Measures Invariant Under Horospherical Subgroups in Positive characteristic

We prove measure rigidity for the action of (maximal) horospherical subgroups on homogeneous spaces obtained by quotient by a uniform (nonuniform) arithmetic lattices over a field of positive characteristic.


Introduction
Let K be a global function field and let S be a finite set of places in K. Let O S be the ring of S-integers. For each ν ∈ S let K ν be the completion of K with respect to ν and let K S = ν∈S K ν . Let G be a connected semisimple group defined over K. Let G = ν∈S G(K ν ). The locally compact topology of K S induces a locally compact topology on G which we will refer to as the Hausdorff topology. Let Γ be a congruence lattice in G(O S ). We let X = G/Γ. By a horospherical subgroup we mean the unipotent radical of a K S -parabolic subgroup of G. If P is a K-parabolic of G, let P = LW be the Levi decomposition of P and let M = [L, L]. Let • P = M W where M = M(K S ) and W = W(K S ). We let P + denote the group generated by all K S -split unipotent subgroups of P = P(K S ), see section 2 for more details. In this paper we prove Theorem 1.1. Let notation be as above. Assume U is a horospherical subgroup of G. Let µ be a U -invariant ergodic probability measure on X = G/Γ.
(i) If Γ is a uniform lattice, then the action of U is uniquely ergodic. Furthermore µ is the probability Haar measure on the closed orbit G + Γ/Γ where the closure is with respect to the Hausdorff topology. (ii) If Γ is non-uniform, then assume that there exists some ν ∈ S such that U contains a maximal horospherical subgroup of G(K ν ). There exists some Kparabolic subgroup P of G and some g ∈ G such that g −1 U g ⊂ P = P(K S ) and µ is the Σ-invariant measure on the closed orbit Σ gΓ where Σ = gP + ( • P ∩ Γ)g −1 and the closure is with respect to the Hausdorff topology.
Using this theorem and the Linearization techniques, see [DM93], we conclude, as a corollary, equidistribution of orbits of subgroups satisfying the assumptions of Theorem 1.1. The statement of this corollary needs some notation which will be fixed in section 2 hence we postpone the discussion to the end of the paper, see Corollary 4.2. It is worth mentioning that the method which is used here in order to prove Theorem 1.1 seems to actually be enough to get the equidistribution result without appealing to linearization techniques, indeed in the course of the proof of W − G (s) = {x ∈ G | s n xs −n → e as n → +∞} Z G (s) = {x ∈ G | sxs −1 = x} If it is clear from the context we sometimes omit the subscript G above. Note that W ± G (s) and Z G (s) are the groups of K S -points of K S -algebraic subgroups W ± G (s) and Z G (s) respectively. The product map is a K S -isomorphism of K S -varieties between W − (s) × Z G (s) × W + (s) and a Zariski open dense subset of G containing the identity, see for example [B91, chapter V] for these statements. Hence if K ℓ is a deep enough congruence subgroup then (1) K ℓ = (K ℓ ∩ W − (s))((K ℓ ∩ Z G (s)))((K ℓ ∩ W + (s))) = (K ℓ ) − (K ℓ ) 0 (K ℓ ) + In the sequel if • ν denotes the ν-component of •. Let s be an element from class A and let U ⊂ W + (s) be a unipotent subgroup which is normalized by s.
These form a filtration of U. We let θ ν be a Haar measure on (U ν ) 0 normalized such that θ ν ((U ν ) 0 ) = 1 and let θ = ν θ ν . Note that {U m } and {(U ν ) m } form an averaging sequence for U and U ν respectively.
Let H be a group defined over K S and let H = H(K S ). We will let H + denote the group generated by all K S -points of all the K S -split unipotent subgroups of H. Thanks to [CGP,theorem,C.3.8] we have H + is the group generated by the K S -points of all the K S -split unipotent radicals of minimal K S -pseudo parabolic subgroups of H. Note that H + is a normal unimodular subgroup of H.
Let H be a K S -algebraic group acting K S -rationally on a K S -algebraic variety M. Assume that H = H(K S ) is generated by one parameter K S -split unipotent algebraic subgroups and elements from class A. The following is proved in [MT94]. It is worth mentioning that the proof in [MT94] follows from the fact that the orbit space in this situation is Hausdorff which is a consequence of [BZ76, appendix A]).
We also need the following general and simple statement.
Lemma 2.2. (cf. [MT94, Lemma 10.1]) Let A be a locally compact second countable group and Λ a discrete subgroup of A. If B is a normal unimodular subgroup of A and µ is a B-invariant ergodic measure on A/Λ. Let Σ = BΛ, where the closure is with respect to the Hausdorff topology. There exists x ∈ A/Λ such that Σ x is closed and µ is the Σ-invariant Haar measure on Σ x.
Let L 2 00 (X) denote the orthogonal complement of G-characters appearing in the regular representation of G on L 2 (X). Hence we have L 2 (X) = L 2 00 (X)⊕(L 2 00 (X)) ⊥ . Indeed L 2 00 (X) can also be characterized as the orthogonal complement of G + -fixed vectors, we will actually use this characterization in this paper. We let C ∞ 00 (X) = C ∞ (X) ∩ L 2 00 (X). We recall the following important and by now folk statement.

Theorem 2.3. (Exponential Decay of Matrix Coefficients)
Let G and Γ be as above. Then there exist positive constants κ and E depending on G and Γ such that for all φ, ψ ∈ C ∞ 00 (X) If for all ν ∈ S the K ν -rank of all of the K ν -simple factors of G is at least 2, then the above is a result of Kazhdan property (T). In this case more is true in particular in this case the constant κ is independent of Γ, see [Oh02]. In general this theorem follows from [Cl03] and [BS91] in the number field case. It is known to experts that the function field case works out much the same way, however we were not able to find an account on this in the literature.
The following is a consequence of the above theorem.
Proposition 2.4. Let s ∈ G + be an element from class A and let U = W + G (s). Then for any f ∈ C ∞ c (U ), compactly supported locally constant function on U, for any φ ∈ C ∞ 00 (X) and any compact set L ⊂ X there exists a constant C = C(f, φ, L) such that for all x ∈ L and for any n ≥ 0 we have Proof. This is proved in [KM96] in the real case and the proof works in S-arithmetic setting also, we recall the proof in the positive characteristic case for the sake of completeness. Note that there is a compact open subgroup U f (resp. K φ ) of U, (resp. G) which leaves f (resp. φ) invariant.
As was mentioned above if g ∈ G is close enough to the identity then g = u − u 0 u + where u ± ∈ W ± G (s) and u 0 ∈ Z G (s). For ℓ large enough let K ℓ denote the ℓ-th congruence subgroup of K and let (K ℓ ) ± = W ± G (s) ∩ K ℓ and (K ℓ ) 0 = Z G (s) ∩ K ℓ . Note that it suffices to prove the statement for f which is the characteristic function of U f , we assume this is the case from now. Now since L and K φ are compact by taking ℓ large, we may assume that the map from the ℓ = ℓ(L, K φ )-th congruence subgroup K ℓ of K into X is injective at x for all x ∈ L and that K ℓ ⊂ K φ . Replacing U f by U f ∩ (K) ℓ we may and will assume that U f = (K ℓ ) + . Hence the map from supp(f ) to X is injective at all x ∈ L.
Let n be a large positive number. Note that the restriction of the module function of the corresponding parabolic (K ℓ ) 0 is trivial, thus we have The last equality follows from the fact that K ℓ ⊂ K φ . Define now ψ = f − f 0 f this is a function supported in a small neighborhood of the identity and thanks to the fact that K ℓ maps injectively to a neighborhood of x ∈ L for any x ∈ L, we may also define ψ x (gx) = ψ(g). This is a (compactly supported) function in C ∞ (X). Let (ψ x ) 00 ∈ C ∞ 00 (X) be the component of ψ x in L 2 00 (X). We now use (4) and Theorem 2.3 and get The proposition is proved.
Remark 2.5. Here we stated Theorem 2.3 and proved Proposition 2.4 in the quantitative form. It is worth mentioning that the qualitative version of Theorem 2.3 i.e. Howe-Moore theorem which is a consequence of Mautner phenomena can be found in the literature, see [BO07] and references in there. Indeed one can then state and prove qualitative version of this proposition assuming only a qualitative version of Theorem 2.3. In other words with notation as above for any ε > 0 there exits n > 0 such that This qualitative statement suffices for the proof of Theorem 1.1 in the way it is stated here.

Non-divergence of unipotent flows
In the proof of Theorem 1.1 we will need results concerning non-divergence of unipotent orbits in X. Results of this sort were first established by Margulis' in the course of the proof of arithmeticity of non-uniform lattices. Later Dani, Dani-Margulis and Kleinbock-Margulis proved quantitative versions of these non-divergence result. We need an S-arithmetic version of [DM91]. In characteristic zero such S-arithmetic statements are proved in [GO,Section 7]. Their proof, which is modeled on [DM91], essentially works in our setting as well. However there are several points which require some explanation. For the sake of completeness we will reproduce the proof in positive characteristic in this section.
Let the notation be as in the introduction. Let A be a maximal K-split torus of G and choose a system {α 1 , . . . , α r } of simple K-roots for (G, A). For each 1 ≤ i ≤ r let P i be the standard maximal parabolic subgroup of G corresponding to α i . The subgroup P = ∩ i P i is a minimal K-parabolic subgroup of G.
We need some results from reduction theory, see [Sp94] and [Ha69] for general results concerning reduction theory in the function field case. In particular we need the following which is Theorem D in [Be87], we thank A. Salehi Golsefidy for pointing out this reference to us. It is worth mentioning that this could also be proved using [Sp94, Proposition 2.6].

(6)
There exists a finite set F ⊂ G(K) such that G(K) = P(K)F Γ Through out this section U = U ν will denote a K ν -split unipotent subgroup of G.
We may and will assume U ⊂ W + G (s) where s is an element from class A. We will further assume that U is normalized by s. As before let U 0 = U ∩ K and define U m = s m U 0 s −m . We will assume that U is an N -dimensional K ν -group, thus there is an s-equivariant isomorphism of K ν -varieties between the Lie algebra of u of U onto U, see [BS91,corollary,9.12]. We will let B be the image of U 0 under this isomorphism. Let θ denote the Haar measure on U normalized so that θ(U 0 ) = 1. Abusing the notation we will let θ also denote the Haar measure on K N ν normalized so that θ(B) = 1. Let Note that there exists some d such that for any g ∈ G the map u → ug is in P N,d .
The following is the main result of this section, see Theorem 2 [DM91] and Theorem 7.1 [GO].
Theorem 3.1. Let the notation be as above and let ε > 0. There exists a compact subset L ⊂ X such that for all x = gΓ ∈ X one of the following holds (1) for all large m, depending on x, We will first make several reductions. Note that if we let p : G → G be the Kcentral isogeny from G to the adgoint form G of G, then the natural map from G/Γ to G(K S )/G(O S ) is a proper map. Thus the statement and the conclusion of Theorem 3.1 are not changed if we replace G by the adjoint form. Hence for the rest of this section we will assume G is adjoint form. We need some more notation. Recall that {α 1 , . . . , α r } is the set of simple simple K-roots for (G, A). Since G is adjoint form they coincide with the fundamental weights. Note now that α i 's uniquely extend to a character of P i which we will also denote by α i . For any 1 ≤ i ≤ r we fix, once and for all, [Sp94], representations ρ i 's and Φ i 's will be used as 'the standard representation of G corresponding to P i " which are used in reduction theory and are also extensively used in [DM91] and [GO].
We have the following Theorem 3.2. For any ε > 0 and α > 0 there exists a compact subset L of G/Γ such that for any Θ ∈ P N,d one of the following holds Proof of Theorem 3.1 module Theorem 3.2. Let ε and α be given and let L be the compact set obtained as in the Theorem 3.2 for this ε and α. Assume now that (1) in the Theorem 3.1 does not hold, thus there exists a sequence m j → ∞ such that loc. cit. fails for U mj . As we mentioned before there exists some d such that the map u → ug is in P N,d for all U mj . We denote these maps by Θ j . Now Theorem 3.2 implies that there exist some 1 ≤ i ≤ r and Note also that gF Γv i is a discrete set hence there are only finitely many λ's module the stabilizer of v i such that Φ i (ugλ) < α. Since U mj 's exhaust U the above discussion implies that there exits some λ ∈ F Γ such that Φ i (ugλ) < α for all u ∈ U. If we utilize the isomorphism f between U and K ν N , then the map ρ i (ugλ)v i is a polynomial map from K N ν into V i . Hence either it is unbounded or constant. Since Φ i (ugλ) < α we get ρ i (ugλ)v i is constant which using (7) implies that λ −1 g −1 U gλ ⊂ P i .
Let us recall the following quantitative non-divergence result.
Using Theorem 3.3 the proof of Theorem 3.2 reduces to the following Theorem 3.4. For any α > 0 there exists a compact subset L of G/Γ such that for any Θ ∈ P N,d one of the following holds The proof of Theorem 3.4 will occupy the rest of this section. Some more notation is needed. For any 1 ≤ i ≤ r we let Q i = {p ∈ P i : α i (p) = 1} and let A i = {a ∈ A : α j (a) = 1, ∀j = i}. For any subset I ⊂ {1, 2 . . . , r} we let Q I is a normal K-subgroup of P I , A I is a K-split torus and P I = A I Q I . Let U I be the unipotent radical of P I and let H I be the centralizer of A I in Q I . We have Q I = H I U I and H I and U I are defined over K and U I is K-split. As usual we let Since G is adjoint for any subset I ⊂ {1, 2 . . . , r} we have The map (α i ) r i=1 : A I → G |I| m is an isomorphism of K-varieties For any ν ∈ S fix a uniformizer ̟ ν ∈ K ∩ O ν and let . . , r} Note that by our choice of ̟ ν and from (9) we get that A 0 ν ⊂ A(K) for all ν ∈ S. Let A 0 = ν∈S A 0 ν . For any subset I ⊂ {1, 2 . . . , r} we let A 0 I = A I ∩ A 0 . We may and will assume that K is chosen so that A = KA 0 I . We let A (1) = a ∈ A 0 : Φ i (a) = 1, for all i = 1, . . . , r and let A (1) Recall that Γ is a congruence subgroup of G(O S ). Hence the choice of ̟ ν , the fact that χ i = α ni i and (9) imply that (10) There is a compact subset Y ⊂ A I such that A (1) We have the following Lemma 3.5. Let I ⊂ {1, 2, . . . , r}, j ∈ {1, 2, . . . , r} \ I and 0 < a < b be given. Then there exists a compact subset M 0 of Q I such that Proof. This is a consequence of the definition together with (11).
We now construct certain compact subsets of G/Γ using the above. These will eventually give us the set L which we need in the proof of Theorem 3.4.
An l-tuple ((i 1 , λ 1 ), . . . , (i l , λ l )) where l ≥ 1, i 1 . . . , i l ∈ {1, . . . , r} and λ 1 , . . . λ l ∈ G(K) is called an called an admissible sequence of length l if i 1 . . . , i l are distinct and λ −1 j−1 λ j ∈ Λ({i 1 , . . . , i j−1 }) for all j = 1, . . . , l and we let λ 0 be the identity element. The empty sequence is called an admissible sequence of length 0. If ξ and η are two admissible sequences of length l and l ′ with l ≤ l ′ we say η extends ξ if the first l terms of η coincide with ξ. For an admissible sequence ξ of length l ≥ 0 we let C(ξ) be the set of all pairs (i, λ) with 1 ≤ i ≤ r and λ ∈ G(K) for which there exists an admissible sequence η of length l + 1 extending ξ, for such η the pair (i, λ) is necessarily the last term. Note that if l = 0 then C(ξ) consists of all pairs (i, λ) with 1 ≤ i ≤ r and λ ∈ Λ(∅).
Let ξ be an admissible sequence of length l ≥ 0. Let α and a < b be positive real numbers define We have the following Proposition 3.7. Let ξ be an admissible sequence of length l ≥ 0. Let α and a < b be positive real numbers. Then W α,a,b (ξ)Γ/Γ is a relatively compact subset of G/Γ.
Proof. The same proof as in [GO,Proposition,7.14] goes through.
We need a few standard facts about polynomial maps. For N, M, d positive integers let us denote by P * N,M,d the space of polynomials from K N ν → K M ν with degree bounded by d. As before we let ̟ ν denote the uniformizer of K ν and we assume |̟ ν | ν = 1/q ν . In non-arthimedean metrics any point of a ball can be considered as the center of the ball thus if B ⊂ K N ν is a ball and t ∈ B we have B = B(t, r) for some r > 0 depending on B, we let 1 qν B(t) denote the ball centered at t with radius r/q ν . We have Lemma 3.8. Let B be a ball in K N ν . Then for any F ∈ P * N,M,d there exists t 0 ∈ B such that |F (t)| ν = sup B |F | ν for all t ∈ 1 qν B(t 0 ).
Proof. Let f and B be given and let t 0 ∈ B such that |F (t 0 )| ν = sup B |F | ν . Expanding F about t 0 we may and will assume t 0 = 0. Now let F = (F 1 , . . . , F M ) we may and will assume sup B |F | ν = |F 1 (0)| ν . We write f = F 1 and let The assumption that Using the ultra metric property, we get |f (t)| ν = |a 0 | ν for all t ∈ 1 qν B as we wanted to show.
Recall from [KT07, Lemma, 2.4] that for any F ∈ P * N,M,d , any ball B ⊂ K N ν and any ε > 0 we have where C > 0 is a constant depending only on N and d. As a corollary we get Lemma 3.9. Given η ∈ (0, 1), there exists R > 1 such that if B 0 ⊂ B are two balls with radius r 0 and r respectively such that r 0 ≥ ηr, then sup B |F | ν ≤ R · sup B0 |F | ν for any F ∈ P * N,M,d .
Using Lemma 3.8 and Lemma 3.9 the same argument as in [GO,Proposition 7.12] gives the following Proposition 3.10. There exists R > 1 such that for any α > 0, any ball B and any subfamily F ⊂ P * N,M,d satisfying The proof of Theorem 3.4 now goes through the same lines as the proof of Theorem 7.3 in [GO], see page 55 in [GO].

Proof of Theorem 1.1 and equidistribution
In this section we will prove Theorem 1.1. Here we give the proof assuming Theorem 2.3. The proofs indeed go through with non-quantitaive version of decay of matrix coefficients as we remarked above. Before starting the proof wee need the following fact which is essentially a restatement of the fact that L 2 00 (X) is the orthogonal complement of the space of G + -invariant vectors.
Let us remark that the converse of the above lemma also holds but we do not need the converse here. We now start the Proof of Theorem 1.1. In the proof we will denote du = dθ(u) for simplicity.
Proof of (i). By our assumption G/Γ is compact hence we may take L = X in the statement of Proposition 2.4. Let φ ∈ C ∞ 00 (X). Recall that loc. cit. asserts that where E ′ is a constant depending on φ and z is any point in X. If one takes z = s −n x one then gets As we mentioned in section 2 the filtration {U m = s m U 0 s −m : m ≥ 0} is an averaging sequence for U . Since µ is U -ergodic (15) implies This and Lemma 4.1 imply that µ is G + -invariant. Now since G + is a normal, unimodular subgroup of G Lemma 2.2 implies that µ is the G + Γ-invariant invariant probability Haar measure on the closed orbit G + Γ/Γ as we wanted to show.
Proof of (ii). We first show that the assertion holds for U = U ν , a maximal horospherical subgroup of G ν .
Let x ∈ X be µ-generic for the action of U i.e. for any bounded function φ ∈ C ∞ (X) and any ε > 0 there exists some n 0 such that if n ≥ n 0 then we have Let ε > 0 be given fix some n 0 as above and let L be the compact set obtained in Theorem 3.1 for this choice of ε. Assume now that x is such that (1) in the Theorem 3.1 holds. Since s normalizes U this implies that (1) in the Theorem 3.1 also holds for s −n x for any n ∈ Z. Hence there exists a positive integer m 0 = m 0 (n), depending on s −n x, such that if m > m 0 , then For the rest of the argument we may and will assume that φ ∈ C ∞ 00 (X). Let K ℓ be a deep congruence subgroup such that (i) (1) holds for K ℓ (ii) φ is fixed by K ℓ (iii) for any x ∈ L the map from k → kx is injective on K ℓ . Fix a large enough n > n 0 such that (5) holds for ψ = 1 θ(K ℓ ∩U) χ K ℓ ∩U in the formulation of Proposition 2.4 this is to say E[K : K ℓ ]s −nκ ′ < ε. Now let m 0 = m 0 (n) be such that (18) holds for all m > m 0 . We have Note that in the above we used the fact that s −m vs m ∈ U 0 and that U 0 is a group to get the second identity. Let B ⊂ U 0 be such that u ∈ B implies that s m us −m s −n x ∈ L and let A = U 0 \ B. By (18) we have |A| < ε. We have The first term is indeed bounded by ε · sup(|φ|). As for the second term recall that since u ∈ B we may utilize Proposition 2.4 and get We now use (17) and get X φdµ < Cε which thanks to Lemma 4.1 says µ invariant under G + -invariant. Hence again by the Lemma 2.2 µ is the G + Γ-invariant invariant probability Haar measure on the closed orbit G + Γ/Γ which finishes the proof if x satisfies (1) in Theorem 3.1.
We will now proceed by induction on the dimension of G. Thanks to the previous paragraph we may and will assume that (2) in Theorem 3.1 holds i.e. if we let x = gΓ, then there exists P i and λ ∈ F Γ such that g −1 U g ⊂ λP i λ −1 where P i = P i (K S ), with the notation as in the Theorem 3.1. Let P i = L · W be the Levi decomposition of P i and let W = W(K S ). Let M = [L, L] be the derived group of L. The group M is semisimple and is defined over K. We let M = M(K S ). Define Note also that g −1 U g ⊂ λ • P i λ −1 . So replacing U by g −1 U g and • P i by λ • P i λ −1 we may and will assume that U ⊂ • P i . Since λ ∈ G(K) we have ∆ = Γ ∩ • P i is a lattice in • P i and also W ∩ Γ is a (uniform) lattice in W hence • P i /∆ is a closed subset of X. These reductions also imply that eΓ is generic for µ with respect to the action of U. As a consequence we may and will consider µ as a measure on • P i /∆. We write U = V W where V is a maximal unipotent subgroup of M. Using the previous notation we let W ν , V ν and M ν be the corresponding ν-components.
Let µ = Y µ y dσ be the ergodic decomposition of µ with respect to W ν . Recall that W ν is a normal and unimodular subgroup of • P i . Hence by Lemma 2.2 we have that for σ-a.e. y ∈ Y the measure µ y is the F = W ν ∆-invariant measure on a closed F -orbit where the closure is taken with respect to the Hausdorff topology. Since Γ is a congruence lattice lattice and W is a K-split unipotent group we deduce that F = W ∆. Hence for σ-a.e. y ∈ Y the measure µ y is invariant under W . Since µ is U ν -invariant ergodic measure we have σ is V ν -invariant and ergodic measure on Y.
Recall that µ is a measure on • P i /∆. Now since σ-a.e. µ y is W -invariant and W is the unipotent radical of • P i = M W we may and will identify Y with M/∆ ∩ M and hence σ is a measure on M/∆ ∩ M.
Let us recall that M is a semisimple group defined over K and ∆∩M is a congruence lattice in M(O S ), furthermore σ is V ν -invariant ergodic measure on M/∆∩M. Hence we may apply the induction hypothesis and get: There exist some g ∈ M and a connected K-parabolic group Q of M such that if we let • Q and Q + be defined as in the introduction then σ is the Haar measure on the closed orbit of gΣ/ • Q∩∆ where Σ = Q + ( • Q ∩ ∆) and the closure is taken with respect to the Hausdorff topology. Hence the measure µ is the Haar measure on the closed orbit gΣW/(Q∩∆)(W ∩∆). Now let H = QW, this is a K-parabolic subgroup of MW. We let H = H(K S ). Since W is a subgroup of the split unipotent radical of all minimal K S -pseudo parabolic subgroups of H we get H + = Q + W and that H + = P + for a K-parabolic subgroup P of G. Note also that since W is the unipotent radical and in particular a normal subgroup we have ΣW = Q + W ( • Q ∩ ∆) = H + ( • P ∩ ∆). This finishes the proof in the case U = U ν .
We now turn to the general case. Hence U ν ⊂ U and µ is U -invariant ergodic measure on X. Let µ = Y µ y dσ be the ergodic decomposition of µ with respect to U ν . For x = gΓ ∈ X we let y(x) denote the corresponding point from (Y, σ). Since µ y(x) 's are U ν -ergodic the above argument says almost all µ y(x) 's are the Σ(x)invariant measure on a closed orbit of Σ(x) · x where Σ(x) = gP + ( • P ∩ Γ)g −1 , and P is a K-subgroup. We will say P is associated to x. For any K-subgroup, P, of G we let P = P(K S ) and let Since there are only countably many such subgroups we have; there exists some P such that σ(S(P )) > 0. Note also that U normalizes U ν , so for every u ∈ U the equality uµ y (x) = µ y(ux) is true for µ-almost all x ∈ X. Furthermore we have Σ(ux) = uΣ(x)u −1 . Thus S(P ) is U -invariant. This and the fact that µ is Uergodic imply that σ(S(P )) = 1. Observe that P + ( • P ∩ ∆) is Zariski dense in • P hence the map x → Σ(x) z , the Zariski closure of Σ(x), is a U -equivariant Borel map from S(P ) to G/N G ( • P ). Lemma 2.1 now implies that this map is constant almost everywhere. Hence µ is the gP + ( • P ∩ Γ)g −1 -invariant measure on a closed orbit gP + ( • P ∩ Γ)Γ/Γ for some g ∈ G. This finishes the proof of Theorem 1.1.
Equidistribution of orbits of horospherical subgroups. As we mentioned in the introduction one application of Theorem 1.1 is to prove equidistribution of orbits of U which satisfy the conditions of Theorem 1.1. It is possible to refine the above proof and get equidistribution statement from the proof. In here however we use the linearization technique in order to get equidistribution result. This seems to be shorter and it is a "standard" method by now. The linearization technique was developed by Dani and Margulis [DM93] in the Lie group case. Similar linearization statements in the S-arithmetic setting in characteristic zero were proved by Tomanov [To00]. In the function field setting such results were proved in [EM, section 5]. The linearization technique is an avoidance principle. Roughly speaking it states that unipotent orbits of algebraically generic points do not spend a "long" time "too close" to proper algebraic varieties. We have the following Corollary 4.2. Let the notation be as before and assume that either (i) or (ii) of Theorem 1.1 is satisfied. Then for any x = gΓ ∈ X and f ∈ C c (X) we have where µ is the gP + ( • P ∩ Γ)g −1 -invariant probability measure on the closed orbit gP + ( • P ∩ Γ)g −1 · x, and P(K S ) is a K-parabolic subgroup P.
Proof. The proof is standard. Let x = gΓ assume P is K-parabolic subgroup of G minimal with the property that g −1 U g ⊂ • P. We will show that the theorem holds with this • P. Let X be the one-point compactification of X if X is not compact and be X if X is compact. For any natural number m define the probability measure µ m on X by where f is a bounded continuous function on X. As X is compact the space of probability measures on X is weak * compact. Let µ be a limit point of {µ n }. By identifying µ we show that there is only one limit points which in return gives convergence. Note that since Y = • P/( • P ∩ Γ) is closed we may and will assume that µ m 's and also µ are supported on Y ∩ {∞}. It follows from nondivergence of unipotent trajectories, see Theorem 3.3, that µ is concentrated on Y . Note that µ is U -invariant. We let µ = Y µ y dσ(y) be a decomposition of µ into U -ergodic components. For any proper K-parabolic subgroup F of P let S(F ) = {p ∈ • P : p −1 U p ⊂ F } By Theorem 1.1 each ergodic component is supported on some S(F ). Minimality of P together with theorem 5.5 in [EM] implies µ(S(F )) = 0 for all proper subgroups F. Since there are only countably many such F we get that σ-almost all µ y(z) 's are the P + -invariant measure on the closed orbit g z P + ( • P ∩ Γ)g −1 z · z. Now since g −1 U g ⊂ P + we see that the support of µ is in gP + ( • P ∩ Γ)g −1 · x which then implies the theorem.