On the dependence of the local Rankin-Selberg gamma factors of $\textrm{Sp}_{2n}\times \textrm{GL}_m$ on $\psi$

0.24 MB·

by  Qing Zhang

ON THE DEPENDENCE OF THE LOCAL RANKIN-SELBERG GAMMA
FACTORS FOR Sp ×GL ON ψ
2n m
QINGZHANG
6 Abstract. Let F be a p-adic field and π be an irreducible smooth representation of Sp2n(F).
1 In this paper, we show that if π and πκ are both generic for a common generic character of the
0 maximal unipotent of afixed Borel, then π∼=πκ, where πκ is the representation induced bythe
2 conjugation action of an element κ ∈ GSp2n(F). This result is a consequence of the standard
localLanglands conjecture andlocal Gan-Gross-Prasadconjecture. Asaconsequence, weextend
n the dependence relationof the local Rankin-Selberggammafactors forSp2n×GLm onψ to the
a generalcase.
J

8
2

Introduction
]
T
Let G be a quasi-split, connected reductive group over a p-adic field F. Let B = TU be a fixed
N
Borel subgroup over F, with unipotent subgroup U and torus T. Let Z be the center of the group
h. G(F). The torus T acts on the group Hom(U,C×). A character θ : U(F)→C× is called generic if
t its centralizer in T(F) is equal to the center Z(F). Let G be the adjoint group of G and T the
a ad ad
m torus of Gad corresponding to T. Then the set D of T(F)-orbits of generic characters is a principal
homogenous space of the abelian group
[
1 E =Tad(F)/Im(T(F))=ker(H1(Z,F)→H1(Z,T)).
v The group of T (F) also acts on G(F) by conjugation and thus acts on the set of isomorphism
ad
8 classes of irreducible smooth representations of G(F). For a Langlands parameter ϕ : WD(F) →
1 Gˆ⋉Gal(F¯/F), let Π be the conjectural (Vogan) L-packets of ϕ. It is part of the local Langlands
6 ϕ
7 conjecture that Πϕ is stable under the action of Tad(F), i.e., if π ∈ Πϕ and κ ∈ Tad(F), then
0 πκ ∈ Πϕ. See §9 of [GGP]. Thus by the local Gan-Gross-Prasad conjecture, Conjecture 17.1 of
. [GGP], if π and πκ are both θ-generic, for some generic character θ, then we should have π ∼= πκ.
1
0 In this paper, we confirm this result for the symplectic group Sp2n(F). More precisely, for κ∈F×,
6 consider the element c =diag(κI ,I )∈GSp (F). For a representation π of Sp (F), denote πκ
κ n n 2n 2n
1 the representation induced by the conjugation by c .
κ
:
v Theorem (Theorem 1.1). Let π be an irreducible smooth representation Sp (F). If π and πκ are
2n
Xi both generic for a common generic character of U, then π ∼=πκ.
r When n = 1, this theorem follows from results of Langlands-Labesse [LL]. In [GeRoS], Gelbart,
a
RogawskiandSoudryusedthisresultinthecasen=1todeducesimilarresultforthegroupU(1,1)
and then proved that in a global endoscopic L-packet of U(2,1), an element π is globally generic if
andonlyifitislocallygeneric. In[Zh],wegiveanewproofofthisresultintheU(1,1)case,basedon
a standardtrick of Gelfand. The proof of the above Theoremfor generalSp is a generalizationof
2n
theproofgivenin[Zh],whichisquitesimilartotheproofoftheuniquenessofWhittakerfunctionals
and should work for more general groups.
One application of the above Theorem is to analyze the dependence of the local Rankin-Selberg
γ factors for Sp ×GL on ψ. Given a nontrivial additive character ψ and the corresponding
2n m
generic character ψ of U, the local Rankin-Selberg type gamma factors γ(s,π ×τ,ψ) is studied
U
in [Ka], where π is an irreducible smooth ψ -generic representation of Sp (F) and τ is a generic
U 2n
representation of GL (F). If κ∈F×,2, it is known that
m
γ(s,π×τ,ψκ)=ωτ(κ)2n|κ|2nm(s−21)γ(s,π×τ,ψ).
2010 Mathematics Subject Classification. 11F70,22E50.
Keywords and phrases. gammafactors,symplecticgroup,dependence onψ.
1
2 QINGZHANG
We will show that the above dependence on ψ relation is true for all κ ∈ F×, once both sides are
well-defined, i.e., when π is both ψ and (ψ ) -generic, see Proposition 3.5.
U κ U
The paper is organized as follows. The main Theorem is proved in §1. In §2, we consider the
κ-action on the Weil representation, and we show the dependence of the γ factor on ψ relation in
§3.
Acknowledgement
IthankmyadvisorProfessorJamesW.Cogdellforhishelp,encouragementandconstantsupport.
Notations
Let F be a p-adic field, O be the ring of integers, P be the maximal ideal of O and ̟ be a
uniformizer of F, i.e., a generator of P. Let q = |O/P|, and | | be the standard valuation of F
F F
with |̟| =q−1.
F F
Let n>1 be an integer and Sp be the rank n symplectic group defined by the matrix
2n
I
s = n ,
n −I
n
where I is the n×n identity matrix. Explicitly,
n
Sp (F)= g ∈ÅGL (F):ãtgs g =s .
2n 2n n n
Denote Gn =Sp2n(F). If the rank n is u(cid:8)nderstood, we will omit the(cid:9)subscript n from the notation,
and just write G=Sp (F).
2n
Let P =M N be the Siegel Levi subgroup of Sp , where
n n n 2n
g
M = m (g):= ,g ∈GL (F),g∗ =g−1 ,
n n g∗ n
and
I X
N = nß(X):= Ån ã,X ∈Mat (F),tX =X™ .
n n I n×n
n
Let U be the upper triangular unipotent subgroup of M , and U = U N , which is the
Mn n Sp2n Mn n
maximalunipotentsubgroupoftheuppertriangularBorelsubgroup. Whentheranknisunderstood,
ß Å ã ™
we will omit the subscript n for simplicity.
Let N¯ be the opposite of N, i.e.,
I
N¯ = n¯ (X):= n ,X ∈Mat (F),tX =X .
n X I n×n
n
LetT bethemaximaltoruswhichconsistselementsoftheformt=diag(a ,...,a ,a−1,...,a−1).
1 n 1 n
The simple roots of Sp2n aßre αi,1≤i≤Ån−1,ãβ, where ™
a
α (t)= i ,1≤i≤n−1,β(t)=a2.
i a n
i+1
Let ∆ = {α ,1≤i≤n−1,β} be the set of simple roots, Σ+ be the set of positive roots of Sp
i 2n
and Σ be the set of roots of Sp . For γ ∈Σ, let U be the root space of γ and let x : F →U be
2n γ γ
the corresponding isomorphism.
Let W be the Weyl group of Sp . For γ ∈ Σ+, let s ∈ W be the simple reflection defined by
2n γ
γ. Then s acts on the set Σ by s (γ′)=γ′−hγ′,γ∨iγ, where γ∨ is the corootof γ, and hγ′,γ∨i is
γ γ
the natural paring between roots and coroots.
The Weyl group W is generated by s and s . We can take representative of s , s , by
αi β αi β
1
..
 . 
1
sαi =mn 1 ,1≤i≤n−1,
 
 .. 
 . 
 
 1
 
 
ON THE DEPENDENCE OF THE LOCAL RANKIN-SELBERG GAMMA FACTORS FOR Sp2n×GLm ON ψ 3
1
where the block is in the (i,i+1)×(i,i+1) position, and
1
I
n−1
Å ã 1
s = .
β I
n−1
−1
Ü ê
We will not distinguish a Weyl element w ∈W and its representative in G by abuse of notation.
We also use the matrix
1
1
Jm = ... ∈GLm
á1 ë
frequently.
1. Dependence on ψ for generic representations for Sp (F)
2n
Let ψ be an additive character of F, and κ ∈ F×, we consider the generic character ψκ of U
U
defined by
n−1
1
ψκ(u)=ψ u +κ u ,u=(u )∈U.
U i,i+1 2 n,2n i,j
!
i=1
X
Note that any generic character ψ′ of U is conjugate to one of ψκ for some κ ∈ F×, and ψκ1 is
U U
conjugate to ψκ2 if and only if κ /κ ∈F×,2. We will write ψ for ψ1. Consider the element
U 1 2 U U
κI
c = n ∈GSp (F).
κ I 2n
n
Forg ∈Sp (F),wehavegκ :=c gc−1 ∈Sp (F),andg 7→gκdefinesanautomorphismofSp (F).
2n κ κ 2n 2n
Moreover,if κ∈F×,2, the automorphismÅ g 7→gκãis inner. In fact, if κ=λ2 for λ∈F×, then
gκ =m (λI )gm (λI )−1.
n n n n
Let (π,V ) be an irreducible smooth representation of Sp (F), we define another representation
π 2n
(πκ,V ) of Sp (F) by
πκ 2n
V =V , and πκ(g)=π(gκ).
πκ π
It is clear that if (π,V ) is ψ -generic, then (πκ,V ) is ψκ-generic. The main theorem is
π U πκ U
Theorem 1.1. Let (π,V ) be an irreducible smooth admissible representation of Sp (F) and κ ∈
π 2n
F×. If π is both ψ and ψκ-generic, then π ∼=πκ.
U U
Note that the condition “π is both ψ and ψκ generic” can be restated as “π and πκ are both
U U
ψ -generic”.
U
Remark: Theorem 1.1 is a simple consequence of the local Langlands conjecture and the local
Gan-Gross-Prasad conjecture. In fact, let Π be the conjectural L-packet which contains π. Then
φ
πκ ∈ Π from the local Langlands conjecture, see §9 of [GGP]. By the local Gan-Gross-Prasad
ϕ
conjecture,Conjecture17.1of[GGP],forafixedψ ,eachL-packetcontainsatmostoneψ generic
U U
element. Thus if πκ is also ψ -generic, then πκ ∼=π.
U
In §1.1. we consider an involution on G which will be used to prove Theorem 1.1. The proof
of Theorem 1.1 is given in §1.2. If κ ∈ F×,2, the assertion of the theorem is trivial. Thus in the
following, we assume that κ∈/ F×,2.
4 QINGZHANG
1.1. An anti-involution on G. To provethis theorem,werecallthe MVW involutionofSp (F),
2n
see [MVW]. Consider the element
−I
δ =c = n ∈GSp (F).
−1 I 2n
n
For an irreducible smooth admissible representation of Sp (F), we consider the representation πδ
2n
on the same space by πδ(g) = π(gδ) withÅgδ = δ−1gãδ, i.e., πδ = π−1, where π−1 denotes πκ when
κ = −1. Let π˜ denote the contragradient representation of π, then we have π˜ ∼= πδ ∼= π−1, see
Th´eoth`em, Chapter 4, II.1, p.91 of [MVW].
I
Let w = n , which is the long Weyl element of Sp (F). We consider a different form
l −I 2n
n
of π˜. Denote gι = w δgδ−1w−1. Then it is clear that π˜ ∼= πδ ∼= πι, where πι(g) = π(gι). We have
l l
Uι = U¯, whÅere U¯ is thãe opposite of U. Define a generic character ψU¯ of U¯ by ψU¯(u¯) = ψU(u¯ι).
Explicitly, we have
ι
A B D C
= ,A,B,C,D ∈Mat (F).
C D B A n×n
In particular
mn(A)ι =mn(tA−Å1),A∈ãGLn(ÅF), andãnn(X)ι =n¯n(tX),X ∈Matn×n(F),tX =X.
Lemma 1.2. An irreducible admissible representation π of G is ψ -generic if and only if π˜ is
U
ψU¯-generic.
Proof. This follows from π˜ ∼=πι, and
HomU(π,ψU)=HomU¯(πι,ψU¯).
(cid:3)
Let gρ =wlc−κgc−κ−1wl−1, and gθ =(g−1)ρ. Explicitly, we have
A B θ tA −κ−1tC
= ,A,B,C,D ∈Mat (F).
C D −κtB tD n×n
It is clear that (g g )θ =gθgθ and (gθ)θ =g, i.e., g 7→gθ is an anti-involution on Sp (F).
1 2 2 1 2n
WewillproveTheÅorem1.1ãfolloÅwingthemethodãof[Sh]. Onekeystepisthefollowingproposition,
which is the analogue of (1.5) in [Sh].
Proposition 1.3. Assume κ∈/ F×,2. Fix s∈NG(T), the normalizer of T in G. If ψU¯(su−1s−1)=
ψκ(u) for all u∈U ∩s−1U¯s, then sθ =s.
U
For w ∈W, denote Σ−w ={γ ∈Σ+ :w(γ)<0} and Uw− = γ∈Σ−wUγ. To prove Proposition 1.3,
we need the following:
Q
Lemma 1.4. (1) There is no w ∈W such that w(β)∈{±α ,±α ,...,±α }.
1 2 n−1
(2) Givenw ∈W,assumethatβ ∈/ Σ− andforeachα ∈Σ−,thepositiveroot−w(α )issimple.
w i w i
Then there exists integers 0=t <t <···<t =n such that w can be representented by
1 2 s
J
t1−t0
J
t2−t1
mn .. .
.
á Jts−ts−1ë
Recall that J is the k×k matrix with 1 on the skew diagonal and 0 elsewhere.
k
Proof Lemma 1.4. (1) Note that there is no i such that ±α has the same length with β. The
i
assertion follows from the fact that w(β) and β have the same root length.
(2) Let w be the long Weyl element in Sp and w′ =w w. Then we get w′(α ) is either simple or
l 2n l i
negative and w′(β) < 0. The Weyl elements w′ ∈ W which satisfies the condition w′( simple root)
are characterizedby Lemma 89 of [St], page 257. Such w′ are exactly Weyl elements which support
Bessel functions. Our assertion follows from Steiberg’s general Lemma directly.
ON THE DEPENDENCE OF THE LOCAL RANKIN-SELBERG GAMMA FACTORS FOR Sp2n×GLm ON ψ 5
Inourcase,wegiveadetailedproofhere. Weintroducethenotationβ =2(α +···+2α )+β
i i n−1
for 1≤i≤n−1 and denote β =β. We can check that
n
β , i=k,
i+1
(1.1) s (β )=β −hβ ,α∨iα = β , i=k+1,
αk i i i k k  i−1
β , otherwise.
 i
In particular, s preserves the set {β } . Moreover,we have
αk i 1≤i≤n 
(1.2) s (β )=β ,1≤i≤n−1, and s (β )=−β .
β i i β n n
Inparticular,everyw ∈W preservesthe set{±β ,1≤i≤n} (we canalsouse the lengthargument
i
toshowthis). Notethatthisgivesanalternativeproofof(1). Nowtakeaw ∈Wasinthecondition,
i.e., β ∈/ Σ− and for each α ∈Σ−, the positive root −w(α ) is simple.
w i w i
Claim 1: w(β )>0 for each i.
i
We prove Claim 1 by descending induction on i. If i = n, w(β ) = w(β) > 0 by assumption. In
n
general, we assume that w(β ) is positive and will show that w(β )>0. By the above argument,
i+1 i
wecanassumethatw(β )=β forsomeω(i+1)∈{1,2,...,n}. Ifw(β )<0,wecanassume
i+1 ω(i+1) i
w(β )=−β for some ω(i). Then we have
i ω(i)
w(2α )=w(β −β )=−β −β .
i i i+1 ω(i) ω(i+1)
If ω(i) = ω(i+1) = n, then w(2α ) = −2β and thus w(α ) = −β, which is impossible by (1). If
i i
at least one of ω(i),ω(i+1) is ≤ n−1, then −w(α ) is not a simple root. This contradicts the
i
assumption. This proves Claim 1.
By Claim 1, we can assume that w(β )=β for each i with ω(i)∈{1,...,n}.
i ω(i)
Claim 2: For i6=j, we have ω(i)6=ω(j). In particular,the map i7→ω(i) is a permutation of the
set {1,2,...,n}.
In fact, if ω(i)=ω(j) for some i6=j, then w(β −β )=0, which is impossible since 1(β −β ) is a
i j 2 i j
nonzero root.
Claim 3: For i ∈ {1,...,n−1}, if ω(i) > ω(i + 1), then ω(i + 1) = ω(i)− 1 and w(α ) =
i
−α =−α .
ω(i)−1 ω(i+1)
Suppose that ω(i)>ω(i+1), then
w(2α )=w(β −β )
i i i+1
=β −β
ω(i) ω(i+1)
=2(α +···+α )−2(α +···+α )
ω(i) n−1 ω(i+1) n−1
=−2(α +···+α ).
ω(i+1) ω(i)−1
Thus w(α ) = −(α + ··· + α ). Since −w(α ) is positive, it must be simple, i.e.,
i ω(i+1) ω(i)−1 i
ω(i)−1=ω(i+1). This proves Claim 3.
Claim 4: For i∈{1,...,n−1}, if ω(i)<ω(i+1), then w(α )= ω(i+1)−1α .
i t=ω(i) t
This follows from the same calculation as in the proof of Claim 3.
P
ByClaim3andClaim4andthefactw(β)=w(β )=β ,theactionoftheWeylelementw on
n ω(n)
theset∆isuniquelydeterminedbyapermutationω oftheset{1,...,n}suchthatifω(i)>ω(i+1)
then ω(i+1) = ω(i)−1. Thus w is uniquely determined by such an permutation. It is easy to
classify such permutations: they are defined by collections of integers 0 = t < t < ··· < t = n,
0 1 s
where the permutationω correspondingto (t ,t ,...,t )acts oneachsegmentI =[t +1,t ]by
0 1 s k k k+1
order reversing translation: t +17→t ,t +27→t −1,...,t 7→t +1, i.e., by the matrix
k k+1 k k+1 k+1 k
1
Jtk+1−tk = ... ,
1
Ö è
6 QINGZHANG
see [BZ], page 59-60. Then it is clear that the Weyl element
J
t1−t0
J
t2−t1
mn ...
á Jts−ts−1ë
is one Weyl element correspondingto the permutation ω, andthus it must be w since w is uniquely
determined by the permutation ω. (cid:3)
Proof of Proposition 1.3. For s ∈ N (T), we can assume that s = tw for some t ∈ T and w ∈ W.
G
Then U ∩s−1U¯s = U−. If w = 1, the assertion is clear. Now we assume that w 6= 1 and hence
w
U− 6=1.
w
We first consider the case that β ∈Σ−. We take u=x (x) for x∈F. Then ψκ(u)=ψ(κx). If
w β U
−w(β)isnotsimple,thenwehaveψU¯(sus−1)=1,andthusitisclearthatwecanfindanxsuchthat
ψU¯(su−1s−1) 6= ψUκ(u). Thus −w(β) must be simple. By Lemma 1.4, we have −w(β) = β. Then
ψU¯(su−1s−1) = ψU¯(twu−1w−1t−1) = ψ(a2nx), where t = mn(diag(a1,...,an)). Since κ ∈/ F×,2,
it is clear that we can choose x ∈ F such that ψ(a2nx) 6= ψ(κx), i.e., ψU¯(su−1s−1) 6= ψUκ(u) for
u = xβ(x). Thus there is no w and t ∈ T such that w(β) < 0 and ψU¯(twu−1w−1t−1) = ψUκ(u) for
all u∈U−.
w
Next we consider the case β ∈/ Σ−. Since Σ− is not empty, it must contains a simple root,
w w
which must be of the form α since β ∈/ Σ−. We first consider the case that there exists an i with
i w
1 ≤ i ≤ n−1 such that −w(α ) is a positive non-simple root, then we have α ∈ Σ−. Moreover,
i i w
if we take u = xαi(x) ∈ Uw−, then ψU¯(su−1s−1) = 1. We can choose x such that ψUκ(u) = ψ(x) is
nontrivial, and thus ψU¯(twu−1w−1t−1)6=ψUκ(u).
Finally we consider the case when β ∈/ Σ− and for each α ∈ Σ−, the positive root −w(α ) is
w i w i
simple. Then by Lemma 1.4, there exists integers 0=t <t <···<t =n such that
0 1 s
J
t1−t0
J
t2−t1
w=mn ... .
á Jts−ts−1ë
Suppose t = mn(diag(a1,...,an)). If ψU¯(twu−1w−1t−1) = ψUκ(u) for all u ∈ Uw−, we claim that ai
is constant for i ∈ I = [t +1,t ] for each k, i.e., a = a for all i,j ∈ I . Take i ∈ I ,i < t
k k k+1 i j k k k+1
andwe havew(α )=−α , whereω is the permutationofthe set{1,2,...,n} correspondingto
i ω(i+1)
w, see the proof of Lemma 1.4. In particular, we have α ∈ Σ−. Take u = x (x) for some x ∈ F.
i w αi
Then ψκ(u)=ψ(x). On the other hand, we have
U
ψU¯(twu−1w−1t−1)=ψ(αω(i+1)(t)x).
By assumption, we have ψ(α (t)x) = ψ(x) for all x ∈ F. Since ψ is nontrivial, we get
ω(i+1)
α (t)=1, i.e., a =a =a . Since ω| is also a permutation, we get a =a for
ω(i+1) ω(i+1) ω(i+1)+1 ω(i) Ik i j
all i,j ∈I . This proves the claim.
k
Thus if ψU¯(twu−1w−1t−1)=ψUκ(u) for all u∈Uw−, we get
a J
t1 t1−t0
a J
t2 t2−t1
tw=mn ... .
á atsJts−ts−1ë
It is clear that (tw)θ =tw. (cid:3)
1.2. Proof of theorem 1.1. Let S(G) be the space of Bruhat-Schwartz functions on G. Consider
the action of G×G on S(G) by ((g ,g ).f)(g) = f(g−1gg ), i.e., (g ,g ).f = l(g )r(g )f, where l
1 2 1 2 1 2 1 2
and r denotes the left and right translation respectively.
ON THE DEPENDENCE OF THE LOCAL RANKIN-SELBERG GAMMA FACTORS FOR Sp2n×GLm ON ψ 7
LetT be a distributiononS(G). As usual,we fix a κ∈F×−F×,2. We call T is quasi-invariant,
if
T((u¯1,u2).f)=ψU¯(u¯1)ψUκ(u2)T(f),∀u¯1 ∈U¯,u2 ∈U2,f ∈S(G).
For a distribution T on S(G), define another distribution Tθ by Tθ(f) = T(fθ), where f ∈ S(G)
and fθ(g)=f(gθ).
Theorem 1.5. If T ∈S(G) is quasi-invariant, then T =Tθ.
Proof. The following proof follows [Sh] closely, and we just give a sketch of the proof. To show
T(f) = Tθ(f) = T(fθ) for all f ∈ S(G), it suffices to show that T(f) = Tθ(f) = T(fθ) for all
f ∈S(U¯TwU) and all w∈W.
Denote C =U¯ ×T ×U and C(w)=U¯TwU, and let p:C →C(w) be the projection defined by
p(u¯,t,u)=u¯twu. Thenpissubmersive(Lemma1.9of[Sh]). ByageneralresultofHarish-Chandra,
for each ξ ∈S(C), there exists a unique function f ∈S(C(w)) such that
ξ
(1.3) h(u¯ twu )ξ(u¯ ,t,u )du¯ dtdu
1 2 1 2 1 2
ZU¯×T×U
= h(u¯ twu )f (u¯ twu )du¯ dtdu ,
1 2 ξ 1 2 1 2
ZU¯×T×Uw+
for all h∈S(C(w)).
Moreover, the assignment ξ 7→ f is surjective, and supp(f ) ⊂ p(supp(ξ)). See Lemma 1.10 of
ξ ξ
[Sh]. From the distribution T on S(C(w)) one can get a distribution T∗ on S(C) by
T∗(ξ)=T(f ),
ξ
which satisfies the condition
T∗((u¯1,u2).ξ)=ψU¯(u1)ψUκ(u2)T∗(ξ),∀u¯1 ∈U¯,u2 ∈U.
For ξ ∈S(C), consider the function F ∈S(T) defined by
ξ
Fξ(t)= ξ(u¯1,t,u2)ψU¯(u¯1)ψUκ(u−21)du¯1du2.
ZU¯×U
Then the assignment ξ 7→F is surjective, and there is a distribution τ on S(T) such that
ξ
(1.4) τ(F )=T∗(ξ)=T(f ),
ξ ξ
see Proposition 1.12 of [Sh] and the references given there.
Let Tψ,χ be the subset of t∈T such that ψU¯(twu−1w−1t−1)=ψUκ(u) for all u∈Uw−.
Claim 1: the distribution τ on T has support in T , cf. Proposition 1.13 of [Sh].
ψ,κ
We give a sketch of the proof of this claim following [Sh]. For u ∈ U−, and ξ ∈ S(C), consider
w
u∗ξ ∈S(C) defined by
u∗ξ(u¯ ,t,u )=ξ(u¯ twu−1w−1t−1,t,uu ).
1 2 1 2
By the defining property of f , Eq.(1.3), it is easy to see that f =f . In particular, we have
ξ u∗ξ ξ
(1.5) τ(F )=T(f )=T(f )=τ(F ).
u∗ξ u∗ξ ξ ξ
On the other hand, by changing variable, we have
Fu∗ξ(t)= ξ(u¯1twu−1w−1t−1,t,uu2)ψU¯(u¯1)ψUκ(u−21)du¯1du2
ZU¯×U
=ψU¯(twuw−1t−1)ψUκ(u)Fξ(t).
For a fixed u∈Uw−, let c(t)=ψU¯(twuw−1t−1)ψUκ(u). By Eq.(1.5) we get (c(t)−1)τ(Fξ)=0 for all
F ∈ S(T). If t ∈/ T , choose u ∈U− such that c(t ) 6= 1. Let D be a small neighborhood of t
ξ 0 ξ,κ w 0 0 0
such that c(t) 6= 1 for all t ∈ D . We can choose ξ such that F be the characteristic function of
0 0 ξ0
D . Thus we get τ(F )=0. This shows that τ has support in T .
0 ξ0 ψ,κ
8 QINGZHANG
Thus if T is empty, we are done. Now we suppose that T is not empty, by Proposition
ψ,κ ψ,κ
1.3, we have wθ = w. Thus (tw)θw−1 = wtθw−1 ∈ T for any t ∈ T. Given ξ ∈ S(C), we define
ξθ ∈S(C) by
ξθ(u¯ ,t,u )=ξ(uθ,(tw)θw−1,u¯θ).
1 2 2 1
Claim 2: We have f = (f )θ. This is a direct but tedious changing variable process using the
ξθ ξ
defining property of f , Eq.(1.3). A slightly simpler way to do this is using the formula for f :
ξ ξ
(1.6) f (u¯ twu )= u∗ξ(u¯ ,t,u )du
ξ 1 2 1 2
ZUw−
= ξ(u¯ twu−1w−1t−1,t,uu )du,
1 2
ZUw−
cf. Eq.(1.11) of [Sh]. From the uniqueness property of f , it is easy to get Eq.(1.6) from Eq.(1.3),
ξ
see [Sh] for more explanations. We now use Eq.(1.6) to check Claim 2. By Eq.(1.6), we get
(1.7) f (u¯ twu )= ξθ(u¯ twu−1w−1t−1,t,uu )du
ξθ 1 2 1 2
ZUw−
= ξ(uθuθ,(tw)θw−1,(twu−1w−1t−1)θu¯θ)du.
2 1
ZUw−
On the other hand, by Eq.(1.6) again, we have
(1.8) fθ(u¯ twu )=f (uθ(tw)θw−1wu¯θ)
ξ 1 2 ξ 2 1
= ξ(uθ(tw)θu˜−1((tw)θ)−1,(tw)θw−1,u˜u¯θ)du.
2 1
ZUw−
Notice that if T is not empty, the Weyl element must be of the form as described in Lemma
ψ,κ
1.4. Then it is easy to check that for u ∈ U−, we have (twu−1w−1t−1)θ ∈ U−, and the map
w w
u7→(twu−1w−1t−1)θ gives an automorphism of U−. Thus if we let u˜=(twu−1w−1t−1)θ, the right
w
side of Eq.(1.7) and Eq.(1.8) are the same, i.e., f =fθ. This proves Claim 2.
ξθ ξ
Claim 3: For t∈T , we have F (t)=F for all ξ ∈S(C).
ψ,κ ξθ ξ
By the definition of F , we have
ξ
Fξθ(t)= ξθ(u¯1,t,u2)ψU¯(u¯1)ψUκ(u−21)du¯1du2
ZU¯×U
= ξ(uθ2,(tw)θw−1,u¯θ1)ψU¯(u¯1)ψUκ(u−21)du¯1du2.
ZU¯×U
Let u¯′1 = uθ2 and u′2 = u¯θ1. It is easy to check that ψU¯(u¯′1) = ψUκ(u−21) and ψUκ((u′2)−1) = ψU¯(u¯1).
Thus by changing variable, we get
Fξθ(t)= ξ(u¯1,(tw)θw−1,u2)ψU¯(u¯1)ψUκ(u−21)du¯1du2.
ZU¯×U
By Proposition1.3,we have(tw)θ =tw fort∈T , andthus we getF (t)=F (t) forall t∈T .
ψ,κ ξθ ξ ψ,κ
This proves Claim 3.
From Claim 1-Claim 3 and the definition of τ, i.e. Eq.(1.5), we get
T(fθ)=τ(F )=τ(F )=T(f ).
ξ ξθ ξ ξ
Since ξ 7→ f is surjective, we get T(f) = T(fθ) for all f ∈ S(C(w)). This completes the proof of
ξ
the theorem. (cid:3)
Proof of Theorem 1.1. Using a Gelfand’s trick, it is not hard to show that Theorem 1.5 implies
Theorem 1.1. We give some details below.
ON THE DEPENDENCE OF THE LOCAL RANKIN-SELBERG GAMMA FACTORS FOR Sp2n×GLm ON ψ 9
Take σ = π˜, then we have σ˜ ∼= π. By assumption, π is ψU¯-generic, and thus σ is ψU¯-generic by
Lemma 1.2. We fix a nonzero element λ∈HomU¯(σ,ψU¯) and a non-zero element µ∈HomU(σ˜,ψUκ).
Note that µ defines a map µ:V →C. Denote its dual by µ∗. Define a distribution T on G by
σ˜
T(f)=λ◦σ(f)◦µ∗ ∈End(C)=C,∀f ∈S(G),
where σ(f) is the canonical operator on V defined by π(f).v = f(g)σ(g)vdg. It is easy to check
σ G
that T(f) is well-defined. From the choice of λ and µ, it is easy to check that
R
T((u¯1,u2).f)=ψU¯(u¯1)ψUκ(u2)T(f),∀u¯1 ∈U¯,u2 ∈U2,f ∈S(G),
i.e., T is quasi-invariant. By Theorem 1.5, we get
Tθ =T.
Consider the bilinear form B :S(G)×S(G)→C defined by
S(f,φ)=T(f ∗φˇ),f,φ∈S(G),
where φˇ(g)=φ(g−1) and ∗ denotes convolution. We have
(1.9) B(f,φ)=T(f ∗φˇ)=Tθ(f ∗φˇ)=T((f ∗φˇ)θ)=T(φˇθ ∗fθ)=T(φρ∗fρ),
recall that gρ =w δc gc−1δ−1w−1 and gθ =(g−1)ρ =(gρ)−1.
l κ κ l
For f ∈ S(G), consider a linear function λ on V by λ (v) = λ(σ(f)v). It is clear that λ ∈
f σ f f
V˜ = V , and f 7→ λ defines a map S(G) → σ˜. As before, let r denote the right translation. We
σ σ˜ f
claim that the map
λ:(r,S(G))7→(σ˜,V ),
σ˜
f 7→λ is in fact intertwining. In fact, for g ∈G, we have
f
hσ˜(g)λ ,vi=hλ ,σ(g−1)vi
f f
=λ(σ(f)σ(g−1)v)
=λ( f(g′)σ(g′g−1)vdg′)
ZG
=λ( f(g′g)σ(g′)vdg′)
ZG
=λ(σ(r(g)f)v)
=hλ ,vi.
σ(r(g)f)
This proves the claim. Since σ˜ is irreducible and f 7→ λ is nontrivial, the map λ : S(G) → σ˜ is
f
surjective.
Similarly, we define a map S(G) →V˜ = V by f 7→ µ , and µ (v˜)= µ(σ˜(f)v˜), for v˜∈V . Let
σ˜ σ f f σ˜
rρ be the representation of G on S(G) defined by rρ(g)f = r(gρ)f. A similar argument as above
shows that the map
µ:(rρ,S(G))→(σρ,V )
σ
f 7→µ
f
is intertwining.
Sinceρisaninvolution((g g )ρ =gρgρ,(gρ)ρ =g),aneasycalculationshowsthattheassignment
1 2 1 2
f 7→fρ defines an isomorphism
(r,S(G))→(rρ,S(G)).
Let J(λ) (resp. J(µ)) be the kernel of the map λ : (r,S(G)) → (σ˜,V ) (resp. µ : (rρ,S(G)) →
σ˜
(σρ,V )). From the irreducibility of our representations σρ and σ˜, it is easy to see that
σ
J(λ)={f ∈S(G)|B(f,φ)=0,∀φ∈S(G)},
and
J(µ)={φ∈S(G)|B(f,φ)=0,∀f ∈S(G)}.
By Eq. (1.9), we have
J(λ)={f ∈S(G)|B(φρ,fρ)=0,∀φ∈S(G)}=J(µ)ρ.
10 QINGZHANG
Now the isomorphism (r,S(G))7→(rρ,S(G)), f 7→fρ gives the following commutative diagram
// // λ // //
0 J(λ) (r,S(G)) σ˜ 0
✤
✤
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)✤
0 // J(µ)=J(λ)ρ //(rρ,S(G)) µ // σρ // 0,
and hence σ˜ ∼= σρ. Recall that gρ = w δc gc−1w−1. It is clear that σρ ∼= σκ ∼= πκ, where we used
l κ κ l
πκ ∼=π˜κ. Thus we get π ∼=πκ, which finishes the proof of Theorem 1.1. (cid:3)
f
f
2. Dependence on ψ for generic representations for Sp (F)
2n
Let Sp (F) be the metaplectic double cover of Sp (F) which is realized by the Rao cocycle,
2n 2n f
[Rao]. More explicitly, let µ ={±1}, then as a set Sp (F)=Sp (F)×µ . The multiplication in
2 2n 2n 2
Sp (Ff) is given by
2n
f
(g ,ǫ )(g ,ǫ )=(g g ,c(g ,g )ǫ ǫ ),g ,g ∈Sp (F),ǫ ,ǫ ∈µ ,
f 1 1 2 2 1 2 1 2 1 2 1 2 2n 1 2 2
where c:Sp (F)×Sp (F)→µ is the Rao cocycle defined in [Rao].
2n 2n 2
It is well-known that the double cover Sp (F) → Sp (F) splits trivially over U . Let U be
2n 2n Sp2n
the preimage of U =U in Sp (F). We have U ∼=U ×µ as a group. For a nontrivial additive
Sp2n 2n 2
character ψ of F and κ∈F×, let ψκ be thfe character of U˜ defined by
U
f
ψκ((u,ǫ))=ǫψκ(u). ‹
e U˜ ‹ U
If κ=1, we will write ψκ as ψ for brevity.
U U
2.1. The κ-action on Sep (Fe). Let (π,V) be an irreducible admissible genuine representation of
2n
Sp (F). In [Sz], it was showed that
2n
f
Hom (π,ψκ)≤1,
f U˜ U
for any κ. If Hom (π,ψκ)6=0, we say that π is ψκ-generic as usual.
U˜ U Ue
FromthetheoryoftheWeilrepresentation(seepage36of[MVW]),thereisamapv :Sp (F)→
κ 2n
µ such that the assignment (g,ǫ) 7→ (g,ǫ)κ := (gκ,ǫv (g)) defines an automorphism of Sp (F).
2 e e κ 2n
Alternatively, we have
(g,ǫ)κ =(c ,1)·(g,1)·(c ,1)−1, f
κ κ
where (c ,1)∈GSp (F) and the product is the product in GSp (F) defined using v , see §2B of
κ 2n 2n κ
[Sz].
The map v is determined in [Sz], Eq.(2-21). We give some details on the formula of v . For an
κ κ
integer j with 0≤j ≤n, let Ω be the subset of Sp (F) which consists elements of the form
j 2n
‹ ‹
A B
,A,B,C,D ∈Mat (F), with dimker(C)=n−j.
C D n×n
Then we have the decomposition Sp (F) = Ω , see Lemma 2.14 of [Rao]. For g ∈ Ω , then
2n j j j
Eq.(2-21) of [Sz] reÅads ã
`
(2.1) v (g)=(x(g),κj+1) (κ,κ)j(j−1)/2,
κ F F
where x is the function x : Sp → F×/F×,2 defined in Theorem 3.5 of [Rao], and ( , ) is the
2n F
Hilbert symbol.