A note on Langlands’ classification and irreducibility of induced representations of p-adic groups

A note on Langlands’ classification and irreducibility of induced representations of p-adic groups

Takuya KONNO ^∗ July 30, 2002

Abstract

In this note, we present a proof of the Langlands classification of the irreducible admissible representations of reductive p-adic groups. Then we deduce certain irre- ducibility result for parabolically induced modules from discrete series representa- tions.

Contents

1 Introduction 2

2 Preliminary 2

2.1 Structure of G(F ) . . . . 3

2.2 Restricted roots . . . . 4

2.3 Unramified quasi-characters . . . . 5

2.4 Representations . . . . 6

2.5 Infinitesimal characters . . . . 8

3 Langlands classification 9 3.1 Standard modules and its matrix coefficients . . . . 9

3.2 The order ≤

_P

and a partition of a

^∗_M

. . . . 10

3.3 Langlands classification . . . . 12

4 Some irreducibility results 15 4.1 A theorem of Waldspurger . . . . 15

4.2 Irreducibility of induced modules from discrete series . . . . 19

∗

Graduate School of Mathematics, Kyushu University, 812-8581 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. 12740018, the

Ministry of Education, Science, Sports and Culture, Japan

1 Introduction

In this note, we shall prove two fundamental results in the representation theory of p-adic groups.

The first is the Langlands classification of irreducible admissible representations of connected reductive p-adic groups (Th. 3.5). This famous theorem had originally been proved by Langlands for real Lie groups [9], then its p-adic group analogue was treated independently in [5] and [11]. But the latter contains no proof (The argument suggested in [5, XI.2] does not work.). Silberger’s article is well-written but the key lemma [11, Lem.5.3] is not true. Since the theorem plays a fundamental role in the harmonic analysis on p-adic reductive groups, I think it is of some value to writing out a complete proof, although it seems to be well-known to experts. The proof given in this note follows Langlands’ original argument in the real case [9], while we rely in an essential way on the infinitesimal characters for p-adic groups introduced by Bernstein [4].

In [12, IV.1], Waldspurger proved that the standard intertwining operators are rational functions on the variety of representations. We combine this with the Langlands classifica- tion, and show that the parabolically induced modules from discrete series representations are irreducible on a Zariski open subset of the variety of such representations (Cor. 4.3).

Again this is well-known to experts. For example, Waldspurger himself mentioned it in his definition of Harish-Chandra’s j and µ-functions in [12, p. 48, IV.3]. Also this was used by Bernstein and Deligne in their analysis of components (under infinitesimal characters) of Hecke algebras [4, Prop. 3.14].

The contents of each section are as follows. In § 2, we collect elementary facts and results on the structure of connected reductive p-adic groups and their representations.

Here we emphasize features caused by the discreteness of the valuation on the base field (see e.g. § 2.3). Also in § 2.2, the geometry of the restricted roots, which are essential in the proof of the Langlands classification, is reviewed from [9, § 4]. In § 3, we prove the Langlands classification. After reviewing Langlands’ lemma on the growth behavior of the matrix coefficients of standard modules and two geometric lemmas, the proof is given in § 3.3. In § 4 we prove the irreducibility result. First in § 4.1, we recollect the proof of Waldspurger’s irreducibility theorem (Th. 4.2) to emphasize the role played by the Langlands classification. Then the irreducibility on Zariski open subsets is proved in

§ 4.2.

Throughout the notes, we use only basic results in the harmonic analysis on p-adic groups, which were proved in [2], [3], [8] and §§ I.1-IV.2 of [12].

2 Preliminary

Let F be a non-archimedean local field of any characteristic. We write O, p

_F

and | |

_F

for the maximal compact subring of F , its unique maximal ideal and the module of F ,

respectively. We write q for the cardinality of the residue field of O.

2.1 Structure of G(F )

Let G be a connected reductive F -group. We fix a maximal F -split torus A

0

so that its centralizer M

₀

is a minimal Levi subgroup of G. Write L, F for the set of F -Levi and F -parabolic subgroups of G, respectively, containing M

₀

. Each P ∈ F has a unique Levi component M in L, while the set P(M ) of P ∈ F having M as a Levi component is finite.

For P ∈ P (M ), we write ¯ P for the element of P (M ) which is opposite to P with respect to M .

Take M ∈ L. As usual, we have the real vector spaces a

M

= Hom(X

^∗

(M)

F

, R ), a

^∗_M

= X

^∗

(M )

_F

⊗ R dual to each other, and the Harish-Chandra map H

_M

: M (F ) → a

_M

given by

exphχ, H

M

(m)i = |χ(m)|

F

, ∀χ ∈ X

^∗

(M )

F

.

Here, X

^∗

(M)

_F

is the group of F -rational characters of M . We write M (F )

¹

for the kernel of H

_M

.

We write A

_G

for the maximal F -split torus in the center Z

_G

of G. The canonical isomorphism X

^∗

(G)

_F

⊗ Q →

^∼

X

^∗

(A

_G

) ⊗ Q (induced by restriction) combined with the commutative diagram

X

^∗

(M )

_F

−−−→ X

^∗

(A

_M

) x







 y X

^∗

(G)

_F

−−−→ X

^∗

(A

_G

)

shows that a

_G

and a

^∗_G

are canonical direct summands of a

_M

and a

_G

, respectively. If we write a

^G_M

for the annihilator of X

^∗

(G)

_F

in a

_M

and a

^G,∗_M

:= X

^∗

(A

_M

/A

_G

) ⊗ R , then we have the direct sum decompositions

a

M

= a

^G_M

⊕ a

G

, a

^∗_M

= a

^G,∗_M

⊕ a

^∗_G

dual to each other. We denote the a

^G_M

and a

_G

-components of H ∈ a

_M

by H

^G

and H

_G

, respectively. Similarly, λ ∈ a

^∗_M

admits a decomposition λ = λ

^G

⊕ λ

_G

, (λ

^G

∈ a

^G,∗_M

, λ

G

∈ a

^∗_G

).

We fix a maximal compact subgroup K of G(F ) which is in good position relative to A

₀

. Then we have the Iwasawa decomposition G(F ) = U (F )M (F )K for any P = M U ∈ F . We write the corresponding decomposition of g ∈ G(F ) as g = u

P

(g )m

P

(g)k

P

(g), where u

_P

(g) ∈ U (F ), m

_P

(g ) ∈ M (F ) and k

_P

(g) ∈ K are of course not unique. We fix various measures as in [12]. In particular, on any subgroup H(F ) ⊂ G(F ), we fix an invariant measure which assigns 1 to the subgroup H(F )∩K. These satisfy the integration formulae

Z

G(F)

f(g) dg = Z

K

Z

M(F)

Z

U(F)

f (umk)δ

_P

(m)

⁻¹

du dm dk

=γ(G/M)

⁻¹

Z

U(F)

Z

M(F)

Z

U(F¯ )

f (um¯ u)δ

_P

(m)

⁻¹

d¯ u dm du

for any continuous compactly supported function f on G(F ) and P = M U ∈ F. Here δ

_P

is the modular character of P (F ) and

γ(G/M ) :=

Z

U(F¯ )

δ

_P

(m

_P

(¯ u)) d¯ u.

This constant is independent of P ∈ P (M) as the notation suggests [12, I.1 (3)].

2.2 Restricted roots

We continue to take M ∈ L. We review some elementary results on restricted roots from [1, § 1], [9, § 4]. We write K

^M

:= K ∩ M for any subgroup K ⊂ G and M ∈ L.

The set of roots of A

_M

in G is denoted by Σ

_M

. P ∈ P (M ) determines the subset Σ

_P

of P -positive elements in Σ

M

. The set of reduced roots in Σ

P

, that is, α ∈ Σ

P

such that α/n / ∈ Σ

_P

for any n ≥ 2, is denoted by Σ

^red_P

. Notice that Σ

_M

spans a

^G,∗_M

.

Suppose M = M

₀

. We know from [6, Cor. 5.8] that (Σ

₀

= Σ

_M0

, a

^G,∗₀

) is a root system.

In particular, the set of coroots Σ

^∨₀

⊂ X

∗

(A

0

) is defined. For P

0

∈ P (M

0

), we have the set of simple roots ∆

_P0

in the positive system Σ

_P0

, and write ∆

^∨_P

0

for the set of corresponding simple coroots. Of course, ∆

P0

and ∆

^∨_P0

are basis of a

^G,∗₀

and a

^G₀

, respectively. We write

∆ b

_P0

= {$

_α

| α ∈ ∆

_P0

} and ∆ b

^∨_P0

= {$

^∨_α

| α ∈ ∆

_P0

} for the basis of a

^G,∗₀

and a

^G₀

dual to

∆

^∨_P

0

and ∆

_P0

, respectively. We write W = W

^G

for the Weyl group of A

₀

in G, which is the Weyl group of the root system (a

^G,∗₀

, Σ

₀

). This acts on a

₀

and hence on F , L. We identify W with its fixed system of representatives in Norm(A

₀

, G(F )).

For general P = M U ∈ F, we choose P

₀

∈ P (M

₀

) contained in P and define ∆

_P

:=

{(α

₀

|

_aM

) | α

₀

∈ ∆

_P0

\ ∆

_P^M

0

}. This is independent of the choice of P

₀

⊂ P . The coroot attached to α = (α

₀

|

_aM

) ∈ ∆

_P

is defined to be (α

^∨₀

)

_M

∈ a

^G_M

. We write ∆

^∨_P

:= {α

^∨

| α ∈

∆

_P

}. It follows from the M

₀

-case that ∆

_P

and ∆

^∨_P

are basis of a

^G,∗_M

and a

^G_M

, respectively.

Notice that their dual basis are given by

∆ b

^∨_P

:= {$

^∨_α

= $

^∨_α0

| (α = α

₀

|

_aM

) ∈ ∆

_P

},

∆ b

_P

:= {$

_α

= $

_α0

| (α = α

₀

|

_aM

) ∈ ∆

_P

}

respectively. In general (Σ

_M

, a

^G,∗_M

) is not necessarily a root system. We write W (M) = W

^G

(M ) := Stab(M, W )/W .

We list some basic properties of restricted roots. First (Σ

₀

, a

^G,∗₀

), a root system, satisfies the following properties.

hα, α

^∨

i = 2, hα, β

^∨

i ≤ 0, ∀α 6= β ∈ ∆

_P0

. (2.1) If we set a

^∗,+_P

0

:= {λ ∈ a

^∗₀

| α

^∨

(λ) > 0, α ∈ ∆

_P0

},

⁺

a

^∗_P0

:= {λ ∈ a

^∗₀

| $

_α

(λ) > 0, α ∈ ∆

_P0

} then we have

a

^∗,+_P

0

⊂

⁺

a

^∗_P

0

. (2.2)

This is an easy consequence of (2.1).

Let us establish analogous properties for general Σ

_M

. (2.2) implies h$

_α

, $

_β^∨

i ≥ 0 for any α, β ∈ ∆

_P0

. This simply restricts to

h$

_α

, $

^∨_β

i ≥ 0, ∀α, β ∈ ∆

_P

.

Setting a

^∗,+_P

:= {λ ∈ a

^∗_M

| α

^∨

(λ) > 0, α ∈ ∆

_P

},

⁺

a

^∗_P

:= {λ ∈ a

^∗_M

| $

^∨_α

(λ) > 0, α ∈ ∆

_P

}, this amounts to the assertion

¯

a

^∗,+_P

⊂

⁺

¯ a

^∗_P

. (2.3)

As opposed to (2.2), the inclusion is between the closures. If α = α

0

|

a_M

, (α

0

∈ ∆

P0

\∆

_P^M

0

), α = (α

₀

)

_M

and α

^M₀

∈ a

^M,∗₀

can be written as

α

^M₀

= X

β∈∆P M 0

x

_β

β, x

_β

∈ R .

Here the coefficient x

_β

is given by hα

^M₀

, $

_β^∨,M

i = hα

₀

, $

_β^∨,M

i. Since $

^∨,M_β

∈ ¯ a

^M,+_PM 0

⊂

⁺

¯ a

^M_PM 0

,

$

^∨,M_β

= P

γ∈∆^M_P

0

y

_γ

γ

^∨

with y

_γ

≥ 0. Thus (2.1) implies x

_β

= X

γ∈∆P M0

y

_γ

hα

₀

, γ

^∨

i ≤ 0.

In particular, we have for (β = β

₀

|

_aM

) 6= α ∈ ∆

_P

hα, β

^∨

i = hα

₀

, β

₀^∨

i − X

γ∈∆P M0

x

_γ

hγ, β

₀^∨

i ≤ 0. (2.4)

Finally we have

hα, α

^∨

i > 0, α ∈ ∆

_P

. (2.5) Otherwise we have hα, β

^∨

i ≤ 0 for any β ∈ ∆

_P

so that α ∈ −¯ a

^G,∗,+_P

∩

⁺

¯ a

^G,∗_P

= {0}. This contradicts α 6= 0.

2.3 Unramified quasi-characters

H

_G

allows us to associate to each λ ∈ a

^∗_G,C

a quasi-character

e

^λ

: G(F ) 3 g 7−→ exphλ, H

_G

(g)i ∈ C

^×

. (2.6) We write X(G(F )) := {e

^λ

| λ ∈ a

^∗_G,

C

} = Hom

_cont

(G(F )/G(F )

¹

, C

^×

) and X

_u

(G(F )) for its subgroup of unitary elements. If we write a

_G(F)

for the lattice H

_G

(G(F )) ⊂ a

_G

and a

^∗_G(F)

⊂ a

^∗_G

for its dual lattice, then we have the isomorphism

a

^∗_G,C

/2πia

^∗_G(F)

3 λ 7−→

^∼

e

^λ

∈ X(G(F )).

This defines a C -torus structure on X(G(F )). For χ ∈ X(G(F )), we write <χ := |χ| and

=χ := (<χ)

⁻¹

χ. <χ is identified with an element of a

^∗_G

by (2.6).

Now we take M ∈ L and consider the restriction homomorphism X(G(F )) → X(M (F )).

Lemma 2.1. For any χ ∈ X

^∗

(G)

_F

, χ(M (F )) = χ(G(F )).

Proof. It suffices to check this in the case M = M

₀

. We write G

_der

for the derived group of G and G

_ab

:= G/G

_der

for its abelianization. If we write M

₀^der

:= M

₀

∩ G

_der

, we have the embedding of exact sequences

1 −−−→ G

der

−−−→ G −−−→ G

ab

−−−→ 1 x





x





1 −−−→ M

₀^der

−−−→ M

₀

−−−→ G

_ab

−−−→ 1

Since X

^∗

(G)

_F

= X

^∗

(G

_ab

)

_F

, we have only to check that the images of G(F ) and M

₀

(F ) in G

_ab

(F ) coincide. For this, we take Galois cohomology to have the commutative diagram

1 −−−→ G

_der

(F ) −−−→ G(F ) −−−→ G

_ab

(F ) −−−→ H

¹

(F, G

_der

) x





x





x





1 −−−→ M

₀^der

(F ) −−−→ M

₀

(F ) −−−→ G

_ab

(F ) −−−→ H

¹

(F, M

₀^der

)

Then, what we have to show is

ker(G

_ab

(F ) → H

¹

(F, G

_der

)) = ker(G

_ab

(F ) → H

¹

(F, M

₀^der

)).

But since the left hand side equals the kernel of G

_ab

(F ) → H

¹

(F, M

₀^der

) → H

¹

(F, G

_der

), this follows from the injectivity of H

¹

(F, M

₀^der

) → H

¹

(F, G

_der

). (Notice that this last statement is equivalent to [6, Th. 4.13, Prop. 4.7] which asserts that the minimal parabolic subgroups are all G(F )-conjugate to each other.)

Since X

^∗

(G)

_F

injects into X

^∗

(M )

_F

by restriction, we can take a basis {χ

_i

}

1≤i≤n

of X

^∗

(M )

F

so that {d

i

χ

i

}

1≤i≤r

for some d

i

∈ N and 0 ≤ r < n is a basis of X

^∗

(G)

F

. If

|χ

_i

(M (F ))|

_F

= q

^miZ

, (1 ≤ i ≤ n, m

_i

∈ N ), then a

^∗_M(F)

=

n

X

i=1

Z (m

_i

log q)

⁻¹

χ

_i

.

Now Lem. 2.1 asserts that |d

_i

χ

_i

(G(F ))|

_F

= |d

_i

χ

_i

(M (F ))|

_F

= q

^dimiZ

, (1 ≤ i ≤ r), so that a

^∗_G(F)

=

r

X

i=1

Z d

_i

d

_i

m

_i

log q χ

_i

is a direct summand of a

^∗_M(F)

. Hence X(G(F )) → X(M (F )) is injective. If we write X

^G

(M (F )) for the group of quasi-characters of M (F ) ∩ G(F )

¹

trivial on M (F )

¹

, then we summarize the argument as the exact sequence of C -tori

1 −→ X(G(F )) −→ X(M (F )) −→ X

^G

(M (F )) −→ 1. (2.7) The maps are restrictions.

2.4 Representations

We freely use the results of [2], [3] and [8] on algebraic (or smooth) representations of reductive p-adic groups, which are summarized in [12, I]. Let us recall some of them. We write Alg(G(F )) for the category of algebraic representations of G(F ). We adopt the convention that the isomorphism class of (π, V ) is denoted by π. If χ ∈ X(G(F )), then we write (π

_χ

, V

_χ

) for the representation π ⊗ χ on the space V . We write

Stab(π, X(G(F ))) = {χ ∈ X(G(F )) | π

_χ

' π}.

By abuse of notation, we write (π

_λ

, V

_λ

) for e

^λ

⊗ π on the space V for λ ∈ a

^∗_M,C

. For P = M U ∈ F, we have the parabolic induction functor Alg(M (F )) 3 (π, V ) 7→ (I

_P^G

(π), I

_P^G

(V )) ∈ Alg(G(F )) and the Jacquet functor Alg(G(F )) 3 (π, V ) 7→ (π

_P

, V

_P

) ∈ Alg(M (F )). They are related by the Frobenius reciprocity

Hom

G(F)

(π, I

_P^G

(τ )) ' Hom

M(F)

(π

P

, τ ).

As for the composition of these functors, we know the following result [3, 2.12]. For P = M U, P

⁰

= M

⁰

U

⁰

∈ F , we take a system of representatives

_P⁰

W

_P

for W

^M0

\W/W

^M

, so that we have the Bruhat decomposition G = `

w∈_P0WP

P w

⁻¹

P

⁰

. We fix a total order

w ≤ w

⁰

on

_P⁰

W

_P

such that

• G(F )

≥w

:= S

v∈_P0WP, v≥w

P (F )w

⁻¹

P

⁰

(F ) is open in G(F );

• P (F )w

⁻¹

P

⁰

(F ) is closed in G(F )

≥w

with respect to the p-adic topology on G(F ). Then for (π, V ) ∈ Alg(M (F )), there is a G(F )-invariant decreasing filtration {F

_w

}

w∈_P0WP

of I

_P^G

(V )

_P⁰

, which we call the Bruhat filtration, such that

F

_w

/F

_>w

' I

_w(P^M0 ₎M0

(w(π

_w⁻¹_(P⁰₎^M

))

as an algebraic representation of M

⁰

(F ). Notice that the isomorphism class w(π

_w⁻¹_(P⁰₎^M

) is independent of the choice of the representative for w. We write (π

^∨

, V

^∨

) for the con- tragredient of (π, V ) ∈ Alg(G(F )). For (π, V ) ∈ Alg(M (F )), we have I

_P^G

(π

^∨

) ' I

_P^G

(π)

^∨

. If (π, V ) ∈ Alg(G(F )) is admissible, (π

_P

)

^∨

is isomorphic to (π

^∨

)

P¯

. In fact, this is valid for B-admissible representations in the sense of [4].

If (π, V ) is an admissible representation of finite length of G(F ), the set JH(π) of the elements of Π(G(F )) which appears as irreducible constituents of (π, V ) is uniquely determined by π. Thus we may consider the Grothendieck group of the category of admissible representations of finite length of G(F ). For such representation (π, V ), we write [π] for its class in the Grothendieck group.

We write Π(G(F )) for the set of isomorphism classes of irreducible admissible rep- resentations of G(F ). We have the subsets Π

temp

(G(F )) ⊃ Π

2

(G(F )) of tempered and square integrable elements of Π(G(F )), respectively. We write ω

_π

for the central character of π ∈ Π(G(F )). For any admissible representation (π, V ) of G(F ), E xp(π) ⊂ Π(A

_G

(F )) denotes the set of its central exponents [12, I.3]. Recall the Langlands-Casselman crite- rion:

(1) An admissible representation (π, V ) of G(F ), having a unitary central character, is square integrable if and only if <E xp(π

P

) ⊂

⁺

a

^G,∗_P

for any P ∈ F .

(2) An admissible representation (π, V ) of G(F ) is tempered if and only if <E xp(π

_P

) ⊂

+

¯ a

^G,∗_P

for any P ∈ F.

In particular, parabolic induction preserves temperedness, while Jacquet functor does not.

We also need the following weak classification of irreducible tempered representation.

Proposition 2.2 ([12] Prop. III.4.1). (i) For any π ∈ Π

_temp

(G(F )), there exist P = M U ∈ F and σ ∈ Π

₂

(M (F )) such that π is a direct summand of I

_P^G

(σ).

(ii) If both (P, σ) and (P

⁰

, σ

⁰

) satisfy (i), then there is w ∈ W such that w(M ) = M

⁰

and w(σ) ' σ

⁰

.

Let (π, V ) be an admissible representation of finite length of M (F ), M ∈ L. For P , P

⁰

∈ P (M ), we have the intertwining integral

J

_P⁰_|P

(π)φ(g) :=

Z

(U∩U⁰)(F)\U⁰(F)

φ(u

⁰

g) du

⁰

, φ ∈ I

_P^G

(V ).

For χ ∈ X(M (F )) with α

^∨

(<χ) >> 0, ∀α ∈ Σ

_P

\ Σ

_P⁰

, the defining integral of J

_P⁰_|P

(π

_χ

)

converges absolutely. Moreover J

_P⁰|P

defined in this way on some open subset of P =

{π

_χ

| χ ∈ X(M (F ))} becomes a rational function on P [12, Th. IV.1.1]. Outside its poles, this defines an element of Hom

_G(F)

(I

_P^G

(V

_χ

), I

_P^G0

(V

_χ

)). Moreover for any χ ∈ X(M (F )), there exists φ ∈ I

_P^G

(V

χ

) such that J

_P⁰|P

(ρ

χ

)φ converges and is not zero [12, IV.1 (10)]. If further π is tempered, then J

_P⁰_|P

(π

_χ

)φ converges at χ ∈ X(M (F )) satisfying α

^∨

(<χ) > 0,

∀α ∈ Σ

_P

\ Σ

_P⁰

[12, Prop. IV.2.1].

2.5 Infinitesimal characters

Recall that an admissible representation (π, V ) of G(F ) is cuspidal if its restriction to G(F )

¹

is finite [2]. A theorem of Harish-Chandra asserts that this is equivalent to π

_P

= {0}

for P 6= G, ∈ F . In particular, if (ρ, E) is a cuspidal representation of M (F ), then for any P , P

⁰

∈ P(M ), the Bruhat filtration simplifies to

[I

_P^G

(ρ)

_P⁰

] = X

w∈W(M)

w(ρ).

We write Π

₀

(G(F )) for the subset of unitarizable cuspidal elements in Π(G(F )). This is contained in Π

₂

(G(F )). For each π ∈ Π(G(F )), there there exist P

_c

= M

_c

U

_c

∈ F and an irreducible cuspidal representation (ρ, V

ρ

) of M

c

(F ) such that π is isomorphic to a subrepresentation of I

_P^G_c

(ρ) [3, Th. 2.5]. Then π appears as a subquotient of I

_P^G0

c

(ρ) for any P

_c⁰

∈ P (M

_c

). Moreover the pair (M

_c

, ρ) is determined uniquely modulo W -conjugacy by π [3, Th. 2.9]. We call the W -conjugacy class the infinitesimal character of π and denote it by X

_π

. Also we write P

_c

(π) := S

(Mc,ρ)∈Xπ

P (M

_c

).

If an admissible representation (π, V ) of finite length of G(F ) admits an irreducible cuspidal subquotient ρ, then ρ appears both as a submodule and a quotient of π [2, 3.30].

From this property we deduce the following.

Lemma 2.3. Let P = M U ∈ F and (ρ, V ) be an irreducible cuspidal representation of M (F ). Then π ∈ Π(G(F )) is a submodule of I

_P^G

(ρ) if and only if it is a quotient of I

_P^G_¯

(ρ).

Proof. π is a submodule of I

_P^G

(ρ) if and only if

{0} 6= Hom

_G(F)

(π, I

_P^G

(ρ)) ' Hom

_M(F)

(π

P

, ρ)

by Frobenius reciprocity. The above remark asserts that this is equivalent to {0} 6=Hom

_M(F)

(ρ, π

_P

) ' Hom

_M(F)

((π

_P

)

^∨

, ρ

^∨

)

'Hom

_M(F)

((π

^∨

)

P¯

, ρ

^∨

) ' Hom

_G(F)

(π

^∨

, I

_P^G_¯

(ρ

^∨

)) 'Hom

_G(F)

(I

_P^G_¯

(ρ), π)

as desired.

Using these, we can strengthen the Langlands-Casselman criterion as follows.

Lemma 2.4. (i) π ∈ Π(G(F )) is square integrable if its central character is unitary and

<E xp(π

_Pc

) ⊂

⁺

a

^∗_Pc

for any P

_c

∈ P

_c

(π).

(ii) π ∈ Π(G(F )) is tempered if its central character is unitary and <E xp(π

Pc

) ⊂

⁺

a ¯

^∗_Pc

for

any P

_c

∈ P

_c

(π).

Proof. (i) The condition is obviously necessary. To see the sufficiency, we take P = M U ∈ F and an irreducible subquotient τ of π

_P

, and show <E xp(τ ) = <(ω

_τ

|

_AM(F)

) ∈

⁺

a

^∗_P

. We take (M

c

, ρ) ∈ X

π

and P

c

∈ P(M

c

) such that π is a submodule of I

_P^G_c

(ρ). Since ρ is cuspidal, the Bruhat filtration simplifies

[I

_P^G_c

(ρ)

_P

] = X

w∈PW_Pc

[I

_w(P^M

c)^M

(w(ρ

_w⁻¹_(Pc)^M

))] = X

w∈W^M\W w(Mc)⊂M

[I

_w(P^M

c)^M

(w(ρ))].

Thus τ is a subquotient of some I

_w(P^M

c)^M

(w(ρ)). In particular, For any irreducible subquotient τ of π

_P

, X

_τ

is a subset of X

_π

.

Now take P

_c^M

∈ P

_c

(τ ) such that τ

_Pc^M

6= {0}. If we write P

_c

:= P

_c^M

U , then π

_Pc

6= {0} and E xp(τ) ={(χ|

_AM(F)

) | χ ∈ E xp(τ

_PM

c

)} = {(ω

_σ

|

_AM(F)

) | σ ∈ JH(τ

_PM c

)}

⊂{(ω

σ

|

_AM(F)

) | σ ∈ JH(π

Pc

)} = {(χ|

_AM(F)

) | χ ∈ E xp(π

Pc

)}.

But by the condition, <χ = P

βc∈∆_Pc

x

_βc

β

_c

, (∃x

_βc

> 0) for χ ∈ E xp(π

_Pc

), so that

<(χ|

_AM(F)

) = (<χ)

_M

= X

βc∈∆_Pc\∆_{P M}

c

x

_β

(β

_c

|

_aM

)

belongs to

⁺

a

^G,∗_P

. The sufficiency is proved. (ii) can be proved in the same way.

3 Langlands classification

In this section we prove the Langlands quotient theorem for p-adic reductive groups.

3.1 Standard modules and its matrix coefficients

Recall that a standard module of G(F ) is a representation of the form (I

_P^G

(π

_λ

), I

_P^G

(V

_λ

)), where P = M U ∈ F, π ∈ Π

_temp

(M(F )) and λ ∈ a

^∗,+_P

. We write

a→

lim

P∞

f(a) = 0

if for arbitrary small , δ > 0, there exists R > 0 such that |f (a)| < for any a ∈ A

_M

(F ) ∩ G(F )

¹

satisfying

• α(H

_M

(a)) < −R, ∀α ∈ Σ

_P

;

• α(H

_M

(a))/β(H

_M

(a)) > η, ∀α, β ∈ Σ

_P

.

The following proposition is the analogue for p-adic groups of [9, Lem. 2.12]. The proof

is completely the same and is omitted.

Proposition 3.1. Let (I

_P^G

(π

_λ

), I

_P^G

(V

_λ

)) be a standard module. Then for φ ∈ I

_P^G

(V

_λ

) and φ

^∨

∈ I

_P^G

(V

_−λ^∨

) we have

a→

lim

P¯

∞

δ

_P

(a)

^1/2

ω

_πλ

(a)

⁻¹

hI

_P^G

(π

_λ

, ma)φ, φ

^∨

i

=γ(G/M )

⁻¹

h(J

P¯|P

(π

_λ

)φ)(m), φ

^∨

(1)i, m ∈ M (F ).

This allows us to define the so called Langlands quotient of a standard module.

Corollary 3.2. Suppose (I

_P^G

(π

_λ

), I

_P^G

(V

_λ

)) is a standard module.

(i) Any φ ∈ I

_P^G

(V

λ

) with J

P¯|P

(π

λ

)φ 6= 0 generates I

_P^G

(V

λ

) as a G(F )-module.

(ii) In particular, the representation J

_P^G

(π

_λ

) on J

_P^G

(V

_λ

) := imJ

P¯|P

(π

_λ

) is irreducible. It is the unique irreducible quotient of I

_P^G

(π

_λ

).

Proof. (i) Take φ as in the statement. It suffices to show that if φ

^∨

∈ I

_P^G

(V

_−λ^∨

) satisfies hI

_P^G

(π

λ

, g)φ, φ

^∨

i = 0, ∀g ∈ G(F ), then φ

^∨

= 0. Applying the proposition to

0 = hI

_P^G

(π

_λ

, g

⁻¹

ah)φ, φ

^∨

i = hI

_P^G

(π

_λ

, ah)φ, I

_P^G

(π

_−λ^∨

, g)φ

^∨

i, we have

0 =γ(G/M ) lim

a→

P

∞

δ

_P

(a)

^1/2

ω

_πλ

(a)

⁻¹

hI

_P^G

(π

_λ

, ah)φ, I

_P^G

(π

_−λ^∨

, g)φ

^∨

i

=hJ

P¯|P

(π

λ

)φ(h), φ

^∨

(g)i, ∀h, g ∈ G(F ).

By assumption, we can take h ∈ G(F ) such that J

P¯|P

(π

_λ

)φ(h) 6= 0. Then we must have 0 = hJ

P¯|P

(π

_λ

)φ(mh), φ

^∨

(g )i = δ

_P

(m)

^−1/2

hπ

_λ

(m)(J

P|P¯

(π

_λ

)φ(h)), φ

^∨

(g)i

for any m ∈ M (F ), g ∈ G(F ). Since π

_λ

is irreducible, this shows φ

^∨

= 0.

(ii) Take any proper maximal subrepresentation V

⁰

of I

_P^G

(V

_λ

). If V

⁰

6⊂ kerJ

P¯|P

(π

_λ

), (i) implies V

⁰

= I

_P^G

(V

_λ

) which contradicts our choice of V

⁰

. Hence V

⁰

⊂ kerJ

P¯|P

(π

_λ

) and any irreducible quotient I

_P^G

(V

_λ

)/V

⁰

has J

_P^G

(V

_λ

) as a quotient.

3.2 The order ≤ _P and a partition of a ^∗ _M

We prepare some geometric properties of restricted roots which play an important role in the proof of Langlands classification [9, § 4].

Take P = M U ∈ F . Define an order λ ≤

_P

µ on a

^∗_M

by µ ∈ λ +

⁺

¯ a

^G,∗_P

. For P

₁

= M

₁

U

₁

⊃ P , ∆

_P1

is obtained by restricting ∆

_P

\ ∆

_PM1

. Thus for λ, µ ∈ a

^∗_M1

, λ ≤

_P

µ is equivalent to λ ≤

_P1

µ. We also set

a

^∗_P

(P

1

) :=

λ ∈ a

^∗_M

(i) α

^∨

(λ) > 0, ∀α ∈ ∆

P1

(ii) $

^∨,M_α ¹

(λ) ≤ 0, ∀α ∈ ∆

_P^M1

= −

⁺

a ¯

^M_PM^1,∗1

⊕ a

^∗,+_P1

. The simplest case is a

^∗_P

(P ) = a

^∗,+_P

, and we have the disjoint decomposition

a

^∗_M

= a

P1∈F(M)

a

^∗,+_P

1

. (3.1)

Here F (M ) is the set of P

₁

∈ F containing M .

Lemma 3.3. a

^∗_M

= `

P1;G⊃P1⊃P

a

^∗_P

(P

₁

).

Proof. First let us show that a

^∗_P

(P

₁

), (P ⊂ P

₁

⊂ G) cover a

^∗_M

by an induction on |∆

_P

|.

If P is maximal, a

^G,∗_M

is 1-dimensional and a

^∗_P

(P ) = a

^∗,+_P

, a

^∗_P

(G) = −¯ a

^∗,+_P

, so that the assertion is clear. For general P ∈ F , we take λ ∈ a

^∗_M

. Suppose λ / ∈ a

^∗_P

(P ) and take α ∈ ∆

_P

such that α

^∨

(λ) ≤ 0. Let P

_α

= M

_α

U

_α

⊃ P , ∈ F be such that ∆

_PMα

= {α}.

Applying the induction hypothesis to λ

_Mα

, we find P

₁

= M

₁

U

₁

⊃ P

_α

such that λ

_Mα

= − X

β∈∆P M1,6=α

x

_β

(β|

_aMα

) + X

γ∈∆_P1

y

_γ

$

_γ

, ∃x

_β

≥ 0, y

_γ

> 0.

On the other hand, since a

^M_M^α,∗

= R α, λ

^Mα

= hλ, α

^∨

i

hα, α

^∨

i α, β|

a_Mα

= β − hβ, α

^∨

i hα, α

^∨

i α.

Thus we have

λ = − X

β6=α,∈∆_{P M1}

x

_β

β − hβ, α

^∨

i hα, α

^∨

i α

+ X

γ∈∆P1

y

_γ

$

_γ

+ hλ, α

^∨

i hα, α

^∨

i α

= − X

β6=α,∈∆_{P M1}

x

_β

β + hλ, α

^∨

i

hα, α

^∨

i + X

β6=α,∈∆_{P M1}

x

_β

hβ, α

^∨

i hα, α

^∨

i

α + X

γ∈∆_P1

y

_γ

$

_γ

.

The coefficient of α is not positive by (2.4) and our assumption, hence λ ∈ a

^∗_P

(P

₁

). Next show that a

^∗_P

(P

₁

), (P ⊂ P

₁

⊂ G) are disjoint. Suppose P

₁

6= P

₂

contain P . We may assume that ∆

_P^M2

\ ∆

_P^M1

is not empty. If α ∈ ∆

_P^M2

\ ∆

_P^M1

, then

h$

_α^∨,M2

a

^∗_P

(P

₁

)i = h$

_α^∨,M2

, a

^M2,∗,+

P₁^M2

i = R

^>0

, h$

^∨,M_α ²

, a

^∗_P

(P

₂

)i = h$

^∨,M_α ²

, −

⁺

¯ a

^M_PM^2,∗2

i = R

^≤0

, hence a

^∗_P

(P

₁

) and a

^∗_P

(P

₂

) are disjoint.

Lemma 3.4. Suppose P , P

⁰

∈ F contain P

_c

∈ F . If λ ∈ a

^∗_Pc

(P ), λ

⁰

∈ a

^∗_Pc

(P

⁰

) satisfy λ ≥

_Pc

λ

⁰

, then λ

_M

≥

_Pc

λ

⁰_M0

.

Proof. The hypothesis amounts to $

_α^∨

(λ) ≥ $

_α^∨

(λ

⁰

), ∀α ∈ ∆

_Pc

. (i) If α ∈ ∆

_Pc

\ ∆

_PM0

c

, λ ∈ a

^∗_P

c

(P ) implies λ

_M

≥

_Pc

λ so that

$

^∨_α

(λ

_M

) ≥ $

_α^∨

(λ) ≥ $

_α^∨

(λ

⁰

) = $

_α^∨

(λ

⁰_M0

).

(ii) Suppose α ∈ ∆

_PM0

c

. Since λ

_M

∈ a

^∗,+_P

, (λ

_M

)

^M0

∈ ¯ a

^M0,∗,+

P_c^M0

⊂

⁺

¯ a

^M0,∗

P_c^M0

and

h$

_α^∨,M0

, λ

_M

i ≥ 0. (3.2)

If we expand $

_α,M^∨ 0

= P

β∈∆_P0

x

β

$

^∨_β

, the coefficient of β = β

c

|

a_M0

, (β

c

∈ ∆

Pc

\ ∆

_PM0 c

) satisfies

x

_β

=hβ, $

^∨_α

i = hβ

_c

− β

_c^M0

, $

^∨_α

i = −hβ

_c^M0

, $

_α^∨,M0

i = −hβ

_c

, $

_α^∨,M0

i

∈ − hβ

_c

, X

γ∈∆P M0 c

R

^≥0

γ

^∨

i = R

^≥0

by (2.4). Combining this with (3.2), we obtain h$

^∨_α

, λ

_M

− λ

⁰_M0

i =h$

_α^∨,M0

, λ

_M

i + X

β∈∆_P0

x

_β

h$

^∨_β

, λ

_M

− λ

⁰_M0

i

≥ X

βc∈∆_Pc\∆

P M0 c

x

_β

h$

_β^∨

c

, λ

_M

− λ

⁰_M0

i.

As was seen in (i), the right hand side is non-negative.

3.3 Langlands classification

Now we prove the result of this section.

Theorem 3.5. (i) For any irreducible admissible representation (π, V ) of G(F ), there exist P = M U ∈ F, τ ∈ Π

_temp

(M(F )) and λ ∈ a

^∗,+_P

such that π ' J

_P^G

(τ

_λ

).

(ii) The triple (P, τ, λ) is uniquely determined by π up to W -conjugacy.

Proof. (i) We first choose P and λ. We fix P

₀

∈ P(M

₀

) and write F (P

₀

) for the set of P

₀

-standard parabolic subgroups of G. Also set P

_c

(π, P

₀

) := P

_c

(π) ∩ F (P

₀

). For each µ ∈ S

Pc∈Pc(π,P0)

<E xp(π

P¯c

) there exists a unique P

_µ

∈ F (P

₀

) such that µ ∈ a

^∗_P0

(P

_µ

) (Lem. 3.3).

Take Λ ∈ S

Pc∈Pc(π,P0)

<E xp(π

P¯c

) such that Λ

_MΛ

∈ a

^∗,+_P

Λ

is maximal with respect to the order ≥

_P0

, and set P := P

_Λ

, λ := Λ

_M

∈ a

^∗,+_P

. Next choose τ . Take P

_c

∈ P

_c

(π, P

₀

) such that <E xp(π

P¯c

) contains Λ. Since P ⊃ P

_c

, we have E xp(π

P¯

) ⊃ {(χ|

_AM(F)

) | χ ∈ E xp(π

P¯c

)}.

Notice that these two sets might not coincide because Jacquet modules along ¯ P

_c^M

of some irreducible constituents of π

P¯

can be zero. Anyway we find χ ∈ Exp(π

P¯

) such that

<χ = Λ|

_aM

= λ. The weak χ-isotypic subspace V

P ,χ¯

of V

P¯

is an M (F )-submodule. Let (τ, V

_τ

) be such that (τ

_λ

, V

_τ,λ

) is an irreducible subrepresentation of V

P ,χ¯

.

Let us prove that (P, τ, λ) satisfies the condition of (i). Combining Frobenius reci- procity and duality for Jacquet modules, we have

{0} 6=Hom

_M(F)

(τ

_λ

, π

P¯

) ' Hom

_M(F)

((π

^∨

)

_P

, τ

_−λ^∨

) ' Hom

_G(F)

(π

^∨

, I

_P^G

(τ

_−λ^∨

)) 'Hom

_G(F)

(I

_P^G

(τ

_λ

), π).

Thus π is an irreducible quotient of I

_P^G

(τ

_λ

). We still have to prove that (τ, V

_τ

) is tempered.

By construction its central character ω

_τ

is unitary. Thanks to Lem. 2.4, it suffices to verify <E xp(τ

_P¯0M

c

) ⊂ −

⁺

¯ a

^M,∗

P_c^0M

for any P

_c^0M

∈ P

_c

(τ, P

₀^M

). For this, we write P

_c⁰

:= P

_c^0M

U ∈ P

_c

(π, P

₀

) and take χ ∈ E xp(τ

_P¯0M

c

). We need to check <χ = P

β∈∆P0

cM

x

_β

β for some x

_β

≤ 0.

Take the parabolic subgroup P ⊃ (Q = LN ) ⊃ P

_c⁰

such that ∆

_P⁰

cL

= {β ∈ ∆

_P⁰

cM

| x

β

> 0}.

Also we find P ⊃ P

₁

⊃ P

_c⁰

for which <χ + λ ∈ a

^∗_P0

c

(P

₁

). From definition, we have

<χ + λ ≥

_Pc⁰

X

β∈∆P0 cM\∆

P0 cL

x

_β

β + λ.

Thus Lem. 3.4 gives

(<χ + λ)

_M1

≥

_Pc⁰

X

β∈∆P0 cM\∆

P0 cL

x

_β

β + λ

M

= λ = Λ

_M

.

But since e

^λ

χ ∈ E xp(τ

_λ,P¯0M

c

) ⊂ E xp(π

P¯_c⁰

), our choice of Λ implies (<χ)

_M1

+ λ = (<χ + λ)

_M1

≤

_Pc⁰

Λ

_M

= λ.

Hence (<χ)

M1

= 0 so that x

β

≤ 0 for any β ∈ ∆

_P⁰

cM

.

(ii) Suppose two triples (P, τ, λ) and (P

⁰

, τ

⁰

, λ

⁰

) as in the theorem satisfy J

_P^G

(τ

_λ

) ' π ' J

_P^G0

(τ

_λ⁰0

). We may assume both P and P

⁰

contain P

₀

. By Prop. 2.2, we have P

_d

= M

_d

U

_d

⊂ P and σ ∈ Π

₂

(M

_d

(F )) such that τ is a direct summand of I

_P^MM

d

(σ). Moreover [3, Th. 2.5] assures that there exists P

_c

= M

_c

U

_c

⊂ P

_d

, ρ ∈ Π

₀

(M

_c

(F )) and µ ∈ a

^M_M^d_c^,∗

such that σ is a submodule of I

^M_¯^d

Pc^Md

(ρ

_µ

), equivalently (Lem. 2.3), a quotient of I

^Md

Pc^Md

(ρ

_µ

).

{0} 6= Hom

_Md(F)

(σ, I

^M_¯^d

Pc^Md

(ρ

_µ

)) ' Hom

_Mc(F)

(σ

_P¯Md c

, ρ

_µ

) combined with the Langlands-Casselman criterion gives µ ∈ −

⁺

a

^Md,∗

Pc^Md

. Also writing Λ :=

λ + µ, π is a quotient of I

_P^G_c

(ρ

Λ

). Similarly for (P

⁰

, τ

⁰

, λ

⁰

) we take P

_c⁰

⊂ P

_d⁰

⊂ P

⁰

, σ

⁰

∈ Π

2

(M

_d⁰

(F )), ρ

⁰

∈ Π

0

(M

_c⁰

(F )) and µ

⁰

∈ −

⁺

a

^M

0 d,∗

P^0M

0 c d

. Since I

_P^G_c

(ρ

µ

) and I

_P^G0

c

(ρ

⁰_µ0

) share the irreducible constituent π, there is w

₁

∈ W such that

w

₁

(M

_c

) = M

_c⁰

, w

₁

(ρ) ' ρ

⁰

, w

₁

(Λ) = Λ

⁰

. (3.3) Next the Bruhat filtration gives

[I

_P^G

d

(σ

_λ

)

P¯_c⁰

] = X

w∈_P¯0 cW_Pd

[I

^Mc0

w(Pd)^Mc0

(w(σ

_λ,w−1( ¯P_c⁰)^Md

))]

= X

w∈W/W^Md w(Md)⊃M_c⁰

[w(σ

_w−1( ¯P_c⁰)^Md

)

_w(λ)

].

Notice that, thanks to (3.3) and the vanishing of the Jacquet modules of cuspidal repre- sentations, the terms of w with w(P

_d

)

^Mc0

6= M

_c⁰

vanish. On the other hand,

{0} 6= Hom

G(F)

(I

_P^G0

c

(ρ

⁰_Λ⁰

), π) ' Hom

G(F)

(π

^∨

, I

_P^G0

c

(ρ

^0∨_−Λ⁰

)) ' Hom

M_c⁰(F)

(π

_P^∨0

c

, ρ

^0∨_−Λ⁰

) ' Hom

_M

c0(F)

(ρ

⁰_Λ0

, π

P¯_c⁰

), so that ρ

⁰_Λ0

∈ JH(π

P¯_c⁰

) ⊂ JH(I

_P^G

d

(σ

_λ

)

P¯_c⁰

). Thus ρ

⁰_Λ0

∈ v(JH(σ

_v−1( ¯P_c⁰)^Md

))

_v(λ)

for some v ∈ W/W

^Md

with v(M

_d

) ⊃ M

_c⁰

. But since I

_P^G

d

(σ) is tempered,

<E xp(I

_P^G_d

(σ)) = [

w∈W/WMd

w(Md)⊃M_c⁰

<E xp(w(σ

_w⁻¹_{( ¯P}⁰

c)^Md

))

is contained in −

⁺

¯ a

^G,∗_P0

c

. Thus Λ

⁰

≤

_Pc⁰

v(λ). Moreover, taking P

₁

⊃ P

_c⁰

such that v(λ) ∈ a

^∗_P0

c

(P

1

), we obtain

λ

⁰

≤

_Pc⁰

v(λ)

_M1

, λ

⁰

≤

_P0

v(λ)

_M1

(3.4)

from Lem. 3.4. Now we claim the following. We fix a W -invariant positive definite

symmetric bilinear form ( | ) and write k k for the associated norm.

Claim 3.5.1. If λ, λ

⁰

∈ ¯ a

^G,∗,+_P

0

satisfy λ ≤

_P0

λ

⁰

, then kλk ≤ kλ

⁰

k.

Proof. Recall that the coroot α

^∨

of α ∈ ∆

_P0

is identified with 2α/kαk

²

by ( | ) [7, VI.1.1 Lem. 2], so that

(α|$

_β

) = kαk

²

2 δ

_α,β

, α, β ∈ ∆

_P0

. (3.5)

λ, λ

⁰

are written as

λ = X

β∈∆P0

y

_β

$

_β

, λ

⁰

= X

β∈∆P0

y

_β⁰

$

_β

, y

_β

, y

⁰_β

≥ 0,

and the assumption on them is

λ

⁰

= λ + X

α∈∆_P

0

x

_α

α, x

_α

≥ 0.

Hence the claim follows from (λ|λ

⁰

) =kλk

²

+ X

α, β∈∆_P

0

x

α

y

β

(α|$

β

) = kλk

²

+ X

α∈∆_P

0

x

α

y

α

kαk

²

2 ≥ kλk

²

, (λ|λ

⁰

) =kλ

⁰

k

²

+ X

α, β∈∆_P

0

x

α

y

⁰_β

(α|$

β

) = kλk

²

− X

α∈∆_P

0

x

α

y

⁰_α

kαk

²

2 ≤ kλ

⁰

k

²

.

Since a

^G,∗_M

and a

^M,∗₀

are spanned by ∆ b

_P0

\ ∆ b

_PM

0

and ∆

_PM

0

, respectively, (3.5) in the above proof assures that a

^G,∗_M

and a

^G,∗₀

are orthogonal to each other under ( | ). Applying this and the claim to (3.4), we obtain

kλ

⁰

k ≤ kv (λ)

M1

k ≤ kv (λ)k = kλk. (3.6) Replacing the role of (P, τ, λ) and (P

⁰

, τ

⁰

, λ

⁰

), we obtain the reverse inequality and hence kλk = kλ

⁰

k. This together with (3.6) implies v(λ) = v(λ)

_M1

∈ a

^G,∗,+_P1

. Thanks to (3.1), this and λ ∈ a

^∗_P

force P

₁

= v(P ). Since P

₁

and P are both P

₀

-standard, we conclude P = P

₁

and v ∈ W

^M

. In particular (3.4) reads λ

⁰

≤

_P0

λ. Again replacing (P, τ, λ) and (P

⁰

, τ

⁰

, λ

⁰

), we also have λ ≤

_P0

λ

⁰

, hence λ = λ

⁰

, P = P

⁰

.

We still have to show τ ' τ

⁰

. Since π ' J

_P^G

(τ

_λ⁰

) is a submodule of I

_P^G_¯

(τ

_λ⁰

), we have Hom

_M(F)

(I

_P^G

(τ

_λ

)

P¯

, τ

_λ⁰

) ' Hom

_G(F)

(I

_P^G

(τ

_λ

), I

_P^G_¯

(τ

_λ⁰

)) 6= {0}.

Comparing this with the Bruhat filtration formula [I

_P^G

(τ

_λ

)

P¯

] = X

w∈P¯WP

[I

_w(P^M ₎M

(w(τ

_w⁻¹_{( ¯P)}M

)

_w(λ)

)], we see

τ

_λ⁰

∈ JH I

_w(P^M ₎M

(w(τ

_w⁻¹_{( ¯P)}M

)

_w(λ)

)

, ∃w ∈

P¯

W

_P

. (3.7)