Discussion on Arthur’s Truncation Operator

Chapter V: Convergence of the Spectral Side

5.4 Discussion on Arthur’s Truncation Operator

In this section we discuss Arthur’s truncation operators in our case. We will use the conventional notations on p.24-29 of [Art05]. For any parabolic subgroupP ofG, letbτ_P be the characteristic function of the subset{t ∈a_P : $(t)> 0, ∀$ ∈b∆_P}of a_P.Leta⁺

0 be the set of positive coroots. LetT ∈a⁺

0,we sayT is suitably regular if α(T)is large, for each simple rootα.For any suitably regular pointT ∈a⁺

0 and any functionφ ∈ B_loc(Z_G(AF)G(F)\G(AF)),define the truncation functionΛ^Tφto be the function inB_loc(Z_G(AF)G(F)\G(AF))such that

Λ^Tφ(x)=Õ

P

(−1)^dim(AP/AG) Õ

δ∈P(F)\G(F)

bτ_P(HP(δx) −T)

∫

[NP]

φ(nδx)dn. (5.13)

The inner sum may be taken over a finite set depending on x,while the integrand is a bounded function ofn.

Before moving on, we still need to choose a nonnegative functionk · konG(AF)to describe properties of the truncation operatorΛ^T quantitatively.

Letx = (x_v)_v ∈G(AF).Recall thatkx_vk_v =maxi,j|xi,j,v|_v ifv <∞; and kx_vk_v = h Õ

i,j

|xi,j,v|²_v i₁/2

, ifv | ∞.

Then kx_vk_v = 1 for almost all v. The height function kxk = Î

vkx_vk_v is therefore well defined by a finite product. Then one has kxyk ≤ kxk · kyk,∀x,y ∈ G(AF). Also one can check that there is some absolute constantsC₀ and N₀ such that for any x ∈G(AF),we havekxk⁻¹ ≤C₀kxk^N0,and

#{x ∈G(F): kxk ≤ t} ≤C₀t^N0, t ≥ 0. Note that the test functionϕis compactly supported. Let

kx·suppϕ· y⁻¹k := max

g∈suppϕkxgy⁻¹k. Then one sees that

|K(x,y)| = |Kϕ(x,y)| ≤ Õ

γ∈ZG(F)\G(F) kγk≤kx·suppϕ·y⁻¹k

sup

g∈ZG(AF)\G(AF)

|ϕ(g)|.

Hence |K(x,y)| ϕ #{γ ∈ G(F) : kγk ≤ kx·suppϕ· y⁻¹k} _ϕ kxk^N0⁰ · kyk^N00, whereN⁰

0is some absolute constant, i.e., independent of the choice of test function ϕ.Thus there exists some constantc(ϕ)such that|K(x,y)| ≤c(ϕ)kxk^N0⁰ · kyk^N0⁰.

Now we consider derivatives of the kernel K(x,y).SupposeXandYare left invariant differential operators on G(AF,∞)of degrees d₁ and d₂. Suppose also that the test functionϕ ∈C_c^r(G(AF)),for some large positiver. For any cuspidal datum χ ∈X, define the corresponding kernel function as

Kχ(x,y)= Õ

P∈P

1 k_P!(2π)^kP

∫

ia^∗_P/ia^∗_G

Õ

φ∈BP,χ

E(x,I_P(λ, ϕ)φ, λ)E(y, φ, λ)dλ, (5.14) which is convergent absolutely. By [Art79] there exists some function ϕX,Y ∈ C_c^r−d1−d2(G(AF))such that its corresponding kernel function Kχ,ϕX,Y(x,y)is equal to XY Kχ(x,y), for all x,y ∈ G(AF). Then one can apply the above estimate for kernel functions to obtain

Õ

χ

XYKχ(x,y)

≤ c(ϕX,Y)kxk^N00· kyk^N00, ∀x,y ∈G(AF). (5.15) Also, for any function H(x,y) in B_loc (G(F)\G(AF)¹) × (G(F)\G(AF)¹),define the partial Fourier transform ofH(x,y)with respect to thex-variable as

F₁H(x,y):=∫

N(F)\N(AF)

H(n₁x,y)θ(n₁)dn₁. (5.16) Letwbe a matrix inK∞,the maximal compact subgroup ofG(AF,∞),representing a Weyl element. Regardwas a matrix inKby setting its components to be the trivial matrix at finite places. Define similarly

F^w

1 H(x,y):=

∫

N(F)\N(AF)

H(w⁻¹n₁wx,y)θ(n₁)dn₁. (5.17) This is well defined since H(x,y) is G(F)-invariant on the x-variable, and the quotientN(F)\N(AF)is compact.

Let AF,∞ = Z_G(AF,∞)\T(AF,∞), where T(AF,∞) is the AF,∞-points of the torus isomorphic to(Gm)ⁿ.For any c> 0,let Ac,∞ be set consisting of all

a=

©

­

­

­

­

­

­

­

­

«

a₁a₂· · ·a_n−1

a₁a₂· · ·a_n−2

. . . a₁

1 ª

®

®

®

®

®

®

®

®

¬

∈ AF,∞ (5.18)

with|a_i|_∞ ≥ cfor 1≤ i ≤ n−1.Clearly Ac,∞ ( Z_G(AF,∞)\T(AF,∞),∀c >0. A Siegel set S_c for c > 0 is defined to be the set of elements of the form nak, n ∈Ω ⊆ N(AF),compact, such that N(F)Ω= N(AF); k ∈ K,anda ∈ A≥c = {a ∈

A(AF) : |α_i(a)| ≥ c, 1 ≤ i ≤ n−1}. Then from reduction theory (see [Bor69]), there exists somec₀ > 0 such thatG(AF) = Z_G(AF)G(F)S_c

0. Denote by S₀ = S_c0, and we may assume that 0 < c₀ < 1. LetRbe a function on Z_G(AF)G(F)\G(AF), we say that Ris slowly increasing if there exists somer > 0 andC > 0 such that R(x) ≤ Ckxk^r,∀ x ∈ G(AF); we say R is rapidly decreasing if for any positive integer N and any Siegel setS_c for G(AF),there is a positive constantCsuch that

|R(x)| ≤Ckxk^−N for every x ∈ S_c.

We will apply the truncation operatorΛ^T to thesecond variableof kernel functions Kχ(x,y)and show that it is absolutely integrable twisted by any slowing increasing functions overAh,∞Aϕ,fin·K for someh> 0,where Aϕ,finbe a compact subgroup of A_fin = Z_G(AF,fin)\T(AF,fin)depending on ϕ.

Lemma 42. Let notation be as above. ThenÍ

χ

F₁Λ^T

2Kχ(x,x)

is rapidly decreas- ing onS₀.Moreover, letRbe a slowly increasing function onZ_G(AF)G(F)\G(AF). Then we have

∫

S₀

Õ

χ

F₁Λ^T

2Kχ(x,x) ·R(x)

dx < ∞, (5.19)

where χruns over all the equivalent classes of cuspidal datum and the truncation operatorΛ^T acts on the second variable of kernel functionsKχ. Moreover,(5.19) holds whenF₁is replaced byF^w

1 (defined in(5.17)), wherewis a Weyl element.

Proof. Clearly for any given x ∈ G(AF), F₁Kχ(x,y) is a well defined function (with respect toy) inB_loc(Z_G(AF)G(F)\G(AF)).Then according to Lemma 1.4 in [Art80] (or a more explicit version given by Proposition 13.2 in [Art05], p.71) one sees that given a Siegel set S, positive integers M and M₁, and an open compact subgroupK₀ofG(AF,fin),one can choose a finite set{X_i}of left invariant differential operators on Z_G(AF,∞)\G(AF,∞) and a positive integerr with the property that if (Ω,dω)is a measure space andφ(ω) 7→ φ(ω,x)is any measurable function fromΩ toC^r(ZG(AF)G(F)\G(AF)/K₀),then

sup

x∈S

kxk^M

∫

Ω

|Λ^Tφ(ω,x)|dω

≤ sup

x∈G(AF)¹

kxk^−M1Õ

i

∫

Ω

|X_iφ(ω,x)|dω

! . (5.20) In particular,{X_i}isindependentof(Ω,dω).

Since our test functions lie in the Hecke algebra, we may assume thatϕis biinvariant underK₀,whereK₀ is an open compact subgroup ofG(AF,fin).Also, set M₁ = M⁰

0

andM large, then we can find a finite set{Y_i}of left invariant differential operators

onG(AF,∞)such that (5.20) holds for the measure space(Ω,dω)= (X,d),whereXis the set of cuspidal datum andd = d_xis a discrete measure depending onx ∈G(AF). Under our current particular choices (5.20) becomes

sup

y∈S₀

kyk^MÕ

χ

Λ^T

2Kχ(x,y)

!

≤ sup

y∈G(A)

Õ

χ

kyk^−M1

Õ

i

(Y_i)₂Kχ(x,y)

!

≤ sup

y∈G(A)

Õ

i

kyk^−M1Õ

χ

(Yi)₂Kχ(x,y)

! ,

where(Y_i)₂above indicates that the differential operator acts on the y-variable; and eachY_i is independent of x.By the estimate (5.15), the right hand side of the above inequality is bounded by

sup

y∈G(A)

Õ

i

kyk^−N1c(ϕI,Y_i)kxk^N0⁰ · kyk^N00

!

≤ Õ

i

c(ϕI,Y_i)kxk^M1, (5.21) for any x ∈ S₀, where I refers to the trivial identity operator. Denote by V₀ = vol([N]).Then by mean value theorem there exists somen₀∈ [N]such that

Õ

χ

F₁Λ^T

2Kχ(x,x)

≤ V₀Õ

χ

Λ^T

2Kχ(n₀x,x)

. (5.22)

Substituting (5.21) into the right hand side of (5.22) we then obtain Õ

χ

F₁Λ^T

2Kχ(x,x)

≤V₀Õ

i

c(ϕI,Y_i)kxk^M1−M,

for anyx ∈ S₀,where χruns over all the equivalent classes of cuspidal datum. Also, since R(x)is slowly increasing, then by taking M to be large enough we conclude thatÍ

χ

F₁Λ^T

2Kχ(x,x)

· |R(x)|is a bounded function onS₀,hence it is integrable.

Note that the above argument still works whenF₁is replaced byF^w

1 .Then Lemma

42 follows.

Let S_n be the permutation group on n letters. Let σ ∈ S_n. For any a ∈ AF,∞, write it in its Iwasawa normal form given in (5.18). Let a⁰_i = a₁a₂· · ·a_n−i, 1 ≤ i ≤ n − 1; and set a⁰_n = 1. For any 1 ≤ i ≤ n, let a_i = a⁻¹

1 · · ·a⁻_σ(¹_n)a⁰_i. Set a_σ = diag(a_σ(1),a_σ(2),· · ·,a_σ(n)). Clearly a_σ(n) = 1, and each a_σ(i) is a rational function ofa₁,· · ·,a_n−1. Moreover, eacha_σ(i) is a monomial, 1 ≤ i ≤ n−1. Note thata_σ is of the form in (5.18). Soσ induces a well defined map AF,∞ −→ AF,∞, a 7→a_σ.Denote byι_σ this map. Thenι_σ is actually a bijection fromA_F,∞to itself.

Writea_σ(i) = a_σ(i)(a₁,a₂,· · · ,a_n−1) to indicate thata_σ(i) is a fractional function of a₁,a₂,· · ·a_n−1. For anyc> 0,let

H_σ^c = (

(a₁,a₂,· · · ,a_n−1):

a_σ(i)(a₁,a₂,· · · ,a_n−1) a_σ(i+1)(a₁,a₂,· · · ,a_n−1)

_∞ ≥ c, 1 ≤i ≤ n−1 )

.

LetS_n^{r eg}:= {σ ∈ S_n : σ(i) ≤ σ(i+1), 3≤ i ≤ n−1}.Let 0 < c ≤ 1.Denote by ι(AF,∞)^{r eg}=

(a₁,a₂,· · · ,a_n−1) ∈Gm(AF,∞)ⁿ⁻¹: a_i

∞ ≥ c, 1≤ i ≤ n−3 . Lemma 43. Let notation be as above. Letn ≥ 4.Then one has

ι(AF,∞)^{r eg} ⊆ Ø

σ∈S^{r eg}n

H_σ^c. (5.23)

Proof. For any σ ∈ S_n^{r eg}, if σ(2) < σ(3), then defineiσ = 2; if σ(i₀) < σ(2) <

σ(i₀+1)for some 3 ≤ i₀ ≤ n−1,then defineiσ = i₀; if σ(2) > σ(n),then define i_σ =n.Since such ani₀(if exists) is unique, theni_σis well defined. It induces ann to 1 surjectionι_{r eg} :S_n^{r eg} → {2,3,· · · ,n},given byσ 7→i_σ.For 2 ≤ i ≤ n,letS_n,i^{r eg} be the fibre atiσ =i,i.e.,S^{r eg}_n,i =ι_{r eg}(i)⁻¹.Then

Ø

σ∈S^{r eg}_n

H_σ^c = Øn

i=2

Ø

σ∈S^{r eg}_n,i

H_σ^c. (5.24)

Let ι₂(AF,∞)^{r eg} =

(a₁,a₂,· · ·,a_n−1) ∈ ι(AF,∞)^{r eg} : a_n−2

_∞ ≥ c . Denote by ι_n(AF,∞)^{r eg} =

(a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} :

a₁a₂· · ·a_n−2

_∞ ≤ 1/c . De- fine, for any 3 ≤ i ≤ n−1, that ι_i(AF,∞)^{r eg} =

(a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} :

an−ian−i+1· · ·an−2

_∞ ≥ c,

an−i+1an−i+2· · ·an−2

_∞ ≤ 1/c . Note that, for any 2≤ i ≤ n, ι_i(A_F,∞)^{r eg}is well defined.

Claim 44. Let notation be as before. Then one has, for any2≤ i ≤ n,that ι_i(A_F,∞)^{r eg} ⊆ Ø

σ∈S_n,i^{r eg}

H_σ^c. (5.25)

A straightforward combinatorial analysis shows that ι(A_F,∞)^{r eg} is contained in the union ofι_i(AF,∞)^{r eg} over 2≤ i ≤ n. Hence (5.23) comes from (5.24) and (5.25):

ι(AF,∞)^{r eg} ⊆

n

Ø

i=2

ι_i(AF,∞)^{r eg} ⊆

n

Ø

i=2

Ø

σ∈S_n,i^{r eg}

H_σ^c = Ø

σ∈S_n^{r eg}

H_σ^c.

Proof of Claim 44. For any 2≤ i ≤ nand anyσ ∈S_n,i^{r eg},recall thatH_σ^c is equal to (

(a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} :

a_σ(i)(a₁,a₂,· · · ,a_n−1) a_σ(i+1)(a₁,a₂,· · ·,a_n−1)

_∞ ≥ c, i ≤ n−1 )

.

Case 1 Let i = 2 and σ ∈ S_n,i^{r eg}. If σ(2) + 1 < σ(1) < n, then there exists a unique jσ such that σ(jσ) < σ(2) < σ(jσ + 1). In this case H_σ^c is equal to

(a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} : |a_n−2|_∞ ≥ c, |a_n−jσ · · ·a_n−1|_∞ ≥ c, |a_n−jσ+1· · ·a_n−1|_∞ ≤ 1/c . Ifσ(1) = n,thenH_σ^c =

(a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} : |a_n−i· · ·a_n−2|_∞ ≥ c, |a₁· · ·a_n−1|_∞ ≤ 1/c . If σ(1) = σ(2)+ 1, then H_σ^c is equal to

(a₁,a₂,· · ·,an−1) ∈ ι(AF,∞)^{r eg} : |an−2an−1|_∞ ≥ c, |a_n−1|_∞ ≤ 1/c, |a_n−1a_n−2|_∞ ≤ 1/c . If σ(1) = σ(2) − 1, then H_σ^c is equal to

(a₁,a₂,· · ·,a_n−1) ∈ ι(A_F,∞)^{r eg} : |a_n−2|_∞ ≥ c, |a_n−1|_∞ ≥ c . Now one sees clearly that the union of H_σ^c over σ ∈ S_n,i^{r eg} does cover the sets

(a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} : a_n−2

_∞ ≥ c, |a_n−1|_∞ ≥ c and (a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} :

a_n−2

∞ ≥ c, |a_n−1|_∞ ≤ 1/c . Therefore it coversι₂(AF,∞)^{r eg}. Hence (5.25) holds in the case wherei =2.

Case 2 Let 3 ≤ i ≤ n−1 andσ ∈ S_n,i^{r eg}. Ifσ(2)+1 < σ(1) < n,then there exists a unique jσ such thatσ(jσ)< σ(2)< σ(jσ+1).In this caseH_σ^c is equal to (a₁,a₂,· · · ,a_n−1) ∈ ι(A_F,∞)^{r eg} : |a_n−i· · ·a_n−2|_∞ ≥ c, |a_n−i+1· · ·a_n−2|_∞ ≤ 1

c,

|a_n−jσ· · ·a_n−1|_∞ ≥ c,ka_n−jσ+1· · ·a_n−1|_∞ ≤ 1/c . Ifσ(1)= n,thenH_σ^c is equal to

(a₁,a₂,· · · ,a_n−1) ∈ ι(A_F,∞)^{r eg} : |a_n−i· · ·a_n−2|_∞ ≥ c,

|a_n−i+1· · ·a_n−2|_∞ ≤ 1/c, |a₁· · ·a_n−1|_∞ ≤ 1/c . Ifσ(1)= σ(2)+1,thenH_σ^c is equal to

(a₁,a₂,· · ·,a_n−1) ∈ι(A_F,∞)^{r eg} : |a_n−i· · ·a_n−2a_n−1|_∞ ≥ c,

|a_n−1|_∞ ≤ 1/c, |a_n−i+1· · ·a_n−2|_∞ ≤ 1/c . Ifσ(1)= σ(2) −1,thenH_σ^c is equal to

(a₁,a₂,· · · ,a_n−1) ∈ ι(A_F,∞)^{r eg} : |a_n−i· · ·a_n−2|_∞ ≥ c,

|a_n−1|_∞ ≥ c, |a_n−i+1· · ·a_n−1|_∞ ≤ 1/c .

If 1 < σ(1) < σ(2) −1,then there exists a uniquejσsuch thatσ(jσ)< σ(2)<

σ(jσ+1).In this caseH_σ^c is equal to

(a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} : |a_n−i· · ·a_n−2|_∞ ≥ c,|a_n−i+1· · ·a_n−2|_∞ ≤ c⁻¹,

|a_n−jσ· · ·a_n−2a_n−1|_∞ ≥ c, |a_n−jσ+1· · ·a_n−1|_∞ ≤ 1/c . Ifσ(1)= 1,then

H_σ^c =

(a₁,a₂,· · ·,a_n−1) ∈ι(AF,∞)^{r eg} : |a_n−ia_n−i+1· · ·a_n−2|_∞ ≥ c,

|a_n−i+1· · ·a_n−2|_∞ ≤ c⁻¹, c ≤ |a_n−2a_n−1|_∞ . Now one sees clearly that the union ofH_σ^c overσ ∈S^{r eg}_n,i does cover the sets

(a₁,a₂,· · · ,a_n−1) ∈ ι(A_F,∞)^{r eg}:

a_n−ia_n−i+1· · ·a_{n−2 ∞} ≥ c,

a_n−i+1a_n−i+2· · ·a_n−2

_∞ ≤ 1/c, |a_n−1|_∞ ≥ c and

(a₁,a₂,· · · ,an−1) ∈ ι(AF,∞)^{r eg} :

an−ian−i+1· · ·an−2

_∞ ≥ c,

a_n−i+1a_n−i+2· · ·a_n−2

_∞ ≤ 1/c, |a_n−1|_∞ ≤ 1/c . Therefore, it coversι_i(AF,∞)^{r eg}.Hence (5.25) holds.

Case 3 Leti = nandσ ∈ S_n,i^{r eg}. Ifσ(1)= σ(2)+1,thenH_σ^c =

(a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} : |a_n−1|_∞ ≤ 1/c, |a₁a₂· · ·a_n−2|_∞ ≤ 1/c .Ifσ(1)= σ(2) −1,then H_σ^c is equal to

(a₁,a₂,· · · ,a_n−1) ∈ ι(A_F,∞)^{r eg}: |a_n−1|_∞ ≥ c, |a₁a₂· · ·a_n−1|_∞ ≤ 1/c . If 1 < σ(1) < σ(2) −1, then there exists a unique j_σ such that σ(j_σ) <

σ(2) < σ(jσ+1). In this caseH_σ^c is equal to

(a₁,· · ·,a_n−1) ∈ ι(AF,∞)^{r eg} :

|a₁· · ·a_n−2|_∞ ≤ 1/c, |a_n−jσ · · ·a_n−1|_∞ ≥ c, |a_n−jσ+1· · ·a_n−1|_∞ ≤ 1/c . If σ(1) = 1,thenH_σ^c =

(a₁,a₂,· · ·,a_n−1) ∈ ι(AF,∞)^{r eg} : |a_n−i+1· · ·a_n−2|_∞ ≤ 1/c, |a₁a₂· · ·a_n−2a_n−1|_∞ ≥ c . Now one sees clearly that the union of H_σ^c over σ ∈ S_n,i^{r eg} does cover the sets

(a₁,a₂,· · · ,a_n−1) ∈ ι(AF,∞)^{r eg} :

a₁a₂· · ·a_n−2

_∞ ≤ 1/c, |a_n−1|_∞ ≥ c and

(a₁,a₂,· · ·,a_n−1) ∈ ι(AF,∞)^{r eg} :

a₁a₂· · ·an−2

_∞ ≤ 1/c, |an−1|_∞ ≤ 1/c . Therefore, it covers ιn(AF,∞)^{r eg}. Hence (5.25) holds fori= n.

Therefore, Claim 44 follows from the above discussions.

Proposition 45. Let notation be as before. Let ι : AF,∞ −→ Gm(AF,∞)ⁿ⁻¹ be the isomorphism given bya 7→ (a₁,a₂,· · ·,a_n−1).Then for any0< c ≤ 1,one has

ι(AF,∞) ⊆ Ø

σ∈Sn

H_σ^c. (5.26)

Proof. When n = 2, (5.26) is immediate. When n = 3, there are six different H_σ^c’s. In this case, one can verify by brute force computation that the union of these hyperboloids does cover ι(AF,∞). Hence (5.26) holds for n = 3. From now on, we may assume n ≥ 4. For any m ∈ Z, let τ_m : Z → Z be the shifting map defined by j 7→ j +m,∀ j ∈ Z. Set S_n−2[2] := {τ₂◦σ ◦τ−2 : σ ∈ S_n−2} as a set of bijections from the set{3,4,· · · ,n} to itself. Regard naturallyS_n−2[2]as the stabilizer of{1,2}ofS_n.Then clearlyS_n−2[2]is isomorphic toS_n−2.Denote by%the natural isomorphismS_n−2

−→∼ S_n−2[2], σ 7→ τ₂◦σ◦τ−2.Note that #S^{r eg}_n =n(n−1), then we have a bijection:

Sn−2×S^{r eg}_n −−→^1:1 Sn, (σ, σ⁰) 7→ %(σ) ◦σ⁰. (5.27) Assume that (5.26) holds for any n₀ ≤ n− 2. Let (a₁,a₂,· · · ,an−1) ∈ ι(AF,∞). Let a ∈ A_F,∞ be such that ι(a) = (a₁,a₂,· · · ,a_n−1). Then (a₁,a₂,· · ·,a_n−3) ∈ ι^≤n−2(A_F,∞), where ι^≤n−2 : A_F,∞ −→ Gm(AF,∞)ⁿ⁻³ is the map given by a 7→

(a₁,a₂,· · · ,a_n−3). Then by our induction assumption, there exists some σ ∈ S_n−2 such that (a₁,a₂,· · ·,a_n−3) ∈ H_σ^c_%, where σ% := %⁻¹(σ). Note that for each 1 ≤ i ≤ n −3, a_σ%(i)(a₁,a₂,· · · ,a_n−1) is independent of a_n−2 and a_n−1, we may write a_σ%(i)(a₁,a₂,· · · ,a_n−1)=a_σ%(i)(a₁,a₂,· · ·,a_n−3),1 ≤i ≤ n−3.Letb₂ =a₁a₂· · ·a_n−2· a_σ%(1)(a₁,a₂,· · · ,a_n−3)⁻¹, b₁ = a₁a₂· · ·a_n−2a_n−1 · b⁻¹

2 , and b_n = 1. Let b_i+2 = a_σ%(i)(a₁,a₂,· · · ,a_n−3), 1 ≤ i ≤ n− 3. Set b = diag(b₁,b₂,· · · ,b_n). Then clearly b = a_σ%. Hence b ∈ AF,∞ and ι(b) = (b₁,b₂,· · ·,bn−1), where bi = b_n−i ·b_n−i+1⁻¹ , 1 ≤ i ≤ n−1. By our definition of b, one sees that ι(b) ∈ ι(A_F,∞)^{r eg}. Then by Lemma 43 there exists someσ⁰∈S_n^{r eg}such thatι(b) ∈ H_σ^c0.Therefore,

|b_σ⁰_(i)(b₁,b₂,· · · ,b_n−1)|_∞ ≥ c· |b_σ⁰_(i+1)(b₁,b₂,· · · ,b_n−1)|_∞, 1≤ i ≤ n−1. Sinceb =a_σ%,the above system of inequalities becomes, for 1≤ i ≤ n−1,that

|a_%(σ)◦σ⁰_(i)(a₁,a₂,· · · ,a_n−1)|_∞ ≥ c· |a_%(σ)◦σ⁰_(i+1)(a₁,a₂,· · ·,a_n−1)|_∞. Then according to (5.27),ι(a) ∈ H^c

σ˜ for ˜σ = %(σ) ◦σ⁰∈ S_n.Hence (5.26) holds for n₀= n.Since it holds forn₀= 2 andn₀= 3,Proposition 45 follows by induction on

initial cases.

Proposition 46. Let notation be as above. Let Aϕ,fin be a compact subgroup of Z_G(AF,fin)\T(AF,fin) depending only on ϕ and F. Let R(x) be a slowly increasing function onS₀.Then we have

∫

K

∫

[N]

∫

AF,∞Aϕ,fin

Õ

χ

F₁Λ^T

2Kχ(nak,nak) ·R(nak)

d^×adndk < ∞, (5.28) where χruns over all the equivalent classes of cuspidal datum.

Proof. LetA^σ_F,∞= {a ∈ A_F,∞ : ι(a) ∈ H_σ^c}.Then Proposition 45 implies that AF,∞ = Ø

σ∈Sn

A^σ_F,∞. (5.29)

The decomposition (5.29) implies that the left hand side of (5.28) is not more than J_c := Õ

σ∈Sn

∫

K

∫

[N]

∫

A^σ_F,∞Aϕ,fin

Õ

χ

F₁Λ^T

2Kχ(nak,nak) ·R(nak)

d^×adndk.

Note that for any σ ∈ S_n, leta ∈ A^σ_F,∞,thena_σ ∈ S_c. Let w ∈ K∞ be the matrix representation of the Weyl element corresponding toσ so thata_σ =w⁻¹aw. Then

F₁Λ^T

2Kχ(nak,nak)= F^w

1 Λ^T

2K^wχ(w⁻¹nwa_σa_finw⁻¹kw,w⁻¹nwa_σa_finw⁻¹kw), where K^wχ refers to the kernel function relative to the test function ϕ^w defined by ϕ^w(x)= ϕ(wxw⁻¹).

Let ι(a) = (a₁,a₂,· · ·,a_n−1),and ι(a_σ) = (aσ,1,aσ,2,· · · ,aσ,n−1),then a straightfor- ward computation shows that each a_i is a rational monomial Pσ of the variables aσ,1,aσ,2,· · · ,aσ,n−1.Since such a monomial is at most polynomially increasing on S_c,one then changes variables to see

∫

K

∫

[N]

∫

A^σ_F,∞Aϕ,fin

Õ

χ

F₁Λ^T

2Kχ(nak,nak) ·R(nak)

d^×adndk

is bounded by the integral overK,N(F)\N(AF)of

∫

A^σ_F,∞Aϕ,fin

Õ

χ

F^w

1 Λ^T

2K^wχ(X,X) ·R(naσa_fink)Pσ(a_σ)

d^×a_σd^×a_fin, (5.30) where X = w⁻¹nwaσa_finw⁻¹kw,w⁻¹nwaσa_finw⁻¹kw. Note that (5.30) is bounded by the integral over Aϕ,finofGσ(n,a∞,k),whereGσ(n,a∞,k)is defined to be

∫

AF,∞∩S_c

Õ

χ

F^w

1 Λ^T

2K^w_χ(X⁰,X⁰) ·R(na∞a_fink)Pσ(a∞) d^×a∞,

where X⁰ = w⁻¹nwa∞a_finw⁻¹kw. Since the functiona∞ 7→ R(na∞a_fink)Pσ(a∞)is slowly increasing onAF,∞∩ S_c,Lemma 42 implies thatGσ(n,a∞,k)is well defined and thus it is continuous. So

∫

K

∫

[N]

∫

Aϕ,fin

Gσ(n,a∞,k)d^×a_findndk < ∞, ∀σ ∈S_n.

Therefore, (5.28) follows from the estimate below:

J_c ≤ Õ

σ∈S_n

∫

K

∫

N(F)\N(AF)

∫

Aϕ,fin

Gσ(n,a∞,k)d^×a_findndk < ∞.

Then Proposition 46 follows.

Corollary 47. Let notation be as above. Let R(x)be a slowly increasing function onZ_G(AF)N(F)\G(AF). Then we have

∫

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

Õ

χ

F₁Λ^T

2Kχ(x,x) ·R(x)

dx < ∞, (5.31) where χruns over all the equivalent classes of cuspidal data.

Proof. To handleF₁Λ^T

2Kχ(x,y)we can substitute the definitions ofF₁andΛ^T into the expansion of Kχ (cf. (5.14)). Note that (5.14) is convergent absolutely andΛ^T is a finite sum for givenx andy.One can thus apply the operatorsF andΛ^T inside the integral overia^∗_P/ia_G^∗ to obtain explicitly that

F₁Λ^T

2Kχ(x,y)= Õ

P∈P

1 c_P

∫

ia^∗_P/ia_G^∗

Õ

φ∈BP,χ

FE(x,I_P(λ, ϕ)φ, λ)Λ^TE(y, φ, λ)dλ.

This an easier analogue of Lemma 37. Then as before, the non-constant Fourier coefficient FE(x,I_P(λ, ϕ)φ, λ)of E(x,I_P(λ, ϕ)φ, λ) becomes a Whittaker function W(x;λ).

Recall that our test function ϕ is K-finite. Hence there is some compact sub- group K₀ ⊂ G(AF,fin)¹ such that ϕ is right K₀-invariant. Then for φ ∈ B_P,χ, FE(x,I_P(λ, ϕ)φ, λ)Λ^TE(y, φ, λ) = 0 unless φ is right K₀-invariant. Let K₀ = Îv<∞K₀,v.Note that the Whittaker functions are decomposable, i.e.,

W(x;λ)= Ö

v∈ΣF

W_v(x;λ).

Then for each finite placev,W_v(x;λ)is rightK₀,v-invariant. So there exists a compact subgroupN₀,v ⊆ K₀,v∩N(F_v),depending only onϕ,such that

W_v(t_vu_v;λ)=W_v(t_v;λ), for allt_v ∈T(F_v)andu_v ∈ N₀,v.

On the other hand, W_v(t_vu_v;λ) = θ_tv(u_v)W_v(t_v;λ), where θ_tv(n_v) = θ(t_vn_vt_v⁻¹), for any n_v ∈ N(F_v). But then, there exists a constantC_v depending only on N₀,v andθ (hence not onλ) such thatθ_tv(uv)=1 if and only if|α_i(tv)| ≤Cv,whereα_i’s are the simple roots ofG(F). Note that for all but finitely manyv < ∞,K₀,v = GL_n(O_F,v), thus we can take the correspondingCv =1. Hence for anytv ∈T(Fv),Wv(tv;λ),0 implies that|α_i(t_v)| ≤ C_v,1 ≤ i ≤ n−1,andC_v = 1 for all but finitely many finite placesv. Set A= Z_G\T,and

Aϕ,fin = n

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

Then suppW(x;λ) |_A(AF)⊆ A(AF,∞)Aϕ,fin, ∀ λ ∈ ia^∗_P/ia_G^∗, 1 ≤ i ≤ 2. So after applying Iwasawa decomposition, we see that the integrand in (5.31) are supported in[N]·A(AF,∞)Aϕ,fin·K,independent of χ.Therefore, (5.31) follows from (5.28).

C h a p t e r 6

RANKIN-SELBERG CONVOLUTIONS FOR GENERIC REPRESENTATIONS

By Theorem G, we see that when Re(s)> 1,I_Whi(s, τ)is equal to Õ

χ

Õ

P∈P

1 c_P

Õ

φ₁∈BP,χ

Õ

φ₂∈BP,χ

∫

Λ^∗

hI_P(λ, ϕ)φ₂, φ₁iΨ_P,χ(s,W₁,W₂;λ)dλ, (6.1)

where Ψ_P,χ(s,W₁,W₂;λ) = ∫

ZG(AF)N(AF)\G(AF)W₁(x;λ)W₂(x;λ)f(x,s)dx, and the Whittaker functionWi(x, λ)=∫

N(AF)φi(w₀nx)e^{(λ+ρP)HP(w0nx)}θ(n)dn,1≤ i ≤ 2,and w₀is the longest Weyl element.

For our purpose, we need to show thatI_Whi(s, τ)is a holomorphic multiple ofL(s, τ). So we have to compute (6.1) explicitly (up to an entire factor), then continue it to a meromorphic function which is a holomorphic multiple of L(s, τ)as we desired.

To achieve that, we start with computing each Ψ_P,χ(s,W₁,W₂;λ) associated to a standard parabolic subgroupPand a cuspidal datum χ=(MP, σ) ∈X.

Let P be a standard parabolic subgroup of G of type (n₁,n₂,· · ·,n_r), 1 ≤ r ≤ n, with n₁+n₂+ · · ·n_r = n. Let χ ∈ X be represented by (M_P, σ). Let B_P,χ be an orthonormal basis of the Hilbert spaceH_P,χ. Forφ_i ∈ B_P,χ,1 ≤ i ≤ 2,define the Whittaker function associated toφ_iparameterized byλ∈ia^∗_P/ia_G^∗ by

WP,χ,i(x, λ)=W(x, φ_i, λ):=

∫

N(AF)

φ_i(w₀nx)e^{(λ+ρP)HP(w0nx)}θ(n)dn,

wherew₀is the longest element in the Weyl groupW_n. Define Ψ_P,χ(s,W₁,W₂;λ,Φ)=∫

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

WP,χ,1(x;λ)WP,χ,2(x;λ)f(x,s)dx.

From now on, we fix such a standard parabolic subgroup P of type (n₁,· · · ,n_r) and a cuspidal datum χ = (M_P, σ) ∈ X,whereσ is a unitary representation of M of central characterω.Then there existr cuspidal representations π_i of GLni(AF), 1 ≤ i ≤ r,such that σ ' π₁⊕ π₂⊕ · · · ⊕π_r. Letπ = Ind^G(_P(A^A_F^F₎⁾(π₁, π₂ · · ·, π_r).For anyλ=(λ₁, λ₂,· · · , λ_r) ∈ia^∗_P/ia^∗_G,denote by

π_λ =Ind^G_P(^(AF)

AF)

π₁⊗ | · |^λ1

AF, π₂⊗ | · |^λ2

AF,· · · , π_r ⊗ | · |^λr

AF

. (6.2)

Thenπλ is also a unitary automorphic representation ofG(AF). Fixφ₁, φ₂ ∈ B_P,χ and a point λ = (λ₁, λ₂,· · ·, λ_r) ∈ ia^∗_P/ia^∗_G. Write W_i(x, λ) = WP,χ,i(x, λ), and Ψ(s,W₁,W₂;λ,Φ)= Ψ_P,χ(s,W₁,W₂;λ,Φ).Sinceλ ∈ia^∗_P/ia_G^∗, λ= −λ,¯ one has

Ψ(s,W₁,W₂;λ,Φ)= ∫

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

W₁(x;λ)W₂ x;−λ¯

f(x,s)dx. (6.3) SinceW₁(x;λ)andW₂ x;−λ¯

are dominant by some gauge, and f(x,s)is slowly in- creasing when Re(s)> 1,thenΨ(s,W₁,W₂;λ,Φ)converges absolutely and normally when Re(s)> 1.Note thatπ_i = ⊗⁰_vπi,v, 1≤ i ≤ r,where, for eachv ∈ Σ_F, πi,vis a uni- tary irreducible representation of GLni(F_v),of Whittaker tape. Then for eachv ∈ Σ_F and λ = (λ₁, λ₂,· · · , λ_r) ∈ ia_P^∗/ia_G^∗, denote by π_v = Ind^G(F_MP(F^v)_v) π₁,v, π₂,v,· · · , πr,v and

πλ,v =Ind^G(F_MP(F^v)_v)

π₁,v ⊗ | · |^λ1

Fv, π₂,v ⊗ | · |^λ2

Fv,· · ·, πr,v ⊗ | · |^λr

Fv

.

Thenπ = ⊗⁰_vπ_v andπλ = ⊗⁰_vπλ,v.Recall that f(x,s)= Î

v f_v(x_v,s),where f_v(x_v,s)=τ_v(detx_v)|detx_v|_F^s

v

∫

ZG(Fv)

Φ_v[(0,· · · ,t_v)x_v]τ_vⁿ(t_v)|t_v|^ns_F

vd^×t_v, ifΦ= ⊗⁰_vΦ_v.Sinceφ₁andφ₂both have central characterωλ =ω,which is unitary.

So isW₁(x;λ)andW₂(x;λ).Hence one can rewriteΨ(s,W₁,W₂;λ,Φ)as

∫

N(AF)\G(AF)

W₁(x;λ)W₂ x;−λ¯Φ(ηx)τ(detx)|detx|^s

AFdx, (6.4) whereη= (0,· · · ,0,1) ∈ Fⁿ. According to the definition we can writeφ_i = ⊗⁰_vφ_i,v, 1≤ i ≤ 2. Thus one can factorW_i(x;λ)asÎ

v∈Σ_FW_i,v(x_v;λ),where Wi,v(xv;λ)=∫

N(Fv)

φ_i,v(w₀nxv)e^{(λ+ρP)HP(w0nxv)}θ(n)dn, xv ∈ Fv, 1 ≤i ≤ 2. We may assumeΦ= ⊗⁰_vΦ_v andφ_i = ⊗⁰_vφi,v,i =1,2. Then one has

Ψ s,W_1,v,W_2,v;λ,Φ = Ö

v∈ΣF

Ψ_v s,W_1,v,W_2,v;λ,Φ_v, where each local factorΨ_v s,W₁,v,W₂,v;λ,Φ_v

is defined to be

∫

N(Fv)\G(Fv)

W₁,v(x_v;λ)W₂,v x_v;−λ¯

Φ_v(ηx_v)τ(detx_v)|detx_v|_F^s

vdx_v, (6.5) where W_i,v(x_v;λ) = ∫

N(Fv)φ_i,v(w₀nx)e^{(λ+ρP)HP,v(w0nx)}θ(n)dn, 1 ≤ i ≤ 2. Since W₁,v(x;λ)andW₂,v x;−λ¯

are dominant by some local gauge, and f_v(x_v,s)is slowly increasing when Re(s) > 1, then Ψ s,W₁,v,W₂,v;λ,Φ_v

converges absolutely and normally when Re(s)> 1,for anyv ∈Σ_F.

6.1 Local Theory forΨ_v s,W₁,v,W₂,v;λ,Φ_v

In this section, we shall compute each local integral representationΨ_v s,W_1,v,W_2,v;λ,Φ_v defined via (6.5). Letv ∈ Σ_F be a place of F. Note thatv may be archimedean or nonarchimedean. Letu = (uj,l)_1≤j,l≤n ∈ N(F_v),the unipotent of GLn,we denote by N⁰_j(u)the matrix

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1 u₁₂ · · · u₁,n−j+2 u₁,n−j+3 · · · u₁,n

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. . . ... ... ... ... ... ...

1 u_n−j+1,n−j+2 u_n−j+1,n−j+3 · · · u_n−j+1,n

1 0 · · · 0

1 · · · u_n−j+3,n

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1 u₁₂ · · · u¹

1,n−j+2 u₁,n−j+3 · · · u₁,n

1 · · · u¹

2,n−j+2 u₂,n−j+3 · · · u₂,n

. . . ... ... ... ... ... ...

1 0 u_n−j+1,n−j+3 · · · u_n−j+1,n

1 0 · · · 0

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. . . ... ...

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associated tou,whereu¹_l,n−j+2 = ul,n−j+2−ul,n−j+1u_n−j−1,n−j+2,1 ≤ l ≤ n− j.For any 2≤ j ≤ n,we denote byN_j⁰(u^∗)the matrix

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1 u₁₂ · · · u₁,n−j+1 u₁,n u_1,n−j+2 · · · u₁,n−1

1 · · · u₂,n−j+1 u₂,n u₂,n−j+2 · · · u₂,n−1

. . . ... ... ... ... ... ...

1 u_n−j+1,n u_n−j+1,n−j+2 · · · u_n−j+1,n−1

1 0 · · · 0

1 · · · un−j+2,n−1

. . . ... ...

1 u_n−2,n−1

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and foru =(uj,l)_1≤j,l≤n ∈ N(F_v),we letN⁰_j+2(u⁰⁰)represent the matrix

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1 u₁₂ · · · u₁,n−j−1 u₁,n u⁰

1,n−j · · · u₁,n−1

1 · · · u₂,n−j−1 u₂,n u⁰

2,n−j · · · u₂,n−1

. . . ... ... ... ... ... ...

1 u_n−j−1,n u⁰_n−j−

1,n−j · · · u_n−j−1,n−1

1 0 · · · 0

1 · · · un−j,n−1

. . . ... ...

1 u_n−2,n−1

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withu⁰_k,n−j =uk,n−j+s_n−j,nuk,n−j−1,1≤ k ≤ n− j−1.

Let w_j be the simple root of GLn corresponding to the permutation (j,j + 1), 1 ≤ j ≤ n−1.Letτ_nbe the longest element in the Weyl groupW_nof GLn,n ≥ 2. Thenτ_n =w_n−1w_n−2w_n−1w· · ·w₁w₂· · ·w_n−1,for anyn ≥ 2.Recall that we writew₀ for the longest element forG; when we highlight the ranknwe then useτ_ninstead.

Theτ_nis only used in this section.

Fix a nontrivial additive character θ = θv on Fv. This notation in only used in this section and should not be thought as a local version of notations in Sec. 3.1.

Let α = (α₁, α₂,· · ·, α_n−1) ∈ F_vⁿ⁻¹. Denote by θ_α the character on F_vⁿ⁻¹ such that θα(x₁,· · · ,x_n−1)= θ(α₁x₁+· · ·+α_n−1x_n−1).Extendingθα to a character onN(F_v) byθα(u)=θ(α₁u₁₂+α₂u₂₃+· · ·+α_n−1u_n−1,n),whereu =(uk,l)_1≤k,l≤n ∈N(F_v).Let

φ_v ∈π_v. Define the Whittaker function associated toφandαby W_v(α, λ)= ∫

N(Fv)

φ_v(τ_nu)e^{(λ+ρ)HB(τnu)}θα(u)du.

Let πv,λ = Ind^GL_B(Fⁿ_v^(F₎^v) χv,1| · |^λ1,· · ·, χv,n| · |^λn

be a principal series. Let u_n−1 = (u₁,u₂,· · ·,c_n−1) ∈ F_vⁿ⁻¹.For anyα= (α₁, α₂,· · · , α_n−1) ∈ F_vⁿ⁻¹,we set

Wv,j(

eα_n−1;eλ)=

∫

Nn−1(Fv)

φv(τ_n−1u)e^{(λ+ρ)HBn−1(τn−1u)}θ

eα_n−1(u)du,

witheα_n−1 = (a(u₁)⁻¹a(u₂)α₁,· · ·,a(u_n−2)⁻¹a(u_n−1)α_n−2) ∈ F_vⁿ⁻². Hence the func- tionW_v,j(eα_n−1;λ)e is a Whittaker function on GLn−1associated to the principal series representation Ind^GL_B ^n−1(Fv)

n−1(Fv) χv,2| · |^λ2,· · · , χv,n| · |^λn

and parameter eα_n−1 ∈ F_vⁿ⁻². Let χ_v,l^ur = | · |_v^iνl, ν_l ∈ R, 1 ≤ l ≤ n. Denote by zk,l = λ_k − λ_l +iν_k −iν_l ∈ C, 1≤ k < l ≤ n.

Letx_v ∈F_v.Ifvis an archimedean place, then define the functionsaandsonF_vas follows:

a(xv)=











(1+|x_v|_v²)^−1/2, s(x_v)= x_va(x_v), ifF_v 'R;

(1+|xv|_v)^−1/2, s(xv)= xva(xv), ifFv 'C.

Ifvis a nonarchimedean place, then define the functionsaandsonF_v as follows:

a(x_v)=











1, if xv ∈ O_Fv; x_v⁻¹, otherwise;

s(x_v)=











0, if xv ∈ O_Fv; 1, otherwise.

Lemma 48. Let notation be as above. Assume thatπ_vis rightK_v-finite. Letv ∈ Σ_F be an arbitrary place and letπ_v = Ind^GL_B(Fⁿ_v^(F₎^v) χv,1,· · · , χv,n

be a principal series.

Then the Whittaker functionW_v(α;λ)is equal to Õ

j∈J

∫

F_vⁿ⁻¹

Wv,j(eα_n−1;λ)eeθ_n(u₁,· · · ,u_n−1) Ön

l=2

|a(u_n−l+1)|_v^1+z1,l

n−1

Ö

j=1

du_j, (6.6) where jruns over a finite index set depending only on theK_v-type ofφ_v;and

eθ_n(u₁,· · · ,u_n−1)=θ(α_n−1u_n−1−α_n−2c(u_n−1)u_n−2−α_n−3c(u_n−2)u_n−3

− · · · −α_n−jc(u_n−j+1)u_n−j− · · · −α₁c(u₂)u₁).

Proof. Assume thatvis an archimedean place. Letr ∈Randβ ∈ [0,2π).Then by a straightforward computation we have the Iwasawa decomposition

1 r e^iβ 1

!

= 1

r e^−iβ 1+r² 1

! e^−iβ

√ 1+r²

e^iβ

√1+r²

! _e^iβ

√1+r²

r e^2iβ

√1+r²

−^{r e√−2iβ}

1+r² e^−iβ

√1+r²

!

. (6.7)

Ifvis a nonarchimedean place ofF.We then fix an uniformizer$_v ofF_v^×. For any u ∈ F_v,one can writeu = u^◦$_v^m,for some m ∈ Z,whereu^◦ ∈ O^×

Fv. If m ≥ 0,then u ∈ O_Fv,implying that 1 u

1

!

∈GL(n,O_Fv).Ifm< 0,then one has that

1 u 1

!

= 1 u⁻¹ 1

! u

u⁻¹

! u⁻¹ 1

1 0

!

. (6.8)

Letvbe arbitrary andu ∈ F_v. For any 2 ≤ j ≤ n,1 ≤ l ≤ 4,letM_l(u) = Ml,j(u)be the matrix defined by

M₁(u)=

©

­

­

­

­

­

« I_n−j

1 u 1

I_j−2

ª

®

®

®

®

®

¬

, M₂(u)=

©

­

­

­

­

­

« I_n−j

a(u)⁻¹ s(u) a(u)

I_j−2

ª

®

®

®

®

®

¬

;

M₃(u)=

©

­

­

­

­

­

« I_n−j

a(u)⁻¹ a(u)

I_j−2 ª

®

®

®

®

®

¬

, M₄(u)=

©

­

­

­

­

­

« I_n−j

1 c(u) 1

I_j−2 ª

®

®

®

®

®

¬ ,

wherec(u)= a(u)s(u). Letτn,j = w_n−1· · ·w_n−j+1,2 ≤ j ≤ n.

Denote by w₀ = I d_n, the identity element. Then one has N⁰

2(u) = u, and for any j ≥ 2, N_j⁰(u) = N¹_j(u)M₁(u_n−j+1,n−j+2) and w_n−j+1N_j¹(u)w_n−j+1 = N_j+1⁰ (u⁰), where u⁰ = (u⁰_k,l)_1≤k,l≤n ∈ N(F_v) is defined by u⁰_k,l = uk,l−1 if l = n − j + 2;

u⁰_k,l = u_k,l+1 ifl = n− j+1; u⁰_k,l = u_k+1,l if k = n− j +2; and u⁰_k,l = uk,l,other- wise. Now applying (6.8) one then has thatM₁(u_n+1−j,n−j+2)= M₂(u_n+1−j,n−j+2)k = M₄(u_n+1−j,n−j+2)M₃(u_n+1−j,n−j+2)k, where k ∈ K(Fv), the maximal compact sub- group of GL(n,Fv). Consequently, we have τ_nN_j⁰(u) = τ_nN_j¹(u)M₁(u_n−j+1,n−j+2) = τ_nw_n−j+1N⁰_j+1(u⁰)w_n−j+1M₁(un−j+1,n−j+2),which is equal to

τ_nw_n−j+1N⁰_j+1(u⁰)w_n−j+1M₄(u_n+1−j,n−j+2)M₃(u_n+1−j,n−j+2)k.

Note that we havew_n−j+1M₄(u_n+1−j,n−j+2)= M₁(u_n+1−j,n−j+2)w_n−j+1and N⁰_j+1(u⁰)M₁(u_n+1−j,n−j+2)= M₁(u_n+1−j,n−j+2)N⁰_j+1(u)˜ ,

where ˜u = (u˜k,l)_1≤k,l≤n ∈ N(Fv) is defined by ˜uk,l = u⁰_k,l + u_n+1−j,n−j+2u⁰_k+1,l if k =n−j+1; ˜uk,l = u⁰_k,l+un+1−j,n−j+2u⁰_k,l−

1ifl =n−j+2; and ˜uk,l =u⁰_k,lotherwise.

Therefore,

τ_nN⁰_j(u)= τ_nw_n−j+1M₁(u_n+1−j,n−j+2)N⁰_j+1(u)w˜ _n−j+1M₃(u_n+1−j,n−j+2)k

= M⁰

1(u_n+1−j,n−j+2)τ_nw_n−j+1N⁰_j+1(u)M˜ ₃(u_n+1−j,n−j+2)⁻¹w_n−j+1k, where M⁰

1(un+1−j,n−j+2) = τ_nw_n−j+1M₁(un+1−j,n−j+2)w_n−j+1τ_n⁻¹. Then one has that M⁰

1(u_n+1−j,n−j+2) ∈ N(F_v).Let 2 ≤ j ≤ nandφ_v,λ = φ_ve^{(λ+ρ)HB(·))}.Then W_v(α;λ)= ∫

N(Fv)

φ_v τ_nN⁰

2(u)

e^{(λ+ρ)HB(τnN20(u))}θα(u)du

=

∫

N(Fv)

φv,λ

M₁⁰(u_n−1,n)τ_nw_n−1N⁰

3(u)˜ M₃(u_n−1,n)⁻¹w_n−1k

θα(u)du

=

∫

N(Fv)

φv,λ

M^τn

3 (un−1,n)τnw_n−1N₃⁰(u^∗)w_n−1k

θ⁽_α²⁾(u)du,

where M^τn

3 (u_n−1,n) = diag(a(u_n−1,n),a(u_n−1,n)⁻¹,I_n−2); θ⁽_α²⁾(u) = θ(α₁u₁₂ + · · · + α_n−3u_n−3,n−2+α_n−2a(u_n−1,n)u_n−2,n−1+α_n−1u_n−1,n) ·θ(−α_n−2c(u_n−1,n)u_n−2,n).

Denote by M^τn

2 (u) = I_n. Let j ≥ 3 and M_l, 3 ≤ l ≤ j, be matrices. Denote by Îj

l=2M_lthe matrixM₂· · ·M_j.Define the matrix

M_j^τn(u)=

j

Ö

l=3

©

­

­

­

­

­

«

a(u_n−l+2,n) I_l−3

a(u_n−l+2,n)⁻¹

I_n−l+1

ª

®

®

®

®

®

¬ .

Write ak,l for a(uk,l); and ck,l for c(uk,l). Let β_k(u) = a⁻_k,n¹a_k+1,n. Denote byθ⁽_α^j)(u) the product of

θ(α₁u₁₂+· · ·+α_n−j−1u_n−j−1,n−j+α_n−ja_n−j+1,nu_{n−j,n−j+1}

+α_n−j+1β_n−j+1(u)u_{n−j+1,n−j+2}+· · ·+α_n−2β_n−2(u)u_n−2,n−1)

andθ(α_n−1u_n−1,n−α_n−2c_n−1,nu_n−2,n−α_n−3c_n−2,nu_n−3,n− · · · −α_n−jc_n−j+1,nu_n−j,n),for any 2 ≤ j ≤ n− 1; and θ⁽_αⁿ⁾(u) = θ(α₁β₁(u)u₁₂ + · · ·+ α_n−jβ_n−j(u)un−j,n−j+1 +

· · ·+α_n−2β_n−2(u)u_n−2,n−1) ·θ(α_n−1u_n−1,n−α_n−2c_n−1,nu_n−2,n−α_n−3c_n−2,nu_n−3,n− · · · − α_n−jc_n−j+1,nu_n−j,n− · · · −α₁c_2,nu_1,n).Let

W_v^(j)(α;λ)=∫

N(Fv)

φv,λ

M_j+1^τn (u)τ_nτn,jN_j+1⁰ (u^∗)k_j+1(u)

θ⁽_α^j)(u)

j

Ö

l=2

a²_n−^−l_l+1,ndu.

where k_j+1(u) = τ_n,j⁻¹k_j(u) and k₂(u) = k. Then W_v(α;λ) = W_v⁽²⁾(α;λ). Let N^∗_j+1(u^∗) = (u⁰⁰_k,l)_1≤k,l≤n such that u⁰⁰_k,l = 0 if (k,l) = (n− j +1,n); and u⁰⁰_k,l = u^∗_k,l

otherwise. LetM₄= M₄(u_n−j+1,n),M₃= M₃(u_n−j+1,n).Then a changing of variables leads to thatW_v^(j)(α;λ)is equal to

∫

N(Fv)

φv,λ

M^τ_j+1ⁿ (u)τ_nτn,jN^∗_j+1(u^∗)M₄M₃k_j+1(u)

θ⁽_α^j)(u)

j

Ö

l=2

a²_n−l+1^−l _,ndu =

∫

N(Fv)

φv,λ

M^τ_j+1ⁿ (u)M₄⁰τnτn,j+1N⁰_j+2(u⁰⁰)w_n−jM₃kj+1(u)

θ_α^(j)(u)

j

Ö

l=2

a_n−l^2−l_+1,ndu,

where W⁰

4 ∈ N(F_v). Since φv,λ is left N(F_v)-invariant, the right hand side of the above equality is equal to

∫

N(Fv)

φv,λ

M_j+1^τn (u)τ_nM₃τ_n,j+1N_j+2⁰ (u)w_n−jk_j+1(u)

θ⁽_α^j+1)(u)

j

Ö

l=2

a²_n−l+1^−l _,ndu,

implying that for any 2 ≤ j ≤ n−2, one hasW_v^(j)(α;λ) = W_v^(j+1)(α;λ). By our definition of θ⁽_αⁿ⁾, a similar computation to the above shows that W_v⁽ⁿ⁻¹⁾(α;λ) = W_v⁽ⁿ⁾(α;λ),namely, one has thatW_v(α;λ)is equal to

∫

N(Fv)

φv,λ

M_n^τ₊₁ⁿ (u)τ_nτn,nN_n⁰₊₁(u^∗)k_n+1(u)

θ⁽_αⁿ⁾(u) Ön

l=2

a²_n−l^−l_+1,ndu. (6.9) By definition, one has, for anyφv,λ ∈πv,λ,that

φv(tvxv)= Ön

j=1

χv,j(tv,j)|tv,j|_v^{n+12 −j+λj} ·φv(xv), xv ∈GL(n,Fv). (6.10) Substituting (6.10) into (6.9) one then sees thatWv(α;λ)is equal to

∫

N(Fv)

φv,λ

τ_nτn,nN_n+1⁰ (u^∗)k_n+1(u)

θ⁽ⁿ⁾_α (u)

n

Ö

l=2

χ₁,l(a_n−l+1,n)a_n¹⁺_−l+1^λ1−λ_,n^ldu, (6.11) where χ_1,l(a_n−l+1,n) = χ_v,1(a_n−l+1,n)χ_v,l(a_n−l+1,n)⁻¹. Since π_v is rightK_v-finite, one then sees, according to (6.11), thatW_v(α;λ)is equal to

Õ

j∈J

∫

N(Fv)

φ⁽_v,λ^j)

τ_nτ_n,nN_n⁰₊₁(u^∗)

θ⁽_αⁿ⁾(u) Ön

l=2

χ_1,l(an−l+1,n)a¹⁺_n−l^λ₊₁^1−λ_,n^ldu, (6.12) whereJis a finite set of indexes, whose cardinality depends only on theK_v-finite type ofπ_v; and each φ⁽_v,λ^j) ∈πv,λ.LetWv,j(α;λ)be the summand of (6.12) corresponding to the index j ∈ J.Leteλ_j =λ_j+λ₁/(n−1),2 ≤ j ≤ n.Denote byB_n−1the standard Borel subgroup of GLn−1andN_n−1the unipotent ofB_n−1.Then a change of variables implies thatWv,j(α;λ)is equal to

∫

Fvⁿ⁻¹

W_v,j(eα_n−1;eλ)eθ_n(u₁,· · · ,u_n−1) Ön

l=2

χ_1,l(a_n−l+1)|a_n−l+1|_v^1+λ1−λl

n−1

Ö

j=1

du_j.

Then Lemma 48 follows.

Letv ∈Σ_F be a fixed nonarchimedean place, leteπλ,v be the contragredient of πλ,v. Let $_v be a uniformizer of O_F,v, the ring of integers of F_v. Let q_v = N_Fv/Qp($_v), wherepis the rational prime such thatvis abovep.Denote by

Rv(s,W₁,v,W₂,v;λ):= Ψ_v s,W₁,v,W₂,v;λ,Φ_v L_v(s, πλ,v ⊗τ_v×

eπ−λ,v), Re(s)> 1.

Proposition 49(Nonarchimedean Case). Let notation be as before. Let s ∈ Cbe such thatRe(s) >1.Then we have

(a) R_v(s,W_1,v,W_2,v;λ)is a polynomial in{q_v^s,q^−s_v ,q^λ_vⁱ,q_v^−λi : 1 ≤ i ≤ r}. (b) We have the local functional equation

Ψ_v s,W₁,v,W₂,v;λ,Φ_v L_v(s, πλ,v ⊗τ_v×

eπ−λ,v) = ε(s, πλ,v ×

eπ−λ,v, θ) · Ψ_v(1−s,We₁,v,We₂,v;−λ,¯ bΦ_v) L_v(1−s,eπ_−λ,v¯ ⊗τ¯_v×πλ,v), whereε(s, π_λ,v ×eπ_−λ,v, θ)is a polynomial in{q^s_v,q^−s_v ,q^λ_vⁱ,q_v^−λi : 1 ≤ i ≤ r}. Proof. We shall only prove Part (a), since Part (b) will follow form [JS81].

LetT(Fv) be the maximal torus ofG(Fv), and for any m ∈ Z,letT^(m)(Fv) = {t ∈ T(Fv) : |dett|_Fv = q_v^−m}. Using Iwasawa decomposition and the fact thatWi,v and Φ_v are rightG(O_F,v)-finite, we can rewriteΨ_v s,W_1,v,W_2,v;λ,Φ_v

as Õ

j∈J

∫

T(Fv)

W^(j)

1,v(a_v;λ)W^(j)

2,v a_v;−λ¯

Φ_j,v(ηa_v)τ_v(deta_v)δ_T⁻¹(a_v)|deta_v|^s_F

vda_v,

where the sum over a finite set J, W_i,v^(j)(av;λ) is a Whittaker function associated to some smooth functions in H_P,χ, 1 ≤ i ≤ 2, andΦ_j,v is some Schwartz-Bruhat function. Note that for 1 ≤ i ≤ 2 and j ∈ J,W_i,v^(j)(x_v;λ)is rightG(O_F,v)-finite. So there exists a compact subgroupN_0,v ⊆ G(O_F,v) ∩N(F_v),depending only onϕ,such thatW_i,v^(j)(t_vu_v;λ)= W_i,v^(j)(t_v;λ),for allt_v ∈T(F_v)andu_v ∈ N₀,v.On the other hand, W_i,v^(j)(t_vu_v;λ) = θ_tv(u_v)W_i,v^(j)(t_v;λ),where θ_tv(n_v) = θ(t_vn_vt_v⁻¹), for any n_v ∈ N(F_v).

But then, there exists a constantC_v depending only onN₀,v andθ (hence not onλ) such that θ_tv(u_v) = 1 if and only if |α_i(t_v)| ≤ C_v, where α_i’s are the simple roots of G(F). Thus each W_i,v^(j)(x_v;λ) is compactly supported for a fixed λ ∈ ia_P/ia_G. Therefore, for a fixed λ, Ψ_v s,W₁,v,W₂,v;λ,Φ_v

is a formal Laurent series in q_v^−s. Indeed, one can chose some nonnegative integerMindependent ofλ(but depending onπandϕ), such that

Ψ_v s,W₁,v,W₂,v;λ,Φ_v

= Õ

m≥−M

Ψ^(m)_v W₁,v,W₂,v;λ,Φ_v

·q_v^−ms,