Reduce to the Kuznetsov Relative Trace Formula

Chapter V: Convergence of the Spectral Side

5.1 Reduce to the Kuznetsov Relative Trace Formula

Lemma 35. Letϕ ∈ H (G(AF), ω). LetK = K^ϕ,K0 = K^ϕ andK^∞ = K∞^ϕ be the corresponding kernel functions. Then

K₀(x,y)= Õ

δ∈N(F)\P₀(F)

∫

[N]

(K(nδx, δx) −K∞(nδx, δx))θ(n)dn. (5.3)

Proof. By the spectral decomposition of K0(x,y)we see it is cuspidal as a function of x. Applying Proposition 26 to the first variable of K₀(x,y)and take y = x we then obtain

K₀(x,y)= Õ

δ∈N(F)\P₀(F)

∫

[N]K₀(nδx,x)θ(n)dn.

Then (5.3) follows from the spectral decomposition K0(x,y) = K(x,y) −K^∞(x,y) and the automorphy of these functions relative to the second variable.

Let Re(s)> 1 in this section. We then plug Lemma 35 into I₀(s, τ)=

∫

ZG(AF)G(F)\G(AF)K₀(x,x)E(x,s)dx and unfold the Eisenstein seriesE(x,s)to obtain

I₀(s, τ)= I_Kl(s, τ) −I_Whi(s, τ), where

I_Kl(s, τ)=

∫

ZG(AF)N(F)\G(AF)

∫

[N]

K(nx,x)θ(n)dn f(x,s)dx. (5.4)

Since K₀is rapidly decaying, then to showI_Whi(s, τ)is well defined, it suffices to show I_Kl(s, τ)converges. We will show I_Kl(s, τ)converges for allϕ ∈ H (G(AF), ω)and Re(s)large enough. Then by Cauchy inequality and the convolution decomposition ofϕwe get the absolute convergence ofI_Whi(s, τ).

By a change of variable one has I_Kl(s, τ)=∫

ZG(AF)N(AF)\G(AF)

J_Kuz(ϕ,x)f(x,s)dx, where

J_Kuz(ϕ,x)=

∫

[N]

∫

[N]

K(n₁x,n₂x)θ(n₁)θ(n₂)dn₁dn₂

is a relative trace formula of Kuznetsov type. Hence, I_Kl(s, τ) is a (multiple) Mellin transform of a Kuznetsov relative trace formula J_Kuz(ϕ,x) since f(x,s) is essentially |detx|^s. Then in principle I_Kl(s, τ) is a sum of Kloosterman sum zeta functions, which should converge when Re(s) is large enough. We verify this intuition by showing that J_Kuz(ϕ,x) is majorized by a gauge. Recall that, for x = diag(x₁· · ·x_n−1,· · ·,x₁x₂,x₁,1) ∈ A(AF),a gauge G is a positive function of the form

G(x)=ξ(x₁,x₂,· · · ,x_n−1) · |x₁x₂· · ·x_n−1|^−M withM ≥ 0 andξ is a Schwartz-Bruhat function on(A^×_F)ⁿ⁻¹.

Proposition 36. Let notation be as above. Then as a function of x ∈ A(AF) = Z_G(AF)\T(AF),J_Kuz(ϕ,x)is a majorized by a finite sum of gauges on A(AF).

Proof. By definition of the kernel function K(x,y)we have J_Kuz(ϕ,x)= ∫

[N]

∫

[N]

Õ

γ∈ZG(F)\G(F)

ϕ(x⁻¹n⁻¹

1 γn₂x)θ(n₁)θ(n₂)dn₁dn₂, which converges absolutely since K(x,y)is continuous and[N]is compact.

Then we consider the double coset Z_G(AF)N(F)\G(F)/N(F),whose element is of the formwa,wherewis a Weyl element anda∈ Z_G(F)\T(F). Let

H_wa:=

(n₁,n₂) ∈ N ×N : n⁻¹

1 wan₂a⁻¹w⁻¹ ∈ Z_G be the stabilizer relative to the representativewa. Then

J_Kuz(ϕ,x)= Õ

wa∈Φ

∫

Hwa(F)\N(AF)×N(AF)

ϕ(x⁻¹n⁻¹

1 wan₂x)θ(n₁)θ(n₂)dn₁dn₂,

whereΦis a set of complete representatives forZ_G(AF)N(F)\G(F)/N(F).Then J_Kuz(ϕ,x)= Õ

wa∈Φ

C_wa

∫

H_wa(AF)\N(AF)×N(AF)

ϕ(x⁻¹n⁻¹

1 wan₂x)θ(n₁)θ(n₂)dn₁dn₂, where

C_wa =∫

[Hwa]

θ(n⁰

1)θ(n⁰

2)dn⁰

1dn⁰

2.

Call wa ∈ Φ relevant if C_wa , 0, i.e., θ(n⁰

1)θ(n⁰

2) is trivial on H_wa(AF). Denote by Φ^∗ the set of relevant elements in Φ. By [JR92] (Prop 1 on p. 272) one can take the following realization: Φ^∗ consists of wa, where w is the longest Weyl element inside a stantard parabolic subgroup P ⊆ G of type (k₁,· · ·,k_r), anda ∈ ZG(F)\diag(Tk₁(F),· · ·,Tkr(F))(modulo some further relations), withTkj

being the maximal split torus of GL(k_j).For instance, whenP = Bthe Borel, then w = I_nanda = I_nandH_wa = N.Therefore,

J_Kuz(ϕ,x)= Õ

wa∈Φ^∗

vol([H_wa])J_Kuz(ϕ,x;wa), where

J_Kuz(ϕ,x;wa)=

∫

Hwa(AF)\N(AF)×N(AF)

ϕ(x⁻¹n₁⁻¹wan₂x)θ(n₁)θ(n₂)dn₁dn₂.

By definition ofΦ^∗eachwcorresponds to a unique (i.e., the minimal one) parabolic subgroupPcontainingw.Supposew, I_n. Then by Levi decomposition it suffices to consider the extreme case whereP= Gandwis the longest.

Recall that the test function ϕis K-finite. Hence there is some compact subgroup K₀ ⊂ G(AF,fin) such that ϕ is right K₀-invariant. Let K₀ = Î

v<∞K₀,v. Note that J_Kuz(ϕ,x;wa)=Î

v≤∞J_Kuz,v(ϕ_v,xv;wa),where J_Kuz,v(ϕv,xv;wa)=

∫

Hwa(Fv)\N(Fv)×N(Fv)

ϕv(x_v⁻¹n⁻₁¹wan₂x)θv(n₁)θv(n₂)dn₁dn₂.

Then for each finite placev,J_Kuz,v(ϕ_v,x_v;wa)is rightK₀,v-invariant. So there exists a compact subgroupN₀,v ⊆ K₀,v∩N(F_v),depending only onϕ_v,such that

J_Kuz,v(ϕv,xvuv;wa)= J_Kuz,v(ϕv,xv;wa), for allxv ∈ A(Fv)anduv ∈N₀,v. On the other hand, J_Kuz,v(ϕ_v,x_vu_v;wa) = θ(x_vu_vx_v⁻¹)J_Kuz,v(ϕ_v,x_v;wa). But then, there exists a constantC_v depending only on N₀,v andθ such that θ(x_vu_vx_v⁻¹) = 1 if and only if |α_i(x_v)|_v ≤ C_v,whereα_i’s are the simple roots ofG(F)relative to B.

Note that for all but finitely many v < ∞, K₀,v = G(O_F,v). Thus we can take the correspondingC_v =1.Hence for anyx_v ∈ A(F_v),J_Kuz,v(ϕ_v,x_v;wa),0 implies that

|α_i(xv)|_v ≤ Cv, 1 ≤ i ≤ n−1,andCv = 1 for all but finitely many finite placesv.

Denote the compact set by Aϕ,fin= n

a=(a_v) ∈ A(AF,fin): |α_i(a_v)|_v ≤ C_v, 1 ≤i ≤ n−1 o.

Then suppJ_Kuz(ϕ,x;wa) ⊆ A(AF,∞)Aϕ,fin.

For any y= ⊗_v(yi,j,v) ∈G(AF). We defineky_vk_v =maxi,j |yi,j,v|_vifv < ∞; and ky_vk_v = h Õ

i,j

|yi,j,v|²_v i₁/2

, ifv | ∞.

Then ky_vk_v = 1 for almost all v. The height function kyk = Î

vky_vk_v is therefore well defined by a finite product. Also, by suppJ_Kuz(ϕ,x;wa) ⊆ A(AF,∞)Aϕ,fin and the compactness of suppϕv, we have kw⁻¹xvwxvak_v ≤ C_v⁰ for some constant C_v⁰ depending only onϕv,v < ∞,andC_v⁰ =1 for almost allv’s.

Now we investigate the archimedean J_Kuz,v(ϕv,xv;wa), i.e., v | ∞. Note that ϕv

is a compactly supported on ZG(Fv)\G(Fv). Then J_Kuz(ϕ_v,xv;wa) = 0 unless n⁻¹

1,vy_vwn_2,v ∈ suppϕ_v, where y_v = x⁻_v¹wax_vw⁻¹. Hence kn⁻¹

1,vy_vwn_2,vw⁻¹k_v ≤ C_v for some constantC_v depending only on ϕ. A straightforward computation shows that kn₁,vk_v + kn₂,vk_v + ky_vk_v ≤ C_v⁰ for some constant C_v⁰ depending only on ϕ. So ϕ_v(n₁,vy_vwn₂,v) has compact support relative to n₁,v and n₂,v. Therefore, J_Kuz(ϕ_v,x_v;wa)= 0 unlessn₁,v,n₂,vrun through a compact set ofN(F_v)andkyk_vis bounded.

Similar to (??) we define an additive character forx ∈ A(AF): ψ_x(u)=

n−1

Ö

i=1

ψ_F/Q x_iu_i,i+1, ∀u= (ui,j)_n×n∈ N(AF).

ThenJ_Kuz(ϕ,x;wa)is equal to δ_w(x)²

∫

Hwa(AF)\N(AF)×N(AF)

ϕ(n⁻¹

1 x⁻¹waxn₂)θ_x(n₁)θ_x(n₂)dn₁dn₂, whereδwis the modular character of the parabolic subgroup associated tow.

Sincen₁andn₂lie in a compact set determined by suppϕ,then for a fixedy ∈ A(AF), v | ∞,thev-th component of

∫

Hwa(AF)\N(AF)×N(AF)

ϕ(n⁻₁¹ywn₂)θx(n₁)θx(n₂)dn₁dn₂

is a Schwartz function ofxsince it is the Fourier transform of a compactly supported smooth function. Hence, J_Kuz(ϕ,x;wa) is majorized by a nonnegative Schwartz- Bruhat functionΞ(x⁻¹waxw⁻¹,x)on A(AF)²with

x⁻¹waxw⁻¹∈ A^∗ := {b∈ A(F): kbk ≤ Ö

v

C⁰_v.}

By properties of the heightk · k(e.g., see [Art05] p. 70) one has

#

w⁻¹x· A^∗·wx⁻¹

≤ C· (|x₁· · ·xn−1|^M+|x₁· · ·xn−1|^−M), for some constantsCandM depending on suppϕ.Therefore,

Õ

a∈A(F)

|J_Kuz(ϕ,x;wa)| ≤ Õ

a∈A^∗

Ξ(w⁻¹xwxa,x)= Õ

a∈w⁻¹x·A^∗·wx⁻¹

Ξ(a,x),

which is majorized by|x₁· · ·x_n−1|^−M·ξ(x₁,· · · ,x_n−1)for someM ≥ 0 and Schwartz- Bruhat functionξ.

The remaining case is thatw = I_n,i.e., P= B.In this case Õ

a∈A(F)

|J_Kuz(ϕ,x;wa)= δ_w(x)

∫

N(AF)

ϕ(an)θ_x(n)dn

is the Fourier transform of a Schwartz-Bruhat function. So it is majorized by a

gauge. Then Proposition 36 follows.

As a consequence of Proposition 36 and the Iwasawa decomposition, we have I_Kl(s, τ) converges absolutely when Re(s) is large enough. Therefore, I_Whi(s, τ) converges when Re(s)is large enough.

To show the absolute convergence of I_Whi(s, τ) and thus to obtain meromorphic continuation, we need to analyze properties of K∞ by its spectral expansion.