Constant term of Eisenstein integrals on
reductive p-adic symmetric spaces
Patrick Delorme, joint work with Jacques Carmona
owes a lot to the work of A.Helminck with S.P. Wang and with
G. Helminck
and to talks with Joseph Bernstein
Goal: Plancherel formula
G F -points of a connected reductive algebraic group G defined
over
F , non archimedean local field, characteristic zero.
X (G ) unramified characters (complex torus).
σ rational involution of G defined over F , H = G σ .
X (G )σ : connected component of 1 in {χ ∈ X (G ) = χ ◦ σ = χ−1 }.
G F -points of a connected reductive algebraic group G defined
over
F , non archimedean local field, characteristic zero.
X (G ) unramified characters (complex torus).
σ rational involution of G defined over F , H = G σ .
X (G )σ : connected component of 1 in {χ ∈ X (G ) = χ ◦ σ = χ−1 }.
Jacquet modules, Second adjointness theorem, Constant
term of H-forms
(π, V ) smooth representation of G , ξ H-form iff ξ ∈ V 0H ( V 0 :
algebraic dual, V̌ : smooth dual, to be introduced later).
P and P − are opposite parabolic subgroup of G iff P ∩ P − Levi
subgroup of P.
G F -points of a connected reductive algebraic group G defined
over
F , non archimedean local field, characteristic zero.
X (G ) unramified characters (complex torus).
σ rational involution of G defined over F , H = G σ .
X (G )σ : connected component of 1 in {χ ∈ X (G ) = χ ◦ σ = χ−1 }.
Jacquet modules, Second adjointness theorem, Constant
term of H-forms
(π, V ) smooth representation of G , ξ H-form iff ξ ∈ V 0H ( V 0 :
algebraic dual, V̌ : smooth dual, to be introduced later).
P and P − are opposite parabolic subgroup of G iff P ∩ P − Levi
subgroup of P.
P σ-parabolic subgroup of G iff P and σ(P) are opposite
(studied by HW , HH).
P ∩ σ(P): The σ-stable Levi subgroup of P.
G F -points of a connected reductive algebraic group G defined
over
F , non archimedean local field, characteristic zero.
X (G ) unramified characters (complex torus).
σ rational involution of G defined over F , H = G σ .
X (G )σ : connected component of 1 in {χ ∈ X (G ) = χ ◦ σ = χ−1 }.
Jacquet modules, Second adjointness theorem, Constant
term of H-forms
(π, V ) smooth representation of G , ξ H-form iff ξ ∈ V 0H ( V 0 :
algebraic dual, V̌ : smooth dual, to be introduced later).
P and P − are opposite parabolic subgroup of G iff P ∩ P − Levi
subgroup of P.
P σ-parabolic subgroup of G iff P and σ(P) are opposite
(studied by HW , HH).
P ∩ σ(P): The σ-stable Levi subgroup of P.
P = MU, U unipotent radical.
(Normalized) Jacquet module: a twist by some character of the
M-module VP := V / < π(u)v − v , u ∈ U, v ∈ V >
jP : V → VP , v 7→ vP , canonical projection.
2 Second adjointness theorem, ξP −
Second adjointness theorem: Casselman, Bernstein
V̌ : smooth dual (linear forms fixed by a compact open subgroup).
Then (V̌ )P − is canonically isomorphic to (VP )ˇfor (P, P − )
opposite parabolic, or, in other words:
There exists a canonical non degenerate pairing (V̌ )P − with VP .
2 Second adjointness theorem, ξP −
Second adjointness theorem: Casselman, Bernstein
V̌ : smooth dual (linear forms fixed by a compact open subgroup).
Then (V̌ )P − is canonically isomorphic to (VP )ˇfor (P, P − )
opposite parabolic, or, in other words:
There exists a canonical non degenerate pairing (V̌ )P − with VP .
Constant term of H-forms: independantly Lagier,
Kato-Takano
(P, P − ) opposite parabolic subgroups of G + HP − open in G , for
each H-form ξ on V , one associates:
jP − ξ ∈ (VP )0M∩H denoted also ξP − ( constant term of ξ) such
that for all v ∈ V , there exists ε s.t.
−1/2
δP
(a) < aξ, v >=< aξP − , vP >, a ∈ A−
M ()
where A−
M () is some translate of a Weyl chamber in AM (maximal
split torus in the center of M), δP modulus function of P.
This is a characteristic property of ξP − which describes the
asymptotics of coefficients.
3
For K compact open subgroup of G , define eK ξ ∈ V̌ by:
< eK ξ, v >:=< ξ, eK v >, eK normalized Haar measure on K .
3
For K compact open subgroup of G , define eK ξ ∈ V̌ by:
< eK ξ, v >:=< ξ, eK v >, eK normalized Haar measure on K .
For K an ”(H, P)-good” compact open subgroup of G , one has :
< ξP − , vP >=< (eK ξ)P − , vP >P , v ∈ V K .
There exists arbitrary small ”(H, P)-good” compact open subgroup
of G (Kato-Takano).
So ξP − is determined by the (eK ξ)P − .
4 First operation on H-forms on induced representations:
jQ − ◦
P = MU σ-parabolic subgroup of G , (δ, E ) a finite length smooth
rep. of M.
iPG E : normalized induced representation.
(Q, Q − ) a pair of opposite parabolic subgroups of M, with
Q − (H ∩ M) open in M.
PQ parabolic with PQ ⊂ P, and P ∩ M = Q. Idem Q − .
4 First operation on H-forms on induced representations:
jQ − ◦
P = MU σ-parabolic subgroup of G , (δ, E ) a finite length smooth
rep. of M.
iPG E : normalized induced representation.
(Q, Q − ) a pair of opposite parabolic subgroups of M, with
Q − (H ∩ M) open in M.
PQ parabolic with PQ ⊂ P, and P ∩ M = Q. Idem Q − .
ǰQ − ◦ : (iPG E )ˇ→ (iPG
Q−
(iPG E )ˇ→ iPG Ě →f iPG
Q−
(Ě )Q − →g iPG
EQ )ˇ
Q−
(EQ )ˇ→ (iPG
Q−
EQ )ˇ
Allows to attach to ξ H-form on iPG E , an H- form ǰQ − ◦ ξ on
iPG − (EQ ).
Q
5 Restriction to open orbits
P 0 σ-parabolic subgroup of G , P ⊂ P 0 , ξ H-form on iPG E . Inducing
in stage
M0 E .
ξ is an H-form on iPG0 E1 , E1 := iP∩M
0
5 Restriction to open orbits
P 0 σ-parabolic subgroup of G , P ⊂ P 0 , ξ H-form on iPG E . Inducing
in stage
M0 E .
ξ is an H-form on iPG0 E1 , E1 := iP∩M
0
As P1 H is open ξ, that may viewed has an E1 -distribution on G , is
simply a function on P1 H. Its value at 1 is denoted
0
rM 0 ξ ∈ (E1 ) M∩H
6 Generic basic geometric lemma
(δ0 , E ) smooth rep. of M, P = MU σ-parabolic subgroup of G .
O = {δ0 ⊗ χ|χ ∈ X (G )σ }.
P 0 an other σ-parabolic subgroup of G .
w in a good set of representatives of P 0 \G /P.
M
−1
Xw := iM
, ...
0 ∩w .P wEM∩w −1 .P 0 where w .P = wPw
6 Generic basic geometric lemma
(δ0 , E ) smooth rep. of M, P = MU σ-parabolic subgroup of G .
O = {δ0 ⊗ χ|χ ∈ X (G )σ }.
P 0 an other σ-parabolic subgroup of G .
w in a good set of representatives of P 0 \G /P.
M
−1
Xw := iM
, ...
0 ∩w .P wEM∩w −1 .P 0 where w .P = wPw
βw : (iPG E )ˇ→ (Xw )ˇ
defined for (δ, E ) generic in O ( the definition of βw will be given
later).
β : (iPG E )ˇ→ (⊕w ∈P 0 \G /P Xw )ˇ
goes through the quotient to an isomorphism:
β : ((iPG E )ˇ)P 0 − → X̌ .
7
Define parabolic subgroups of G : P̃w ⊂ P, P̃w0 ⊂ P
0
0−
with
P̃w ∩ M = M ∩ w −1 .P − , P̃w0 ∩ M 0 = M 0 ∩ w .P.
7
Define parabolic subgroups of G : P̃w ⊂ P, P̃w0 ⊂ P
0−
with
0
P̃w ∩ M = M ∩ w −1 .P − , P̃w0 ∩ M 0 = M 0 ∩ w .P.
βw = rM 0 ◦ t A(w .P̃w , P̃w0 , w .jM∩w −1 P 0 δ) ◦ λ(w ) ◦ ǰM∩w −1 P 0 − ◦
A(w .P̃w , P̃w0 , w .jM∩w −1 P 0 δ): intertwining integrals. t A transposed
map.
λ(w ): left translation by w .
7
Define parabolic subgroups of G : P̃w ⊂ P, P̃w0 ⊂ P
0−
with
0
P̃w ∩ M = M ∩ w −1 .P − , P̃w0 ∩ M 0 = M 0 ∩ w .P.
βw = rM 0 ◦ t A(w .P̃w , P̃w0 , w .jM∩w −1 P 0 δ) ◦ λ(w ) ◦ ǰM∩w −1 P 0 − ◦
A(w .P̃w , P̃w0 , w .jM∩w −1 P 0 δ): intertwining integrals. t A transposed
map.
λ(w ): left translation by w .
From our isomorphism β : ((iPG E )ˇ)P 0 − → X̌ , X identifies with
(iPG E )P 0 (by second adjointness theorem). So ξP 0 − identifies with a
linear form denoted again ξP 0 − on X . By reduction to eK ξ one
proves:
Theorem
ξP 0 − ,w = rM 0 ◦ t A(w .P̃w , P̃w0 , w .jM∩w −1 P 0 δ) ◦ λ(w ) ◦ ǰM∩w −1 P 0 − ◦ ξ
0
(choice of w such that (M ∩ w −1 .P − )(M ∩ H) open).
8 Proof of the generic basic geometric lemma and of the
Theorem, ξ(P, δ, η)
1)Study the generic bijectivity of intertwining integrals + notion of
infinitesimal character (w.r.t. to Bernstein’s center) + basic
geometric lemma ( gives the graded of a filtration of the Jacquet
modules of induced representations).
8 Proof of the generic basic geometric lemma and of the
Theorem, ξ(P, δ, η)
1)Study the generic bijectivity of intertwining integrals + notion of
infinitesimal character (w.r.t. to Bernstein’s center) + basic
geometric lemma ( gives the graded of a filtration of the Jacquet
modules of induced representations).
2) Formula for ξP 0 − ,w : unwind the definitions by reduction to the
eK ξ for K an (H, P 0 ) good open compact open subgroup of G .
8 Proof of the generic basic geometric lemma and of the
Theorem, ξ(P, δ, η)
1)Study the generic bijectivity of intertwining integrals + notion of
infinitesimal character (w.r.t. to Bernstein’s center) + basic
geometric lemma ( gives the graded of a filtration of the Jacquet
modules of induced representations).
2) Formula for ξP 0 − ,w : unwind the definitions by reduction to the
eK ξ for K an (H, P 0 ) good open compact open subgroup of G .
0
By Blanc Delorme: there exists , for η ∈ E M∩H , a rational map
0
δ → ξ(P, δ, η) ∈ (iPG δ) H , supported on the closure of PH, with
ξ(P, δ, η)(1) = η.
8 Proof of the generic basic geometric lemma and of the
Theorem, ξ(P, δ, η)
1)Study the generic bijectivity of intertwining integrals + notion of
infinitesimal character (w.r.t. to Bernstein’s center) + basic
geometric lemma ( gives the graded of a filtration of the Jacquet
modules of induced representations).
2) Formula for ξP 0 − ,w : unwind the definitions by reduction to the
eK ξ for K an (H, P 0 ) good open compact open subgroup of G .
0
By Blanc Delorme: there exists , for η ∈ E M∩H , a rational map
0
δ → ξ(P, δ, η) ∈ (iPG δ) H , supported on the closure of PH, with
ξ(P, δ, η)(1) = η.
ASSUME FOR SIMPLICITY: PH is the only (P, H)-open double
coset. Q σ-parabolic subgroup of G with Levi M.
0
B-matrix : B(Q, P, δ) ∈ End(E M∩H ) rational in δ s.t.
t
A(P, Q, δ)ξ(P, δ, η) = ξ(P, δ, B(Q, P, δ)η)
9 Cuspidal H-forms
Definition
(π, V ) finite length rep. of G . ξ H-form on V is said cuspidal or
H-cuspidal if one of the equivalent statements holds
(Kato-Takano):
1) ξP = 0 for every proper σ-parabolic subgroup of G
2) For all v ∈ V , cξ,v ∈ C ∞ (H\G ) is compactly supported
mod-center where:
cξ,v (Hg ) =< ξ, π(g )v >
10 Main theorem
Let ξ = ξ(P, δ, η) with η M ∩ H-cuspidal.
For δ generic ,if ξP 0 −,w is non zero, one may choose w such
that:
(i) M 0 ∩ w .P is a σ-parabolic subgroup of M 0 , P̃w0 and w .P̃w are
σ-parabolic subgroups of G .
10 Main theorem
Let ξ = ξ(P, δ, η) with η M ∩ H-cuspidal.
For δ generic ,if ξP 0 −,w is non zero, one may choose w such
that:
(i) M 0 ∩ w .P is a σ-parabolic subgroup of M 0 , P̃w0 and w .P̃w are
σ-parabolic subgroups of G .
0
(ii) w .M ⊂ M 0 , M ∩ w −1 .P − = M, so that P̃w = P.
10 Main theorem
Let ξ = ξ(P, δ, η) with η M ∩ H-cuspidal.
For δ generic ,if ξP 0 −,w is non zero, one may choose w such
that:
(i) M 0 ∩ w .P is a σ-parabolic subgroup of M 0 , P̃w0 and w .P̃w are
σ-parabolic subgroups of G .
0
(ii) w .M ⊂ M 0 , M ∩ w −1 .P − = M, so that P̃w = P.
(iii) Assuming only one (P, H) open orbit, and M 0 = M. Then w
normalizes M and
ξP 0 − ,w = B(P̃w0 , w .P, w δ)η
is M 0 ∩ H cuspidal.
Remark : as expected, when P 0 is too small, ξP 0 − vanishes, by (ii).
11 First key lemma
First key lemma Let P be a parabolic subgroup of G , not
necessarily σ-parabolic. Let A0 be a σ-stable maximal split torus of
P and M a Levi subgroup of G with A0 ⊂ M
Let (ξχ ) be a smooth family of H-forms on iPG (δ0 ⊗ χ), χ in a
neighborhood of 1 in a complex subtorus of X (M), X .
Assume that there there exists χ strictly P-dominant in X .
Assume PH is in the support of ξχ .
Then P is a σ-parabolic subgroup of G .
11 First key lemma
First key lemma Let P be a parabolic subgroup of G , not
necessarily σ-parabolic. Let A0 be a σ-stable maximal split torus of
P and M a Levi subgroup of G with A0 ⊂ M
Let (ξχ ) be a smooth family of H-forms on iPG (δ0 ⊗ χ), χ in a
neighborhood of 1 in a complex subtorus of X (M), X .
Assume that there there exists χ strictly P-dominant in X .
Assume PH is in the support of ξχ .
Then P is a σ-parabolic subgroup of G .
Corollary
Applying to the family ξP 0 − ,w ( with G replaced by M 0 ), one gets:
In the theorem one can take M 0 ∩ w .M σ-parabolic subgroup of M 0 .
12 Second and third key Lemmas
The cuspidality of η is not enough to prove that if
M ∩ w −1 .P 0 6= M then ξP 0 − ,w = 0.
What it is proved directly is that ǰM∩w −1 .P 0 ◦ ξ vanishes on the
open (P̃w , H)-double cosets, because in that case :
12 Second and third key Lemmas
The cuspidality of η is not enough to prove that if
M ∩ w −1 .P 0 6= M then ξP 0 − ,w = 0.
What it is proved directly is that ǰM∩w −1 .P 0 ◦ ξ vanishes on the
open (P̃w , H)-double cosets, because in that case :
Second key lemma
jM∩w −1 .P 0 η = 0 if M ∩ w −1 .P 0 6= M.
12 Second and third key Lemmas
The cuspidality of η is not enough to prove that if
M ∩ w −1 .P 0 6= M then ξP 0 − ,w = 0.
What it is proved directly is that ǰM∩w −1 .P 0 ◦ ξ vanishes on the
open (P̃w , H)-double cosets, because in that case :
Second key lemma
jM∩w −1 .P 0 η = 0 if M ∩ w −1 .P 0 6= M.
Third key lemma: the intertwining integrals of the definition of
ξP 0 − ,w do not increase the dimension of the support.
ξP 0 − ,w = rM 0 ◦ t A(w .P̃w , P̃w0 , w .jM∩w −1 P 0 δ) ◦ λ(w ) ◦ ǰM∩w −1 P 0 − ◦ ξ.
12 Second and third key Lemmas
The cuspidality of η is not enough to prove that if
M ∩ w −1 .P 0 6= M then ξP 0 − ,w = 0.
What it is proved directly is that ǰM∩w −1 .P 0 ◦ ξ vanishes on the
open (P̃w , H)-double cosets, because in that case :
Second key lemma
jM∩w −1 .P 0 η = 0 if M ∩ w −1 .P 0 6= M.
Third key lemma: the intertwining integrals of the definition of
ξP 0 − ,w do not increase the dimension of the support.
ξP 0 − ,w = rM 0 ◦ t A(w .P̃w , P̃w0 , w .jM∩w −1 P 0 δ) ◦ λ(w ) ◦ ǰM∩w −1 P 0 − ◦ ξ.
The philosophy comes of a Lemma of Matsuki on orbit closures of
orbits of parabolic subgroups on a real G /H.
rM 0 :
Applying rM 0 to something which vanishes on open (P˜w0 , H)-double
cosets, you get something which vanishes on open
(M 0 ∩ w .P, M 0 ∩ H)-double cosets.
12 Second and third key Lemmas
The cuspidality of η is not enough to prove that if
M ∩ w −1 .P 0 6= M then ξP 0 − ,w = 0.
What it is proved directly is that ǰM∩w −1 .P 0 ◦ ξ vanishes on the
open (P̃w , H)-double cosets, because in that case :
Second key lemma
jM∩w −1 .P 0 η = 0 if M ∩ w −1 .P 0 6= M.
Third key lemma: the intertwining integrals of the definition of
ξP 0 − ,w do not increase the dimension of the support.
ξP 0 − ,w = rM 0 ◦ t A(w .P̃w , P̃w0 , w .jM∩w −1 P 0 δ) ◦ λ(w ) ◦ ǰM∩w −1 P 0 − ◦ ξ.
The philosophy comes of a Lemma of Matsuki on orbit closures of
orbits of parabolic subgroups on a real G /H.
rM 0 :
Applying rM 0 to something which vanishes on open (P˜w0 , H)-double
cosets, you get something which vanishes on open
(M 0 ∩ w .P, M 0 ∩ H)-double cosets.
Last ingredient: by Blanc-Delorme for generic δ, it implies that if
M ∩ w −1 P 0 6= M, ξP 0 − ,w = 0
This proves the first part of the Theorem.
13 B-matrices preserve cuspidal forms
The computation of ξP 0 − ,w when P, P 0 have the same Levi is
straightforward.
The last statement, which says that B-matrix preserves cuspidal
M ∩ H-forms, comes from the hereditary properties of jP 0 − and the
fact observed in the remark, that for P 0 small enough jP 0 − ξ = 0.