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.
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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.,
(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
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 \(\beta \in \mathfrak {p},\)
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
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,
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
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
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
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
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
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:
and this action also induces a G-action on \(S(\mathfrak {p})\) given by
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
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
The cocycle condition d) can be rewritten as
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
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,
.
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,
Furthermore, the two quantities are increasing in s, while the third in t. Hence,
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 \(\lambda (x,\beta ,t) = \langle \mu _\mathfrak {p}(\exp (t\beta )x), \beta \rangle .\)
where \(\Phi : X\times G\rightarrow \mathbb {R}\) is the Kempf-Ness function. By the properties of the Kempf-Ness function,
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
Note that
and so
Lemma 4.3
Let \(\beta ,\beta '\in \mathfrak {p}\). If \(\beta \in \mathfrak {g}_x\) and \([\beta ,\beta ']=0\), then
Proof
By the cocycle condition, keeping in mind that \([\beta ,\beta ']=0\), we have
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
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
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
Lemma 4.5
Let \(\beta \in \mathfrak {p}-\{0\}\) and let \(v=\frac{\beta }{\parallel \beta \parallel }\). Then
Proof
\(\Phi _x (\exp (tv))=\Phi _x (\exp \big (\frac{t}{\parallel \beta \parallel } \beta \big ))\). Then
\(\square \)
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
and
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
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
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
and
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
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,
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
Since \(\lambda _y(e(\xi )) = \lambda _y(e(\beta ')) = 0,\) it follows by Proposition 4.4 that
Applying Lemma 4.5, we have
and so the vector
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
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
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,
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
is achieved. We claim that
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
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
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
This proves
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,
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:
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
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
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
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
is the Kempf-Ness function for \((\mathscr {P}(X),G,K)\) with gradient map
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
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
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
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,
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
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
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
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 \(\lambda _{\nu '}(\xi )=0\), \(\beta \in \mathfrak {g}_{\nu '}\)
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. \(\square \)
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Biliotti, L., Windare, O.J. A Hilbert–Mumford criterion for polystability for actions of real reductive Lie groups. Annali di Matematica (2024). https://doi.org/10.1007/s10231-024-01480-w
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DOI: https://doi.org/10.1007/s10231-024-01480-w