Translates of horospherical measures and counting problems. Amir Mohammadi & Alireza Salehi Golsefidy Draft Abstract We show translates of certain homogenous measures converge to a homogenous measure and give two applications. 1

1

Introduction and statement of the results.

As it has been observed by A. Eskin, S. Mozes, and N. Shah, in order to count number of integer points on certain homogeneous varieties, one needs to understand the possible limiting measures of the translates of certain probability measures on the space of probability measures on G/Γ for certain Lie group G and Γ a lattice in G. Let G be a simply connected absolutely almost simple Q-group with a fixed embedding in GLl , Γ = G(Z) = G(Q)∩GLl (Z), P a minimal parabolic Q-subgroup, S a maximal Q-split torus Q-subgroup of P, U the unipotent radical of P, and M the Q-anisotropic kernel of the centralizer of S. We let G = G(R), A = S(R), U = U(R) and M = M(R). We will further assume that the Q-rank of G is equal to the R-rank of G. In particular, M is compact. For any topological space X, let P(X) be the space of probability measures on X. Let µU be the U -invariant measure in P(G/Γ) supported on U Γ/Γ. Let ∆ = {α1 , · · · , αr } be the set of S-relative simple roots corresponded to P, λi ’s the fundamental weights. Since G is simply connected, it is well-known that (λi )ri=1 : S → Grm defines an isomorphism as Q-groups. For any subset F (1) of {1, · · · , r}, let PF be the corresponding parabolic. Let PF = ∩i∈F ker(λi ), (1) (1) where λi ’s are viewed as characters of PF . Let SF = S ∩ PF . Let QF be the product of non-compact factors of the semisimple part of the Levi component of PF by Ru (PF ). It is in fact, the smallest normal subgroup of PF which contains Ru (P∆ ). Note that QF ∩ Γ is a lattice in QF . 1 2000

Mathematics Subject Classification ?

1

In this article, we would like to study the possible limiting measures of µU , and as applications solve two counting problems. One of geometric nature and the other of number theoretic nature. By Iwasawa decomposition, it is easy to see that we have to turn our focus on the A-translates of µU . Theorem 1. Let {an } be a sequence of elements in A. Then i) If λi (an ) goes to zero for some i, then an µU diverges in P(G/Γ). (1)

ii) If an ’s belong to SF and λi (an ) goes to infinity for any i 6∈ F for F a subset of {1, · · · , r}, then an µU converges to µQF . In particular, when F = ∅, it converges to the Haar measure on G/Γ. Remark 2. One can replace the maximal unipotent radical with any other unipotent radical, and get some result! However the cone should be define with respect to the characters which appear in the space of stabilizer of U , and analysis of the possible probability measures on the walls is more complicated. We should add that it is the correct cone. In the sense that as one goes to infinity in the interior of this cone, we get the full Haar measure, and outside of this cone the sequence diverges.

2

Preliminary and Notations.

We let G be as in the introduction and let g denote the Lie algebra of G and a be the Lie algebra of A. Let θ be a Cartan involution on g chosen such that if g = k + p is the corresponding Cartan decomposition of g then a is a maximal abelian subspace of p. Let now K be the analytic subgroup of G with Lie algebra k. Then K is a maximal compact subgroup of G and one has G = KAU (Iwasaw decomposition) and G = KAK (Cartan decomposition). In what follows k · k denotes a K-invariant Euclidean norm on g. We extend this to a norm on the exterior algebra ∧g of g, which we also denote by k · k. With this choice of norm the A-weight spaces are orthogonal to each one another with respect to the corresponding inner product. Recall that by a theorem of Borel and Harish-Chandra there exists a finite set F ⊂ G(Q) such that G(Q) = P(Q) · F · Γ. Throughout the paper we let π : G → G/Γ be the natural projection. We now define certain functions di : G → R. These functions were considered by Dani and Margulis in [6] in order to study the recurrence properties of unipotent flows on homogenous spaces. Let ∆ be as before and if F = {i} ⊂ ∆ is a singleton we let Pi = PF . The subgroups Pi are standard maximal parabolic subgroups of G. Let Ui = Ru (Pi ) and let ui denote the Lie algebra of Ui . We let `i = dim ui and let ϑi = ∧`i Ad denote the i-th exterior power of the Ad representation. Note that ∧`i ui defines a Q-rational one dimensional subspace of ∧`i g. Let vi ∈ ∧`i ui (Q) \ {0}. Note hat if g ∈ Pi then ϑi (g)vi = det(Ad(g)|ui )vi .

2

Define di : G → R by di (g) = kϑi (g)vi k for all g ∈ G. We also recall the following definition from [17]. For d, N ∈ N let PN,d be the set of continuous maps Θ : Rd → G such that for all c, a ∈ Rd and X ∈ g, the map t 7→ Ad(Θ(tc + a)(X)) ∈ g is a polynomial of degree at most N in each coordinate of g. We will use the following two theorems in the sequel. These two theorems provide us with geometric and algebraic formulations of conditions which will guarantee a quantitative recurrence property of “polynomial like” flows. Theorem A. (cf. [9, Theorem 3.4]) Let G, Γ and π be as before. Then given a compact set C ⊂ G/Γ, an  > 0 and an N ∈ N there exists a compact set C¯ 0 ⊂ G/Γ with the following property; For any Θ ∈ PN,d and any open convex set Ω in Rd one of the following holds 1.

∈ Ω : π(Θ(t)) ∈ C¯ 0 }) > 1 − , where m denotes the Lebesgue measure on Rd . 1 m(Ω) m({t

2. π(Θ(Ω)) ∩ C = ∅ Theorem B. (cf. [9, Theorem 3.5]) Let d, N ∈ N, α > 0 be given. Then there exists a compact set C ⊂ G/Γ such that for any Θ ∈ PN,d and any bounded convex set Ω ⊂ Rd one of the following conditions is satisfied 1. There exist f γ ∈ F Γ and i ∈ ∆ such that sup di (Θ(t)f γ) < α t∈Ω

2. π(Θ(Ω)) ∩ C 6= ∅

3

Translates of horospherical measures.

In this section we would like to prove theorem 1. Part (i) is an easy consequence of lemma 4. We will prove the second part in two steps. First using the recurrence properties of unipotent flows, we will show that under the given conditions an µU is a convergent sequence in P(G/Γ). Second we will prove that it converges to the claimed probability measure. Lemma 3. For any i, there is a (positive) natural number ki such that: X ξi := β = ki λi . β∈Φ+ ,(λi ,β)6=0

3

Proof. Let ( , ) denote the duality between root and weight space and σj is the reflection with respect to αj . Then for any j 6= i we have (λi , σj (β)) = (λi , β) and σj (Φ+ \ {αj }) = Φ+ \ {αj }. Therefore σj (ξi ) = ξi , and so ξi is orthogonal to αj , for any j 6= i. Hence ξi is a multiple of λi . It is also in the root space and clearly has the same direction. Lemma 4. For any C compact subset of G, if π(au) is in π(C) for some a ∈ A and u ∈ U , then for any i, |λi (a)| ≥ inf {kϑ(c−1 )k−1/ki : c ∈ C}, where ϑ = ∧Ad : G → GL(∧g), and ki is the constant in the lemma 3. Proof. If π(au) ∈ π(C), then au = cγ for some γ ∈ Γ and c ∈ C. Let vi be a vector with the shortest length in ∧`i ui ∩ ∧glN (Z) where ui = Lie(Ru (Pi )), `i = dim(ui ). By lemma 3 and the definition of ϑ, λi (a)ki = ϑ(au)vi = ϑ(cγ)vi . Because of the arithmetic structure of Γ, kϑ(cγ)(vi )k ≥ kϑ(c−1 )k−1 . Altogether |λi (a)|ki ≥ kϑ(c−1 )k−1 . ¯ tends to zero as n Proof of part(i). It is enough to show that (an µU )(C) ¯ goes to infinity, for any C compact subset of G/Γ. One can consider C¯ as π(C) where C is a compact subset of G. Since supp(an µU ) = an U , to complete the proof it is enough to note that π(an U ) ∩ π(C) is empty for large enough n, as a consequence of hypothesis and lemma 4. e be the one point compactification Step 1 of the proof of part(ii). Let X e of G/Γ and µ ∈ P(X) the limit of an µU . We would like to show that for any ε > 0, there is C 0 a compact subset of G such that (an µU )(π(C 0 )) > 1 − ε. In particular, µ(G/Γ) ≥ µ(π(C)) > 1 − ε, and so µ ∈ P(G/Γ). Let x1 , · · · , xd be a basis of u = Lie(U) such that xi ’s are in glN (Z) and [xi , xj ] in Span{xk |k ≥ max(i, j)}. Define Θ0 : Rd → U ⊂ G by Θ0 (t1 , · · · , td ) = exp(t1 x1 ) · · · exp(td xd ), and Θn (t1 , · · · , td ) = an Θ0 (t1 , · · · , td ), for any n. Clearly for any n, all the matrix entries of Θn (t1 , · · · , td ) are polynomials in ti ’s of degree at most N , and as a corollary Θn ∈ PN,d . For any i, let Wi be the sum of the highest weight spaces in the representation Vi = ∧`i g. By lemma 1.4 in [17], we know that for any ball Ω in Rd there is a constant e such that: kvk ≤ e · sup kprWi (ϑi (Θ0 (t))v)k. t∈Ω

As a consequence we get the following lemma. Lemma 5. For any a ball in Rd , and F a finite subset of G(Q), there is θ > 0 such that if |λi (a)| ≥ 1 for any i, then sup di (aΘ0 (t)f γ) ≥ θ, t∈Ω

for any γ ∈ Γ and f ∈ F . 4

Proof. By the definition of di , we have: di (aΘ0 (t)f γ) = kϑi (aΘ0 (t)f γ)vi k. It is clear that there is a constant e0 depending on the set F (and independent of γ) such that kϑi (f γ)vi k ≥ e0 . On the other hand, because the weight spaces are assumed to be orthogonal to one another, we have kϑi (aΘ0 (t)f γ)vi k ≥ kϑi (a)(prWi (ϑi (Θ0 (t)f γ)vi ))k ϑi (a) acts by the product of highest weights of the irreducible components of ϑi on Wi . Since any highest weight is product of fundamental weights and the fundamental weights at a are at least 1, a enlarges the length of prWi (ϑi (Θ0 (t)f γ)vi ) So one can finish the argument using Shah’s mentioned result. Now using lemma 5 and Theorem B (see also theorem 2.1 [17]) which is the multidimensional version of Dani-Margulis’ recurrence theorem, for a given convex bounded set Ω in Rd , we can find a compact subset C of G, such that π(Θn (Ω)) ∩ π(C) is not empty. When a (multi-dimensional) unipotent comes to a compact set, it comes back to a possibly larger compact set with a positive density. More precisely, by Theorem A (see also theorem 2.1 of [17]), for any ε > 0 and a given convex bounded set Ω in Rd , we can find C¯ 0 a compact subset of G/Γ such that: m({t ∈ Ω|π(Θn (t)) ∈ C¯ 0 }) ≥ (1 − ε)m(Ω), for any n, where m is the Lebesgue measure on Rd . Let Ω be the convex hull of a strongly Malcev basis strongly based on U ∩ Γ (see theorem 5.1.6 [4]). Hence if we denote the Haar measure on U also by µU , then µU (Ω0 ) = µU (π(Ω0 )), for any Ω0 ⊆ Ω. On the other hand, note that µU (Ω) := m(Θ−1 0 (Ω)) is a Haar measure on U (e.g. see [16]). Therefore (an µU )(C¯0 ) ≥ (1 − ε)µU (Ω) = 1 − ε, and the proof of the first step is done. Step 2 of the proof of part(ii). To show this step, we will heavily use the main theorems of Mozes and Shah [14], specially theorem 1.1. Let {u(i) (t)} be a finite set of one-parameter unipotent subgroups of U , such that µU is u(i) (t)-ergodic for any i, and the group generated by them is U . We can find Y a subset of µU -full measure in π(U ) such that for any y ∈ Y, Z lim

T →∞

T

f (u(i) (t)y)dt =

0

Z f dµ, π(U )

5

for any (bounded) continuous function f on π(U ). For any n and i, µn = an µU is un (t) = an u(i) (t)a−1 n -invariant and ergodic, and µn converges to µ ∈ P(π(G)), as n goes to infinity. Therefore by corollary 1.1 of Mozes and Shah [14] and work of Dani on U -invariant measures, µ is algebraic, i.e. there is H a closed subgroup of G such that π(H) is a closed subset of π(G), and g ∈ G such that µ = gµH . Therefore Λ = g(H ∩ Γ)g −1 is a lattice in gHg −1 . It is also clear that µ is U -invariant, and so U ⊆ gHg −1 . Thus by [1, Proposition 8.6] there is a standard parabolic such that QE ⊆ gHg −1 ⊆ PE , where QE is the smallest normal subgroup of PE which contains U . As it is mentioned in the proof of the same proposition, QE and PE share the same unipotent radical and the Levi component of QE is a semisimple group which consists of all non-compact factors of PE . It is clear that no element in the Levi component of PE commutes with Ru (PE ). Therefore by a corollary of Auslander-Wang’s theorem [15, Corollary 8.28] Λ intersects the radical of gHg −1 in a lattice. The radical of gHg −1 con0 sists of Ru (PE ) and SE some part of torus. Since the radical has a lattice, it is unimodular, and therefore the radical is the same as the unipotent radical. (1) Overall we have that QE ⊆ gHg −1 ⊆ PE , for some E. (1)

On the other hand, since an ’s are in SF , an U ⊆ QF and π(QF ) is a closed subset of π(G). Therefore one can view an µU ’s as elements of P(π(QF )). Again using Mozes-Shah’s result, we can deduce that µ is in P(π(QF )). In particular, gH ⊆ QF , and so g ∈ QF and H ⊆ QF . To finish the proof of the theorem, we have to show that E = F , and so far we know F ⊆ E. In the rest of this section, we will show that E cannot be larger than F . After going to a subsequence, it is easy to find {gn } a sequence of elements in G with the following properties: i) gn → e, as n → ∞. (i)

ii) π(gn g) is a generic point for all the unipotent flows un (t) with respect to µn . Now we can conclude the following lemma. Lemma 6. In the above setting, we have: i) There is n0 such that for n ≥ n0 , π(an U ) ⊆ π(gn gH). (i)

ii) For any n and i, gn−1 un (t)gn ⊆ gHg −1 . In particular, gn−1 U gn ⊆ gHg −1 . Proof. The first part is a consequence of part-2 of theorem 1.1 form [14]. The second part is a consequence of part-3 of the same theorem, the fact that u(i) (t)’s generate U , and an normalizes U .

6

Lemma 7. If U is the unipotent radical of P∆ the standard minimal parabolic and P is another standard parabolic, then T (U, P ) := {q ∈ G | q −1 U q ⊆ P } = P. Proof. After using Bruhat decomposition, without loss of generality, we can assume that q ∈ NG (A). Therefore q −1 ZG (A)U q ⊆ P . On the other hand, P∆ = ZG (A)U, and so q ∈ T (P∆ , P ). However the later is equal to P (See [1, proposition 4.4]). Since gHg −1 ⊆ PE , by the second part of lemma 6, gn ∈ T (U, PE ), for any n. Hence by lemma 7, gn ∈ PE . From the first part of lemma 6, for any n, there are hn ∈ H and γn ∈ Γ such that an = gn ghn γn = gn (ghn g −1 )(gγn ).

(1)

Since an , gn , ghn g −1 ’s are in PE , so are gγn ’s and γ1−1 γn . Now assume that αi ∈ E \ F , and apply λi to the both sides of equation (1). (1) We note that λi (ghn g −1 ) = 1 since gHg −1 ⊆ PE , and |λi (gγn )| = |λi (gγ1 )| · |λi (γ1−1 γn )| = |λi (gγ1 )|, as λi is a Q-character and Γ preserves the Z-structure. Altogether we have |λi (an )| = |λi (gn )| · |λi (gγ1 )|, which is a contradiction since the right hand side tends to infinity, and the left hand side remains bounded, as n goes to infinity. Corollary 8. For any compactly supported continuous function Ψ on G/Γ, M > 0, and ε0 > 0, there is (xΨ,ε0 ,F )F ⊆{1,··· ,r} such that for any F a subset of {1, · · · , r} we have Z Z | Ψ d(aµU ) − Ψ d(aF µQF )| < ε0 (2) G/Γ

G/Γ

if λi (a) ≥ xΨ,ε0 ,F for any i 6∈ F and e−M ≤ λi (a) ≤ max|F 0 |<|F | (xΨ,ε0 ,F 0 ) for any i ∈ F . Proof. We start with F = ∅. By theorem 1, there is xΨ,ε0 ,∅ such that the inequality 2 holds for F = ∅ if λi (a) ≥ xΨ,ε0 ,∅ for any i. Now let F be any subset of {1, · · · , r}, and assume that we have already found xΨ,ε0 ,F 0 for any subset F 0 of size less than |F |. Again by theorem 1, for any aF , we can find x0 depending on Ψ, ε0 and aF such that if λi (aF c ) ≥ x0 for any i 6∈ F , then the inequality 2 holds for a = aF · aF c . If aF comes from a compact set, one can find x0 which simultaneously works for all of them. Hence we can find xΨ,ε0 ,F which satisfies the desired properties.

7

4

Counting horospheres.

Eskin and McMullen [8] considered the following counting problem, among others. Let x0 be a fixed point in H the two dimensional hyperbolic space, c a horocycle, and Γ a lattice in G = PSL2 (R) such that C = π(c) is a closed horocycle in Σ = H/Γ. Then N (R) the number of Γ translates of c which intersects B(x0 , R) ball of radius R centered at x0 is asymptotically 1 leungth(C) area(B(x0 , R)). π area(Σ) They also remark that their method does not work for the maximal unipotent subgroups in higher-rank groups. Indeed they used mixing and a key lemma called “Wavefront lemma” which can show a special case of 1. Namely, if an ’s go to infinity in the positive Weyl chamber (instead of the interior of the dual of the Weyl Chamber), then an µU converge to the Haar measure of G. In this section, we would like to address the higher rank version of this problem. Let G be a (semi)simple Lie group (without compact factor), X = G/K its symmetric space, x0 ∈ X whose stabilizer is K and A · x0 is a maximal flat in X, B(x0 , R) ball of radius R centered at x0 in X with the Reimannian metric, U = U g0 K/K a maximal horosphere in X, Γ a(n irreducible) lattice in G such that U = π(U) is a closed horosphere in M = Γ\X, and N (R) = #{γ ∈ Γ | γU ∩ B(x0 , R) 6= ∅}. We can and will assume that U g0 K = U a0 K, where a0 ∈ A. In this section, we prove the following theorem. Theorem 9. As R goes to infinity,  r−1  2 √ r−1 vol(U ∩ Γ\U) π √ N (R) ∼ · ρ(a0 )2 · R 2 · e2 rR . 2 vol(M) r Without loss of generality we can assume that G is the real points of a eR = {g ∈ G | d(x0 , gx0 ) ≤ R}, simply connected algebraic R-group. Let B e e AR = A ∩ BR , and B R = {gU | g ∈ BR }. Define FR ∈ L2 (G/Γ) as following, X FR (gΓ) := χR (gγg0 U ), γ∈Γ/Γ∩U

where χR is the characteristic function of B R . Let FeR (gΓ) = fa 1(R) FR (gΓ), 0 R where fa0 (R) = A> a0 ρ(a)2 da and A> a0 = {a ∈ AR a0 | ∀i λi (a) ≥ 1}. Here we R R give a lower and an upper bound for the growth of integrals similar to fa0 (R). Lemma 10. Let x = (x1 , · · · , xn ) ∈ Rn and R ∈ R+ . There there is a constant c, depending on n and x, and P a polynomial of degree n − 1 depending on x such that Z Pn √ √ √ 2 nR ≤ kyk ≤ R e2 i=1 yi dy ≤ P ( nR) e2 nR c·e xi ≤ yi 8

Proof. The Q lower√bound is an easy consequence of the fact that integral over the box [xi , R/ n] is less than the above Pnintegral.√For the upper bound, we notice that for any y in the desired region i=1 yi ≤ nR, by Cauchy-Schwartz inequality. Hence Z Z P P 2 n 2 n Pn √ i=1 yi dy ≤ i=1 yi dy. e kyk ≤ R y ≤ nR e i

i=1

xi ≤ yi The later equals to pletes our proof.

xi ≤ yi R √nR Pn

i=1

xi

vol({y |

Pn

i=1

yi = s, xi ≤ yi }) e2s ds, which com-

Now we describe the asymptotic behavior of fa0 (R). Lemma 11. For any x = (x1 , · · · , xn ) ∈ Rn , Z Z Z P P yi yi 2 n 2 n i=1 i=1 dy ∼ dy ∼ kyk ≤ R e kyk ≤ R e

Pn

i=1

yi

dy.

kyk≤R

0 ≤ yi

xi ≤ yi

e2

Proof. Let us start with the first two terms. Let Bj = {y ∈ Rn |kyk ≤ R, −|xj | ≤ yj ≤ |xj |, ∀i 6= j min{xi , 0} ≤ yi } and B 0 = {y0 ∈ Rn−1 |ky0 k ≤ R, min{x1 , · · · , xn , 0} ≤Pyi0 }. RIt is clear that Pn n the difference between the first two terms is bounded by j=1 Bj e2 i=1 yi dy, which is clearly at most Z n Z X Pn 0 ( e2yj dyj ) · e2 i=1 yi dy0 . B0

−|xj |≤yj ≤|xj |

j=1

2 kyk ≤ R e 0 ≤ yi zero as R goes to infinity, which gives us the claim.

By lemma 10, the ratio of the above value by

R

Pn

i=1

yi

dy tends to

Pn To i=1 yi is at most √ compare the second and the third terms. Note that n − 1R for y’s which are in the ball of radius R centered at the origin with one non-positive coordinate. So Z Z √ P Pn 2 n yi i=1 dy − kyk ≤ R e2 i=1 yi dy| < vol(B(0, R)) · e2 n−1R . | e kyk≤R

0 ≤ yi

By lemma 10, we can finish the proof. Lemma 12. Z e kyk≤R

2

Pn

i=1 yi

1 dy ∼ 2



as R goes to infinity. 9

π √ n

 n−1 2 ·R

n−1 2

· e2

√

nR

,

Proof. By Fubini’s theorem, Z Z Pn e2 i=1 yi dy =

√

nR

√

vol{y|kyk ≤ R,

− nR

kyk≤R

n X

yi = s} · e2s ds.

(3)

i=1 Pn i=1

yi = s and the ball On the other hand, intersection of the hyperplane of radius R centered at the origin is a ball of dimension n − 1 and of radius q R2 −

s2 n.

Hence the right hand side of equation 3 is equal to

√

Z νn−1

nR

√ − nR

(R2 −

√ s2 n−1 2s ) 2 · e ds = νn−1 nRn n

Z

1

(1 − s2 )

n−1 2

√

e2

nRs

ds,

−1

where νn−1 is the volume of a ball of radius 1 in Rn−1 . Let gm (x) = m s2 ) 2 exs ds. Using integration by part, it is easy to see that

R1 −1

(1 −

x2 gm (x) = −m(m − 1)gm−2 (x) + m(m − 2)gm−4 (x), (4) Q m for m ≥ 3. Let g m (x) = x gm (x)/$m , where $m = 0≤k
(5)

By induction, it is clear that there are sequences of polynomials Pm and Qm which satisfy similar recursive formulas as in equation 5, and   Pm (x)g 2 (x) + Qm (x)g 0 (x) if 2|m. g m (x) =  Pm (x)g 1 (x) + Qm (x)g −1 (x) if 2 6 |m. We can further determine the leading terms of Pm and Qm .

l.t. of Pm &Qm =

 −k(2k + 1)x2k−2 , x2k         −2k(k + 1)x2k  x2k ,   x2k ,        −2(k + 1)2 x2k , x

if m = 4k. if m = 4k + 1.

−k(2k + 3)x2k

if m = 4k + 2.

x2k+2

if m = 4k + 3.

One also can easily see that g 0 (x) ∼ ex and g 2 (x) ∼ ex . By the definition of modified Bessel functions, one can see that g1 (x) = πx I1 (x) and g−1 (x) = πI0 (x). x x Hence g 1 (x) ∼ π √e2πx and g −1 (x) ∼ πx · √e2πx as x goes to infinity. Altogether  2k−1 x x e if m = 4k.       p π 2k− 1 x   2e if m = 4k + 1.  2x g m (x) ∼   x2k ex if m = 4k + 2.       p  π 2k+ 12 x e if m = 4k + 3. 2x 10

Therefore one can finish the proof as follows   n−1 2 √ √ √ n n−1 1 π √ νn−1 nR gn−1 (2 nR) ∼ · R 2 · e2 nR . 2 n

Next lemma gives a decomposition for B R . Lemma 13. B R = KAR U/U . Proof. Let gU ∈ B R . By Iwasawa decomposition, without loss of generality, we can and will assume that g = ka, where k ∈ K and a ∈ A. Let u ∈ U such that d(x0 , kaux0 ) ≤ R. Therefore d(k −1 x0 , aux0 ) = d(x0 , aux0 ) ≤ R. Let at x0 be a geodesic in Ax0 , and dist(t) = d(at x0 , auat x0 ) = d(x0 , a(a−1 t uat )x0 ). So choosing the contracting direction, we can make sure that limt→∞ dist(t) = d(x0 , ax0 ). On the other hand, dist is a convex function [2, Theorem 3.6] and so its value at each point is at least the value of its limit in the infinity if finite. Therefore for any t, d(at x0 , auat x0 ) ≥ d(x0 , ax0 ). In particular, d(x0 , aux0 ) ≥ d(x0 , ax0 ), and so d(x0 , ax0 ) ≤ R, which finishes proof of the lemma. Since G(R) = G, where G is simply connected, (λi )ri=1 : S → Grm is a Qisomorphism. Let us remind that A = S(R)◦ . For any F a subset of {1, · · · , r} and a ∈ A, let aF ∈ A such that λi (aF ) = λi (a) for any i 6∈ F and λi (aF ) = 1 for any i ∈ F . Clearly a = aF · aF c . Lemma 14. For any Ψ compactly supported continuous function on G/Γ, hFeR , Ψi →

vol(U/U ∩ Γ) h1, Ψi, vol(M)

as R goes to infinity. Proof. Without loss of generality we can assume that h1, Ψi = 1. Moreover in this proof ν always denote a measure coming from the Riemannian geometry and µ is a probability measure on the corresponded space. R P hFeR , Ψi = fa 1(R) G/Γ γ∈Γ/Γ∩U χR (gγg0 U )Ψ(gΓ) dνG 0

(lemma 13);

=

1 fa0 (R)

=

ν(π(U )) fa0 (R)

R

=

ν(π(U )) fa0 (R)

R

=

ν(π(U )) fa0 (R)

R R

R G/Γ∩U

R

G/U

G/U

K

χR (ga0 U )Ψ(gΓ) dνG U/Γ∩U

χR (gua0 U )Ψ(guΓ) dµU (u)dνG (g)

χR (ga0 U )

AR a0

R G/Γ

11

R G/Γ

Ψ(g 0 Γ) d(gµU )(g 0 )dνG (g)

Ψ(g 0 Γ) d(kaµU )(g 0 )ρ(a)2 da dk.

Let C be a compact subset of G such that π(C) = supp(Ψ). If supp(kaµU ) intersects supp(Ψ), then kau = cγ for some c ∈ C and γ ∈ Γ, and by lemma 4, there is a constant M depending on Ψ such that for any i, |λi (a)| ≥ e−M . Now, by corollary 8, we define the regions TΨ,ε0 ,F in A in which we get the desired estimates of the translated measures from the corollary, and follow the above equations by Z Z Z X ν(π(U )) Ψ(kaF qF Γ) dµQF ρ(a)2 da dk (6) fa0 (R) K AR a0 ∩TΨ,ε0 ,F QF Γ/Γ F ⊆{1,··· ,r}

±

ε0 · ν(π(U )) · vol(K) · fa0 (R)

Z

ρ(a)2 da.

{a∈AR a0 |∀iλi (a)≥e−M }

By lemma 11, it is clear that the second term is of order O(ε0 ) for large enough R. In particular, we can make it as small as we wish by decreasing ε0 , and increasing R. So for a given F , we follow just the first part, use log λi (a) as a coordinate Q system for Lie(A), and let x0i = log λi (a0 ). It is well-known that ρ(a) = λi (a). So in this coordinates, we get aR,F = aR,x0 ,Ψ,ε0 ,F the set of elements (xi )ri=1 with the following three properties. P i) (xi − x0i )2 ≤ R2 . ii) −M ≤ xi ≤ maxF 0 (F log xΨ,ε0 ,F 0 (∀i ∈ F ). iii) log xΨ,ε0 ,F ≤ xi (∀i 6∈ F ). Let prF be the projection onto the F components, and {xF } × a(xF ) the section of aR,F over xF in the vector space of F components. Hence the first term of 6 is as follows Z Z K

Z

prF (aR,F )

Ψ(kexF qF Γ)e2

P

i∈F

xi

Z

e2

dµQF dxF

QF Γ/Γ

P

i6∈F

xi

dxF c dk.

a(xF )

Note that for any xF ∈ prF (aR,F ), by lemma 10, there is P a polynomial (independent of xF ) such that Z √ c P p e2 i6∈F xi dxF c ≤ P ( |F c |R) e2 |F |R . a(xF )

In particular, if F is not empty, again by lemma 10, Z P e2 i6∈F xi dxF c /fa0 (R) a(xF )

tends to zero as R goes to infinity. Overall we have R )) R R hFeR , Ψi = ν(π(U Ψ(kaF q∅ Γ) dµQ∅ ρ(a)2 da dk fa (R) K AR a0 ∩T π(Q∅ ) 0 Ψ,ε ,∅

0

+ε00

P

F 6=∅

±

R R K

R

prF (aR,F ) π(QF )

Ψ(kexF qF Γ) e2

O(ε0 ), 12

P

i∈F

xi

dµQF dxF dk

for large enough R. In the second term, note that prF (aR,F ) is a bounded region. Therefore the integral is bounded by a function of a0 , ε0 , and Ψ. So by fixing them and increasing R, we can make the second term as small as we wish. Thus we have to focus on the first term. Since G is a (semi)simple Lie group without compact factor Q∅ = G. Hence the first term equals to R ν(π(U )) 1 2 vol(G/Γ) vol(K) · fa0 (R) AR a0 ∩TΨ,ε0 ,∅ ρ(a) da. We get the desired result by again using lemma 11. We would like to prove a pointwise convergence. Similar to [7], it is enough to take Ψε an approximation of of the identity near π(g), and then show what happens to FeR (g) after a small perturbation of g. Let Ψε be such that supp(Ψε ) ⊆ {π(g 0 )|d(gx0 , g 0 x0 ) ≤ ε & d(g −1 x0 , g 0−1 x0 ) ≤ ε}. Then it is easy to see that fa (R + ε) e fa0 (R − ε) e · FR−ε (π(g)) ≤ hFeR , Ψε i ≤ 0 · FR+ε (π(g)). fa0 (R) fa0 (R)

(7)

So the following lemma together with lemma 14 and (7) show that FeR (π(g)) tends to vol(U/U∩Γ) vol(M) , as R goes to infinity, for any g ∈ G. Lemma 15. We have fa0 (ε + R) , R→∞ fa0 (R)

b(ε) = lim with b(ε) → 1 as ε → 0.

Proof. This is a direct corollary of lemmas 11 and 12, and the coordinate system introduced in the proof of lemma 14.

5

Batyrev-Tschinkel-Manin’s conjecture for flag variety.

Let P be a Q-parabolic of G. For any Q-character χ of P, X = G × Ga / ∼, where (g, x) ∼ (gp, χ(p)x) for any p ∈ P, is a line bundle on the flag variety G/P. When G is simply-connected, any line bundle is of the above form [11]. It is well-known that the anti-canonical line bundle corresponds to ρ2 , and the cone of the effective divisors corresponds to the integral vectors in C + positive Weyl chamber (here we will call the linear function on the Lie algebra of A coming from logarithm of a character with the same notation.) For any weight χ, there is η : G → GL(V) a unique irreducible representation of G whose highest weight is χ. Let v be a primitive integral vector in the highest

13

weight space. Clearly the stabilizer of [v] in the projective space Stab([v]) contains P, and so it is a parabolic subgroup, and it acts via a character on v which is an extension of χ. Hence if χ has non-trivial component in the direction of each fundamental weight λi , Stab([v]) = P, and we can identify G/P with the G-orbit of [v] in P(V) the projective space of V. Let H be the pull back of the usual height on P(V)(Q) to (G/P)(Q). Franke, Manin, and Tschinkel [11] essentially proved that #{x ∈ (G/P)(Q) | H(x) ≤ T } ∼ CT aχ (log(T ))bχ −1 , where aχ = inf{a ∈ R; aχ − 2ρ ∈ C + } and bχ is the codimension of the face containing aχ χ − 2ρ. (Indeed they just proved it for the anti-canonical line bundle, and later Batyrev and Tschinkel [3] formulated and observed this general statement.) In this section, we will give an alternative proof of this statement using theorem 1. The general approach is very similar to the previous section. However there are also major differences. By a theorem of Borel-Harish-Chandra there is F a finite subset of G(Q) such that G(Q) = Γ · F · P(Q). It is also easy to see that (G/P)(Q) = G(Q)/P(Q). So it is enough to understand the asymptotic behavior of NT = #{γ ∈ Γ/Γ ∩ P | kη(γ)vk ≤ T }, where k.k is a K-invariant real norm on V(R). Without loss of generality we can assume that kvk = 1 and P is the standard minimal parabolic. Let BT be the ball of radius T centered at the origin in V(R), and similar to the previous eT in G/U . This time, eT the pull back of BT on G, and B T image of B section, B it is much easier than the previous section to give a decomposition of B T . Lemma 16. Let AT = {a ∈ A | χ(a) ≤ T }. Then B T = KAT U/U. Proof. It is clear since k · k is K-invariant, U fixes v, A acts by the character χ on v, and G = KAU . Similar to the previous section, let X FT (gΓ) :=

[η(gγ)v ∈ BT ],

γ∈Γ/Γ∩P

and FeT (gΓ) =

1 f (T ) FT (gΓ),

Z f (T ) = A+ T

where

ρ(a)2 da and A+ T = {a ∈ AT | ∀i λi (a) ≥ 1}.

We would like to prove that FeT (e) tends to a constant as T goes to infinity. In the analogues case in the previous section, we proved a much stronger statement. We proved both weak and pointwise convergence of FeR to a constant 14

function. However in this section, we cannot prove such statements. Instead we only prove what we need for the counting problem, namely pointwise convergence at the identity. By the definition of FeT andP going to the Lie algebra of A, we need to estimate r certain integrals. Let χ = i=1 ci λi (Again let us emphasize that we denote the logarithm of characters of A, with the same notation as the characters themselves.) Lemma 17. Let c1 , . . . , cn be positive real numbers, and CT = {x | ∀i 0 ≤ xi ,

n X

ci xi ≤ log T }.

i=1

Then

Z

e2

Pn

i=1

xi

dx ∼ C T a (log T )b−1 ,

x∈CT 2 , b = #Fmin , Fmin = {k | ck = min(ci )} and C = C(c1 , . . . , cn ) where a = min(c i) is some positive number, as T goes to infinity.

Proof. See either [13] or [12]. Corollary 18. Let c1 , . . . , cn be positive real numbers, y = (y1 , . . . , yn ) ∈ Rn , and n X CT,y = {x | ∀i yi ≤ xi , ci xi ≤ log T }. i=1

Then

Z

e2

Pn

i=1

xi

dx ∼ e

Pn

i=1 (2yi −aci yi )

C T a (log T )b−1 ,

x∈CT ,y

where a, b, and C = C(c1 , . . . , cn ) are as in lemma 17. Let {Ψε } be a family of continuous functions on G/Γ which approximates the dirac function at π(e) ∈ G/Γ. Moreover assume that supp(Ψε ) ⊆ π({g ∈ G | kϑ(g −1 )k ≤ eε }) ∩ π(KAF,ε QF ), where AF,ε = {aF ∈ AF |∀i e−ε ≤ λi (a) ≤ eε }. By corollary 8, for any ε, ε0 , and F ⊆ {1, . . . , r}, there is xΨε ,ε0 ,F which R satisfies the conditions of corollary 8 for M = ε, and moreover by lemma 4 | G/Γ Ψε d(aµU )| = 0 if λi (a) < e−ε for some i. Lemma 19. There is a function E such that hFeT , Ψε i ∼ E(ε), as T goes to infinity. Moreover E(ε) is convergence as ε goes to zero.

15

Proof. Similar to the proof of lemma 14, using lemma 16, we have Z Z Z 1 e Ψε (g 0 Γ) d(kaµU )(g 0 )ρ(a)2 da dk. hFT , Ψε i = f (T ) K AT G/Γ Again as before using corollary 8 and the way we chose {Ψε }, we have Z Z Z X 1 Ψ(kaF qF Γ) dµQF ρ(a)2 da dk f (T ) K AT ∩TΨ ,ε0 ,F QF Γ/Γ F ⊆{1,··· ,r}

±

(8)

ε

ε0 · vol(K) · f (T )

Z

ρ(a)2 da.

{a∈AT |∀i λi (a)≥e−ε }

By corollary 18, the second term is of type O(ε0 ) for large enough R, where the constant depends on ε and ci ’s. So as in the previous section, we focus on the first term for a given F . Going to the Lie algebra this time we get the following sets. aT,F = aT,ε,ε0 ,F the set of elements (xi )ri=1 with the following three properties. P i) ci xi ≤ log T. ii) −ε ≤ xi ≤ maxF 0 (F log xΨ,ε0 ,F 0 =: zF (∀i ∈ F ). iii) yF := log xΨε ,ε0 ,F ≤ xi (∀i 6∈ F ). Hence as before the F -term of the first term of (8) is Z Z Z Z P Ψε (kexF qF Γ)e2 i∈F xi dµQF dxF K

prF (aT ,F )

QF Γ/Γ

e2

P

i6∈F

xi

dxF c dk.

a(xF )

(9) For a given xF ∈ prF (aT,F ), by lemma 18 Z P P P e2 i6∈F xi dxF c ∼ CeyF i6∈F (2−aF ci ) · e−aF i∈F ci xi · T aF (log T )bF −1 , a(xF )

(10) 2 c c where aF = mini∈F and b = #{k ∈ F | c = min (c )}. Again applyF k i∈F i c (ci ) ing lemma 18 to estimate f (T ) and combining with the estimate of (10), we can easily see that if min(ci ) 6= mini∈F c (ci ) or F ∩ Fmin 6= ∅, Z P e2 i6∈F xi dxF c /f (T ) → 0, a(xF )

as T goes to infinity. So we have to only study the terms corresponded to subsets F such that Fmin ⊆ F c . In fact, we claim that if F c 6= Fmin , we can ignore theP corresponding F -term. To see the claim, it is enough to notice that otherwise i∈F c 2 − aF ci < 0, we

16

can choose yF to be very large in the proof of corollary 8, and −ε ≤ xi for any i ∈ F , and thus we can make eyF

P

i6∈F (2−aF ci )

· e−aF

P

i∈F

ci xi

,

smaller than ε0 , and we get the claim. Note that for large enough T , Rε,ε0 ,F = prF (aT,F ) is a bounded region independent of T . If F c = Fmin , then again by estimate (10) for large enough T , (9) is almost Z Z Z P Ψε (kexF qF Γ)e i∈F (2xi −aF ci xi ) dµQF dxF dk + O(ε0 ). CeO(ε) Rε,ε0 ,F

K

QF Γ/Γ

(11) As ε0 goes to zero Rε,ε0 ,F covers the cone C>ε = {x ∈ R|F | |∀i : −ε ≤ xi }. Let Z Z Z P g(ε) = Ψε (kexF qF Γ)e i∈F (2xi −aF ci xi ) dµQF dxF dk K

C>ε

QF Γ/Γ

If we show that g(ε) is finite, (11) converges to E(ε) = CeO(ε) g(ε) a function of ε, as ε0 goes to zero. Here we would like Qto show that g(ε) is finite. Let ηF be a representation whose highest weight is i∈F λ2i and vF an integral highest weight vector. For a given T 0 , let ST 0 = {γ(Γ ∩ Q) | kηF (γ)vF k ≤ T 0 }, where k · k is a K-invariant norm. If kaqΓ ∈ supp(Ψε ) = π(Cε ), a ∈ AF , and ρ(a)2 ≤ T 0 , then kaq(Γ ∩ Q) ∈ Cε ST 0 . On the other hand, Z Z Z P Ψε (kexF qF Γ)e i∈F (2xi −aF ci xi ) dµQF dxF dk = K

C>ε

QF Γ/Γ

Z

Ψε (gΓ)e(−aF ci )·H(g) dg,

(12)

G/Γ∩Q

where H(g) ∈ Lie(AF ) is definedP as g = keH(g) q for k ∈ K and q ∈ Q. Let |F | CT 0 ,ε = {x ∈ R |∀i : −ε ≤ xi , i∈F 2xi ≤ T 0 }, Gm = K(C2m+1 ,ε \ C2m ,ε )Q for m > 0 and G0 = K(C1,ε )Q. Also assume δ is a positive number such that 2 + 2δ < aF ci for any i ∈ F . Consequently g(ε) =

∞ Z X m=0

Ψε (gΓ)e(−aF ci )·H(g) dg ≤

Gm (Γ∩Q)/Γ∩Q

∞ X

2(−1−δ)m |S2m+1 |.

m=0

Using the anticanonical case, we know that |ST 0 | ∼ T 0 log(T 0 )|F |−1 . Hence ∞ X m=0

2(−2−δ)m |S2m+1 | ≤

∞ X

2(−1−δ)m 2m+1 (m + 1)|F |−1 < ∞.

m=0

17

At this point, we would like to show that g(ε) is convergence as ε goes to zero. By (12) we know that Z Z X g(ε) = Ψε (gΓ)e(−aF ci )·H(g) dg = Ψε (gΓ) e(−aF ci )·H(gγ) dg. G/Γ∩Q

G/Γ

γ∈Γ/Γ∩Q

P Let ξ(gΓ) = γ∈Γ/Γ∩Q Ψ(gΓ)e(−aF ci )·H(gγ) , where Ψ is a compactly supported continuous function on G/Γ which is non-negative, at most one, and equal to one over the support of Ψε ’s. The proof will be completed after we show that ξ is a continuous function. Let P ηe be the irreducible representation of G whose highest weight is λ = i∈F aF ci λi and v0 an integral highest vector. We would like to show that if ηe(g ±1 ) are ηe(g 0±1 ) are close, so are ξ(gΓ) and ξ(g 0 Γ). Letke η (g ±1 )−e η (g 0±1 )k ≤ ε0 . 0 Then ke η (gγ)v0 − ηe(g γ)v0 | ≤ δkηF (γ)v0 k for any γ. Hence 0

|eλ(H(gγ)) − eλ(H(g γ)) | ≤ δke η (γ)v0 k. Therefore 0

|e−λ(H(gγ)) − e−λ(H(g γ)) | ≤

2ε0 kg −1 k ε0 ke η (γ)v0 k ε0 kg 0−1 k ≤ ≤ . 0 eλ(H(gγ)) eλ(H(g γ)) eλ(H(gγ)) eλ(H(gγ))

Accordingly |ξ(gΓ) − ξ(g 0 Γ)| ≤ 2kg −1 kξ(gΓ)ε0 , which finishes the proof.

References [1] [BoT65] A. Borel, J. Tits, Groupes r´eductifs, ´ math´ematiques de l’I.H.E.S. 27 (1965) 55-151.

Publications

[2] [Bu48] H. Busemann, Spaces with non-positive curvature, Acta Mathematica. 80 (1948) 259-310. [3] [BaT98] V. V. Batyrev, Y. Tschinkel, Tamagawa numbers of polarized algebraic varieties, Asterisque 251 (1998) 299-340. [4] [CG04] L. Corwin, F. P. Greenleaf, Representations of nilpotent Lie groups and their applications Part1: Basic theory and examples, Cambridge studies in advanced mathematics, Cambridge, 2004. [5] [D81] S. G. Dani, Invariant measures and minimal sets of horospherical flows, Inventiones math. 64 (1981) 357-385 [6] [DM91] S. G. Dani, G. A. Margulis, Asymptotic behavior of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. (Math. Sci.) 101 no. 1(1991) 1-17.

18

[7] [DRS93] W. Duke, Z. Rudnik, P. Sarnak, Density of integer points on affine homogeneous varieties, Duke mathematical journal 71 no. 1 (1993) 143-179. [8] [EM93] A. Eskin, C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke mathematical journal 71 no. 1 (1993) 181-209. [9] [EMS97] A. Eskin, S. Mozes, N. Shah, Non-divergence of translates of certain algebraic measures, GAFA 7 (1997) 48-80. [10] [EMS96] A. Eskin, S. Mozes, N. Shah, Unipotent flows and counting lattice points on homogenous varieties, Annals of math., 2nd ser., 143, no. 2 (1996) 253-299. [11] [FMT89] J. Franke, Y. I. Manin, Y. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), no. 2, 421435. [12] [GOS08]A. Gorodnik, H. Oh, N. Shah, Strong wavefront lemma and counting lattice points in sectors, 2008 Preprint. [13] [GW07]A. Gorodnik,b. Weiss, Distribution of lattice orbits on homogeneous varieties. Geom. Funct. Anal. 17 (2007), no. 1, 58–115. [14] [MS93] S. Mozes, N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 149–159 [15] [Rag72] M. S. Raghunathan, Discrete subgroups of Lie groups, SpringerVerlag, New York, 1972. [16] [Ra90] M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. math 101 (1990) 449-482. [17] [Sh95] N. A. Shah, Limit distribution of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 2, 105–125.

19