The space of homogeneous probability measures on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Gamma \backslash X}^S_\mathrm{max}$$\end{document}Γ\X¯maxS is compact

In this paper we prove that the space of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space is compact. As an application, we explain some consequences for the distribution of weakly special subvarieties of Shimura varieties.


Introduction
In this paper, we study the behaviour of sequences of homogeneous measures. More specifically, given a sequence of such measures, we will be interested in describing its limit points. This problem has been studied by Eskin, Mozes, and Shah [9,10,15], who showed that, under certain conditions, any limit point is either a homogeneous measure itself or a zero measure. The later case amounts to the existence of a subsequence of measures diverging to infinity and, in [10], a non-divergence condition was established. Such results concerning the convergence of measures have found several remarkable applications in arithmetic geometry (see, for instance, [6,9,11]). However, the applicability of these tools have so far been limited to the case in which divergence to infinity can be ruled out. The goal of the present paper is to investigate limits of divergent sequences by considering them inside a Satake compactification. Ultimately, we show that any limit point is also a homogeneous measure supported on precisely one of the boundary components of the compactification.
We conjectured this result in our previous paper [8], wherein we developed several tools with which to study it and also proved some particular cases, including the locally symmetric space associated with SL 3 (R). We refer to the introduction of [8] for some further historical background in homogeneous dynamics. We simply recall here the importance, for our purposes, of the seminal works of Ratner [21,22] on the dynamics of unipotent flows, and of some of its developments by Dani-Margulis [7] and Eskin-Mozes-Shah, as alluded to above.

Formulating the main result
Let G be a semisimple algebraic group defined over Q and let G denote the connected component of G(R) containing the identity. Let K be a maximal compact subgroup of G and let ⊂ G(Q) ∩ G be an arithmetic lattice. Denote by X the associated Riemannian symmetric space G/K , denote by x 0 the point in X with stabilizer equal to K , and denote by S the associated arithmetic locally symmetric space \X . Let P(S) denote the set of Borel probability measures on S.
A Q-algebraic subgroup H ⊂ G is said to be of type H if the radical R H of H is unipotent and the real Lie groups underlying the Q-simple factors of H are not compact. With an algebraic subgroup H ⊂ G of type H and some g ∈ G we can associate a probability measure μ H,g ∈ P(S) with support equal to \ Hgx 0 ⊂ S. Such a measure is called homogeneous and we denote by The maximal Satake compactification of S has a decomposition \X S max = \X P∈E X P \X P (1) where P varies among a (finite) set of representatives E of the -conjugacy classes of proper Q-parabolic subgroups of G and the boundary component X P \X P is the arithmetic locally symmetric space associated with P. As a consequence, for any boundary component X P \X P of \X S max , we can define the set Q( X P \X P ) of homogeneous probability measures on X P \X P as we (ii) Let (H n ) n∈N be a sequence of algebraic subgroups of G of type H and let (g n ) n∈N be a sequence of elements of G. Let μ ∈ P( \X S max ) be a weak limit of the associated sequence (μ H n ,g n ) n∈N of homogeneous measures on S = \X . Then μ is a homogeneous measure on \X S max . In (ii), if μ H n ,g n → μ and μ is supported on the boundary component X P \X P , then there exists a connected algebraic subgroup H of P of type H and an element g ∈ P such that μ = μ H,g and H n is contained in H for n large enough.
Note that, in Theorem 1.1, (ii) is an immediate consequence of (i), but (ii) (for any G) implies (i) by simple properties of Satake compactifications (see Proposition 2.3 (ii)).

Sequences of weakly special subvarieties
If we assume, moreover, that X is hermitian, then, by a fundamental result of Baily-Borel [1], the hermitian locally symmetric space S = \X has the structure of a quasi-projective algebraic variety. Such varieties have been studied extensively by Shimura and Deligne as a generalization of the modular curve. As such they are usually called Shimura varieties and they now play a central role in the theory of automorphic forms (in particular, the Langlands program), the study of Galois representations, and Diophantine geometry. The main examples of Shimura varieties are given by the moduli spaces A g of principally polarized abelian varieties of dimension g, in which case G = Sp 2g and = Sp 2g (Z). In general, Shimura varieties are moduli spaces of Hodge structures of a restricted type. They are endowed with special points, special subvarieties, and weakly special subvarieties that play a central role in their theory and are the central objects in the André-Oort and Zilber-Pink conjectures [19,20,29].
Special points of S parametrize 'maximally symmetric' Hodge structures (more precisely, Hodge structures whose Mumford-Tate groups are tori). In the case of A g , they correspond to abelian varieties with complex multiplication. The weakly special subvarieties of S are the totally geodesic subvarieties, and a special subvariety is a weakly special subvariety containing a special point. A special subvariety can also be described in Hodge theoretic terms as a certain locus of 'non-generic' Hodge structures. The relevance of these notions for our purposes is due to the fact that any weakly special subvariety of S is the support of a homogeneous measure.
The equidistribution of sequences of homogeneous measures associated with special subvarieties of Shimura varieties (in situations where there is no escape of mass) has been studied by Clozel and the third author [6,25], and played a central role in the proof of the André-Oort conjecture under the Generalized Riemann Hypothesis [14,26]. The very successful strategy of Pila-Zannier [16,18], which has yielded unconditional cases of the André-Oort and Zilber-Pink conjectures, has highlighted the importance of understanding the distribution of weakly special varieties. The Ax-Lindemann conjecture [13,17,27], at the heart of their strategy, asserts that the Zariski closure of an algebraic flow is weakly special.
The main result of this paper has implications on the equidistribution properties of sequences of weakly special subvarieties, even in the situation when there is escape of mass. In the Shimura case, the Baily-Borel compactification S B B of S has the form where E max is a set of representatives for the -conjugacy classes of maximal Qparabolic subgroups of G and each boundary component ) of probability measures on S B B any weak limit μ of (μ n ) n∈N is homogeneous.
One can show that the weak limits in Theorem 1.2 need not be supported on a weakly special subvariety.

Applications and developments
The main construction of the paper, Theorem 5.3 (which, via Theorem 4.2, yields Theorem 1.1), has recently been applied in a new proof (due to Richard and the third author) of the so-called géométrique André-Oort conjecture (see [23,Théorème 1.3]). The proof passes through the following "dynamic alter ego" of the aforementioned conjecture. In fact, the authors obtain the natural generalization of Theorem 1.3 in the setting for which S is replaced by a general arithmetic quotient and V is a real analytic subvariety definable in an o-minimal structure (see [23,Théorème 1.8]). This allows the authors, in an appendix with Chen, to complete the work of the latter author [5] on the geometric André-Oort conjecture for variations of Z-Hodge structures (a conjecture due to Klingler [12]).
In a recent preprint, Zhang has announced an extension of Theorem 1.1 in the Borel-Serre compactification, in which the type H assumption on the H n , for H n = H fixed, is relaxed (see [30,Theorem 1.2]).

Organisation of the paper
In Sect. 2, we give the necessary preliminaries to clarify the statement of the main result. In particular, we explain why Theorem 1.1 (ii) implies Theorem 1.1 (i). In Sect. 3, we give some further definitions and prove some useful results on parabolic subgroups. In Sect. 4, we state the three main tools used in the proof of Theorem 1.1, namely, two criteria for convergence proved in [8], and an inequality between simple roots and dual weights due to Li. In Sect. 5, we explain an algorithm combining these three tools, which proves Theorem 1.1 and, in the process, determines the boundary component supporting the limit of a convergent sequence of homogeneous measures. The appendix, due to Li, contains a proof of the aforementioned result regarding root systems.

Preliminaries
In this section, we collect the definitions required in order to explain the main results. We repeat several definitions established in [8].

Borel probability measures
Let S be a metrizable topological space and let be its Borel σ -algebra. By a Borel probability measure on S, we mean a probability measure on . We let P(S) denote the space of all Borel probability measures on S. We say that a sequence (μ n ) n∈N in P(S) converges (weakly) to μ ∈ P(S) if we have

Algebraic groups
By an algebraic group G, we refer to a linear algebraic group defined over Q and by an algebraic subgroup of G we again refer to an algebraic subgroup defined over Q. We will use boldface letters to denote algebraic groups (which, again, are always defined over Q). By convention, semisimple and reductive algebraic groups are connected.
If G is an algebraic group, we will denote its radical by R G and its unipotent radical by N G . We will write G • for the (Zariski) connected component of G containing the identity. We will denote the Lie algebra of G by the corresponding mathfrak letter g, and we will denote the (topological) connected component of G(R) containing the identity by the corresponding Roman letter G. We will retain any subscripts or superscripts in these notations.
If M and A are algebraic subgroups of G, we will write Z M (A) for the centralizer of A in M and N M (A) for the normalizer of A in M. We will denote by G Q the intersection G(Q) ∩ G and we will refer to an arithmetic group of G(Q) contained in G Q as an arithmetic subgroup of G Q .

Groups of type H
We say that an algebraic group G is of type H if R G is unipotent and the quotient of G by R G is an almost direct product of almost (Q-simple) algebraic groups whose underlying real Lie groups are non-compact. In particular, an algebraic group of type H has no rational characters. Note that the image under a morphism of algebraic groups of any group of type H is a group of type H.

Homogeneous probability measures on 0\G
Let G denote an algebraic group and let denote an arithmetic subgroup of G Q . If H is a connected algebraic subgroup of G possessing no rational characters, then there is a unique Haar measure supported on H whose pushforward μ to \G is a Borel probability measure on \G. For g ∈ G, we refer to the pushforward of μ under the right multiplication-by-g map as the homogeneous probability measure on \G associated with H and g. More explicitly, this is the g −1 Hg-invariant probability measure supported on Hg.

Parabolic subgroups
A parabolic subgroup P of a connected algebraic group G is an algebraic subgroup such that the quotient of G by P is a projective algebraic variety. In particular, G is a parabolic subgroup of itself. However, by a maximal parabolic subgroup, we refer to a maximal proper parabolic subgroup. Note that R G is contained in every parabolic subgroup of G.

Cartan involutions
Let G be a reductive algebraic group and let K be a maximal compact subgroup of G. Then there exists a unique involution θ on G such that K is the fixed point set of θ . We refer to θ as the Cartan involution of G associated with K .

Boundary symmetric spaces
Let G be a semisimple algebraic group and let K be a maximal compact subgroup of G. Let P be a parabolic subgroup of G. As in [2, (I.1.10)], we have the real Langlands decomposition (with respect to K ) where L P := M P A P is the unique Levi subgroup of P such that K P := L P ∩ K = M P ∩ K is a maximal compact subgroup of L P , and A P is the maximal split torus in the centre of L P . We denote by X P the boundary symmetric space M P /K P , on which P acts through its projection on to M P .

Maximal Satake compactifications
Let G be a semisimple algebraic group and let K be a maximal compact subgroup of G. Denote by X the symmetric space G/K and let denote an arithmetic subgroup of G Q . We let where P varies over the (rational) parabolic subgroups of G. We endow Q X S max with the topology defined in [2,III.11.2]. This topology is defined by a convergence class of sequences, from which one obtains a closure operator, which in turn induces a topology; see [2, p 113-114]. By [2, Proposition III.11.7], the action of G Q on X extends to a continuous action on Q X S max and, by [2, Theorem III.11.9], the quotient endowed with the quotient topology, is a compact Hausdorff space, inside of which \X is a dense open subset. We refer to \X S max as the maximal Satake compactification of \X .
If E is any set of representatives for the proper (rational) parabolic subgroup of G modulo -conjugation, the maximal Satake compactification \X S max is equal to the disjoint union where X P is the projection of P = ∩ P to M P .

Homogeneous probability measures on 0\X
S max Consider the situation described in Sect. 2.8 (in particular, X denotes the symmetric space G/K , where K is a maximal compact subgroup of G). If H is a connected algebraic subgroup of G of type H and g ∈ G, the homogeneous probability measure on \G associated with H and g pushes forward to \X S max under the natural maps We refer to this probability measure as the homogeneous probability measure on \X S max associated with H and g. Similarly, if P is a parabolic subgroup of G, H is a subgroup of P of type H and g ∈ P, we can define the homogeneous probability measure on \X S max associated with P, H and g in precisely the same way via the natural maps (Recall that X P = M P /K P and the first map is induced by the projection from P to M P .) We say that a Borel probability measure on \X S max is homogeneous if is a homogeneous probability measure.

Properties of Satake compactifications
Consider the situation described in Sect. 2.8. We first make two elementary remarks.

Remark 2.1
Consider another maximal compact subgroup gK g −1 of G, for some g ∈ G (recall that all maximal compact subgroups of G are of this form). The maximal Satake compactifications of \X corresponding to K and gK g −1 are homeomorphic. It follows that Theorem 1.1 is equivalent to the same statement in which K is replaced with gK g −1 and the g n are replaced with g n g −1 .

Remark 2.2
Similarly, for any c ∈ G Q , we obtain a homeomorphism of compactifications induced by the homeomorphism x → cx on Q X S max (recall that the action is continuous). It follows that Theorem 1.1 is equivalent to the same statement in which we replace with c −1 c and we replace the H n with c −1 H n c and the g n with c −1 g n .
Next we prove a result regarding the structure of the Satake compactifications that is presumably well-known to experts.
Then the closure of X Q \X Q in \X S max with respect to the above topology is homeomorphic to the maximal Satake compactification of X Q \X Q .
Therefore, to define the maximal Satake compactification, we consider the disjoint union where P varies over the (rational) parabolic subgroups of M der Q , X P = M P /K P , and K P = K der Q ∩ M P . Note that for each parabolic P of M der Q we have a corresponding parabolic subgroup P of G contained in Q (see [2, p 276, (III.1.13)]) and X P = X P . In particular, Q X Q S max is a subset of Q X S max and it follows from [2, III.11.2] that it is set-theoretically equal to the closure X Q of X Q in Q X S max . Furthermore, from the description of convergent sequences given in [2, III.11.2], the Satake topology on Q X Q S max coincides with the induced topology from Q X S max . Replace X Q with its intersection with M der Q . Then we have It follows from the definition of the Satake topology that X Q \X Q is dense in P⊂Q \ X P and that P⊂Q \ X P is closed in \X S max . This proves the result.
(ii) Let (μ n ) n∈N be a convergent sequence of homogeneous probability measures in P( \X S max ) supported on a boundary component X Q \X Q . We want to prove that the limit of (μ n ) n∈N is homogeneous. By definition, there exist sequences (H n ) n∈N and (g n ) n∈N with H n a connected algebraic subgroup of Q of type H and g n ∈ Q such that μ n is the homogeneous probability measure on \X S max associated with Q, H n , and g n . By Remark 2.1, we may assume that M Q corresponds to a group M Q defined over Q. Since X Q is equal to M der Q /K der Q , as in (i), we may replace H n with its image in M der Q and g n with its image in M der Q and then the result follows from (i).

Further preliminaries
In this section, we collect preliminaries used in the proof of Theorem 1.1. Many of these also appeared in [8].

Rational Langlands decomposition
Let G be a connected algebraic group and let K be a maximal compact subgroup of G. Let P be a parabolic subgroup of G. As in [2, (III.1.3)], we have the rational Langlands decomposition (with respect to K ) P = N P M P A P .
Since G = P K , the rational Langlands decomposition of P yields In particular, if g ∈ G, we can write g as g = nmak ∈ N P M P A P K .
Note that the product N P M P is always associated with a connected algebraic group over Q, which we denote H P (see [8, Section 2.6]).

Standard parabolic subgroups
Let G be a reductive algebraic group and let A be a maximal split subtorus of G. The non-trivial characters of A that occur in the adjoint representation of G restricted to A are known as the Q-roots of G with respect to A.
Let P 0 be a minimal parabolic subgroup of G containing A. We let (P 0 , A) denote the set of characters of A occurring in its action on n, where N = N P 0 . (Recall our convention that n = Lie(N).) As explained in [2, III.1.7], (P 0 , A) contains a unique subset = (P 0 , A) such that every element of (P 0 , A) is a linear combination, with non-negative integer coefficients, of elements belonging to . On the other hand, P 0 is determined by A and . We refer to as a set of simple Q-roots of G with respect to A.
For a subset I ⊂ , we define the subtorus Then the subgroup P I of G generated by Z G (A I ) and N is a parabolic subgroup of G. We refer to P I as a standard parabolic subgroup of G. Every parabolic subgroup of G containing P 0 is equal to P I for some uniquely determined subset I ⊂ .
Let K be a maximal compact subgroup of G such that A is invariant under the Cartan involution of G associated with K . Then, as in [7, Section 1], Z G (A I ) is the Levi subgroup of P appearing in the rational Langlands decomposition of P with respect to K . Note that A I is the maximal split subtorus of the centre of Z G (A I ) and we can write Z G (A I ) as an almost direct product M I A I , where M I is a reductive group with no rational characters. The rational Langlands decomposition with respect to K is then where N I = N P I . We will also write H I = N I M I . For ease of notation, when I = \ {α} for some α ∈ , we will write P α , A α , N α , M α , and H α instead of P I , A I , N I , M I , and H I , respectively.
The set I restricts to a set of simple roots of M I for its maximal split torus A I = (A ∩ M I ) • . Therefore, for any subset J ⊂ I , we obtain, as before, a standard parabolic subgroup of M I , which we denote P I J . We let K I denote the maximal compact subgroup K ∩ M I of M I and we obtain a rational Langlands decomposition with respect to K I :

The d P,K functions
Let G be a reductive algebraic group and let K be a maximal compact subgroup of G. Let P be a proper parabolic subgroup of G and let n P denote the dimension of n P . Consider the n th P exterior product V P = ∧ n P g of g and let L P denote the one-dimensional subspace given by ∧ n P n P . Then the adjoint representation induces a linear representation of G on V P and, since P normalizes N P , we obtain a linear representation of P on L P . That is, P acts on L P via a character χ P .
Fix a K -invariant norm · P on V P ⊗ Q R and let v P ∈ L P ⊗ Q R be such that v P P = 1. We obtain a function d P,K on G defined by d P,K (g) = g · v p P .
Note that, for any g ∈ G, we can write g = kp, where k ∈ K and p ∈ P. Therefore, d P,K (g) = g · v p P = p · v p P = χ P ( p) · v p P = χ P ( p) (note that χ P is necessarily positive on the connected component P). In particular, d P,K is a function on G of type (P, χ P ), as defined in [3, Section 14.1]. Furthermore, it does not depend on the choices of · P and v P .

Remark 3.7 Suppose that ⊂ G(Q)
is an arithmetic subgroup. Let P denote ∩ P and let H P denote ∩ H P . We claim that P = H P . To see this, write P as the product of H P with a maximal split subtorus S P of the centre of a Levi subgroup of P (defined over Q but not necessarily arising from the rational Langlands decomposition associated with K ). Let (S P ) denote a basis for the character group of S P and, for each χ ∈ (S P ), letχ denote the morphism P → G m defined by (h, s) → χ(s) n for some sufficiently large n ∈ N such that the morphism is well-defined. By [3, Section 8.11],χ( P ) is an arithmetic subgroup of Q × for any χ ∈ (S P ). On the other hand, it is contained in R >0 and, therefore, it is trivial. Hence, P is contained in G (S P ) = ∩ χ ∈ (S P ) kerχ and, since G (S P ) = H P , the claim follows.

The ı-functions
Let G be a connected algebraic group with no rational characters and let L be a Levi subgroup of G. Then G is the semidirect product of L and N = N G . We denote by π the natural (surjective) morphism from G to L.
Let P 0 be a minimal parabolic subgroup of L and let A be a maximal split subtorus of L contained in P 0 . Let K denote a maximal compact subgroup of L such that A is invariant under the Cartan involution of G associated with K . For any proper parabolic subgroup P of L, we obtain a function d P,K on L, as defined in Sect. 3.3, and, for each α ∈ = (P 0 , A), we write d α = d P α ,K .
Let be an arithmetic subgroup of G Q and let L = π( ), which is arithmetic subgroup of L Q . By [3, Théorème 13.1], there exists a finite subset F of L Q and a t > 0 such that L = K A t ωF −1 L , where ω is a compact subset of H P 0 and A t = {a ∈ A : α(a) ≤ t for all α ∈ }.

Main tools
In this section, we describe the three main tools used in the proof of Theorem 1.1.

Theorem 4.1 For each n ∈ N, let H n be a connected algebraic subgroup of G of type
H, let g n ∈ G and let μ n be the homogeneous probability measure on \G associated with H n and g n . Assume that lim inf n→∞ δ(H n , g n ) > 0.

Then the set {μ n } n∈N is sequentially compact in P( \G).
Furthermore, if μ is a limit point in {μ n } n∈N , then μ is the homogeneous probability measure on \G associated with a connected algebraic subgroup H of G of type H and an element g ∈ G, and H n is contained in H for all n large enough. Theorem 4.2 For each n ∈ N, let H n denote a connected algebraic subgroup of G of type H, let g n denote an element of G and let μ n denote the homogeneous probability measure on \X S max associated with H n and g n . Suppose that there exists a parabolic subgroup P of G such that, (i) for all n ∈ N, H n is contained in H P , (ii) we can write g n = h n a n k n ∈ H P A P K , such that α(a n ) → ∞, as n → ∞, for all α ∈ (P, A P ), and, (iii) if we denote by ν n the homogeneous probability measure on H P \H P associated with H n and h n , then (ν n ) n∈N converges to ν ∈ P( H P \H P ).

The criterion for convergence in 0\X
Then there exists a connected algebraic subgroup H of P of type H and an element g ∈ P such that (μ n ) n∈N converges to the homogeneous probability measure on \X S max associated with P, H and g, and, furthermore, H n is contained in H for n large enough.
Note that, by the results of Mozes-Shah [15] and Ratner [21] condition (iii) is equivalent to the stronger statement that ν not only exists but is also homogeneous (see [8,Theorem 2.9]). This is used crucially in the proof of Theorem 4.2 (see [8]).

A key result on root systems
Let G be reductive algebraic group, let P 0 be a minimal parabolic subgroup of G, and let A be a maximal split subtorus of G contained in P 0 . Let X * (A) denote the character module of A and let X * (A) Q denote the Q-vector space X * (A) ⊗ Z Q. Fix a non-degenerate scalar product (·, ·) on X * (A) Q that is invariant under the action of N G (A)(Q). Then the Q-roots of G with respect to A equipped with the inner product (·, ·) constitute a root system in X * (A) Q . We refer the reader to [24, Section 3.5] for further details. Let = (P 0 , A) and, for each α ∈ , let w α ∈ X * (A) Q be the unique element such that (w α , β) = δ αβ for any β ∈ . These elements are usually called dual weights. They are a particular choice of quasi-fundamental weights, as defined in [8,Section 2.9]. It follows from [8, Lemma 4.1] that w α is a positive rational multiple of the character χ α defined therein (namely, the restriction of χ P α to A). We therefore deduce the following fact.

Proof
Since H I has no rational characters and A I is contained in H I , the result follows from the fact that χ α is the restriction of a character on P I . Notice that and {w α } α∈ constitute two bases of X * (A) Q . By construction, we have and it is a simple calculation to see that the matrices with coefficients (w α , w β ) and (α, β), respectively, are inverse to one another. It follows from [28, Chapter 3, Proposition 1.16] that (w α , w β ) ≥ 0 for all α, β ∈ (see also the paragraph following Lemma 6.5 in the appendix). We define t α = β∈ (w α , w β ). In particular, t α > 0.

Definition 4.4
For each α ∈ , we writew α = 1 t α w α . We refer to the set {w α } α∈ as a set of weighted dual weights of G with respect to A. From the above and [8, Section 4.1 (3)], we see thatw α is a positive rational combination of positive roots.
The key result we need is the following. A proof (due to Jialun Li) is provided in the appendix (see Corollary 6.2).
Note that, in the appendix, the conditionw α (a n ) ≥ 0 appears. The calculations there take place in the Lie algebra a I of A I . Passing to A I is via the exponential map, hence the conditionw α (a n ) ≥ 1 above.
The condition "α is connected to \ (I ∪ {α}) in the Dynkin diagram" is to say that if we write as the disjoint union 1 ∪ · · · ∪ m according to the decomposition of the root system into irreducible root systems, and α ∈ k , then, among the roots in \ (I ∪ {α}), there exists at least one root that also belongs to k . In other words, I ∪ {α} does not contain k . Proof Let G ad denote the quotient of G by its center, and let ad : G → G ad denote the natural morphism. Recall that the adjoint representation factors through ad. Furthermore, G ad is equal to a product G 1 × · · · × G m of Q-simple groups, P ad 0 = ad(P 0 ) is a product P 1 × · · · × P m of minimal parabolic subgroups, and A ad = ad(A) is a product A 1 × · · · × A m of maximal split tori. The decomposition = 1 ∪ · · · ∪ m above is The image of A J in A ad is therefore equal to the product A J 1 1 × · · · × A J m m , where J i = J ∩ i . The assumption that α is not connected to \ (I ∪ {α}) in the Dynkin diagram is equivalent to the statement that I ∪ {α} contains k . Therefore, the image of A J (I ∪{α})∩J in A ad is also a product and its k th component is A J k k,J k = {1}. Since α ∈ k is trivial outside of A k , we conclude that it is trivial on A J (I ∪{α})∩J .

The proof
In this section, we will give the proof of Theorem 1.1. First we will define a useful notion.

Maximal couples
Consider the situation described in Sect. 3.4. For each n ∈ N, let H n be a connected algebraic subgroup of G of type H and let g n ∈ G.
Definition 5.1 A maximal couple for (H n ) n∈N and (g n ) n∈N with respect to G, K , , and F is a couple (α, (λ n ) n∈N ) for some α ∈ and λ n ∈ L F such that H n ⊂ Nλ n P α (λ n ) −1 for every n ∈ N and, whenever H n ⊂ Nλ n P β (λ n ) −1 for some β ∈ and λ n ∈ L F and we write λ −1 n g n = h n a n k n ∈ N H α A α K and (λ n ) −1 g n = h n a n k n ∈ N H β A β K according to the rational Langlands decompositions, then w α (a n ) ≥w β (a n ), wherew α andw β are weighted dual weights for L, as in Definition 4.4.
We have the following important properties of maximal couples.

Proposition 5.2 (i)
Suppose that lim inf n→∞ δ(H n , g n ) < ∞. After possibly extracting subsequences, a maximal couple for (H n ) n∈N and (g n ) n∈N with respect to G, K , , and F exists. (ii) Suppose that lim inf n→∞ δ(H n , g n ) = 0. If (α, (λ n ) n∈N ) is a maximal couple for (H n ) n∈N and (g n ) n∈N with respect to G, K , , and F and we write λ −1 n g n = h n a n k n ∈ N H α A α K according to the rational Langlands decomposition, then α(a n ) → ∞, as n → ∞.

Proof
(i) As explained in Sect. 4.3, eachw β is a positive rational multiple of the corresponding χ β . Since there are only finitely many of them, the claim follows from [8,Lemma 4.3].
Therefore, by [8,Lemma 4.2], if we write (λ n ) −1 g n = h n a n k n ∈ N H β A β K , according to the rational Langlands decomposition, we have β(a n ) → ∞, as n → ∞.
Now, since (α, (λ n ) n∈N ) is a maximal couple for (H n ) n∈N and (g n ) n∈N with respect to G, K , , and F, we have α(a n ) =w α (a n ) 1 nα ≥w β (a n ) for some n α , n β > 0 (see the description of weighted dual weights in Sect. 4.3). Therefore, α(a n ) → ∞, as n → ∞.

Proof of Theorem 1.1
Let P 0 be a minimal parabolic subgroup of G and let A be a maximal split subtorus of G contained in P 0 . By Remark 2.1, we may assume that A is invariant under the Cartan involution of G associated with K . Let I 0 = = (P 0 , A). By Proposition 2.3 (ii), it suffices to prove Theorem 1.1 (ii). By Remark 2.2 and Theorem 4.2, it suffices to prove the following theorem. Theorem 1.1 (ii). After possibly extracting a subsequence, there exists a parabolic subgroup P of G, an element c ∈ G Q , and, for each n ∈ N, an element γ n ∈ such that, (i) for all n ∈ N, c −1 γ −1 n H n γ n c is contained in H P , (ii) we can write c −1 γ −1 n g n = h n a n k n ∈ H P A P K , such that α(a n ) → ∞, as n → ∞, for all α ∈ (P, A P ),

and,
(iii) if we denote by ν n the homogeneous probability measure on (c −1 c ∩ H P )\H P associated with c −1 γ −1 n H n γ n c and h n , then (ν n ) n∈N converges to ν ∈ P((c −1 c ∩ H P )\H P ).

Outline of the proof of Theorem 5.3
The essence of the proof is to select successive maximal couples. At each stage, up to conjugation and taking subsequences, we have H n ⊂ H I and h I ,n ∈ H I for some fixed I . We apply the criterion given by Theorem 4.1 and, if it fails, we obtain, by Proposition 5.2, a maximal couple (α, (λ n ) n∈N ). After conjugation and taking a subsequence, we find H n ⊂ H I ∪α and h I ∪α,n ∈ H I ∪α and we repeat the procedure. After a finite number of steps, the criterion given by Theorem 4.1 holds, which guarantees (iii), namely, convergence of the associated homogeneous measures in a space of the form H J \H J for some J . The fact that we have chosen maximal couples allows us to apply Theorem 4.5, which in turn yields (ii).

Proof of Theorem 5.3
The proof is via an iterative procedure.
If (4) and, by Proposition 5.2 (ii), if we write n ∈ N I r H I r I r +1 A I r I r +1 K I r = H I r +1 A I r I r +1 K I r (the equality justified by Corollary 3.4), then We define and we write λ (r +1) n = γ (r +1) n c r ∈ r F r . After possibly enlarging F r and extracting a subsequence, we may assume that (this is possible because r is an arithmetic subgroup of M I r contained, and hence of finite index, in r ). We now return to the start of Step r + 1.

Completing the proof
Iterating the previous step, we will eventually obtain (3)  n is sequentially compact. Then, Theorem 5.3 follows (with P = P I r , c = c 0 · · · c r −1 , and γ n = γ (1) n · · · γ (r ) n ) from the following proposition.
Proof We will prove the proposition by induction. The base case is the following.
Lemma 5. 5 We have where the outer inclusions are part of the definitions, and the middle inclusion is justified by Lemma 3.1. We conclude that and so the result follows from the fact that α i r (a (r ) n ) → ∞, as n → ∞ (by Step r ).
The inductive step is given by the following lemma.

Lemma 5.6
Let 0 ≤ l ≤ r − 2 and assume that Then Proof For s ≥ l + 2, we have where the outer inclusions are part of the definitions, and the middle inclusion is justified by Lemma 3.1. We conclude that α i r −l−1 (a (r ) n · · · a (1) n ) = α i r −l−1 (a (r ) n · · · a (r −l−1) n ).
Now we are in the situation of Sect. 4.3, where the ambient group is M I r −l−2 , the maximal split torus is A I r −l−2 , and the set of simple roots is I r −l−2 . Note that Therefore, if α i r −l−1 is not connected to in the Dynkin diagram of M I r −l−2 , then we can apply Lemma 4.6 (with I = I r , α = α i r −l−1 , and J = I r −l−1 ) to conclude that α i r −l−1 (a (r ) n · · · a (r −l−1) Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Jialun Li
The purpose of this appendix, is to prove Theorem 4.5. The main ingredient is an inequality between simple roots and dual weights. For this discussion, we refer without further mention to [4]. Let E be a linear space. Let be a root system in E, which generates E, and let (·, ·) be the inner product on E invariant under the Weyl group. Fix a set of simple roots in . Let {w α } α∈ be the set of dual weights in E, which are defined by the relations (w α , β) = δ αβ for β ∈ , where δ αβ is the Kronecker symbol. The set of dual weights and the set of simple roots form two bases of E. Using the inner product, we can easily compute the coefficients in the transition matrix. Then we have a relation between simple roots and dual weights, α = (α, α)w α + β∈ \{α} (α, β)w β for α ∈ and w α = β∈ (w α , w β )β for α ∈ .
It remains to prove Lemma 6.3. We first recall two facts.

Lemma 6.4 A connected subgraph of a Dynkin diagram is a Dynkin diagram.
Lemma 6.5 The inverse of the Gramm matrix (α, β) α,β∈ of an irreducible root system is a matrix with positive entries.
Lemma 6.5 is usually formulated in terms of the Cartan matrix of the root system that has entries 2(α,β) (β,β) with α, β ∈ . Since the Cartan matrix is the product of the Gramm matrix and the diagonal matrix with the positive entries 2 (β,β) with β ∈ , the claim is also true for the Gramm matrix as well.