K}$-equivariant map such that $\textrm{grad}\, \mu _\mathfrak {p}^\beta =\beta _X$. In [4] the authors introduced an abstract setting for actions of noncompact real reductive Lie groups on topological spaces that admit functions similar to the Kempf-Ness function.

Let $\Phi :X \times G \longrightarrow X$ be the Kempf-Ness function such that

$$\begin{aligned} \langle \mu _\mathfrak {p}(x), \beta \rangle = \dfrac{\textrm{d} }{\textrm{dt}}\bigg \vert _{t=0} \Phi (x, \exp (t\beta )), \end{aligned}$$

for any $\beta \in \mathfrak {p}$. As before, we have fixed B an $\textrm{Ad}(U^{\mathbb {C}})$-invariant inner product on $\mathfrak {u}^{\mathbb {C}}$ and $\langle \cdot , \cdot \rangle$ denotes the real part of B restricted on $\mathfrak {g}$.

Proposition 6.2

[4, Proposition 31] The function

$$\begin{aligned} \Phi ^{\mathscr {P}}: \mathscr {P}(X) \times G \longrightarrow \mathbb {R}, \qquad \Phi ^{\mathscr {P}} (\nu ):=\int _X \Phi (x,g) \mathrm d \nu (x), \end{aligned}$$

is the Kempf-Ness function for $(\mathscr {P}(X),G,K)$ with gradient map

$$\begin{aligned} {\mathscr {F}} (\nu )=\int _X \mu _\mathfrak {p}(x) \mathrm d \nu (x). \end{aligned}$$

Definition 6.1

Let $\nu \in \mathscr {P}(X)$. Then

• a) $\nu$ is called polystable if $G\cdot \nu \cap \mathscr {F}^{-1}(0)\ne \emptyset$.

• b) $\nu$ is called stable if it is polystable and $G_\nu$ is compact.

• c) $\nu$ is called semistable if $\overline{G\cdot \nu } \cap \mathscr {F}^{-1}(0)\ne \emptyset$.

• d) $\nu$ is called unstable if it is not stable, polystable and semistable.

In [4], see also [3], the authors construct the maximal weight function

$$\begin{aligned} \lambda _\nu : \partial _\infty (G/K) \longrightarrow \mathbb {R}\cup \{+\infty \}, \end{aligned}$$

for any $\nu \in \mathscr {P}(X)$ proving that the maximal weight is G-equivariant. The main goal of this section is to show that the Mundet criterion for polystability holds for the G-action on the measure. The same proof of Theorem 5.5 works.

Let $\beta \in \mathfrak {p}$. Then $\mu _\mathfrak {p}^\beta : X \longrightarrow \mathbb {R}$ is a Morse-Bott function with $\textrm{Crit}\, \mu _\mathfrak {p}^\beta =X^\beta .$ Let $C_1,\ldots ,C_k$ be the connected components of $X^\beta$ and let

$$\begin{aligned} W_i^\beta :=\{x\in X:\, \lim _{t\rightarrow +\infty } \exp (t\beta )x \in C_i\}. \end{aligned}$$

By Theorem 2.3, $W_i^\beta$ is an immersed submanifold and $X=\bigcup W_i^\beta$ is a disjoint union.

Lemma 6.3

Let $\nu \in \mathscr {P}(X)$ and let $\beta \in \mathfrak {p}$. If $\beta \in {\mathfrak {g}}_\nu$ then $\nu (X^\beta )=\nu (X)$ and so $\nu (X-X^\beta )=0$

Proof

$\exp (\mathbb {R}\beta )$ fixes pointwise $C_i$ and so $C_i\subseteq W_i^\beta$. Let $L_n=\exp (n\beta ) (W_i^\beta )$ for any $n\in {\mathbb {N}}$. Since $\exp (t\beta )$ fixes $\nu$ it follows that $\nu (L_n)=\nu (W_i^\beta )$ for any $n\in {\mathbb {N}}$. Since $W_i^\beta$ is $\exp (\mathbb {R}\beta )$-invariant, it follows that $L_{n+1} \subset L_{n}$ and $C_i =\bigcap _{n=1}^{+\infty } L_n$. Therefore

$$\begin{aligned} \nu (C_i)=\lim _{n\mapsto +\infty } \nu (L_n)=\nu (W_i^\beta ). \end{aligned}$$

Hence $\nu (X)=\sum _{i=1}^k \nu (W_i^\beta )=\sum _{i=1}^k \nu (C_i )=\nu (X^\beta )$, concluding the proof. $\square$

Corollary 6.1

Let $\nu \in \mathscr {P}(Z)$ and let $\beta \in \text {i}\mathfrak {u}$. If $\beta \in \mathfrak {u}^{\mathbb {C}}_\nu$ then $\text {i}\beta \in {\mathfrak {u}}^{\mathbb {C}}_\nu$.

Proof

Since $(\text {i} \beta )_Z=J (\beta _Z)$ it follows that $Z^{\text {i} \beta }=Z^\beta$. Let $U\subset Z$. Then $U=(U\cap Z^{ \text {i} \beta } ) \cup (U{-} Z^{\text {i}\beta })$ and both set are $\exp (t\text {i}\beta )$-invariant. Therefore,

$$\begin{aligned} \begin{array}{lcl} \nu (\exp (-t\text {i}\beta ) (U))&{}=&{}\nu (\exp (-t\text {i}\beta ) (U\cap Z^{\text {i} \beta }))+\nu (\exp (-t\text {i}\beta ) (U-Z^{\text {i}\beta })) \\ &{}=&{} \nu (\exp (U\cap Z^{\text {i} \beta })) \\ &{}=&{} \nu (U), \end{array} \end{aligned}$$

concluding the proof. $\square$

If $\beta \in \mathfrak {u}$ and $\beta \in \mathfrak {u}^{\mathbb {C}}_\nu$ then it is not true that $\text {i} \beta \in \mathfrak {u}^{\mathbb {C}}_\nu$. Indeed, the volume form $\omega$ on the unit sphere $S^2$ is invariant with respect to $\textrm{SO}(3)$. In particular the Killing field X generated to the one parameter subgroup $t\mapsto \left[ \begin{array}{ccc} \cos t &{} -\sin t &{} 0 \\ \sin t &{} \cos t &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right]$ preserves $\omega$. The vector field J(X) is the gradient of the the height function

$$\begin{aligned} S^2 \longrightarrow \mathbb {R}, \qquad p \mapsto \langle p, \left[ \begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \rangle \end{aligned}$$

where $\langle \cdot , \cdot \rangle$ is the euclidean scalar product. We claim that J(X) does not preserve $\omega$. Indeed, keeping in mind that $(S^2)^{J(X)}=\{e_3,-e_3\}$, if the flow of J(X) fixes the Borel measure $\nu$ associated to $\omega$, then by Lemma 6.3 it follows that $1=\nu (S^2)=\nu (\{e_3\})+\nu (\{-e_3\})$. A contradiction.

Proposition 6.4

Let $\nu \in \mathscr {P}(X)$ and let $\mathfrak {a}\in \mathfrak {p}$ be an Abelian subalgebra. Let $\nu \in \mathscr {P}(X)$. If $\mathfrak {a}\subset \mathfrak {g}_\nu$ then ${\mathfrak {F}} (\nu ) \in \mathfrak {p}^{\mathfrak {a}}$.

Proof

By [6, Theorem 1,1] there exists $\beta \in \mathfrak {a}$ such that

$$\begin{aligned} X^\mathfrak {a}=\{p\in X:\, \gamma _X (p)=0, \quad \text {for any} \ \gamma \in \mathfrak {a}\}=X^\beta . \end{aligned}$$

By change of variable formula, we get ${\mathscr {F}} (\nu )={\mathscr {F}}(\exp (t\beta ) \nu )=\int _X \mu _\mathfrak {p}(\exp (t\beta )x) \mathrm d \nu (x)$. Taking the limit for $t\mapsto +\infty$ we get

$$\begin{aligned} {\mathscr {F}} (\nu )= & {} \sum _{i=1} \int _{W_i^\beta } \mu _\mathfrak {p}(x) \mathrm d \nu (x)=\sum _{I=1}^k \lim _{t\rightarrow +\infty } \int _{W_i^\beta } \mu _\mathfrak {p}(\exp (t\beta )x) \textrm{d} \nu (x)\\= & {} \sum _{i=1}^k \int _{C_i} \mu _\mathfrak {p}(x) \mathrm d \nu (x), \end{aligned}$$

where $C_1,\ldots ,C_k$ are the connected components of $X^\mathfrak {a}$. By [5, Proposition 2.10], the image of $(\mu _\mathfrak {p})_{\vert _{C_i}}$ lies in $\mathfrak {p}^{\mathfrak {a}}$ and so the result follows. $\square$

Finally, one can characterize the stability condition in terms of the maximal weight functions, see for instance [4, Theorem13].

Theorem 6.5

A measure $\nu$ is stable if and only if $\lambda _\nu >0$.

Theorem 6.6

A measure $\nu$ is polystable if and only if $\lambda _\nu \ge 0$ and for any $p\in Z(\nu )$ there exists $p'\in Z(\nu )$ such that p and $p'$ are connected.

Proof

If $Z(\nu )=\emptyset$ then $\nu$ is stable. Otherwise, by Lemma 5.2 and 5.3 there exists $\nu '\in G\cdot \nu$ and $\xi \in S(\mathfrak {p})$ such that $\lambda _{\nu '}(\xi )=0$, $\beta \in \mathfrak {g}_{\nu '}$

$$\begin{aligned} \dim T_\xi = \dim T_\beta = \max _{\eta \in e^{-1}(Z(\nu '))} \dim T_\eta . \end{aligned}$$

Let ${\mathfrak {t}}_\xi$ be the Lie algebra of $T_\xi$. Let $i:X \hookrightarrow Z$ be the inclusion and let $\nu ''=i_{\#} \nu '$. Since $\xi \in \mathfrak {p}_{\nu '}$ it follows that $\xi \in \mathfrak {p}_{\nu ''}$. By Corollary 6.1, $\text {i} \xi \in {\mathfrak {u}}^{{\mathbb {C}}}_{\nu ''}$ and so $T_\xi$ fixes $\nu ''$. Therefore ${\mathfrak {a}}= \text {i} {\mathfrak {t}}_\xi \cap \mathfrak {p}^\xi \cap \mathfrak {p}_{\nu '}$ is an Abelian subalgebra contained in $\mathfrak {p}^\xi$ and different from zero since $\xi \in \mathfrak {a}$. From now on, the proof of Theorem 5.5 holds for the G-action on the measure. $\square$