Degenerate Whittaker models for p-adic groups after C. Mœglin and J-L. Waldspurger ∗

Degenerate Whittaker models for p-adic groups after C. Mœglin and J-L. Waldspurger ^∗

Takuya KONNO ^†, Kazuko KONNO ^‡ 20, July 1998

Abstract

In this note we present a detailed account for a work of Mœglin-Waldspurger on the dimension of the space of degenerate Whittaker vectors [MW]. We hope to extend this to the twisted case. This is in order to prove that certain twisted endoscopy forGL(n) implies the (local) generic packet conjecture for many reductive groups.

Contents

1 Degenerate Whittaker models 2

1.1 Some estimations . . . . 2

1.2 The lattice Λ⁰ . . . . 4

1.3 The subgroup K_n . . . . 6

1.4 The unipotent subgroup attached to (N, φ) . . . . 6

1.5 Remark . . . . 7

1.6 The character χ_n . . . . 8

1.7 Degenerate Whittaker models . . . 10

1.8 Character expansion . . . 11

1.9 System of isotypic spaces . . . 12

1.10 . . . 13

1.11 . . . 14

1.12 . . . 15

∗Note of the Automorphic Seminar at Kyushu University, 1998.

†Graduate School of Mathematics, Kyushu University, 812-80 Hakozaki, Higashi-ku, Fukuoka, Japan E-mail: [email protected]

URL: http://knmac.math.kyushu-u.ac.jp/^∼tkonno/

The author is partially supported by the Grants-in-Aid for Scientific Research No. 12740019, the Ministry of Education, Science, Sports and Culture, Japan

‡JSPS Post Doctoral Fellow at Graduate School of Human and Environmental Studies, Kyoto Uni- versity, 606-8501 Yoshida, Sakyo-ku, Kyoto, Japan

E-mail: [email protected]

URL: http://knmac.math.kyushu-u.ac.jp/^∼kkonno/

1.13 . . . 17

1.14 Key injectivity . . . 21

1.15 Injectivity continued . . . 23

1.16 The result . . . 25

1.17 . . . 26

1 Degenerate Whittaker models

1.1 Some estimations

Let F ⊃ O ⊃ p_F = ($) be the usual notation for a non-archimedean local field of characteristic zero. Let p be the residual characteristic of F, and write [F : Qp] = ef where e is the order of ramification andf is the modular degree. Forx∈Q_p, we have

ord_F(x) =−log_p^f |x|^ef_p =e·ord_Qp(x).

On the other hand, if a∈N then ord_Qp(a!) =

[logX_pa]

k=1

k¡· a p^k

¸

−

· a p^k+1

¸¢

=

[logX_pa]

k=1

· a p^k

¸

≤ X∞

k=1

a

p^k = a p−1. These yield

ordF(a!)≤ ae

p−1. (1.1.1)

Let us consider a connected reductive F-group G. Let g be its Lie algebra. We fix a non-degenerate symmetric bilinear form B(, ) on g(F). We need an F-group embedding ι:G ,→GL(n). This gives a LieF-algebra embedding ι:g(F),→gl(n, F) =M_n(F). Let Λ^ι be the O-lattice ι⁻¹(M_n(O)) of g(F). Clearly we have [Λ^ι,Λ^ι] ⊂ Λ^ι. Choose A⁰ ∈ N such that B(X, Y)∈ O, for any X,Y ∈$^A0Λ^ι, and set Λ := $^A0Λ^ι, Λ :=e $^A0Mn(O). Λ is an O-lattice in g(F) satisfying

(i)B(Λ,Λ)⊂ O, (ii) [∗,∗]⊂ ∗, (iii) ι(∗)ι(∗)⊂e∗.

We shall use the followings.

Lemma . (a) There exists A∈N such that exp|_$^A_Λ is injective.

(b) For c∈N there exists A₁ := sup(A,_p−1^3e +c+ 2) such that

∀X ∈$ⁿΛ, ∀Y ∈$^mΛ, with n, m≥A₁ (1.1.2)

log(expXexpY)−(X+Y +1

2[X, Y])∈$^n+m+cΛ,

∀X ∈$ⁿΛ with n≥A₁, ∀Y ∈$^mΛ (1.1.3)

Ad(expX)Y −(Y + ad(X)Y)∈$^2n+m−[p−1^2e ]Λ.

The term − h 2e

p−1

i

is unnecessary if the residual characteristic of F is not 2.

Remark . In [MW, 1.1.3], it was stated that

Ad(expX)Y −(Y + ad(X)Y)∈$^n+m+cΛ, ∀X ∈$ⁿΛ, ∀Y ∈$^mΛ, instead of (1.1.3). But this is insufficient.

Proof. (a) is well-known (cf. [H]).

To prove (1.1.2) we identify X ∈g(F) withι(X)∈M_n(F). By symmetricity, we may assume that n≥m≥A₁. We fixk ∈N such that

(k+ 1)¡

m− e p−1

¢ ≥n+m+c+ 1.

It follows from (1.1.1) that X^a

a! ∈$^na−[p−1^ae ]Λe ⊆$[⁽ⁿ⁻p−1^e )a]Λ.e In particular we have

expXexpY ∈ µ

1 +X+ X²

2 +$[³⁽ⁿ⁻p−1^e )]Λe

¶ ÃX_k

j=0

Y^j

j! +$[^(k+1)(m−p−1^e )]Λe

! . 2n−m ≥n ≥A₁ gives

·

3(n− e p−1)

¸

>3 µ

n− e p−1

¶

−1> n+m+c, and our choice of k implies

·

(k+ 1)(m− e p−1)

¸

>(k+ 1)(m− e

p−1)−1≥n+m+c.

Thus we obtain expXexpY ∈

µ

1 +X+X²

2 +$^n+m+cΛe

¶ µ

1 +Y +Y²

2 +· · ·+Y^k

k! +$^n+m+cΛe

¶

= 1 +X+Y + X²

2 +XY + Y²

2 +X²Y

2 +XY² 2 + Y³

3! +. . . + X²Y^k−2

2(k−2)! + XY^k−1 (k−1)! + Y^k

k! +$^n+m+cΛe

= 1 +X+Y + X²

2 +XY + Y² 2 +

Xk

j=3

Y^j

j! +$^n+m+cΛ.e

(*) Note that if j ≥1,` ≥2, the choice ofA1 gives

· j

µ

n− e p−1

¶¸

+

·

` µ

m− e p−1

¶¸

> j µ

n− e p−1

¶ +`

µ

m− e p−1

¶

−2

≥n− e p−1 + 2

µ

m− e p−1

¶

−2 = (n+m+c) +m− µ 3e

p−1 +c+ 2

¶

≥n+m+c.

On the other hand, similar estimation gives exp(X+Y + 1

2[X, Y])

∈1 +X+Y +[X, Y]

2 +

Xk

j=2

1 j

µ

X+Y + [X, Y] 2

¶_j

+$^n+m+cΛe

= 1 +X+Y + XY −Y X

2 +X²+XY +Y X+Y²

2 +

Xk

j=3

Y^j

j! +$^n+m+cΛ.e

(∗∗)

(∗) and (∗∗) imply

expXexpY ∈exp(X+Y + [X, Y]

2 +$^n+m+cΛ),e and the assertion follows.

For (1.1.3), we still identify X with ι(X). Note that n ≥A₁ implies

·

3(n− e p−1)

¸

>3n− 3e

p−1−1≥2n+c+ 1.

Thus we have Ad(expX)Y ∈

µ

Y +XY + X²Y

2 +$^2n+m+c+1Λe

¶ µ

1−X+X²

2 +$^2n+c+1Λe

¶

=Y +XY −Y X+X²Y

2 −XY X+Y X²

2 +$^2n+m+c+1Λe

=Y + [X, Y] + X

2[X, Y]−[X, Y]X

2 +$^2n+m+c+1Λ.e Noting X/2∈$ⁿ⁻[p−1^2e ]Λ, [X, Ye ]∈$^n+mΛ, we get (1.1.3).e

1.2 The lattice Λ⁰

LetN ∈Λ be a nilpotent element. We writeg^N andG^N for the centralizer ofN ingand G, respectively.

Lemma . X ∈g^N(F) if and only if B(N,[X, Y]) = 0 for any Y ∈g(F).

Proof. We may assume that g = gl(n) and B(X, Y) = tr(XY). Then the statement becomes that XN = NX if and only if tr(NXY) = tr(XNY) (= tr(NY X)) for any Y ∈M_n(F). This is clear since the latter condition reads

B(NX, Y) =B(XN, Y), ∀Y ∈gl(n, F) and B is non-degenerate.

Let φ be arbitrary element in X_∗(G)_F such that

Ad(φ(t))N =t⁻²N, ∀t∈G_m. We putgi :={X ∈g|Ad(φ(t))X =tⁱX, ∀t∈Gm} so that

g=M

i∈Z

g_i.

Since g^N is Ad(φ(G_m))-stable, we may choose m⊂g such that

g=g^N ⊕m, (1.2.1)

m is Ad(φ)-stable. (1.2.2)

We putmi =m∩gi: m=L

i∈Z mi.

Now it follows from the above lemma that

B_N :g×g3(X₁, X₂)7−→B(N,[X₁, X₂])∈G_a

is an alternating form andB_N|_m is non-degenerate. Since the definition ofφ implies B_N(Ad(φ(s))X₁,Ad(φ(s))X₂) =B(N,Ad(φ(s))[X₁, X₂])

=B(Ad(φ(s⁻¹))N,[X₁, X₂]) = B(s²N,[X₁, X₂]) =s²B_N(X₁, X₂), the duality BN is betweenmi and m2−i. Thus we can choose

• Basis{X₁, . . . , X_k}of⊕^−∞_i=0 m_i and{X₁^∗, . . . , X_k^∗}of⊕^∞_i=2m_idual to each other under B_N, and

• A Witt basis{Z1, . . . , Z`;Z<sub>1</sub><sup>∗</sup>, . . . , Z<sub>`</sub>^∗} of m1 for BN, so that

B_N(X_i, X_i^∗) = 1, 1≤i≤k, B_N(Z_i, Z_i^∗) = 1, 1≤i≤`.

We put

m^Λ0 = Xk

i=1

(OX_i+OX_i^∗) + X`

i=∞

(OZ_i+OZ_i^∗) and let

Λ⁰ :=m^Λ0 +X

i∈Z

Λ∩g^N_i (F), g^N_i :=gi∩g^N. This is a lattice in g(F), and we can take d∈N such that

$^dΛ ⊂Λ⁰ ⊂$^−dΛ.

1.3 The subgroup K_n

Lemma . Fix C ≥d−1. For D≥sup(A1+d, C+ 3d), we have the followings.

(1) exp|_$^D_Λ⁰ : $^DΛ⁰ → exp($^DΛ⁰) is a bijection. K_n := exp($ⁿΛ⁰) is a subgroup of G(F) for n≥D.

(2) For n, m≥D, X ∈$ⁿΛ⁰ and Y ∈$^mΛ⁰, we have log(expXexpY)−(X+Y + 1

2[X, Y])∈$^n+m+CΛ.

(3) For n≥D, m∈Z, X ∈$ⁿΛ⁰ and Y ∈$^mΛ⁰, we have

Ad(expX)Y −(Y + ad(X)Y)∈$^2n+m−3d−[p−1^2e ]Λ.

The term − h 2e

p−1

i

is not necessary if the residual characteristic of F is odd.

Proof. Putting c = 2d+C, we deduce (2), (3) and the first assertion of (1) from the lemma in 1.1. We shall check that exp($ⁿΛ⁰) is a group. For X, Y ∈$ⁿΛ⁰ with n≥D, we need to show that log(expXexpY)∈$ⁿΛ⁰. We have from (2) that

log(expXexpY) =X+Y +1

2[X, Y] +Z, ∃Z ∈$^2n+C−dΛ⁰. Then the assertion follows from

1

2[X, Y]∈$[²(^n−d−p−1^e )]Λ⊂$²(^n−d−p−1^e )^−d−1Λ⁰, and

2 µ

n−d− e p−1

¶

−d−1 =n+n−3d− 2e p−1 −1

≥n+ e

p−1 +C+ 1−d > n.

1.4 The unipotent subgroup attached to (N, φ)

We now put

u:=X

i>0

g_i, ,u:=X

i<0

g_i, p:=X

i≤0

g_i,

and set U := exp(u) and P := Stab(u, G). P is a parabolic subgroup of G such that LieP =p. For n ≥D, we set K⁰_n := Ad(φ($⁻ⁿ))K_n.

Lemma .

K_n= (K_n∩P(F))(K_n∩U(F)) = (K_n∩U(F))(K_n∩P(F)).

Proof. Lemma.1.3 implies

K_n∩U(F) = exp($ⁿΛ⁰ ∩u(F)), K_n∩P(F) = exp($ⁿΛ⁰∩p(F)).

Also the definition of Λ⁰ gives Λ⁰ = Λ⁰∩p(F)⊕Λ⁰∩u(F).

Now let g = expX ∈ K_n (X ∈ $ⁿΛ⁰). We can choose Y₀ ∈ $ⁿΛ⁰ ∩ p(F) and Z₀ ∈ $ⁿΛ⁰ ∩u(F) such that X = Y₀ +Z₀. If we set y₀ := expY₀ and z₀ := expZ₀, then Lemma.1.3 (2) asserts that g1 :=y₀⁻¹gz₀⁻¹ ∈ Kn+1. Thus we can take X1 ∈$ⁿ⁺¹Λ⁰ such that g₁ = expX₁. Take Y₁ ∈ $ⁿ⁺¹Λ⁰ ∩p(F) and Z₁ ∈ $ⁿ⁺¹Λ⁰ ∩u(F) satisfying X₁ =Y₁ +Z₁. Again Lemma.1.3 (2) gives

g1 :=y₁⁻¹g1z₁⁻¹ ∈Kn+2, y1 := exp(Y1), z1 := exp(Z1).

By repeating this process, we can take {y_m ∈ K_n+m ∩P(F)}_m∈N and {z_m ∈ K_n+m ∩ U(F)}_m∈N such that

g_n+m := (y₀· · ·y_m)⁻¹g(z_m· · ·z₀)⁻¹ ∈K_n+m, ∀m∈N.

We may choose subsequences of {(y₀· · ·y_m)}_m∈N and {(z_m· · ·z₀)}_m∈N which are conver- gent. We write the limit of these sequences as y and z, respectively. Then

y⁻¹gz⁻¹ ∈ \

m≥0

K_n+m={1},

and hence g =yz with y∈K_n∩P(F), z ∈K_n∩U(F).

1.5 Remark

(1) Put

Q_n := Ad(φ($⁻ⁿ))(K_n∩P(F)), n≥D V_n := Ad(φ($⁻ⁿ)) exp($ⁿΛ⁰∩U(F)), n ≥0.

Then we have

K⁰_n=Q_nVn. (1.5.1)

Consider the two groups on the right hand side.

It follows from the definition that

$ⁿΛ⁰ ∩p(F) = $ⁿm^Λ0∩p(F) +X

i≤0

$ⁿΛ∩g^N_i (F)

=$ⁿ³X^k

j=1

OX_|+X

i≤0

∗ ∩g^N_i (F)´ .

Thus{Q_n}_n≥D is a system of fundamental neighborhoods of 1 in P(F).

Next comes V_n. expX belongs to V_n if and only if X lies in Ad(φ($⁻ⁿ))($ⁿΛ⁰ ∩u(F))

= Ad(φ($⁻ⁿ))$ⁿ

³X^k

j=1

OX_|^∗+ X`

|=∞

(OZ_|+OZ_|^∗) + X

i≥∞

∗ ∩g^N_i (F)

´

which clearly contains Xk

j=1

$⁻ⁿOX_|^∗+ X`

|=∞

(OZ_|+OZ_|^∗) +∗ ∩g^N_∞(F) +X

i≥∈

$^−\∗ ∩g^N_i (F)

=$⁻ⁿ



X^k

j=1

OX_|^∗+X

i≥∈

∗ ∩g^N_i (F)



+ Λ∩g^N₁ (F) + X`

j=1

(OZ_|+OZ_|^∗).

ThusV_n becomes bigger and bigger as n increases. Since X`

j=1

(OZ|+OZ_|^∗) +∗ ∩g^N_∞(F) = ∗⁰ ∩g∞(F),

we have [

n≥0

V_n= exp¡

Λ⁰ ∩g₁(F)¢

·U²(F). (1.5.2)

with U²(F) := exp¡P

i≥2 g_i(F)¢ .

(2) It follows form Lemma.1.3 (3) that K_m CK_n if m≥n ≥D.

1.6 The character χ_n

We fix a non-trivial character ψ of F of order zero.

Lemma . Suppose that n ≥D and the residual characteristic of F is not 2.

(1) χn:Kn3expX 7→ψ(B($⁻²ⁿN, X))∈C¹ is a character.

(2) Let n ≥m≥D. Then Stab(χ_n,K_m) is contained in K_nexp($^mΛ∩g^N(F)).

Proof. (1) If we write X, X⁰ ∈$ⁿΛ⁰ in the form X =

Xk

j=1

$ⁿ(x^∗_jX_j^∗+x_jX_j) + X`

j=1

$ⁿ(z_jZ_j+z_j^∗Z_j^∗) +$ⁿZ,

X⁰ = Xk

j=1

$ⁿ(x^0∗_jX_j^∗+x⁰_jX_j) + X`

j=1

$ⁿ(z_j⁰Z_j +z^0∗_jZ_j^∗) +$ⁿZ⁰, with xj,x^∗_j, zj, z_j^∗,x⁰_j, x^0∗_j, z_j⁰, z^0∗_j ∈ O and Z, Z⁰ ∈Λ∩g^N(F), then we have

BN(X, X⁰) =$²ⁿ Ã _k

X

j=1

(xjx^0∗_j −x^∗_jx⁰_j) + X`

j=1

(zjz⁰_j^∗−z_j^∗z_j⁰)

!

∈p²ⁿ_F .

That is,ψ(B($⁻²ⁿN,[X, X⁰])) = 1 forX, X⁰ ∈$ⁿΛ⁰. Now for X,Y ∈$ⁿΛ⁰, we have χ_n(expXexpY)^{1.3 (2)}= ψ◦B($⁻²ⁿN, X+Y +1

2[X, Y] +$^2n+CΛ)

=ψ◦B($⁻²ⁿN, X +Y)ψ◦B(N, $^CΛ).

The assertion follows from B(N,Λ)∈ O.

(2) Suppose that expX0 ∈ Stab(χn,Km), (X0 ∈$^mΛ⁰). We can write X0 =Y0 +Z0, Y₀ ∈$^mm^Λ0,Z₀ ∈$^mΛ∩g^N(F). We have

χ_n(Ad(exp(−Z₀)) expX) =ψ◦B($⁻²ⁿN,Ad(exp(−Z₀))X)

=ψ◦B($⁻²ⁿAd(expZ₀)N, X) =ψ◦B($⁻²ⁿN, X)

=χ_n(expX)

Thus we can replace expX₀ by expX₀exp(−Z₀). It follows from 1.3 (2) that expX₀exp(−Z₀) = exp¡

Y₀+[Y0,−Z0]

2 +Z), ∃Z ∈$^2m+CΛ

∈exp(Y₀+$^2m−3dΛ⁰).

m≥C+ 3d assures that 2m−3d > m+C. Thus if we set X1 := log(expX0exp(−Z0)), it can be written as X₁ = Y₁+Z₁ with Y₁ ∈ $^mm^Λ0 and Z₁ ∈ $^m+CΛ∩g^N(F). Again we can replace expX₁ by

expX1exp(−Z1) = exp(Y1+[Y1,−Z1]

2 +Z⁰), ∃Z⁰ ∈$^2m+2CΛ

∈exp(Y1+$^2m+C−3dΛ⁰).

We haveX2 := log(expX1exp(−Z1)) =Y2+Z2 withY2 ∈$^mm^Λ0,Z2 ∈$^m+2CΛ∩g^N(F).

By repeating this process we obtain, for any r∈N,

expX₀ = expX_rexpZ_r−1· · ·expZ₀,

X_r =Y_r+Z_r, Y_r∈$^mm^Λ0, Z_r∈$^m+rCΛ∩g^N(F).

Thus, if we takeZ ∈$^mΛ∩g^N(F) such that expZ is the limit of a convergent subsequence of {expZr· · ·expZ0}r∈N, then Y := log(expX0exp(−Z)) is in$^mm^Λ0.

Now let Y ∈$^mm^Λ0 be such that expY ∈Stab(χ_n,K_m). Since 1.3 (3) gives χ_n(Ad(expY) expX) = ψ¡

B($⁻²ⁿN, X+ [Y, X] +Z_X,Y)¢

, ∃Z_X,Y ∈$^2m+n−3dΛ, this is equivalent to

BN(Y, X) +B(N, ZX,Y)∈p²ⁿ_F , ∀X ∈$ⁿΛ⁰. (∗) If 2m−3d < n, this forces us to have B_N(Y, X) ∈ −B(N, Z_X,Y) +$²ⁿO ⊂ p^∈m+\−3d_F . m^Λ0 is self-dual with respect to B_N and this implies that Y ∈ $^2m−3dm^Λ0. By repeating this argument, one may assume that Y ∈$^km^Λ0 with 2k−3d ≥n. Then B(N, ZX,Y)∈ B(N, $²ⁿΛ)⊂$²ⁿO. Thus (∗) becomes

B_N(Y, X)∈p²ⁿ_F , ∀X ∈$ⁿΛ⁰. Now the self-duality of m^Λ0 gives Y ∈$ⁿΛ⁰.