the hardy–littlewood property of flag varieties

THE HARDY–LITTLEWOOD PROPERTY OF FLAG VARIETIES

Takao Watanabe

The notion of Hardy–Littlewood varieties was first introduced by Borovoi and Rudnick [BR]. An affine variety X in an affine space V defined over Q is called a strongly Hardy–

Littlewood variety if the asymptotic behavior

|(B_T ×B_f)∩X(Q)| ∼ω_X(AQ)(B_T ×B_f) as T → ∞

holds for any open compact subset B_f of the finite adele space X(AQ,f), whereB_T denotes the set ofx∈X(R) with||x|| ≤T for a Euclidean norm||·||onV(R) andω_X(AQ) denotes the measure onX(AQ) attached to a gauge form onX. It is known that many affine symmetric spaces have the strongly Hardy–Littlewood property. After Borovoi and Rudnick, Mor- ishita and Watanabe [MW1] introduced more generally the notion of S-Hardy–Littlewood homogeneous spaces.

In this note, we see that some generalized flag varieties have a property like the strongly Hardy-Littlewood property. The details will be given in [W2].

In the following, letGbe a connected reductive algebraic group defined over an algebraic number field k. For a connected k-subgroup R of G, X_k^∗(R) stands for the module of k- rational characters ofR. We writeR(A) for the group of adele points ofRandR(A)¹ for the subgroup {g∈R(A) : |χ(g)|A = 1 for all χ∈X_k^∗(R)}. The Tamagawa measure on R(A)¹ is denoted byωR. The Tamagawa numberτ(R) ofRis the volume ofR(A)¹/R(k) with respect to ω_R. We fix a minimal k-parabolic subgroup P of G and a Borel subgroup B of P. If R is a standardk-parabolic subgroup of G, then we denote by UR the unipotent radical ofR, MR the standard Levi subgroup of R, ZR the greatest central k-split torus of MR and AR

the split component ofM_R(A). The groupA_R is isomorphic to (R>0)^dimZR, and one has a direct product decomposition ARMR(A)¹ of MR(A).

LetQbe a maximal standard k-parabolic subgroup ofGand K a good maximal compact subgroup of G(A). We set X =Q\G and Y =Q(A)¹\G(A)¹. Then it is known that

(1) X(k) is contained in Y,

(2) Y has a right G(A)¹-invariant measure ωY =ωQ\ωG, and (3) the mapping

ι: AG\AQ×K −→Y : (AGz, h)7→Q(A)¹z⁻¹h⁻¹ is surjective.

SinceQ is maximal, the moduleX_k^∗(ZG\ZQ) is of rank one. Hence, there is a Z-basisαQ of X_k^∗(Z_G\Z_Q) and the characterz 7→ |α_Q(z)|AofZ_Q(A) induces an isomorphism fromA_G\A_Q onto R>0. By this way, A_G\A_Q is identified with R>0. Now we set for T >0

BT =ι((0, T]×K_∞) and Yf =ι({1} ×Kf) =Q(Af)\G(Af),

whereK_∞ andKf denotes the infinite and the finite part ofK, respectively. Then the main theorem is stated as follows.

Theorem. The asymptotic behavior

|(B_T ×B_f)∩X(k)g| ∼ τ(Q)

τ(G)ω_Y(B_T ×B_f) as T → ∞ holds for any open subset B_f of Y_f and any g ∈G(A)¹.

We explain the meaning of the cardinality of (B_T×B_f)∩X(k) before we mention an outline of the proof. It is known that there exists ak-rational absolutely irreducible representation π: G −→ GL(V), where V is a representation space defined over k of finite dimension n, such that

(1) its highest weight line xπ with respect to B is defined over k, and (2) the stabilizer of xπ in G is equal to Q.

(There are infinitely many k-rational absolutely irreducible representations of G satisfying (1) and (2). We fix one of such representations.) We use a right action ofGonV defined by a·g=π(g⁻¹)a for g∈Gand a∈V. Then the mappingg 7→xπ·ggives rise to a k-rational embedding fromX into the projective spacePV. We fix ak-basise₁,· · · ,e_n of thek-vector space V(k), and define a local height Hv on V(kv) for each placev of k as follows:

Hv(a1e1+· · ·+anen) =







√a²₁+· · ·+a²_n (if v is real) a1a1+· · ·+anan (if v is imaginary) sup(|a1|v,· · · ,|an|v) (if v is finite) The product H_π = ∏

vH_v of all H_v defines a height on PV(k), and on X(k) by restric- tion. By using this height, the set (BT ×Yf)∩X(k) is described as {x ∈ X(k) : Hπ(x) ≤ Hπ(xπ)T^eπ}, whereeπ is a positive rational number such thateπαQ represents the dominant highest weight ofπ in X_k^∗(Z_G\Z_Q)⊗ZQ. Therefore, the cardinality of (B_T ×B_f)∩X(k) is equal to the number of points x ∈ X(k) satisfying Hπ(x) ≤ Hπ(xπ)T^eπ together with the congruent condition x∈Bf. We illustrate Theorem with a simple example.

Example 1. Let V be ann-dimensional vector space defined over k, Ga group of linear automorphisms of V and π: G−→ G the natural representation. We fix a free O-lattice L in V(k) and its O-basis e1,· · · ,en. Here O stands for the ring of integers in k. Then V(k)

and G are identified with the column vector space kⁿ and the general linear group GLn, respectively. LetP be the subgroup of upper triangular matrices and Q the stabilizer in G of the line spanned by e₁. Then the map g 7→ e₁·g = g⁻¹e₁ yields an isomorphism from X = Q\G to the projective space PV = Pⁿ⁻¹. Let Hπ be a height on X(k) defined as above. We take a maximal compact subgroup K =∏

vKv as follows:

K_v =







GLn(Ov) (v is finite) O(n) (v is a real place)

U(n) (v is an imaginary place)

For each finite place v, pv and fv stand for the maximal ideal of Ov and the residual field O_v/p_v, respectively. If we set

D_v ={g∈K_v: g≡







∗ ∗

0... ∗

0





 mod p_v},

then Dv\Kv is isomorphic to Pⁿ⁻¹(fv) by the reduction homomorphism. For every x ∈ Pⁿ⁻¹(k_v), there is an h_x ∈K_v such that x= k_v(e₁ ·h_x). We denote by [x]_v the reduction of xmodulo pv, i.e., [x]v =fv(e1·hx mod pv). Let Sbe a finite subset of finite places. We fix a point (av)v∈S in ∏

v∈SPⁿ⁻¹(kv) and set

N(Pⁿ⁻¹(k), T,(av)v∈S) =|{x ∈Pⁿ⁻¹(k) :Hπ(x)≤T and [x]v = [av]v for all v ∈S}|. It is obvious that

N(Pⁿ⁻¹(k), T,(av)v∈S) =|(BT ×ι({1} ×Df ·hf))∩X(k)|, whereDf =∏

v∈SDv×∏

v̸∈SKv andhf = (ha_v)v∈S×(e)v̸∈S ∈Kf. By Theorem and the calculation of [W1, Example 2], we have

N(Pⁿ⁻¹(k), T,(a_v)_v∈S)∼ ∏

v∈S

|f_v| −1

|fv|ⁿ−1 · Res_s=1ζ_k(s)

|Dk|^(n−1)/2nZk(n) ·Tⁿ as T → ∞. Here ζ_k(s) is the Dedekind zeta function ofk,Z_k(s) = (π^−s/2Γ(s/2))^r1((2π)^1−sΓ(s))^r2ζ_k(s) andr1 (resp. r2) denotes a number of real (resp. imaginary) places of k.

We mention an outline of the proof. We may assume without loss of generality thatBf is the image of an open subgroupD_f ofK_f, i.e.,B_f =ι({1}×D_f). Letχ_T be the characteristic function ofBT ×Bf and FT the function on G(k)\G(A)¹ defined by

F_T(g) = 1

ω_Y(B_∞×B_f)

∑

x∈X(k)

χ_T(x·g) = |(BT ×Bf)∩X(k)g| ω_Y(B_T ×B_f) . The assertion of Theorem is equivalent to the pointwise convergence

(∗) lim

T→∞FT(g) = τ(Q) τ(G)

for any g∈G(A)¹. The next lemma is an analogue of [MW1, Lemma 5.2].

Lemma. If one has

(∗∗) lim

T→∞

∫

G(k)\G(A)¹

(

F_T(g)− τ(Q) τ(G)

)

θ(g)dω_G(g) = 0

for all pseudo-Eisenstein series θ on G(k)\G(A)¹, then (∗) holds for every g ∈G(A)¹.

Thus, we need to compute the integral

< FT, θ >=

∫

G(k)\G(A)¹

FT(g)θ(g)dωG(g).

By the unfolding argument, we have

< FT, θ >= eQ

T^eQ

∫ T 0

t^eQ {∫

Q(k)\Q(A)¹

θ(m·ι(t× {1}))dωQ(m) }

dt t ,

where e_Q is a positive integer determined by the relation δ_Q(ι(t × {1})) = t^−eQ for the modular character δ_Q⁻¹ of Q(A) and dt the Lebesgue measure on R. The inner integral is calculated by using the theory of constant terms of Eisenstein series. As a consequence of calculations, we can show (∗∗).

If B_f is the image of an open subgroup D_f of K_f, then we have the following formula of the volume of BT ×Bf:

ω_Y(B_T ×B_f) = [D_f(K_f ∩Q(Af)) :D_f]C_Gd_Q [Kf :Df]CQdGeQ

T^eQ,

where d_Q = [X_k^∗(Z_Q) : X_k^∗(M_Q)], d_G = [X_k^∗(Z_G) : X_k^∗(G)] and C_G/C_Q is a ratio of two different Haar measures on G(A). There is an explicit formula of CG/CQ if G is one of k-split groups, special orthogonal groups or unitary groups.

Example 2. Let V, L and e1,· · ·,en be the same as in Example 1. Let Φ be a non- degenerate isotropic quadratic form on V(k), G = SO_Φ the special orthogonal group of Φ andπ: G−→GL(V) the natural representation. The height Hπ is the same as Example 1.

We assumen≥3 and Φ has the following matrix form with respect to the basis e1,· · · ,en:

Φ =



 1

Φ₀ 1



 ,

where Φ₀ is a non-degenerate (n−2)×(n−2) symmetric matrix. Thus e₁ is an isotropic vector of Φ. Let Q be the stabilizer in G of the isotropic line spanned by e1. The map

g 7→ e1 ·g = g⁻¹e1 gives rise to a k-rational embedding from XΦ = Q\G into Pⁿ⁻¹. The image of X_Φ(k) is the set of all Φ-isotropic lines in Pⁿ⁻¹(k). We put

N(XΦ(k), T) =|{x∈XΦ(k) : Hπ(x)≤T}|.

Since the Levi-subgroup MQ is isomorphic to GL1×SOΦ0, we have τ(G) = τ(Q) = 2 and d_G = d_Q = 1, and furthermore, e_Q = dimU_Q = n−2 and e_π = 1. Therefore, Theorem implies

N(XΦ(k), T)∼ CG

(n−2)C_QTⁿ⁻² as T → ∞.

The constant C_G/C_Q was computed by Ikeda in general. If Φ has a maximal Witt index [n/2], then one has

C_G CQ

=









Ress=1ζk(s)

|Dk|^(n−2)/2Zk(n−1) (n is odd) Ress=1ζk(s)

|Dk|^(n−2)/2Zk(n−2) (n is even)

References

[BR] M. Borovoi and Z. Rudnick,Hardy–Littlewood varieties and semisimple groups, Invent. Math. 119 (1995), 37 - 66.

[DRS] W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J.71(1993), 143 - 179.

[FMT] J. Franke, Y. I. Manin and Y. Tschinkel,Rational points of bounded height on Fano varieties, Invent.

Math.95(1989), 421 - 435.

[MW1] M. Morishita and T. Watanabe,OnS-Hardy-Littlewood homogeneous spaces, Int. J. Math.9(1998), 723 - 757.

[MW2] ,Adele geometry of numbers, Class Field Theory -Its Centenary and Prospect, Adv. Studies in Pure Math., Math. Soc. Japan, 2001, pp. 509 - 536.

[W1] T. Watanabe,On an analog of Hermite’s constant, J. Lie Theory10(2000), 33 - 52.

[W2] ,The Hardy–Littlewood property of flag varieties, preprint.

Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043 Japan

E-mail address: watanabe@math.wani.osaka-u.ac.jp