The second moment of the Siegel transform in the space of symplectic lattices
Dubi Kelmer Shucheng Yu


Using results from spectral theory of Eisenstein series, we prove a formula for the second moment of the Siegel transform when averaged over the subspace of symplectic lattices. This generalizes the classical formula of Rogers for the second moment in the full space of unimodular lattices. Using this new formula we give very strong bounds for the discrepancy of the number of lattice points in an Borel set, which hold for generic symplectic lattices.

Topological waves in fluids with odd viscosity
    Vincenzo Vitelli
    December 30, 2023
==============================================






§ INTRODUCTION

Given a function f:^n→ of sufficiently fast decay, its modified Siegel transform[In the standard Siegel transform the sum is over all lattice points.] F_f is a function on the space, X_n, of unimodular lattices in ^n, defined by

    F_f(Λ)=∑_v⃗∈Λ_ prf(v⃗),

where Λ_ pr denotes the set of primitive vectors in a lattice Λ. The space of lattices comes with a natural _n()-invariant probability measure ν_n and Siegel's mean value theorem states that

    ∫_X_nF_f(Λ)dν_n(Λ)=1/ζ(n)∫_^nf(x⃗)dx⃗.

In <cit.> Rogers proved beautiful and useful formulas for higher moments of F_f. In particular, his second moment formula states that, for n≥ 3

    ∫_X_n|F_f(Λ)|^2dν_n(Λ)=1/ζ(n)^2|∫_^nf(x⃗)dx⃗|^2+1/ζ(n)∫_^n|f(x⃗)|^2dx⃗+1/ζ(n)∫_^nf(x⃗)f(-x⃗)dx⃗.

Rogers' formula was since used in many applications of metric Diophantine approximations. For example,  Schmidt <cit.> used it to study the lattice point counting problem, counting the number of lattice points in a Borel set B⊆^n{0}, proving an optimal bound for the discrepancy for a generic lattice. The expected number of lattice points (respectively primitive lattice points) in B is (B) (respectively (B)/ζ(n)) and the discrepancies are defined by

    D(Λ,B)=|#(Λ∩ B)(B)-1|,    D_ pr(Λ,B)=|ζ(n)#(Λ_ pr∩ B)(B)-1|.

For n≥ 3, applying Rogers' second moment formula to the indicator function of B gives the mean square bound,

    ∫_X_n|D_ pr(Λ,B)|^2dν_n(Λ)≤2ζ(n)/(B).

Using this mean square bound, Schmidt <cit.> showed that for any linearly ordered (with respect to inclusion) family, , of Borel sets in ^n{0}, for ν_n-a.e. lattice Λ∈ X_n there is a constant C_Λ such that for all B∈ with (B)>C_Λ both discrepancies D_ pr(Λ,B) and D(Λ,B) are bounded by log^2((B))(B)^1/2.




We remark that the term log^2((B)) above can be replaced by log((B))ψ^1/2(log(B)) for any positive, non-increasing function ψ with ∫_1^∞ψ(t)dt<∞, and we made a choice of ψ(t)=1/t^2 to simplify the statement. We further note that, while Schmidt stated his result for the larger space of all lattices (not just unimodular lattices) his proof actually implies the above stated result on unimodular lattices as well.


While Rogers' second moment formula is valid only for n≥ 3, by using a different argument, Schmidt was able to prove a weaker variant 
also for n=2 after adding another factor of log((B)).  Moreover,  it was later shown in <cit.>, using spectral theory of Eisenstein series of _2(), that for primitive lattice points (<ref>) still holds (up to a constant factor) also for n=2, and hence D_ pr(Λ,B)≤log^2((B))(B)^1/2 also holds for ν_2-a.e. Λ∈ X_2. 

In addition to Schmidt's classical result, Rogers' formula was also used  in <cit.> to prove a random version of Minkowski theorem studying the set of lattices missing a large set,  and more recently in <cit.> to prove an effective and quantitative version of Oppenheim conjecture for generic forms. Rogers' formulas for higher moments were also used in <cit.> to study values of Epstein zeta functions.



The starting point of our work is the observation that the Siegel transform can be interpreted as an incomplete Eisenstein series for a suitable maximal parabolic subgroup of _n(), and Rogers' formula can be seen as formula for moments of these incomplete Eisenstein series. It is thus possible to use the spectral theory of Eisenstein series to give another proof of Rogers' formula. The advantage of this approach is that it could be generalized to other semisimple groups and other maximal parabolic subgroups, leading  to other summation formulas. To demonstrate this approach, instead of giving another proof of Rogers' original formula, we shall prove a second moment formula with _n() replaced by the symplectic group. We will then use this formula to give bounds for the discrepancy for a generic symplectic lattice, a problem recently studied by Athreya and Konstantoulas in <cit.>.


To describe the space of symplectic lattices, fix a symplectic form on ^2n and let (2n,) denote the group of linear transformations preserving this form. The space of symplectic lattices in ^2n is then parameterized by the homogenous space Y_n=(2n,)(2n,) via the map sending g∈(2n,) to the lattice Λ=^2ng, and it has a natural probability measure μ_n coming from Haar measure on (2n,).  There is a natural generalization of Siegel's mean value theorem to the subspace of symplectic lattices, stating that

    ∫_Y_nF_f(Λ)dμ_n(Λ)=1/ζ(2n)∫_^2nf(x⃗)dx⃗.

Our main result is the following generalization of Rogers' second moment formula to this space. We say that a function f on ^2n is even if f(x⃗)=f(-x⃗) for almost every x⃗∈^2n and denote by
L^2_even(^2n) the space of even functions in L^2(^2n).

For any n≥ 1, there is an isometry ι:L^2_even(^2n)→ L^2_even(^2n) such that for any bounded compactly supported even f:^2n→

    ∫_Y_n|F_f(Λ)|^2dμ_n(Λ)=|1/ζ(2n)∫_^n f(x⃗)dx⃗|^2+2/ζ(2n)∫_^2n(|f(x⃗)|^2+ f(x⃗)ι(f)(x⃗))dx⃗.



For n=1 we have that (2n,)=_2() and this result follows from the analysis in <cit.>, we will thus concentrate below on the case when n≥ 2.


The restriction to even function is just cosmetic. Any function can be decomposed as f=f_even+f_odd and since lattices are invariant under inversion we always have that F_f=F_f_even. We thus get a similar moment formula for all functions.
The condition that f being compactly supported can also be relaxed, and replaced with the condition that f decays sufficiently fast to insure that the series defining F_f absolutely converges, or alternatively, with the condition that f∈ L^2(^2n) is bounded and nonnegative. In particular, it applies for f the indicator function of any finite-volume set in ^2n.






We now describe an application of this formula to the problem of symplectic lattice point counting. The space of symplectic lattices, Y_n, naturally sits inside the space, X_2n, of all unimodular lattices. But since it has ν_2n-measure zero,  one can not say anything about symplectic lattices from statements on generic lattices in X_2n.  Instead, it is more natural to work with the measure μ_n on Y_n directly.
In <cit.>, Athreya and Konstantoulas asked if one can give good bounds for the discrepancy, D_ pr(Λ,B), which hold for any Borel set B and for μ_n-almost every symplectic lattice. They actually studied the slightly larger space Ỹ_n=G_n() G_n() of general symplectic lattices. In this space, under some mild geometric assumptions on the target sets, they managed to prove a power saving bound for the mean square of the discrepancy, as well as for the discrepancy of a generic general symplectic lattice.



As a first application of our moment formula we can get the following square root bound for the mean square of the discrepancy, in the primitive, as well as general lattice point counting problems.
 
For n≥ 2, for any Borel set B⊂^2n∖{0} of finite volume

    ∫_Y_n|D_ pr(Λ,B)|^2dμ_n(Λ)≤4ζ(2n)/(B) ,

and

    ∫_Y_n|D(Λ,B)|^2 dμ_n(Λ)≤4ζ(n)^2/ζ(2n)(B).



When averaging over all unimodular lattices, for symmetric sets there is an equality, ∫_X_n|D_ pr(B,Λ)|^2dν_n(Λ)=2ζ(n)/(B), so the mean square bound is optimal.
For Y_n, however, we can only show an inequality. Since, we can not exclude additional cancelation between ∫ |f(x⃗)|^2dx⃗ and ∫f(x⃗)ι(f)(x⃗)dx⃗, it might be possible to construct a set B for which the mean square is much smaller.

Next, using this mean square bound we can adapt Schmidt's original arguments <cit.>  to give the following square root bound for the discrepancy of a generic symplectic lattice.

Let n≥ 2. For any linearly ordered family, , of finite-volume Borel sets in ^2n∖{0}, for μ_n-a.e. Λ∈ Y_n there is C_Λ such that for any B∈ with (B)>C_Λ

    D_ pr(Λ,B)≤log^2((B))/√((B)),    D(Λ,B)≤log^2((B))/√((B)).














In many applications one is interested in a family of sets obtained by dilation of a fixed initial set, that is ={tB:t>0} for some fixed Borel set B.
If the set B is not star shaped, (e.g., if B is an annulus or even a ball not containing the origin) the family of dilations is not linearly ordered with respect to inclusion and Theorem <ref> does not apply directly.
Nevertheless, we show that the same result still applies also for dilations of sets that can be obtained as differences and unions of star shaped sets. Explicitly, we show

Let B⊆^2n∖{0} be a Borel set with (B)=1. Assume that B can be written as a finite disjoint union of sets B_1,… B_k with each B_j=B_j^+∖ B_j^- with B_j^-⊆ B_j^+ star shaped. Then for μ_n-a.e. Λ∈ Y_n there is a constant C_Λ such that for all t> C_Λ we have

    D_ pr(Λ,tB)≤log^2(t)/t^n,   D(Λ,tB)≤log^2(t)/t^n.



We note that the same result (with the same proof) also holds for ν_n-a.e. lattice in X_n. This is closely related to a recent result of Athreya and Margulis <cit.>  who proved a weaker power saving bound for this problem, but with no restriction on the set B⊆^n.
We further note that, since their proof relied on Rogers' formula, using our moment formula it could also be adapted to give the same result for μ_n-a.e. lattice in Y_n.



 §.§ Acknowledgements

The authors would like to thank Yuanqing Cai, Solomon Friedberg and Spencer Leslie for many helpful conversations.










§ PRELIMINARIES AND NOTATION



 §.§ Notations

For any integer n≥ 1, let J_n denote the n× n exchange matrix, and let G_n= (2n,) be the group of linear transformations preserving the symplectic form on ^2n with coordinate matrix
[       J_n; -J_n      ].  We denote by _n=(2n,) the lattice of integer points, by Y_n=_n G_n and let μ_n denote the probability measure on Y_n coming from the Haar measure of G_n. For any F∈ L^2(Y_n) and f∈ L^2(^2n) we denote by F_2 and f_2 their L^2-norms with respect to μ_n and Lebesgue measure respectively. For x⃗∈^2n we denote by x⃗ the standard Euclidean norm. We denote by ^+ the set of positive real numbers.




 §.§ Coordinates on the symplectic group

We will use the following coordinates on the symplectic group G_n.
Let P_n≤ G_n denote the identity component of the maximal parabolic subgroup preserving the line spanned by e⃗_2n=(0,⋯,0,1)∈^2n.

When n=1, (2,)=_2() and P_1 has a Langlands decomposition P_1=U_1M_1A_1 with M_1 the trivial group and

    U_1={u_t=[ 1 t; 0 1 ] | t∈}, A_1= {a_y=[    y    0;    0 y^-1 ] | y>0 },

while for n≥ 2,
P_n=U_nM_nA_n with

    A_n= {a_y=[      y      0      0;      0 I_2n-2      0;      0      0   y^-1 ] | y>0 },


    M_n={m=[ 1 0 0; 0 m 0; 0 0 1 ] |  m∈(2n-2,)}

and

    U_n={u_t⃗=[      1   t⃗'⃗   t_2n;      0 I_2n-2 t⃗'⃗^∗;      0      0      1 ] | t⃗'⃗=(t_2,…,t_2n-1)∈^2n-2 
    }
    ,

where t⃗'⃗^∗= (t_2n-1, …,t_n+1, -t_n,…, -t_2)^t.






Fix a maximal compact subgroup, K_n=(2n,)∩(2n), of (2n,) and  note that K_n is isomorphic to the unitary group U(n). We further note that for n=1,  M_n∩ K_n is the trivial group, while for n≥ 2, M_n∩ K_n is a maximal compact subgroup of M_n≅(2n-2,). Thus with slight abuse of notation, for any n≥ 1, we denote K_n-1:=M_n∩ K_n. For future reference, we note that the map sending k∈ K_n to its last row induces an identification between K_n-1 K_n and the unit sphere S^2n-1:={x⃗=(x_1,…,x_2n)∈^2n | ∑_j=1^2nx_j^2=1}.



 §.§ Haar measures

We now wish to express the Haar measure μ_n explicitly in our coordinates.  To simplify notations, we denote by Q_n=U_nM_n. Let e⃗_2n=(0,…,0,1)∈^2n. For any g∈ G_n, note that x⃗(g)=e⃗_2ng equals the last row of g. The map G_n→^2n sending g to e⃗_2ng gives an identification between the homogeneous space Q_n G_n and ^2n:=^2n{0}. Let dx⃗ denote Lebesgue measure on ^2n that we think of also as a measure on Q_n G_n under this identification. We will also use the following polar coordinates. By Iwasawa decomposition any element in Q_n G_n can be represented uniquely by a_yk with a_y∈ A_n and k∈ K_n-1 K_n≅ S^2n-1. In these coordinates we have that

    dx⃗(a_yk)=2π^n/Γ(n)dy/y^2n+1dσ_n(k),

where σ_n denotes the right K_n-invariant probability measure on K_n-1 K_n (which is the surface measure on the sphere).


Next, let Q_n()=Q_n∩_n and denote by μ_Q_n the probability Haar measure on Q_n() Q_n. Explicitly, when n=1 we have that Q_1=U_1 and

    dμ_Q_1(q)=dμ_Q_1(u_t)=dt,

while for n≥ 2, M_n is isomorphic to (2n-2,), and

    dμ_Q_n(q)=dμ_Q_n(u_t⃗m)=dt⃗dμ_n-1(m).


Using this decomposition we see that for any F∈ C_c^∞(G_n),

    ∫_G_nF(g)dμ_n(g)=ω_n∫_Q_n G_n∫_Q_nF(qa_yk)dμ_Q_n(q)dy/y^2n+1dσ_n(k),

for a suitable constant ω_n.













To compute ω_n, after unfolding, (<ref>) implies that

    dμ_n(g)=1/ζ(2n)dμ_Q_n(q)dx⃗(a_yk),

and comparing (<ref>), (<ref>) and (<ref>) we get

    ω_n=2/ξ(2n),

where ξ(s)=π^-s/2Γ(s/2)ζ(s) is the Riemann Xi function.









 §.§ Mellin transform

Let C_c^∞(^+) denote the space of smooth compactly supported functions on ^+. For any ρ∈ C_c^∞(^+), its Mellin transform is defined by

    ρ̂(s):=∫_0^∞ρ(y)y^-(s+1)dy.

We note for ρ smooth and compactly supported, ρ̂(s) is an entire function with |ρ̂(s)| decaying super polynomially in s as (s)→±∞, and it satisfies the inversion formula (for any σ)

    ρ(y)=1/2π i∫_(σ)ρ̂(s)y^sds.





We will need to work with a slightly larger family of functions: We denote by 𝒞(^+), the family of functions ρ with Mellin transform ρ̂(s) analytic for (s)>1 with super polynomial decay on vertical strips, and note that these are all smooth functions with ρ(y)→ 0 as y→ 0. For ρ∈𝒞(^+), we still have the inversion formula (<ref>) for any σ >1.






 §.§ Eisenstein series

Using the Iwasawa decomposition G_n= U_nM_nA_nK_n,  for any g∈ G_n we write

    g=uma_yk

with u∈ U_n, m∈ M_n, a_y∈ A_n and k∈ K_n. Note that this decomposition is not unique, however a_y is uniquely determined by (<ref>) and 1/y^2 is given by the sum of squares of the last row of g. Thus for each parameter s∈, the map

    φ_s: G_n⟶

sending g to y^s is well-defined. 
Identifying K_n-1 K_n with Q_nA_n G_n, we can think of ϕ∈ L^2(K_n-1 K_n) as a left Q_nA_n-invariant function on G_n. The corresponding Eisenstein series, E_n(s,g,ϕ), attached to this data is then defined as

    E_n(s,g,ϕ)=∑_γ∈_P_n_nφ_s(γ g)ϕ(γ g),

where _P_n=_n∩ P_n. We note that E_n(s,g,ϕ) converges absolutely for (s)> 2n and has an analytic continuation and functional equation relating s and 2n-s. When ϕ=1, the corresponding Eisenstein series E_n(s,g,1) is right K_n-invariant, and we abbreviate it by E_n(s,g).

Given a bounded compactly supported function f on Q_n G_n (that we think of as a left Q_n-invariant function on G_n),
the corresponding incomplete Eisenstein series Θ_f∈ L^2(Y_n,μ_n) is defined by

    Θ_f(g)=∑_γ∈_P_n_nf(γ g).



For functions f on Q_n G_n that factor as f(a_yk)=ρ(y)ϕ(k) with ρ∈𝒞(^+) and ϕ∈ L^2(K_n-1 K_n) the Eisenstein series and incomplete Eisenstein series can be related via the Mellin transform as follows: For σ>2n we have

    Θ_f(g)= 1/2π i∫_(σ)ρ̂(s)E_n(s,g,ϕ)ds.

For future reference we record the following identity (essentially equivalent to (<ref>))

Let σ_0<2n and let υ̂ be a meromorphic function that is analytic in the half plane (s)>σ_0, and satisfies that |υ̂(σ+ir)|≪_σ,k (1+r^2)^-k for any σ>σ_0 and any k∈. Then for any
σ>2n, we have

    1/2π i∫_Y_n∫_(σ)v̂(s)E_n(s,g)dsdμ_n(g)= ω_nυ̂(2n),

with ω_n=2/ξ(2n) as before.


Let υ(y)=1/2π i∫_(σ)v̂(s)y^sds and note that this does not depend on the choice of σ>σ_0, that |υ(y)|≪_σ y^σ for any σ>σ_0 and that υ̂(s)=∫_0^∞υ(y)y^-sdy/y for (s)>σ_0. Now let f(a_yk)=υ(y) and let Θ_f(g)  denote the corresponding incomplete Eisenstein series, then by (<ref>) for σ>2n

    ∫_Y_n1/2π i∫_(σ)υ̂(s)E_n(s,g)dsdμ_n(g)=∫_Y_nΘ_f(g)dμ_n(g).

Next note that the condition |f(a_yk)|≪_σ y^σ with σ>2n implies that the series defining Θ_f absolutely converges. Thus we can use unfolding and (<ref>) to get that

    ∫_Y_nΘ_f(g)dμ_n(g)   =ω_n∫_Q_n G_nf(a_yk)dy/y^2n+1dσ_n(k)∫_Q_n() Q_ndμ_Q_n(q)
       =ω_n∫_0^∞υ(y)y^-(2n+1)dy
       =ω_nυ̂(2n).





§ SIEGEL TRANSFORM AND INCOMPLETE EISENSTEIN SERIES

As mentioned in the introduction the Siegel transform F_f can be identified as an incomplete Eisenstein series corresponding to a maximal parabolic subgroup of _n(). We now show that it can also be identified as an incomplete Eisenstein series for the symplectic group and use it to prove some preliminary identities for the second moment.


 §.§ Transitivity

The symplectic group acts transitively on ^2n leading to the identification with Q_n G_n. We now note that the
 integer points _n also act transitively on _ pr^2n (the set of primitive points in ^2n), leading to the following bijection.

There is a bijection between _P_n_n and _ pr^2n sending _P_nγ∈_P_n_n to the last row of γ.


We first show that _P_n=Q_n(). Indeed, for any g∈_P_n=_n∩ P_n, using Langlands decomposition P_n=U_nM_nA_n, we can write g=u_t⃗ma_y; by direct computation it is easy to see that the (1,1)-entry of g equals y and the (2n,2n)-entry of g equals 1/y; but all entries are integers so y=1. Now for any g∈_P_n=Q_n(), gγ has the same last row as γ. Hence the map _P_n_n→_ pr^2n sending _P_nγ∈_P_n_n to the last row of γ is well-defined.

For injectivity, we need to show that if γ_1,γ_2∈_n have the same last row, then γ_1γ_2^-1∈Γ_P_n. Since γ_1γ_2^-1∈_n, it suffices to show that γ_1γ_2^-1∈ Q_n. Recall that for any g∈ G_n its last row is given by e⃗_2ng, and Q_n consists exactly of matrices in G_n with the last row equaling e⃗_2n. Hence it suffices to show that e⃗_2nγ_1γ_2^-1=e⃗_2n. But since γ_1 and γ_2 have the same last row, we have e⃗_2nγ_1=e⃗_2nγ_2 and the map is injective. For surjectivity, see <cit.>.


Using this bijection we get the following identification between the Siegel transform and the incomplete Eisenstein series.

For any function f on ^2n, let f̃ denote the corresponding function on Q_n G_n given by f̃(g)=f(e⃗_2ng).
Then for any g∈ G_n and any symplectic lattice of the form Λ=^2ng we have that F_f(Λ)=Θ_f̃(g).


Since the primitive vectors in Λ=^2ng are exactly  ^2n_ prg, by Lemma <ref>, there is a unique γ∈_P_n_n such that any primitive v⃗∈Λ is of the form
v⃗=e⃗_2nγ g. We thus get that

F_f(Λ)=∑_v⃗∈^2n_prgf(v⃗)=∑_∈_P_n_nf(e⃗_2nγg)=Θ_f̃(g).


With this identification we can rewrite the second moment as

    ∫_Y_n|F_f(Λ)|^2dμ_n(Λ)=Θ_f̃_2^2.

From here on we will work directly with a function f on Q_n G_n and find a formula for Θ_f_2^2.

The formula for Θ_f_2^2 is a special case of more general inner product formula between incomplete Eisenstein Series. While such formulas are known in great generality (see e.g. <cit.>), it is not always easy to translate the general results to an explicit formula such as (<ref>).  Instead we will give here a completely classical and self contained proof for a formula for Θ_f_2^2, from which (<ref>) easily follows.




 §.§ Period formula

Following the identification of the Siegel transform with the incomplete Eisenstein series we prove the following preliminary identity for the second moment in terms of an inner product with a certain period of the incomplete Eisenstein series.

For any measurable, bounded and compactly supported function, f, on Q_n G_n we have

    Θ_f_2^2= ω_n∫_Q_n G_nf(a_yk)𝒫_f(a_yk)   dy/y^2n+1dσ_n(k),

where 𝒫_f is the period

    𝒫_f(a_yk)=∫_M_n() M_n∫_U_n() U_nΘ_f(u_t⃗ma_yk)dt⃗dμ_n-1(m),

with M_n()=M_n∩_n and U_n()=U_n∩_n.


First, using unfolding we have

    Θ_f_2^2=∫__P_n G_nf(g)Θ_f(g)dμ_n(g).

Next, recalling that _P_n=Q_n(), by (<ref>) and (<ref>) we have

    Θ_f_2^2   = ω_n∫_Q_n G_n∫_Q_n() Q_nf(qa_yk)Θ_f(qa_yk)dμ_Q_n(q)dy/y^2n+1dσ_n(k)
       =ω_n∫_Q_n G_nf(a_yk)∫_M_n() M_n∫_U_n() U_nΘ_f(u_t⃗ma_yk)dt⃗dμ_n-1(m)dy/y^2n+1dσ_n(k),

where for the second equality we used the assumption that f is left Q_n-invariant.


Here the condition that f is compactly supported can be relaxed and replaced with the condition that f(a_yk)→ 0 sufficiently fast as y→ 0 to insure that the series defining Θ_f and the integrals in the formula all absolutely converge.


Having this preliminary identity, in order to prove our moment formula we need to compute the period 𝒫_f explicitly in terms of f. We will do it first in the special case where f is spherical. We  then use raising operators to get it for other K_n-types and finally combine all K_n-types to get it for general functions.









§ PERIOD COMPUTATION FOR SPHERICAL FUNCTIONS

Let f∈ C^∞(Q_n G_n/K_n) be smooth and spherical in the sense that f(a_yk)=ρ(y) with ρ∈(^+).  In this case by (<ref>)  we have for σ>2n

    𝒫_f(a_yk)=1/2π i∫_M_n() M_n∫_(σ)ρ̂(s)∫_U_n() U_nE_n(s,u_t⃗ma_y)dt⃗dsdμ_n-1(m).

Here the inner most integral is the constant term of the Eisenstein series  along the unipotent radical of P_n. We will first compute this constant term explicitly and then use it to compute the period.










 §.§ Constant term formula

The constant term of E_n(s,g) along the unipotent radical of P_n is defined as

    c_E_n(s,g):= ∫_U_n() U_nE_n(s,u_t⃗g)dt⃗.

For n=1 the formula c_E_1(s,a_y)=2(y^s+ξ(s-1)/ξ(s)y^2-s) is well known,
we now compute c_E_n(s, ma_y) explicitly for n≥ 2 in terms of the coordinates (s,y,m).










Let n≥ 2.
Keep the notation as above. Then

    c_E_n(s, ma_y)= 2y^s+ 2ξ(s-2n+1)/ξ(s)y^2n-s+ ξ(s-1)/ξ(s)yE_n-1(s-1,m),

where ξ(s)= π^-s/2(s/2)ζ(s) is the Riemann Xi function.



Using the expansion ζ(s)=∑_k=1^∞1/k^s for the Riemann zeta function and noting that any lattice point can be written in a unique way as a multiple of a primitive lattice point, Lemma <ref> for (s)>2n implies that

    ζ(s)E_n(s,g)= ∑_v⃗∈^2n{0}1/||v⃗g||^s.


Next, for v⃗=(v_1,…,v_2n), denote v⃗^M_n=(v_2,…, v_2n-1), 
and integrate (<ref>) over U_n() U_n to get that

    ζ(s)c_E_n(s, ma_y)   =∫_U_n() U_n∑_v⃗∈^2n{0}dt⃗/||v⃗u_t⃗ma_y||^s
       =∫_[0,1]^2n-1(∑_(v_1,v⃗^M_n)=0_v_2n≠ 0+∑_v_1≠ 0_(v⃗^M_n,v_2n)∈^2n-1+∑_v_1=0, v⃗^M_n≠ 0 _v_2n∈)dt⃗/||v⃗u_t⃗ma_y||^s.

By direct computation  we have

    v⃗u_t⃗ma_y=(v_1y, (v_1t⃗'⃗+v⃗^M_n)m, y^-1(v_1t_2n+v⃗^M_nt⃗'⃗^∗+v_2n)),

where for t⃗=(t_2,…, t_2n-1,t_2n)∈^2n-1, t⃗'⃗=(t_2,…, t_2n-1)∈^2n-2.

First, when (v_1,v⃗^M_n)=0, we have v⃗u_t⃗ma_y= (0,…,0,y^-1v_2n). Thus

    ∫_[0,1]^2n-1∑_(v_1,v⃗^M_n)=0_v_2n≠ 0dt⃗/||v⃗u_t⃗ma_y||^s=∫_[0,1]^2n-1∑_v_2n≠ 0y^s/|v_2n|^sdt⃗=2ζ(s)y^s.


For the second term, when v_1≠ 0, we have

    v⃗u_t⃗ma_y= v_1(y, (t⃗'⃗+v⃗^M_n/v_1)m, y^-1(t_2n+ v⃗^M_nt⃗'⃗^∗+v_2n/v_1)).

Thus

    ∫_[0,1]^2n-1∑_v_1≠ 0_(v⃗^M_n,v_2n)∈^2n-1dt⃗/||v⃗u_t⃗ma_y||^s
       = ∫_[0,1]^2n-2∑_v_1≠ 0_v⃗^M_n∈^2n-2∫_0^1∑_v_2n∈dt_2ndt⃗'⃗/|v_1|^s(y^2+||(t⃗'⃗+v⃗^M_n/v_1)m||^2+ y^-2(t_2n+v⃗^M_nt⃗'⃗^∗+v_2n/v_1)^2)^s/2
       = ∫_[0,1]^2n-2∑_v_1≠ 0∑_v⃗^M_n∈^2n-2∫_dt_2ndt⃗'⃗/|v_1|^s-1(y^2+||(t⃗'⃗+v⃗^M_n/v_1)m||^2+ y^-2t_2n^2)^s/2
       = ∑_v_1≠ 0∫_^2n-1dt⃗/|v_1|^s-2n+1(y^2+||t⃗'⃗m||^2+y^-2t_2n^2)^s/2
       =2ζ(s-2n+1)∫_^2n-1dt⃗/(y^2+||t⃗'⃗m||^2+y^-2t_2n^2)^s/2.

Since m∈(2n-2,) has determinant one, the above integral equals

    2ζ(s-2n+1)∫_^2n-1dt⃗/(y^2+||t⃗'⃗||^2+y^-2t_2n^2)^s/2
       = 2ζ(s-2n+1)y∫_^2n-1dt⃗/(y^2+ t_2^2+⋯+ t_2n-1^2+ t_2n^2)^s/2
       =2ζ(s-2n+1)y^2n-s∫_^2n-1dt⃗/(1+t_2^2+⋯+t_2n-1^2+t_2n^2)^s/2
       =2π^2n-1/2(s-2n+1/2)/(s/2)ζ(s-2n+1)y^2n-s.



Finally, when v_1=0 and v⃗^M_n≠ 0, say v_2n+1-i≠ 0 for some 2≤ i≤ 2n-1, then we have

    v⃗u_t⃗ma_y   =(0, v⃗^M_nm, y^-1(∑_j=2^2n-1δ_jv_2n+1-jt_j+v_2n))
       =v_2n+1-i(0, v⃗^M_nm/v_2n+1-i, y^-1(δ_it_i+∑_2≤ j≤ 2n-1_j≠ iδ_jv_2n+1-jt_j/v_2n+1-i+ v_2n/v_2n+1-i)),

where δ_j equals  -1 for 2≤ j≤ n and 1 for n+1≤ j≤ 2n-1. Thus for such v⃗^M_n we have

    ∫_[0,1]^2n-1∑_v_2n∈dt⃗/||v⃗u_t⃗ma_y||^s
       = ∫_[0,1]^2n-3∫_0^1∑_v_2n∈dt_i∏_2≤ j≤ 2n-1 
     j≠ idt_j/|v_2n+1-i|^s(||v⃗^M_nm/v_2n+1-i||^2+ y^-2(δ_it_i+∑_2≤ j≤ 2n-1 
     j≠ iδ_jv_2n+1-jt_j/v_2n+1-i+ v_2n/v_2n+1-i)^2)^s/2
       =∫_[0,1]^2n-3∫_dt_i∏_2≤ j≤ 2n-1 
     j≠ idt_j/|v_2n+1-i|^s-1(||v⃗^M_nm/v_2n+1-i||^2+ y^-2t_i^2)^s/2
       =y/|v_2n+1-i|^s-1∫_dt_i/(||v⃗^M_nm/v_2n+1-i||^2+ t_i^2)^s/2
       =y√(π)(s-1/2)/(s/2)1/||v⃗^M_nm||^s-1.

Thus by (<ref>) we have

    ∫_[0,1]^2n-1∑_v_1=0, v⃗^M_n≠ 0 _v_2n∈dt⃗/||v⃗u_t⃗a_ym||^s   =y√(π)(s-1/2)/(s/2)∑_v⃗^M_n∈^2n-2{0}1/||v⃗^M_nm||^s-1
       =y√(π)(s-1/2)/(s/2)ζ(s-1)E_n-1(s-1,m),

where E_n-1(s-1,m) is the Eisenstein series on M_n≅(2n-2,) defined as above. Combining these three terms and dividing both sides by ζ(s) we get (<ref>) as claimed.





 §.§ Period computation

With the above constant term formula, we can compute 𝒫_f for  f smooth and spherical as follows.

For any function f on Q_n G_n such that f(a_yk)= ρ(y) with ρ∈𝒞(^+), we have

    𝒫_f(a_yk)=2ρ(y)+1/π i∫_(n)ρ̂(s)ξ(s-2n+1)/ξ(s)y^2n-sds+ 2ρ̂(2n)/ξ(2n).



Again, for n=1 this is well known and we will consider here the case of n≥ 2.
First we note that ξ(s)/ξ(s+1) is analytic in the half plane (s)>1 and |ξ(s)/ξ(s+1)| grows polynomially in s as (s)→±∞. Thus ρ̂(s+1)ξ(s)/ξ(s+1) is analytic in the half plane (s)>1 and |ρ̂(s+1)ξ(s)/ξ(s+1)| decays super polynomially in s as (s)→±∞. Thus it satisfies the conditions in Lemma <ref>.



Moreover, note that M_n() M_n=Y_n-1 and for σ> 2n we have σ-1>2(n-1). Hence applying Lemma <ref> to ρ̂(s+1)ξ(s)/ξ(s+1) and recalling that ω_n-1=2/ξ(2n-2) we get

    1/2π i∫_M_n() M_n∫_(σ)ρ̂(s)ξ(s-1)/ξ(s)E_n-1(s-1,m)dsdμ_n-1(m)
       =1/2π i∫_Y_n-1∫_(σ-1)ρ̂(s+1)ξ(s)/ξ(s+1)E_n-1(s,m)dsdμ_n-1(m)
    
       =2/ξ(2n-2)ρ̂(2n-1)ξ(2n-2)/ξ(2n-1)=2ρ̂(2n-1)/ξ(2n-1).

Thus by (<ref>), Proposition <ref> and the above equation, we have for σ> 2n,

    𝒫_f(a_yk)   =1/π i∫_(σ)ρ̂(s)(y^s+ξ(s-2n+1)/ξ(s)y^2n-s)ds+2ρ̂(2n-1)/ξ(2n-1)y
       =2ρ(y)+1/π i∫_(σ)ρ̂(s)ξ(s-2n+1)/ξ(s)y^2n-sds+2ρ̂(2n-1)/ξ(2n-1)y.

Recall the Riemann Xi function ξ(s)=π^-s/2Γ(s/2)ζ(s) is a meromorphic function with two simple poles at s=0 (with residue -1) and s=1 (with residue 1) (see <cit.>). Moreover, for n≥ 2, ξ(s) has neither zeros nor poles in the half plane (s)≥ n. Thus in this half plane, the function ρ̂(s)ξ(s-2n+1)/ξ(s)y^2n-s has only two simple poles at s=2n-1 (with residue -ρ̂(2n-1)/ξ(2n-1)y) and s=2n (with residue ρ̂(2n)/ξ(2n)). 
Shifting the contour of integration from the line (s)=σ to the line (s)=n (picking up the contribution of the two poles) we get

    𝒫_f(a_yk)   =2ρ(y)+1/π i∫_(n)ρ̂(s)ξ(s-2n+1)/ξ(s)y^2n-sds+2ρ̂(2n)/ξ(2n)-2ρ̂(2n-1)/ξ(2n-1)y+2ρ̂(2n-1)/ξ(2n-1)y
       =2ρ(y)+1/π i∫_(n)ρ̂(s)ξ(s-2n+1)/ξ(s)y^2n-sds+ 2ρ̂(2n)/ξ(2n).














§ RAISING OPERATORS

When the function f is not spherical, even if it is of a rather simple form f(a_yk)=ρ(y)ϕ(k),  we don't have such a nice formula for the constant term that will allow us to compute the period _f as we did for the spherical case. Instead of taking this direct approach, borrowing ideas from <cit.>,  we will start with the formula for _f for spherical f and apply raising operators to obtain similar formulas for functions of the form f(a_yk)=ρ(y)ϕ(k) with ϕ of different K_n-types. Before we can do this we need some more background on L^2(K_n-1 K_n) and how it decomposes into irreducible K_n-representations. We will assume here that n≥ 2 and note that for n=1 we have that K_0 is trivial, K_1=(2) and the decomposition is the standard Fourier decomposition.



 §.§ Root-space decomposition

Let _n be the Lie algebra of G_n and _n be the Lie algebra of K_n. Let _n,=_n⊗_ and _n,:=_n⊗_ be their complexifications respectively. Explicitly,

    _n={[ A B; C D ]∈𝔰𝔩_2n() | A^tJ_n+ J_nD= 0, C^tJ_n= J_nC, B^tJ_n= J_n B},

and


    _n= {[          A          B;       -B^t -J_nA^tJ_n ]∈𝔰𝔩_2n() | A^t+ A=0, B^tJ_n=J_nB}.

Let E_ℓ,j be the 2n× 2n matrix with one in the (ℓ,j)^th entry and zero elsewhere. Let  be the real vector space spanned by the set

    {E_2n+1-j,j-E_j,2n+1-j | 1≤ j≤ n}.

Note that  is the Lie algebra of a maximal torus of K_n. Let _ be the complexification of  and for each 1≤ j≤ n, let

    H_j:=i(E_2n+1-j,j-E_j,2n+1-j)∈_

and let ε_j: _→ be the linear functional on _ characterized by ε_ℓ(H_j)=δ_ℓ j, where δ_ℓ j is the Kronecker symbol. Then there is a root-space decomposition of _n, with respect to _:

    _n,=_⊕⊕_α∈Φ(_n,,𝔥)_α

with Φ=Φ(_n,,_)={ε_ℓ-ε_j | 1≤ℓ≠ j≤ n} the corresponding set of roots, and for any α∈Φ,

    _α:={X∈_n, | [H, X]=α(H)X for any H∈_}

is the corresponding root-space. Fix a choice of simple roots

    {ε_j-ε_j+1 | 1≤ j< n}

such that the set of positive roots is given by Φ^+={ε_ℓ-ε_j | 1≤ℓ< j≤ n}. For any finite-dimensional _n,-module V and any linear functional λ: _→, we say a nonzero vector v∈ V is of K_n-weight β if H· v=β(H)v for any H∈_ and we say v∈ V is a highest K_n-weight vector if X· v=0 for any X∈_α with α∈Φ^+.



 §.§ Induced representations

Let P_n= U_nM_nA_n, Q_n=U_nM_n and K_n-1=M_n∩ K_n as before. Let π denote the right regular action of G_n on functions on Q_n G_n. For each parameter s∈, the induced representation, I^s, is the representation of G_n consisting of measurable functions f: Q_n G_n→ satisfying

    f(Q_na_yg)= y^sf(Q_ng) for μ_n-a.e. g∈ G_n and for any a_y∈ A_n,

with G_n acting on I^s via the right regular action.




By restricting to K_n, each I^s can be viewed as a representation of K_n. Moreover, for each s∈, there is a natural isomorphism between I^s and L^2(K_n-1 K_n) as representations of K_n, given by the restriction map sending f∈ I^s to f|_K_n-1 K_n.

Recall the identification between Q_n G_n and ^2n sending Q_ng to x⃗(g)∈^2n with x⃗(g)=(x_1,…,x_2n)=e⃗_2ng the last row of g. For 1≤ j≤ n, let z_j:=x_j+ ix_2n+1-j and z̅_j:=x_j- ix_2n+1-j. Thus functions on Q_n G_n can be realized as functions in coordinates (z_j,z̅_̅j̅). For any pair of nonnegative integers (p,q), we say a polynomial P in (z_j,z̅_̅j̅)_1≤ j≤ n is bihomogeneous of degree (p,q) if

    P(λ z_1,…,λ z_n,λ̅z̅_1,…,λ̅z̅_n)= λ^pλ̅^qP(z_1,…,z_n,z̅_1,…,z̅_n) for any λ, z_i∈.

We say a polynomial is harmonic if it is annihilated by the Euclidean Laplacian

    Δ:=4∑_j=1^n∂^2/∂ z_j∂z̅_j,

where ∂/∂ z_j:=1/2(∂/∂ x_j-i∂/∂ x_2n+1-j) and ∂/∂z̅_j:=1/2(∂/∂ x_j+i∂/∂ x_2n+1-j). For each pair of nonnegative integers (p,q), let H^p,q be the space of bihomogeneous harmonic polynomials of degree (p,q) and ℋ^p,q:={P|_K_n-1 K_n |    P∈ H^p,q}. As a function space, L^2(K_n-1 K_n) decomposes as

    L^2(K_n-1 K_n)=⊕̂_̂(̂p̂,̂q̂)̂∈̂^̂2̂ℋ^p,q,

where ⊕̂ denotes the Hilbert direct sum. Moreover, one can check that each ℋ^p,q is invariant under the action of K_n. Thus this is a decomposition as K_n-representations. Correspondingly, for each parameter s∈, let ℋ^s,p,q⊂ I^s be the preimage of ℋ^p,q under the restriction map from I^s to L^2(K_n-1 K_n) described above. Then I^s has a corresponding decomposition (as K_n-representations)

    I^s=⊕̂_̂(̂p̂,̂q̂)̂∈̂^̂2̂ℋ^s,p,q.

Moreover, there is a natural _n,-module structure on I^s_∞:=⊕_(p,q)∈^2ℋ^s,p,q by taking Lie derivatives: For any f∈ I_∞^s and any X∈_n, the Lie derivative π(X), is defined by


    (π(X)f)(g):=d/dtf(gexp(tX))|_t=0.

This defines a _n-module structure on I_∞^s and it induces a _n,-module structure on I_∞^s via complexification: For any X_1+iX_2∈_n, with X_1,X_2∈_n, define π(X_1+iX_2):=π(X_1)+iπ(X_2). Similarly, there is a _n,-module structure on each ℋ^s,p,q.

For each (s,p,q) as above, define h_s,p,q by the formula

    h_s,p,q(z_j,z̅_j):=z_1^pz̅_n^q/(∑_j=1^nz_jz̅_j)^s+p+q/2.

One can check that h_s,p,q is an element in ℋ^s,p,q and is of K_n-weight pε_1-qε_n. Moreover, using Weyl dimension formula for compact connected Lie groups (<cit.>) we can compute that the highest weight _n,-module of weight pε_1-qε_n is of dimension

    d_p,q:=(n+p-2)!(n+q-2)!(n+p+q-1)!/(n-1)!(n-2)!p!q!

and it is exactly the dimension of ℋ^s,p,q (see <cit.>). Thus ℋ^s,p,q is a highest weight _n,-module with the highest _n,-weight vector given by h_s,p,q.


 §.§ Raising operators

Define the raising operators ℛ^(2,0),ℛ^(0,2)∈π(_n,) by

    ℛ^(2,0):=π((E_1,1-E_2n,2n)+i(E_1,2n+E_2n,1))
 and

    ℛ^(0,2):=π((E_n,n-E_n+1,n+1)-i(E_n,n+1+E_n+1,n)).


We will apply ℛ^(2,0) and ℛ^(0,2) to functions in I^s_∞. By examining commutator relations, one can check that ℛ^(2,0) and ℛ^(0,2) commute (as left-invariant differential operators), and that ℛ^(2,0) (resp. ℛ^(0,2)) sends the highest K_n-weight vector h_s,p,q to a multiple of the highest K_n-weight vector h_s,p+2,q (resp. h_s,p,q+2). Using the coordinates (z_j,z̅_j) on Q_n G_n as above, by direct computation we have

    ℛ^(2,0)=2z_1∂/∂z̅_1  and  ℛ^(0,2)=2z̅_n∂/∂ z_n.




Thus for any s∈ and any nonnegative pair (p,q), we have

    ℛ^(2,0)h_s,p,q=-(s+p+q)h_s,p+2,q

and

    ℛ^(0,2)h_s,p,q=-(s+p+q)h_s,p,q+2.



 §.§ Parity operator

Using these raising operators and starting from a spherical function we can get functions of type (p,q) for any even integers (p,q). In order to be able to represent all even functions we also need functions of type (p,q) with p and q both odd. For this reason we introduce the following auxiliary left-invariant differential operator ℛ∈π(_n,) defined by

    ℛ:=π((E_1,n+E_n,1-E_n+1,2n-E_2n,n+1)+i(E_1,n+1+E_n+1,1+E_n,2n+ E_2n,n)).

Using the  (z_j,z̅_j) coordinates on Q_n G_n this operator is given by

    ℛ=2z_n∂/∂z̅_1+2z_1∂/∂z̅_n.




By examining commutator relations we see that ℛ sends a vector of K_n-weight β to a vector of K_n-weight β+ ε_1+ε_n, but ℛ is not a raising operator in the sense that it does not send a highest K_n-weight vector in I^s_∞ to another highest K_n-weight vector in I^s_∞. Nevertheless, using (<ref>) by direct computation we have








    ℛh_s,0,2(z_j,z̅_j)   =-2(s+2)z_1z_nz̅_n^2/(∑_j=1^nz_jz̅_j)^s+4/2+4z_1z̅_n/(∑_j=1^nz_jz̅_j)^s+2/2
       =-2(s+2)(z_1z_nz̅_n^2-2/n+2(∑_j=1^nz_jz̅_j)z_1z̅_n)/(∑_j=1^nz_jz̅_j)^s+4/2+4(n-s)/n+2z_1z̅_n/(∑_j=1^nz_jz̅_j)^s+2/2.

In other words, using the polar coordinates (a_y,k) on Q_n G_n, we can write this as

    ℛh_s,0,2(a_yk)= (s+2)φ_s(a_y)ψ_2,2(k)+(n-s)φ_s(a_y)ψ_1,1(k),

where
ψ_2,2(z_j,z̅_j)=-2(z_1z_nz̅_n^2-2/n+2(∑_j=1^nz_jz̅_j)z_1z̅_n)|_K_n-1 K_n∈ℋ^2,2 and ψ_1,1(z_j,z̅_j)=4z_1z̅_n/n+2|_K_n-1 K_n∈ℋ^1,1.

Using this operator allows us to pass from a function of type (0,2) (which can be obtained from a spherical function via the raising operator) to a function of type (1,1). Then, starting from ψ_1,1 and applying the raising operators we can construct functions of type (p,q) also for odd p and q.




§ PERIOD FORMULA FOR NON-SPHERICAL FUNCTIONS

We can now use the raising operators to boot-strap the spherical period formula to obtain period formulas for other K_n-types.
For any nonnegative even integer m, we define

    P_m(s):= {[                       1                  if m=0; ∏_j=0^m-2/22n-s+2j/s+2j                 if m>0, ].

and Z_m(s)=P_m(s)ξ(s-2n+1)/ξ(s). Note that, using the functional equation ξ(s)=ξ(1-s) (<cit.>) and the definition of P_m(s), we see that Z_m(s) satisfies the functional equation Z_m(2n-s)Z_m(s)= 1. In particular, when (s)=n we have

    |Z_m(s)|^2=Z_m(s)Z_m(s)=Z_m(s̅)Z_m(s)= Z_m(2n-s)Z_m(s)=1.


For any pair of nonnegative integers (p,q) with the same parity, let 𝒜^p,q denote the family of functions f∈ L^2(Q_n G_n) of the form f(a_yk)=ρ(y)ϕ(k) with ρ∈𝒞(^+) and ϕ∈ℋ^p,q.
By using the raising operators we can obtain the following period formula for f∈^p,q.

Let (p,q) be a pair positive integers with the same parity. For any f=ρϕ∈𝒜^p,q we have

    𝒫_f(a_yk)=2ρ(y)ϕ(k)+(-1)^p/π i∫_(n)ρ̂(s)Z_p+q(s)y^2n-sdsϕ(k).




When n=1 we have a similar formula for functions of the form f(a_yk_θ)=ρ(y)e^i mθ with m nonzero and even given by

    _f(a_yk_θ)=2ρ(y)e^i m θ+1/π i∫_(1)ρ̂(s)Z_|m|(s)y^2-sds e^i mθ,

where k_θ=[  cosθ  sinθ; -sinθ  cosθ ].



 §.§ Preliminary lemmas

We first prove the following two preliminary lemmas showing that having  (<ref>) for one function implies that (<ref>) holds for many functions.

Let (p,q) be as in Proposition <ref>. Suppose that there exists some ρ∈𝒞(^+) and nonzero ψ∈ℋ^p,q such that  (<ref>) holds for f=ρψ, then (<ref>) holds for f=ρϕ for any ϕ∈ℋ^p,q.


Since ℋ^p,q is an irreducible _n,-module, for any ϕ∈ℋ^p,q there is some left-invariant differential operator 𝒟 generated by π(_n,) such that 𝒟ψ=ϕ. Recall the function φ_s: G_n→ defined in (<ref>). Since 𝒟 is generated by π(_n,), 𝒟 acts trivially on φ_s. Moreover, 𝒟 commutes with the left regular action of G_n on φ_sψ. Thus 𝒟φ_s(γ g)ψ(γ g)= φ_s(γ g)𝒟ψ(γ g)=φ_s(γ g)ϕ(γ g) for any γ∈_n and g∈ G_n. Here as before we view ψ,ϕ∈ℋ^p,q⊂ L^2(K_n-1 K_n) as left U_nA_nM_n-invariant functions on G_n. Hence using (<ref>) and applying 𝒟 to Θ_ρψ we have for any σ> 2n,

    𝒟Θ_ρψ(g)   =1/2π i∫_(σ)ρ̂(s)∑_γ∈_P_n_n𝒟φ_s(γ g)ψ(γ g)ds
       =1/2π i∫_(σ)ρ̂(s)E_n(s,g,ϕ)ds=Θ_ρϕ(g).

We thus have

    𝒟𝒫_ρψ(a_yk)   =∫_M_n() M_n∫_U_n() U_n𝒟Θ_ρψ(u_t⃗ma_yk)dt⃗dμ_n-1(m)
       =∫_M_n() M_n∫_U_n() U_nΘ_ρϕ(u_t⃗ma_yk)dt⃗dμ_n-1(m)=𝒫_ρϕ(a_yk).

Since we assume  (<ref>) holds for ρψ, we have

    𝒫_ρψ(a_yk)=2ρ(y)ψ(k)+(-1)^p/π i∫_(n)ρ̂(s)Z_p+q(s)y^2n-sdsψ(k),

and applying 𝒟 to both sides gives

    𝒫_ρϕ(a_yk)   =2ρ(y)𝒟ψ(k)+(-1)^p/π i∫_(n)ρ̂(s)Z_p+q(s)y^2n-sds𝒟ψ(k)
       =2ρ(y)ϕ(k)+(-1)^p/π i∫_(n)ρ̂(s)Z_p+q(s)y^2n-sdsϕ(k).




Fix a pair of positive integers (p_0,q_0) with the same parity. Suppose (<ref>) holds for all functions f∈𝒜^p_0,q_0, then for any pair of positive integers (p,q) such that p≥ p_0, q≥ q_0, p≡ p_0 (mod 2) and q≡ q_0 (mod 2),  (<ref>) holds for any functions f∈𝒜^p,q.


For any such (p,q), let h_p,q:=h_s,p,q|_K_n-1 K_n be the unique (up to scalars) highest K_n-weight vector in ℋ^p,q. In view of Lemma <ref>, it suffices to show that (<ref>) holds for any ρ h_p,q with ρ∈𝒞(^+). Fix σ>2n and for any y>0 define

    υ(y):=1/2π i∫_(σ)ρ̂(s)/(s+p_0+q_0)(s+p_0+q_0+2)⋯(s+p+q-2)y^sds.

Then υ∈(^+) with Mellin transform

    υ̂(s)= ρ̂(s)/(s+p_0+q_0)(s+p_0+q_0+2)⋯(s+p+q-2)

for (s)>1. Applying Proposition <ref> to υ h_p_0,q_0 and using Mellin inversion for v, we have

    𝒫_υ h_p_0,q_0(a_yk)=1/π i∫_(n)υ̂(s)(y^s+(-1)^p_0Z_p_0+q_0(s)y^2n-s)dsh_p_0,q_0(k).





Let 𝒟_p,q:=(-1)^(p+q-p_0-q_0)/2(ℛ^(2,0))^(p-p_0)/2(ℛ^(0,2))^(q-q_0)/2. Note that φ_sh_p_0,q_0=h_s,p_0,q_0 where φ_s is as in (<ref>). Thus (<ref>), (<ref>) and commutativity of ℛ^(2,0) and ℛ^(0,2) imply that

    𝒟_p,qφ_sh_p_0,q_0= (s+p_0+q_0)(s+p_0+q_0+2)⋯ (s+p+q-2)φ_sh_p,q.

Moreover, since 𝒟_p,q commutes with the left regular action of G_n on φ_sh_p_0,q_0, using (<ref>), (<ref>) and (<ref>) we have for any σ>2n

    (𝒟_p,qΘ_υ h_p_0,q_0)(g)   =1/2π i∫_(σ)υ̂(s)∑_γ∈_P_n_n𝒟_p,qφ_s(γ g)h_p_0,q_0(γ g)ds
    
       =1/2π i∫_(σ)ρ̂(s)E_n(s,g,h_p,q)ds=Θ_ρ h_p,q(g).

Thus applying 𝒟_p,q to the left-hand side of (<ref>) we get

    𝒟_p,q𝒫_υ h_p_0,q_0(a_yk)   =∫_M_n() M_n∫_U_n() U_n𝒟_p,qΘ_υ h_p_0,q_0(u_t⃗ma_yk)dt⃗dμ_n-1(m)
       =∫_M_n() M_n∫_U_n() U_nΘ_ρ h_p,q(u_t⃗ma_yk)dt⃗dμ_n-1(m)=𝒫_ρ h_p,q(a_yk).

Similarly, using (<ref>) and (<ref>), after applying 𝒟_p,q the right-hand side of (<ref>) becomes

    1/π i∫_(n)ρ̂(s)(y^s+(-1)^p_0Z_p+q(s)y^2n-s)dsh_p,q(k)
       =2ρ(y)h_p,q(k)+(-1)^p/π i∫_(n)ρ̂(s)Z_p+q(s)y^2n-sdsh_p,q(k),












completing the proof.



 §.§ Proof of Proposition <ref> for p,q even.

Starting from Proposition <ref> for spherical functions and applying ^(0,2) and ^(2,0) respectively, note that the constant term in (<ref>) is killed by these two differential operators, to see that (<ref>)
holds for any  ρ h_0,2∈^0,2 and ρ h_2,0∈^2,0 with ρ∈(^+). From this using Lemma <ref> and Lemma <ref> we see that Proposition <ref> holds for any even p and q.









 §.§ Proof of Proposition <ref> for p,q odd

Let ℛ denote the left invariant auxiliary operator defined in (<ref>), and recall that

    ℛh_s,0,2(a_yk)= (s+2)φ_s(a_y)ψ_2,2(k)+(n-s)φ_s(a_y)ψ_1,1(k),

with ψ_2,2∈ℋ^2,2 and ψ_1,1∈ℋ^1,1 given in (<ref>).
Now, in view of the Lemma <ref> and Lemma  <ref> to prove Proposition <ref> for all p,q odd, it is enough to show that (<ref>) holds for f=ρψ_1,1∈^1,1 that we show as follows.


Let ψ_1,1∈ℋ^1,1 be as above. For any f∈𝒜^1,1 of the form f=ρψ_1,1 with ρ∈𝒞(^+) we have

    𝒫_f(a_yk)=2ρ(y)ψ_1,1(k)-1/π i∫_(n)ρ̂(s)Z_2(s)y^2n-sdsψ_1,1(k).













For f=ρψ_1,1, fix σ> 2n and for any y>0 define

    υ(y):=1/2π i∫_(σ)ρ̂(s)/n-sy^sds.

As before υ is independent of the choice of σ>2n and for (s)>n we have

    υ̂(s)=ρ̂(s)/n-s.

We note that if ρ̂(n)≠ 0, then ρ̂(s)/n-s has a pole at s=n. Thus υ h_0,2 is not necessarily contained in 𝒜^0,2. However, since υ̂(s) is analytic in the half plane (s)>n and satisfies the Mellin inversion formula v(y)=1/2π i∫_(σ)υ̂(s)y^sds for σ>2n, using the same arguments[The only difference is that in Proposition <ref>, instead of shifting the contour of integration from (s)=σ to (s)=n, we shift the contour from (s)=σ to (s)=η.] as in Proposition <ref>, Lemma <ref> and Lemma <ref>, one can deduce the formula

    𝒫_υ h_0,2(a_yk)=1/π i∫_(η)υ̂(s)(y^s+ Z_2(s)y^2n-s)dsh_0,2(k)

for  η∈(n,2n-1). With similar computations as in Lemma <ref> and <ref>, using (<ref>), (<ref>), and applying ℛ to the left-hand side of (<ref>) we get

    ℛ𝒫_υ h_0,2(a_yk)= 𝒫_υ_1 ψ_2,2(a_yk)+ 𝒫_ρψ_1,1(a_yk),

where υ_1(y):=1/2π i∫_(σ)υ̂(s)(s+2)y^sds for y>0 and σ> 2n. We note that for (s)> n, υ̂_1(s)=υ̂(s)(s+2) and hence for η∈ (n,2n-1) we have

    𝒫_υ_1 ψ_2,2(a_yk)=1/π i∫_(η)υ̂(s)(s+2)(y^s+ Z_4(s)y^2n-s)dsψ_2,2(k).

On the other hand, using (<ref>) and applying ℛ to the right-hand side of (<ref>) we get

    1/π i∫_(η)υ̂(s)(ℛy^sh_0,2(k)+ Z_2(s)ℛy^2n-sh_0,2(k))ds
       =1/π i∫_(η)υ̂(s)((s+2)φ_s(a_y)ψ_2,2(k)+(n-s)φ_s(a_y)ψ_1,1(k)
       + Z_2(s)((2n-s+2)φ_2n-s(a_y)ψ_2,2(k)+(s-n)φ_2n-s(a_y)ψ_1,1(k)))ds
       =𝒫_υ_1ψ_2,2(a_yk)+1/π i∫_(η)ρ̂(s)(y^s-Z_2(s)y^2n-s)dsψ_1,1(k)
       =𝒫_υ_1ψ_2,2(a_yk)+1/π i∫_(n)ρ̂(s)(y^s-Z_2(s)y^2n-s)dsψ_1,1(k)
       =𝒫_υ_1ψ_2,2(a_yk)+2ρ(y)ψ_1,1(k)-1/π i∫_(n)ρ̂(s)Z_2(s)y^2n-sdsψ_1,1(k),

where for the second equality we used the relations (<ref>) and υ̂_1(s)=υ̂(s)(s+2) for (s)=η> n, and equation (<ref>), for the third equality we shift the contour of integration from (s)=η to (s)=n (noting that ρ̂(s)(y^s-Z_2(s)y^2n-s) is analytic on the strip n≤(s)≤η), and for the last equality we used Mellin inversion formula for ρ. Comparing (<ref>) and the above equations completes the proof.



























 §.§ The isometry

Before completing the proof of the second moment formula, we need to define the isometry of L^2_ even(^2n) which is the same as L^2_ even(Q_n G_n).
For any even bounded compactly supported f on Q_n G_n, we define ι(f)
by

    ι(f)=1/2𝒫_f-f-1/2ζ(2n)∫_^2nf(x⃗)dx⃗.

We now show that this is indeed an isometry.

The map ι:L^2_ even(Q_n G_n)→ L^2_ even(Q_n G_n) is an isometry.


We first show that for any (p,q) with same parity and any f∈^p,q we have ι(f)_2=f_2.
Indeed, writing f(a_yk)=ρ(y)ϕ(k)  we have

    ||f||_2^2   =2π^n/Γ(n)∫_0^∞|ρ(y)|^2dy/y^2n+1ϕ_2^2
       =2π^n/Γ(n)1/2π i∫_(n)|ρ̂(s)|^2ds ϕ_2^2,

where the first line is just integration in polar coordinates and the second is Plancherel's theorem.
Now by Proposition <ref>, Proposition <ref> and the relation (<ref>)  we have

    ι(f)(a_yk)   =(-1)^p/2π i∫_(n)ρ̂(s)Z_p+q(s)y^2n-sdsϕ(k)
       =(-1)^p/2π i∫_(n)ρ̂(2n-s)Z_p+q(2n-s)y^sdsϕ(k)=v(y)ϕ(k),

where v(y) has Mellin transform v̂(s)=(-1)^pρ̂(2n-s)Z_p+q(2n-s) for 1<(s)<2n-1. In particular, by (<ref>) for (s)=n we have that
|v̂(s)|=|ρ̂(2n-s)| and hence

    ι(f)^2   =2π^n/(n)1/2π i∫_(n)|ρ̂(2n-s)|^2dsϕ_2^2
       =2π^n/(n)1/2π i∫_(n)|ρ̂(s)|^2dsϕ_2^2=f^2.




























Now since the different spaces ^p,q are orthogonal, and any even smooth compactly supported functions can be decomposed as
f=∑_p≡ q2 f_p,q with f_p,q∈^p,q, we get that ι(f)_2=f_2 for all even, smooth, compactly supported functions. Since even, smooth, compactly supported functions are dense in L^2_ even(^2n) this concludes the proof.

















 §.§ Proof of second moment formula

Let f be an even, compactly supported function on ^2n and let f̃ be the corresponding function on Q_n G_n given by f̃(g)=f(e⃗_2ng).
Using Proposition <ref>, Proposition <ref> and the relation (<ref>) we see that

    F_f_2^2=Θ_f̃_2^2=1/ζ(2n)∫_^2nf(x⃗)𝒫_f(x⃗)dx⃗,

Now from the definition of the isometry ι we have

    _f=2ι(f)+2f+1/ζ(2n)∫_^2nf(x⃗)dx⃗,

and plugging this in we get that

    F_f_2^2   =1/ζ(2n)∫_^2nf(x⃗)(2ι(f)(x⃗)+2f(x⃗)+1/ζ(2n)∫_^2nf(x⃗)dx⃗)dx⃗
       =|1/ζ(2n)∫_^2nf(x⃗)dx⃗|^2+2/ζ(2n)∫_^2n|f(x⃗)|^2dx⃗+2/ζ(2n)∫_^2nf(x⃗)ι(f)(x⃗)dx⃗,

thus concluding the proof.
 

 
For an even bounded non-negative function f∈ L^2(^2n) that is not compactly supported, we can take a sequence f_j(x⃗)=f(x⃗)χ_B_j(x⃗) with B_j⊆^2n the ball of radius j. Then f_j monotonously converges to f pointwise, as well as in L^2. Moreover |F_f_j(Λ)|^2 also monotonously converges to |F_f(Λ)|^2, hence, by monotone convergence ∫_Y_n|F_f(Λ)|^2dμ_n=lim_j→∞∫_Y_n|F_f_j(Λ)|^2dμ_n giving the same formula for f.



Given two even compactly supported functions, f,g by computing the mean square of F_f+g  the second moment formula is equivalent to the following inner product formula

    F_f,F_g_Y_n=f,11,g/ζ(2n)^2+2/ζ(2n)(f,g+ι(f),g),

where ·,· and  ·,·_Y_n are the corresponding inner products on L^2(^2n) and L^2(Y_n,μ_n).






§ APPLICATIONS TO COUNTING


In this section, we apply our second moment formula to get results on lattice point counting problems for a generic symplectic lattice.


 §.§ Mean square bound

Our first simple application gives a mean square bound for the discrepancy. For the primitive lattice points this is almost immediate while for all regular lattice points we follow the standard argument converting results from primitive lattice points to all lattice points.









For any Borel set B⊆^2n let χ_B be the characteristic function of B and let f(x⃗)=χ_B(x⃗)+χ_B(-x⃗)/2∈ L^2_ even(^2n) its even part so that ∫_^2nf(x⃗)dx⃗=(B) and f_2^2≤(B). Note that for any Λ∈ Y_n, F_f(Λ)=F_χ_B(Λ)=#(Λ_ pr∩ B) and by (<ref>)

    ∫_Y_nF_f(Λ)dμ_n(Λ)=1/ζ(2n)∫_^2nf(x⃗)dx⃗=vol(B)/ζ(2n).

Thus by Theorem <ref> and Cauchy-Schwartz we have

    ∫_Y_n|#(Λ_ pr∩ B)-(B)/ζ(2n)|^2dμ_n(Λ)   =∫_Y_n|F_f(Λ)-vol(B)/ζ(2n)|^2dμ_n(Λ)
       =∫_Y_n|F_f(Λ)|^2dμ_n(Λ)-(vol(B)/ζ(2n))^2
       =2/ζ(2n)∫_^2n(|f(x⃗)|^2+f(x⃗)ι(f)(x⃗))dx⃗
    
    
       ≤2/ζ(2n)(f_2^2+f_2ι(f)_2)
       =4f_2^2/ζ(2n)≤4(B)/ζ(2n),




which gives (<ref>) after multiplying both sides by ζ(2n)^2/(B)^2.





Next for the regular lattice point counting problem, consider the dilated functions f^k(x⃗)=f(k x⃗) and χ_B^k(x⃗)=χ_B(kx⃗).
Since, by our assumption, 0∉B we have that

    #(Λ∩ B)=∑_k=1^∞#(k Λ_ pr∩ B)=∑_k=1^∞ F_f^k(Λ).

 Integrating over Y_n and using  (<ref>) we get that

    ∫_Y_n#(Λ∩ B)dμ_n(Λ)   =∑_k=1^∞∫_Y_nF_f^k(Λ)dμ_n(Λ)
    =1/ζ(2n)∑_k=1^∞∫_^2nf^k(x⃗)dx⃗
       =1/ζ(2n)∑_k=1^∞1/k^2n∫_^2nf(x⃗)dx⃗=(B).

Next, using the moment formula in the form of (<ref>), gives

    ∫_Y_n|#(Λ∩ B)|^2dμ_n(Λ)   =∑_k,l⟨ F_f^k,F_f^l⟩_Y_n
       = ∑_k,lf^k,11,f^l/ζ(2n)^2+∑_k,l2/ζ(2n)(f^k,f^l+ι(f^k),f^l)
       =(B)^2+2/ζ(2n)∑_k,l(f^k,f^l+ι(f^k),f^l),

and hence

    ∫_Y_n|#(Λ∩ B)-(B)|^2dμ_n(Λ)   =∫_Y_n|#(Λ∩ B)|^2dμ_n(Λ)-(B)^2 
       =∑_k,l2/ζ(2n)(f^k,f^l+ι(f^k),f^l)
       ≤2/ζ(2n)∑_k,l(f^k_2f^l_2+ι(f^k)_2f^l_2)
       =4 f_2^2/ζ(2n)∑_k,l1/k^nl^n≤4ζ(n)^2(B)/ζ(2n).

 Dividing both sides by (B)^2 concludes the proof of (<ref>).



 §.§ Schmidt's argument

We can now use the above mean square estimate together with Schmidt's argument from <cit.> to prove Theorem <ref>, which  follows from the following by taking ψ(x)=c/x^2 below with an appropriate choice of constant c.

Let  be a linearly ordered  family of Borel sets in ^2n. Let ψ be a positive, non-increasing function such that e^tψ(t) is eventually non-decreasing and ∫_1^∞ψ(t)dt<∞. Then for μ_n-a.e. Λ∈ Y_n there is C_Λ such that for all B∈ with (B)>C_Λ

    |#(Λ_ pr∩ B)-(B)/ζ(2n)|≤√((B))log((B)) /ψ^1/2(log(B)),

and

    |#(Λ∩ B)-(B)|≤√((B))log((B)) /ψ^1/2(log(B)).



The arguments are identical to the ones given in <cit.>, and we include the details for the readers' convenience.
Since the proofs for the primitive lattice point counting and for the regular lattice point counting follow from the exact same argument, we will give the details only for first one.

First note that if the set of volumes of sets in  is bounded the statement holds vacuously by taking C_Λ larger than the volume of any set in , so we can assume there are arbitrarily large volumes. With this assumption, by  <cit.>, we can assume without loss of generality (after perhaps adding more sets to ) that {(B) | B∈}=^+. Thus for any positive integer N, there exists some B_N∈ with (B_N)= N. For any Λ∈ Y_n, and N≥ 1 we denote

    S_N(Λ)=#(Λ_ pr∩ B_N)-N/ζ(2n)

and for any 1≤ N_1< N_2

    _N_1S_N_2(Λ)=#(Λ_ pr∩ (B_N_2 B_N_1))-N_2-N_1/ζ(2n).

For any integer T≥ 3 we denote by  _T the set of all pairs of integers N_1, N_2 of the form 0≤ N_1< N_2≤ 2^T, N_1=ℓ2^t and N_2=(ℓ+1)2^t, for integers ℓ and t≥ 0. Applying (<ref>) to the sets B_N_2∖ B_N_1 and repeating the exact same arguments as in
 <cit.>  we get that for any T≥ 3

    ∑_(N_1,N_2)∈𝒦_T∫_Y_n|_N_1S_N_2(Λ)|^2dμ_n(Λ)≤ C_n(T+1)2^T,

where C_n=4/ζ(2n). Next, let _T⊆ Y_n denote the set of all lattices Λ∈ Y_n for which

    ∑_(N_1,N_2)∈𝒦_T|_N_1S_N_2(Λ)|^2> (T+1)2^T/800ψ(log(2)(T-1)).

Then (<ref>) implies that

    μ_n(_T)<  800C_nψ(log(2)(T-1)).

Consider the limsup set

    _∞=lim sup_T→∞_T:=⋂_j≥ 3⋃_T≥ j_T.

Since the right-hand side of (<ref>) is summable we have that μ_n(_∞)=0, and we will take its complement Y_n∖_∞ to be the full measure set of lattices for which the discrepancy is small.


Now, note that for N≤ 2^T, the interval [0,N) can be expressed as a disjoint union of at most T intervals of the form [N_1,N_2) with (N_1,N_2)∈𝒦_T.
We can thus write

    S_N(Λ)=∑_[N_1,N_2)∈ℐ_N_1S_N_2(Λ),

where ℐ is a set consisting of at most T intervals of the form [N_1,N_2) with (N_1,N_2)∈𝒦_T. Using Cauchy-Schwartz and (<ref>) we have for any Λ∉_T and any N<2^T

|S_N(Λ)|^2≤(T+1)^22^T/800ψ(log2(T-1)).

Now, for any Λ∉_∞ there is some T_Λ such that for all T≥ T_Λ we have that Λ∉_T and hence   |S_N(Λ)|^2≤(T+1)^22^T/100ψ((T-1)log 2)  for all N<2^T.

Now, for any Λ∉_∞ let C_Λ= max{2^T_Λ+1,N_0} with N_0 sufficiently large that for all N≥ N_0 we have

    √(N+1)log(N+1)/2ψ^1/2(log(N+1))+1≤√(N)log(N)/ψ^1/2(log(N)),

where we used that Nψ(log(N)) is eventually non-decreasing to make sure such N_0 exists.
Then, for any integer N> C_Λ-1, choose integer T such that 2^T-1≤ N<2^T. In particular we have that T≥ T_Λ and N<2^T so,

    |S_N(Λ)|^2   ≤(T+1)^2 2^T/800 ψ(log 2(T-1))
       ≤(log N/log 2+2)^2 N/400ψ(log(N))<Nlog^2 N/4ψ(log(N)) ,

where we used that (log N/log 2+2)≤ 10log(N) for all N≥ 2.
We have thus verified that for all N>C_Λ-1 we have |S_N(Λ)|≤√(N)log(N)/2ψ^1/2(log(N)).

Next, for any set B∈ with (B)> C_Λ, there exists an integer N> C_Λ-1 such that B_N⊆ B⊆ B_N+1.
We can interpolate, to bound

    |#(Λ_ pr∩ B)-(B)/ζ(2n)|≤max{|S_N(Λ)|, |S_N+1(Λ)|}+1,

and since N,N+1≥ C_Λ-1 we can bound

    |#(Λ_ pr∩ B)-(B)/ζ(2n)|≤√(N+1)log(N+1)/2ψ^1/2(log(N+1))+1≤√((B))log((B))/ψ^1/2(log((B))) ,

where we used (<ref>) recalling that N≥ N_0.

The same proof with the obvious modifications give the same result for the general lattice point counting problem.





 §.§ Dilations

We now want to apply our result for the special case where our family is given by a dilation of a fixed set B⊆^2n.

Write B=_j=1^k B_j with B_j=B_j^+∖ B_j^- and note that any dilation is of the form tB=_j=1^k tB_j and that tB_j=tB_j^+∖ tB_j^-.
Moreover,

    #(Λ∩ tB)=∑_j=1^k#(Λ∩ tB_j^+)-∑_j=1^k#(Λ∩ tB_j^-),

and similarly for the primitive lattice points.
Considering the finitely many linearly ordered families _j^±={tB_j^±:t∈^+} and applying Theorem <ref> to each one with ψ(t)=64n^4k^2/t^2, we get that, for each j, for μ_n-a.e. Λ∈ Y_n there is C_Λ,j^± such that for all t>C_Λ,j^±

    | #(Λ∩ tB_j^±)-(tB_j^±) |≤t^nlog^2(t)/2k.

The intersection of these finitely many full measure sets is still of full measure and taking C_Λ=max{ C_Λ,j^±: 1≤ j≤ k} we get that for all t>C_Λ

    |#(Λ∩ tB)-t^2n|
       ≤∑_j=1^k|#(Λ∩ tB_j^+)-(tB_j^+)|+|#(Λ∩ tB_j^-)-(tB_j^-)|≤ t^nlog^2(t),

so that D(Λ,tB)≤log^2(t)/t^n as claimed.
A similar argument gives the same bound for the primitive lattice points.








































































alpha


