P(F) -conjugacy Classes - A Coarse Jacquet-Zagier Trace Formula for GL( n ) with Applications

For anyG(F)-conjugacy classC,denote by C_r.e.^P(F) the component C₀given in (2.3) ifCis regular, and takeC_r.e.^P(F) to be an empty set ifCis irregular. SinceC_r.e.^P(F)does not intersect any standard maximal parabolic subgroups and is nontrivial only when C is regular, for convenience, we call C_r.e.^P(F) the regular elliptic component of C, despite of the fact that it might not be elliptic.

LetC_r.e.^P⁽^F⁾be the union of regular elliptic components of allG(F)-conjugacy classes in G(F). Then C_r.e.^P⁽^F⁾ is a disjoint union of P(F)-conjugacy classes in G(F) by Proposition 12. Moreover, Proposition 13 and Proposition 12 give a decomposition ofG(F)asP(F)-conjugacy classes

G(F)=C_r.e.^P(F)ÞØⁿ⁻¹

k=1

Q_k(F)^P⁽^F⁾. (4.1)

For any 2 ≤ k ≤ n,let P_k be the standard maximal parabolic subgroup of GLk of type (k −1,1). In the following, we identify P_k with diag(P_k,I_n−k) when view it as a subgroup of G = GL_n. Write W_k the Weyl group of GLk with respect to the

standard Borel subgroup and its Levi component. Let∆_k be the set of simple roots.

LetS_k be the subgroup of symmetric groups S_ngenerated by permutations among {1,2,· · · ,k} ⊂ {1,2,· · ·,n}. For any α ∈ ∆_k, via the isomorphisms and natural inclusionW_k −→^∼ S_k ,→S_n−→^∼ W_n,we identify it with its natural extension in∆,the set of simple roots forG(F)=GLn(F).Henceforth, write∆_k ={α_i,i₊₁: 1 ≤ i ≤ k−1}, and for each simple rootα_i,i₊₁,writew_i^kfor the corresponding simple reflection and identify it withw_iby the natural embedding.

Denote by C_r.e.^P^k^(F) the union of regular elliptic components of all G(F)-conjugacy classes in GLk(F),2≤ k ≤ n.LetR_kbe a set consisting of exactly all representatives of theP_k(F)-conjugacy classesC_r^P_.e.^k^(F).

To compute I∞,Reg(s, τ), an explicit choice of representatives of C_r^P_.e.⁽^F⁾ in Bruhat normal form needs to be taken. We will find at the end of this section that for each 2 ≤ k ≤ n,there exists a particular choice of eachR_k,such that R_k is determined byR_k−₁.Thus a desiredR_ncould be obtained by induction. This will be illustrated in Proposition 29, to prove which, we start with the following result to narrow the candidates of representatives forC_r.e.^P(F).

Lemma 27. Let notation be as before. Set R_P = {w_n−₁w_n−₂· · ·w₁b: b ∈ B(F)}.

Denote byR^P(F)

P the union of P(F)-conjugacy classes of elements inR_P.Then one has

C_r.e.^P(F) =R^P(F)

P . (4.2)

Proof. By Bruhat decomposition, one has G(F)= P(F)Þ

P(F) ·w_n−1·P(F).

For anyg₁ ∈ P(F)andg₂ ∈ P(F) ·w_n−1·P(F),since different Bruhat cells do not intersect, the P(F)-conjugacy class of g₁ does not intersect with that of g₂. Also note thatP(F)-conjugacy classes ofP(F)lie inP(F),so they are not regular elliptic.

Hence we reject all representatives in P(F), and see clearly that P(F)-conjugacy classes inC_r.e.^P⁽^F⁾are represented by elements inw_n−₁P(F).

For any g = w_n−1

A_n−₁ b d_n

∈ w_n−1P(F) ∩C_r^P(F)_.e. , by Bruhat decomposition, either An−1 ∈ Pn−1(F)or An−1 ∈ Pn−1(F)w_n−₂Pn−1(F),wherePn−1is the standard maximal parabolic subgroup of GLn−1(F) of type (n− 2,1). If A_n−₁ ∈ P_n−₁(F), then g ∈ Q_n−₂(F) ⊂ Ð

1≤k≤n−1Q_k(F)^P(F). Thus g < C_r.e.^P⁽^F⁾. Therefore, A_n−₁ ∈

P_n−1(F)w_n−2P_n−1(F). For any 1 ≤ k ≤ n− 1, write R^∗_k the standard parabolic subgroup ofG= GL_nof type(k,1,· · ·,1).So we can write

g⁽⁰⁾ =g = w_n−1

In−2 c b₁ 1 b₂ 1

¬ w_n−2

An−2 cn−2

d_n−₁ d_n

∈w_n−1R^∗_n−₁(F),

which is conjugate byw_n−2

A_n−2 c_n−2

d_n−1

d_n ª

∈P(F)to

g⁽¹⁾ =w_n−₂©

A_n−₂ c_n−₂ d_n−₁

d_n ª

¬ w_n−₁©

I_n−₂ c b₁ 1 b₂ 1

=w_n−₂w_n−₁©

A_n−2 c_n−2

d_n d_n−1

I_n−2 c b₁ 1 b₂ 1

∈w_n−₂w_n−₁R_n^∗₋

2(F).

Again, apply Bruhat decomposition to GLn−2(F) ,→ GL_n(F)to see either A_n−2 ∈ Pn−2(F) or An−2 ∈ Pn−2(F)w_n−₃Pn−2(F), where Pn−2 is the standard maximal parabolic subgroup of GLn−2(F)of type (n−3,1). If A_n−₂ ∈ P_n−₂(F),then g⁽¹⁾ ∈ Q_n−₃(F) ⊂ Ð

1≤k≤n−1Q_k(F)^P(F). Thusg⁽¹⁾ <C_r.e.^P⁽^F⁾.Therefore, A_n−₂ ∈P_n−₂(F)w_n−₃P_n−₂(F). So we can write

g⁽¹⁾ = w_n−2w_n−1

I_n−₃ c⁰ c⁽_n−¹⁾

2 b⁰

1 c⁽_n−²⁾

2 b⁰

1 b⁰

1 ª

¬ w_n−3

A_n−₃ c_n−₃ d_n−₂

d_n d_n−₁

¬ ,

which is conjugate byw_n−₃

A_n−3 c_n−3

d_n−2

d_n dn−1

∈ P(F)to

g⁽²⁾ =w_n−₃

An−3 cn−3

d_n−₂ d_n

d_n−₁ ª

w_n−₂w_n−₁

In−3 c⁰ c_n−⁽¹⁾

2 b⁰

1 c_n−⁽²⁾

2 b⁰

1 b⁰

1 ª

=w_n−3w_n−2w_n−1

A_n−₃ c_n−₃ d_n

d_n−1

d_n−2

I_n−₃ c⁰ c_n⁽¹₋⁾

2 b⁰

1 c_n⁽²₋⁾

2 b⁰

1 b⁰

1 ª

¬ .

Clearly,g⁽²⁾ ∈w_n−3w_n−2w_n−1R_n−^∗

3(F). Continue this process inductively to see that gisP(F)-conjugate to some elementg⁽ⁿ⁻²⁾ ∈w₁w₂· · ·w_n−1R^∗

1(F). Therefore,C_r.e.^P(F) ⊆ {γ^P⁽^F⁾ : γ ∈w₁w₂· · ·w_n−₁R^∗

1(F)}.So we have {g⁻¹: g∈C_r^P_.e.⁽^F⁾} ⊆ {γ^P(F) : γ ∈R^∗₁(F)w_n−1· · ·w₂w₁}

= {γ^P⁽^F⁾ : γ ∈ w_n−1w_n−2· · ·w₁B(F)}, since R^∗

1(F) = B(F) ⊆ P(F). Denote byι :G(F) −→^∼ G(F),g 7→ g⁻¹,the inversion isomorphism. ThenC_r.e.^P⁽^F⁾is stable underι,sinceÐ

1≤k≤n−1Q_k(F)^P(F)is stable under ι.Hence,

C_r^P(F)_.e. = {g⁻¹ : g ∈C_r.e.^P(F)} ⊆ {γ^P⁽^F⁾ : γ ∈w_n−1w_n−2· · ·w₁B(F)}= R^P(F)

P .

Now we show that R^P⁽^F⁾

P ∩Ð

1≤k≤n−1Q_k(F)^P(F) = ∅, which implies by (4.1) that R^P⁽^F⁾

P ⊆ C_r^P_.e.⁽^F⁾.Therefore,C_r.e.^P⁽^F⁾ = R^P⁽^F⁾

P .

Assume that R^P(F)

P ∩Q_k(F)^P(F) , ∅ for some 1 ≤ k ≤ n−1. If k = n−1,then Q_k(F)^P(F) = P(F).Then the assumption forces thatw_n−1w_n−1· · ·w₁ ∈P(F),which is obviously a contradiction. Thus we may assume that 1 ≤ k ≤ n−2. Then by Bruhat decomposition, one has

P(F)= Þ

w∈Wn−1

N(F)wB(F), andQ_k(F)= Þ

w⁰∈Wk

N(F)w⁰B(F).

Forw ∈Wn,denote byC(w)= B(F)wB(F),the Bruhat cell with respect tow.Then the assumptionR^P⁽^F⁾

P ∩Q_k(F)^P(F) ,∅leads to that

C(w)C(w_n−1w_n−2· · ·w₁)C(w⁻¹) ∩C(w⁰),∅. (4.3)

However, Lemma 28 below shows that for any 1 ≤ k ≤ n − 2, any (w,w⁰) ∈ W_n−1×W_k, the intersection in the left hand side of (4.3) is always empty, which

gives a contradiction and thus ends the proof.

Lemma 28. Let notation be as before,1≤ k ≤ n−2,then one has

C(w)C(w_n−₁w_n−₂· · ·w₁)C(w⁻¹) ∩C(w⁰)= ∅, ∀w ∈W_n−₁, w⁰∈W_k. Proof. Recall that for anyw ∈W_nandα ∈∆,we have (see [Spr09], Lemma 8.3.7)

C(sα)C(w)=











C(sαw) ifl(sαw)=l(w)+1,

C(w) tC(sαw) ifl(sαw)=l(w) −1, (4.4) wherel :W →Zis the length function. Also, a similar computation shows that

C(w)C(sα)=











C(ws_α) ifl(ws_α)= l(w)+1,

C(w) tC(ws_α) ifl(ws_α)= l(w) −1. (4.5) Then by (4.4) and (4.5), one obtains that

C(w)^s^α =













C(sαwsα), ifl(sαwsα)=l(w)+2;

C(sαw) tC(sαwsα), ifl(sαw)< l(w),l(sαwsα)> l(sαw); C(ws_α) tC(s_αws_α), ifl(ws_α)< l(w),l(s_αws_α)> l(ws_α); C(w) tC(sαw) tC(ws_α) tC(sαws_α), otherwise,

(4.6)

where we useC(w)^s^α to denote byC(s_α)C(w)C(s_α).

Letw⁰ ∈W_k andw ∈W_n−₁. Letl(w)be the length ofw. Thenw could be written as a products ofl(w)simple reflectionss_i,1 ≤i ≤ n−2,and eachs_icorresponds to the associated reflection of some simple roots inW_n−1.

• Assume that l(w_n−1· · ·w₂w₁w⁻¹) = l(w_n−1· · ·w₂w₁)+ l(w⁻¹). Take w = s_l(w)· · ·s₂s₁ to be a reduced representation by simple reflections and apply (4.4) and (4.5) inductively one then sees that

C(w)C(w_n−1w_n−2· · ·w₁)C(w⁻¹)=C(ww_n−1w_n−2· · ·w₁w⁻¹).

We will simply identify Weyl elements in W_n with translations on the set {1,2,· · · ,n} under the isomorphism W_n −→^∼ S_n. Then the cycle type decomposition ofww_n−₁w_n−₂· · ·w₁w⁻¹ is the same as that ofw_n−₁w_n−₂· · ·w₁, which is ann-cycle. However, since elements inW_k can never ben-cycles, C(ww_n−1w_n−2· · ·w₁w⁻¹) ∩C(w⁰)= ∅,∀w⁰∈W_k.

• Assume that l(w_n−1· · ·w₂w₁w⁻¹) < l(w_n−1· · ·w₂w₁)+ l(w⁻¹). Denote by D(w)the set of all possible reduced representations of w⁰ by simple reflections. Then by our assumption, one can take a reduced representation of w = s⁰_l_(w)· · ·s⁰

2s⁰

1such thats⁰

1 =w₁.Hence one can well define j_w := max

1≤j≤l(w)

s_l_(w)· · ·s₂s₁∈ D(w): s_i= w_i, 1 ≤i ≤ j .

Letw = s_l(w)· · ·s₂s₁be a reduced representation such thats_i = w_i,1≤ i ≤ j_w. Thenw⁻¹ = s₁s₂· · ·s_l(w).Also, by (4.4) or (4.5) we have

C(w)=C(s_l(w)) · · ·C(s₂)C(s₁),andC(w⁻¹)=C(s₁)C(s₂) · · ·C(s_l(w)), so

C(w)C(w)C(we ⁻¹)=C(s_l_(w)) · · ·C(s₂)C(s₁)C(w)C(se ₁)C(s₂) · · ·C(s_l_(w)), where we denote by we = w_n−₁w_n−₂· · ·w₁ for convenience. According to (4.6), a brute force computation shows that

C(w)C(

ew)C(w⁻¹)=C(w^∗)Þ Þ

1≤i≤jw

C(w⁽ⁱ⁾),

where w^∗ = ww_n−1· · ·w_j_w₊₂w_j_w₊₁s_j_w+1· · ·s_l(w), and for 1 ≤ i ≤ jw, w⁽ⁱ⁾ = ww_n−₁· · ·w_i₊₁w_iw_i₊₁· · ·w_j_wsjw+1· · ·s_l(w).

Letw(jw) = s_l(w)· · ·s_j_w+1,w₍^∗

jw) = w_j_w· · ·w₁w_n−1· · ·w_j_w₊₂w_j_w₊₁.Then w^∗₍

jw)

is ann-cycle, and thusw^∗ = w(jw)w₍^∗

jw)w₍⁻¹

jw) is also ann-cycle. Sow^∗ <Wk, implying thatC(w^∗) ∩C(w⁰)= ∅,∀w⁰ ∈W_k.

For each 1≤ i ≤ jw,let

w^∗_(i) = w_i−₁· · ·w₁w_n−₁w_n−₂· · ·w_i₊₁,

w_(i) = s_l(w)· · ·s_j_w₊₁w_j_w· · ·w_i. Then w⁽ⁱ⁾ = w_(i)w_(i)^∗ w⁻_(i)¹. One can check that w_(i)^∗ = (1,2,· · · ,i)(i+1,· · · ,n),i.e. the cycle type ofw^∗_(i) is(i,n−i). Sow⁽ⁱ⁾ also has cycle type (i,n−i). Since elements in W_k can never have type of the form(i,n−i),w⁽ⁱ⁾ < W_k. Therefore, C(w^∗) ∩C(w⁽ⁱ⁾) = ∅,∀1 ≤ i ≤ j_w, w⁰ ∈W_k.

This completes the proof.

Now we consider P(F)-conjugation among elements in R_P = {w_n−1w_n−2· · ·w₁b : b ∈ B(F)} to determine representatives of R^P(F)

P . Define a relation R on the set

w_n−1w_n−2· · ·w₁tu : t∈T(F^×), u ∈ N(F)

such thatw_n−1w_n−2· · ·w₁tuis related tow_n−1w_n−2· · ·w₁t⁰u⁰if and only ifu= u⁰and dett= dett⁰,which is equivalent to that there are elementsa₁,· · · ,a_n= a₀∈ F^×such that













a₀t₁a⁻¹

1 =t⁰

a₁t₂a⁻¹

2 =t⁰ ... 2

a_n−₁t_na⁻_n¹=t_n⁰.

(4.7)

One can check easily thatR^P⁽^F⁾

P forms an equivalence relation.

Proposition 29. Let notation be as before. Set Re_P =

w_n−₁w_n−₂· · ·w₁tu : t ∈T F^×, u ∈ N_P(F)

/R.

ThenRe_Pforms a family of representatives ofR^P(F)

P .

Proof. Let w_n−1w_n−2· · ·w₁b and w_n−1w_n−2· · ·w₁b⁰ be two elements in R_P, and writeb =t_nu,b⁰= t_n⁰u⁰,the corresponding Levi decomposition. Assume that there exists some p_n∈ P_n(F)= P(F)such that

p_nw_n−1w_n−2· · ·w₁bp⁻_n¹ =w_n−1w_n−2· · ·w₁b⁰. (4.8) Thenw_n−₁p_nw_n−₁= w_n−₂· · ·w₁b⁰p_nb⁻¹w₁· · ·w_n−₂ ∈P(F) =Q_n−₁(F). Sincep_n ∈ P(F),it is necessary of the following form

p_n= ©

A_n−₂ c_n−₁ c_n a_n−1 0 a_n

∈©

GL_n−₂(F) ∗ ∗ F^× 0 F^×

⊂Q_n−₂(F).

Hence,w_n−₂w_n−₁p_nw_n−₁w_n−₂= w_n−₃· · ·w₁b⁰p_nb⁻¹w₁· · ·w_n−₃ ∈Q_n−₂(F),i.e.,

w_n−2

A_n−2 c_n c_n−1

a_n 0 a_n−1

w_n−2 ∈©

GL_n−2(F) ∗ ∗ F^× 0 F^×

⊂ Q_n−2(F).

ThenA_n−2must lie in a maximal parabolic subgroup of GLn−2(F)of type(n−3,1), and the last component ofc_nmust be vanishing. Thus we can write

p_n =

A_n−₃ c_n−₂ c_n−⁽ⁿ⁻³⁾

1 c_n⁽ⁿ⁻³⁾ a_n−₂ c_n−₂,n−1 0

a_n−₁ 0 a_n

∈

GL_n−₃(F) ∗ ∗ ∗ F^× ∗ 0 F^× 0 F^×

¬ ,

where for any column vectorc_i = (c₁,i,c₂,i,· · · ,cm,i)^T,any 1 ≤ k ≤ m, writec^(k)

i =

(c₁,i,c₂,i,· · ·,ck,i)^T,namely, the firstk-entries. Now a similar analysis on the identity w_n−₃w_n−₂w_n−₁p_nw_n−₁w_n−₂w_n−₃=w_n−₄· · ·w₁b⁰p_nb⁻¹w₁· · ·w_n−₄ ∈Q_n−₃(F) leads toc_n=c_n⁽ⁿ⁻⁴⁾,namely, the last 4 elements ofc_nare all zeros. Likewise, continue this process(n−4)-more times to getc_n= 0.Now (4.8) becomes

a_n 0 0 . . . 0

a₁ c₁,2 · · · c₁,n−1

. . . ...

a_n−₂ c_n−₂_,n−₁ a_n−₁

¬ b

a₁ c₁,2 · · · c₁,n−1 0 a₂ · · · c₂,n−1 0 . . . ... ...

a_n−₁ 0 a_n

−1

= b⁰. (4.9)

When expanded, (4.9) becomes (4.13), which will be investigated below. Before seeking for a solution to (4.9), we will simplify it by taking showing that one can actually only consider some specialbandb⁰. This is justified by Claim 30 below.

Writet_n =diag(t₁,· · · ,t_n),and sett_i,j = t_it⁻_j¹; for anyn−2 ≤ k ≤ n−1,define

n_n−k(F)= (

« Ik uk

I_n−k−₁ ª

: u_k ∈ M_k×1(F) )

Letu_n−k =u∩n_n−k(F),n−2≤ k ≤ n−1.Thenu= u₁u₂.

Claim 30. For any b ∈ B(F), there exists a unique u ∈ n₂(F) ⊂ P(F),such that u⁻¹w_n−₁w_n−₂· · ·w₁bu ∈w_n−₁w_n−₂· · ·w₁T(F)n₁(F).

So we only need to consider the P(F)-conjugacy of among elements in R = {w_n−₁w_n−₂· · ·w₁tu : t∈T(F^×), u ∈ NP(F)}.

Let w_n−1w_n−2· · ·w₁t_nu₁, w_n−1w_n−2· · ·w₁t⁰_nu⁰

1 ∈ Rbe P(F)-conjugate. Then there exists some p_n=diag(b,a_n) ∈diag(B(F),F^×)such that

p_nw_n−1w_n−2· · ·w₁t_nu₁p⁻_n¹= w_n−1w_n−2· · ·w₁t⁰_nu⁰

1. (4.10)

Writeb=diag(a₁,a₂,· · · ,a_n−1)u,where we identityuwith diag(u,1) ∈n₂(F).Then comparing the Levi components of both sides in (4.10) leads exactly the system of relations (4.7); while the unipotent radical gives the equation (4.13) with c_i,j⁰ = 1, 1≤ i < j ≤ n−1.By the uniqueness of solution (shown in the proof of Claim 30), u = In. Therefore,u₁=u⁰

1∈n₁(F)= NP(F).Then the proof follows.

Proof of Claim 30. Let notation be as in the proof of Proposition 29. Let u = {ui,j}₁_≤_i,j_≤_n ∈ n₂(F),andc = {ci,j}₁_≤_i,j_≤_n = w₁· · ·w_n−1u⁻¹w_n−1· · ·w₁t_nut_n⁻¹. De- note byu^× = {u_i,j⁰ }₁_≤_i,_j_≤_n ∈n₂(F).Then one has, for any 1 ≤i < j ≤ n,that

ui,j+u⁰_i,i+1u_i+1,j+u⁰_i,i+2u_i+2,j+· · ·+u⁰_i,j₋₁u_j−1,j+u⁰_i,j = 0. (4.11) Also, an elementary computation shows that for any 1 ≤i < j ≤ n,one has

ci,j =t_it⁻_j¹ui,j+t_i+1t⁻_j¹u_i−⁰ ₁_,iu_i+1,j+· · ·+t_j−1t⁻_j¹u⁰_i−₁_,j−₂u_j−1,j +u⁰_i−₁_,j−₁. (4.12) Now fix c_i,j⁰, 1 < i < j < n,andt_n,then we show by a double induction that there exists uniquelyui,j,1 < i < j < n,such thatci,j = c_i,j⁰,1 < i < j < n,i.e., we want to solve the system of equations, for fixedt₁,· · · ,t_n,

t_it⁻_j¹u_i,_j+t_i₊₁t⁻_j¹u⁰_i−

1,iu_i₊₁_,j+· · ·+t_j−₁t⁻_j¹u_i−⁰

1,j−2u_j−₁_,j+u_i−⁰

1,j−1 =c⁰_i,j. (4.13) Whenn ≤ 4, one can check directly by hand that the solution to (4.13) exists and is unique. So from now on we assume thatn > 4. LetD_i = {Di,j = u_j+1_,j+i+1 : 1 ≤ j ≤ n−2−i},1 ≤ i ≤ n− 3. By (4.12), c₁,j = t₁t⁻_j¹u₁,j, so to make c¹^,j = c⁰

1,j, one takesu₁,j = t⁻¹

1 t_jc⁰

1,j,1 ≤ j ≤ n−1. Also, by (4.11),u_i,i⁰₊₁ = −u_i,i+1,so (4.12) shows thatci,i+1= ui,i+1+u⁰_i−

1,i= ui,i+1−ui−1,i,2≤ i ≤ n−2.Letci,i+1= c_i,i⁰₊₁,then u_i,i₊₁ = u_i−₁_,i+c⁰_i,i₊₁,2 ≤ i ≤ n−2. Sinceu₁₂ = t⁻¹

1 t₂c⁰

1,2,then a simple induction shows that elements in D₁ are uniquely determined by the equation c_i,j = c_i,⁰_j, 1< i < j < n.

Now, let 1 < i₀ ≤ n −3, assume that D_i are uniquely solved out by (4.13) for any 1 < i < i₀. By our assumption and (4.11), u⁰

1,i are now uniquely determined, 1 ≤ i ≤ i₀−1. Then according to (4.12), Di₀,1 = ui₀+1,i₀+2 = c_i⁰

0+1,i₀+2+u₁,i₀+1+ u⁰

1,2u₃_,i

0+2+u⁰

1,3u₄_,i

0+2+· · ·+u⁰

1,i₀u_i

0+1,i₀+2,whereu_i,i

0+2∈ D_i

0+2−i,3≤ i ≤ i₀+1.So D_i

0,1is uniquely determined. Assume that we have solved out allD_i

0,j,j < j₀,inD_i

0. Then by (4.12),D_i₀,j₀ = u_j₀₊₁,j₀+i0+1completely depends onu⁰_j

0,j,j₀+1 ≤ j ≤ j₀+i₀, and u_j₀_+k,_j₀_+i₀₊₁ ∈ D_i

0+1−k, 2 ≤ k ≤ i₀. Again, by (4.11), we can inductively compute each u⁰_j

0,j in terms of u_i⁰,j⁰, j₀ ≤ i⁰ < j⁰ ≤ j such that (i⁰,j⁰) , (j₀,j).

By our inductive assumption, all theseu_i⁰,j⁰’s andu_j₀_+k,j₀_+i₀₊₁have been solved out uniquely. Then D_i₀,j₀ is thus obtained. By induction, elements in D_i

0 are uniquely determined. Therefore, by induction on the indexi₀,one verifies that the solution to (4.13) does exist and in fact is unique.

Denote by u₀ = u_c0,t_n ∈ n₂(F) the solution to (4.13). Then u₀ depends only on c⁰ = {c_i,j⁰ }₁_≤i,_j≤n and t_n, where we define c_i,n⁰ = δ_i,n, 1 ≤ i ≤ n, here δ is the Kronecker symbol. Letb = u₂u₁t_n be an arbitrary element in B(F). Takec⁰ = u₂, u₀ ∈ n₂(F)the solution to (4.13), and definen_P = u₀t_n⁻¹u₁t_nu⁻¹

0 ∈n₁(F).Then the following conjugacy equation holds:

u⁻¹

0 w_n−₁w_n−₂· · ·w₁t_nn_Pu₀ =w_n−₁w_n−₂· · ·w₁u₂u₁t_n. Therefore, one can take representatives ofP(F)-conjugacy classesR^P(F)

P in the set R = {w_n−1w_n−2· · ·w₁tu : t ∈T(F^×), u ∈ N_P(F)}. Remark 31. Letγ = w_n−₁w_n−₂· · ·w₁tu ∈GL_n(F),t ∈T(F^×),u ∈ N(F),then the P(F)-conjugacy class ofγ is thoroughly determined bydetγ andu∩N_P(F).

Now we consider for our purpose the decomposition of ZG(F)\G(F) into P(F)- conjugacy classes. By (4.1) one has the following decomposition

Z_G(F)\G(F)= Z_G(F)\C_r.e.^P(F)ÞØⁿ⁻¹

k=1

(Z_G(F)\Q_k(F))^P⁽^F⁾. (4.14) Corollary 32. Let notation be as before. Set(F^×)ⁿ = {tⁿ : t ∈F^×},and let

Re^∗_P = (

« I_n−₃

t I₂

u : t ∈ F^×/ F^×n, u ∈ N_P(F) )

. (4.15)

ThenRe^∗

Pforms a family of representatives of Z_G(F)\C_r^P(F)_.e. . Proof. By Lemma 27 and Proposition 29,

(

« In−2

t 0 1 ª

u: t ∈ F^×/ F^×n, u ∈N_P(F) )

(4.16)

forms a family of representatives of Z_G(F)\C_r.e.^P⁽^F⁾. Then the inverse of elements in the set defined in (4.16) also form a family of representatives ofZ_G(F)\C_r.e.^P(F).Note that these inverses are bijectivelyP₀(F)-conjugate toRe^∗

P,then the proof follows.

Dalam dokumen A Coarse Jacquet-Zagier Trace Formula for GL( n ) with Applications (Halaman 50-60)