Automorphic Forms with Anisotropic Periods on a Unitary Group · 2014-08-28 · automorphic forms...

Journal of Algebra 220, 636–663 (1999)

Article ID jabr.1998.7928, available online at http://www.idealibrary.com on

Automorphic Forms with Anisotropic Periods on aUnitary Group

Yuval Z. Flicker*

Department of Mathematics, The Ohio State University, 231 West 18th Avenue,Columbus, Ohio 43210-1174

E-mail: [email protected]

Communicated by Walter Feit

Received January 29, 1999

0. STATEMENT OF RESULTS

Let G be an algebraic group over a global field F with ring ! of adeles,denote by Z the center of G and by C an algebraic subgroup of G overF such that the cycle C!F"\C!!" has finite volume. Fix unitary characters!# Z!!"/Z!F" $ "1 != unit circle in "%" and "# C!F"\C!!" $ "1, anddenote by ## G!F"\G!!" $ " a cusp form in the cuspidal representation$ of G!!", whose central character is !$ = !. By a cuspidal representationwe mean an irreducible one. We say that $ is C!!"-cyclic if it has a non-zero C!!"-period PC!!"!#" =

!C!F"\C!!" #!c""!c"dc. The overbar indicates

complex conjugation. Studies of cyclic automorphic forms have applicationsto special values of L-functions (Waldspurger [W1,W2], Jacquet [J1, J2]),lifting problems [F], and studies of cohomology of symmetric spaces, inparticular the Tate conjecture on algebraic cycles on some Shimura surfaces[FH]. The purpose of this paper—inspired by the applications to the Tateconjecture of [FH]—is to compare the notion of cyclicity by C!!", withcyclicity by an inner form of C!!". We let G be the quasi-split unitary groupU!2% 1" = U!2% 1&E/F" in three variables defined by means of a quadraticseparable extension E/F of global fields. The subgroup C is taken to be the

*Present address until November 1999: Department of Mathematics, National Universityof Singapore, 2 Science Drive 2, Singapore 117543.

636

0021-8693/99 $30.00Copyright © 1999 by Academic PressAll rights of reproduction in any form reserved.

automorphic forms with anisotropic periods 637

quasi-split unitary group U!1! 1" = U!1! 1#E/F" in two variables defined byE/F (from Section 1 on—where more precise definitions are introduced—C and " of this section will be denoted by C1

1 and "2).The notation U!2! 1" and U!1! 1" is borrowed from the theory of real

groups, but we do not discuss that theory in this paper, and for us G =U!2! 1" and C = U!1! 1" are the uniquely defined (up to isomorphism)quasi-split E/F-unitary groups in three and two variables. The anisotropicinner form of C over F will be denoted by U!2" = U!2#E/F". To be ableto concentrate on the aspects of the p-adic theory, we assume either thatF is a function field (of odd characteristic), or that at each place v of Fwhere Ev/Fv = !/" the group U!2"!"" (= the group of real points ofU!2!E/F"" is quasi-split (hence isomorphic to U!1! 1"!""". Then U!2" isnot isomorphic to U!1! 1" only at a finite set $ of finite places of F , whichstay prime in E. Indeed, at a place v which splits in E the anisotropic innerform D!2!Fv" of U!1! 1#Fv" = GL!2!Fv" is not a subgroup of U!2! 1#Fv" =GL!3!Fv". Moreover, neither GL!2!Fv" nor D!2!Fv" is a subgroup of theanisotropic inner form D!3!Fv" of U!2! 1#Fv". Hence our question cannotbe asked with any inner form of U!2! 1" other than that which is split ateach place v of F which splits in E. So we stick to our group G. It is wellknown that the cardinality %$% of $ is even.

We shall also consider the local analogous question of cyclicity. At aplace v of F which splits in E we fix characters #v! #v of F&

v , and saythat an admissible irreducible GL!3!Fv"-module $v with central charac-ter #v is GL!2!Fv"-cyclic if there is a non-zero linear form l' $v ( !with l!$v!h"w" = "v!h"l!w" !h ) GL!2!Fv"!w ) $v", where "v = #v * det!namely HomCv

!$v! "v" += ,0- (we put Cv for C!Fv" = U!1! 1#Fv", which isGL!2!Fv" in our case). If v is a place of F which stays prime in E thenwe similarly say that the irreducible admissible representation $v of Gv =G!Fv" = U!2! 1#Fv" is Cv-cyclic if HomCv

!$v! "v" += ,0-. Here Cv can bethe quasi-split unitary group U!1! 1#Fv", or the anisotropic form U!2#Fv".In all cases a result of the Appendix to [F6] asserts that dim!!$v! "v" . 1.Moreover it is easy to see that all local components $v of a C!$"-cycliccuspidal representation $ are Cv-cyclic (with C = U!1! 1" or U!2"". How-ever, a cuspidal $ all of whose components are Cv-cyclic, need not beC!$"-cyclic. It is the global obstruction to the global cyclicity of an every-where locally cyclic $, which is of interest. Thus our question is: WhenHomC!$"!$! "" is non-zero, is it generated by PC!$"?

0.1. Theorem. (1) Every U!2#$"-cyclic cuspidal generic representationof U!2! 1#$" is U!1! 1#$"-cyclic.

(2) If Ev/Fv is a local field extension, every U!2#Fv"-cyclic (irreducibleadmissible) generic representation of U!2! 1#Fv" is U!1! 1#Fv"-cyclic.

638 yuval z. flicker

The identification of those U!1! 1"-cyclic generic U!2! 1"-modules whichare also U!2"-cyclic cannot be stated simply in terms of the representationtheory of the group U!2! 1", since this last theory is described by meansof liftings. Two important such liftings are introduced in [F1] and studiedin [F2,F3], namely the "-endoscopic lifting from U!1! 1" to U!2! 1", whichdepends on a choice of a character "# !$

E /E$NE/F!

$E % "1 whose restric-

tion to !$/F$N!$E is non-trivial, and the base-change lifting from U!2! 1"

to GL!3!E".In [F6] the U!1! 1"-cyclic generic U!2! 1"-modules are identified as the

image of the "-endoscopic lifting from U!1! 1". In fact [F4] establishes acorrespondence between the set of packets of U!1! 1&!"-cyclic generic cus-pidal U!2! 1&!"-modules, and the set of generic GL!2!!E"-modules whichare cuspidal and GL!2!!"-cyclic (the adjective “distinguished”—instead of“cyclic”—is used in [F4] and [F6] in this context, and with # = 1, andso will it be here), or are normalizedly induced I '!µ'

1!µ'2" from the char-

acter! a (0 b

")% µ'

2!a"µ'2!b" of the upper triangular subgroup B'!!E" of

GL!2!!E", where µ'i# !$

E /!$E$ % "1 and µ'

1 *= µ'2. In [F4] it is shown

that this last set of GL!2!!"-distinguished cuspidal GL!2!!E"-modules,and the I '!µ'

1!µ'2"! µ'

i# !$E /!

$E$ % "1!µ'1 *= µ'

2, is the image of theunstable base-change lifting (this lifting also depends on a choice of ")from U!1! 1&!" to GL!2!!E". The composition of the "-base-change lift-ing from U!1! 1" to GL!2!E", and the correspondence from (distinguishedgeneric representations on) GL!2!E" to (cyclic generic representations on)U!2! 1", is the "-endoscopic lifting. The analogous local results are also es-tablished in [F6] (and [F4]).

We repeat that by a GL!2!!"-distinguished cuspidal representa-tion of GL!2!!E" we mean one which is cyclic, with # = 1. SinceHomGL!2!!"!$! #" = HomGL!2!!"!$ + #,1! 1", where # denotes also an ex-tension of # from !$/F$ to !$

E /E$, there is no loss of generality in taking

# = 1. We denote -z . E$v & zz = 1/ by E1

v .

0.2. Theorem. (1) A U!1! 1&!"-cyclic generic cuspidal representa-tion $ of U!2! 1&!" is U!2&!"-cyclic precisely when for each v . 0(= the set of finite F-places where U!2&Fv" is anisotropic) the component$v of $ does not correspond (in the sense of [F6]) to the GL!2!Ev"-module I '!µ'

1!µ'2"!µ'

i# E$v /F

$v % "1, namely $v is not the "v-endoscopic

lift of the U!1! 1&Fv"-packet $0!µ1!µ2" of [F2, Sections 3.7/8, page 49],µi# E1

v % "1!µ'i!z" = µi!z/z"!z . E$

v " (where $0 is denoted by %).(2) A U!1! 1&Fv"-cyclic generic admissible irreducible U!2! 1&Fv"-

module is U!2&Fv"-cyclic precisely when it does not correspond (a la [F6]) toI '!µ'

1!µ'2"!µ'

i# E$v /F

$v % "1!µ'

1 *= µ'2, namely it is not the "v-endoscopic lift

of $0!µ1!µ2"! µi# E1v % "1!µ1 *= µ2.


Another way of stating these results is by means of the base change theoryof [F2], from U!2! 1" to GL!3!E". Thus [F6] asserts that a generic admissi-ble irreducible U!2! 1#Fv"-module is U!1! 1#Fv"-cyclic when its base changeis generic but not discrete series (= square-integrable), and Theorem 0.2(2)asserts that it is also U!2#Fv"-cyclic when its base-change is also not in-duced of the form I!µ$

1!µ$2!µ

$3"! µ$

i% E&v /F

&v ' !1, with distinct µ$

i. Theo-rem 0.2(1) asserts that the U!1! 1#""-cyclic " is also U!2#""-cyclic if each"v!v ( )" is U!2#Fv"-cyclic and not of the form I!µ$

v"! µ$v% E&

v /F&v ' !1.

We denote by I!µv" the U!2! 1#Fv"-module with central character #v nor-malizedly induced from the character diag!a! b! 1/a" *' µv!a"!#v/µv"!b"of the upper triangular subgroup B!Fv" of Gv = U!2! 1#Fv".

As a notational convention, representations of U!1! 1" will have anindex zero (e.g., "0, I0!µ"", those of GL!2" will carry a prime (e.g.," $! I $!µ1!µ2"", while those of U!2! 1" are denoted simply by " or I!µ".Also by a U!1! 1"-cyclic generic U!2! 1"-module we mean the set ofU!1! 1"-cyclic generic elements in its packet (as defined in [F2]). It is ex-pected that this set consists of a single element, but this has not beenshown as yet.

For an archimedean analogue of our local results the reader may like toconsult Oshima and Matsuki [OM] and Kobayashi [Ko].

The statement of Theorem 0.2 suggests that its proof would be relatedto the variation of the notion of distinguishability (= cyclicity) of a genericcuspidal GL!2!"E"-module, with respect to inner forms of GL!2!"". Thisquestion is studied in [FH], whose result which is needed for the proof ofTheorem 0.2 (and Theorem 0.1) is reviewed next.

Denote by D the inner form of GL!2" over F which is ramified preciselyat the places of ). Then D!"" is a subgroup of D!"E" = GL!2!"E".

0.3. Theorem (0.2 of [FH]). A cuspidal (irreducible) representation " $ ofGL!2!"E" is D!""-distinguished precisely when it is GL!2!""-distinguishedand its components " $

v at v ( ) are not of the form I $!µ$1!µ

$2" with µ$

i trivialon F&

v .

Our proof of Theorems 0.1 and 0.2 consists of a comparison of U!2"-cyclic U!2! 1"-modules with D-distinguished GL!2"-modules (same ) forU!2" and D), and an application of Theorem 0.3. We then state this com-parison next. For this purpose we recall that the correspondence of [F6] re-lates almost everywhere locally distinguished automorphic representations" $ of GL!2!"E" with almost everywhere locally cyclic automorphic repre-sentations " of U!2! 1#"", and the relation is that " $ and " correspond if

(1) at almost all places v of F which split, the GL!2!Ev" =GL!2!Fv" &GL!2!Fv"-module " $

v = " $1v &" $

2v is distinguished (has a non-zero GL!2!Fv"-invariant form; then " $

2v + " $1v(= contragredient of " $

1v)),


and then !v is the GL!2"Fv"-cyclic GL!3"Fv"-module I!! #1v $ #v/#! #

1v"

normalizedly induced from the maximal parabolic of type (2,1) as indi-cated, where #v = #!v

is the central character of !v and #! #1v

is that of! #1v, and

(2) at almost all places v of F which stay prime in E, the GL!2"Fv"-distinguished component ! #

v is of the form I #!µv" = I #!µv"µ%1v ", and the

corresponding U!1" 1&Fv"-cyclic component !v is I!µv".Note that the central character of !v is #v, and that of ! #

v is ##v/$

#v,

where $#v!z" = $v!z/z"!z ' E$v " is well defined since $v is a character of

E1v = (z ' E$

v & zz = 1).

The theorems of [F6] establish that the correspondence relates cuspi-dal distinguished ! #, or ! # of the form I!µ#

1"µ#2""µ#

1 *= µ#2, with cuspidal

cyclic !. Since the groups U!2&E/F" and D are not isomorphic over Fv toU!1" 1&E/F" and GL!2" only for v in the finite set +, and the definitionof correspondence depends only on almost all places, the definition applieswith these anisotropic groups, and we can state the following.

0.4. Theorem. The correspondence is a bijection from the set of D!!"-distinguished cuspidal GL!2"!E"-modules with central character ##/$#, tothe set of packets of U!2&!"-cyclic generic cuspidal U!2" 1&!"-modules withcentral character #.

This global result permits extending the definition of the local correspon-dence (the definition depends on a certain relation of Whittaker–Period dis-tributions) in the case that Ev/Fv is a field to show that the correspondenceis a bijection from the set of Dv-distinguished generic GL!2"Ev"-modules! #v to the set of packets of U!2&Fv"-cyclic generic U!2" 1&Fv"-modules !v

(with #! #v= ##

v/$#v if #!v

= #v", such that ! #v is square-integrable precisely

when !v is, and ! #v is supercuspidal precisely when !v is, and !v = I!µv"

when ! #v = I #!µv", as defined above.

Theorems 0.1 and 0.2 follow on combining the correspondence of Theo-rem 0.4 with that of Theorem 0.3 ([FH]) and that of [F6]. This paper willthen be concerned with the proof of Theorem 0.4, which is an anisotropicanalogue of the work of [F6]. Most of the technical difficulties in ourpresent project have already been overcome in [F6], and those of the pas-sage from an inner form to the quasi-split form, in [FH]. Theorem 0.1is easy to prove by a direct comparison of U!1" 1"-cyclic and U!2"-cyclicforms, but to prove Theorem 0.2 by such a direct comparison we wouldneed to compute directly the Whittaker–Period distributions of the localrepresentations mentioned in Theorem 0.2. We prefer to deduce these lo-cal computations from the global comparisons; see Proposition 11 below.Analogous local computations have been carried out for the comparison


of GL!2!Fv"- and Dv-distinguished representations of GL!2!Ev", in [FH],except that [FH] considers the bi-period distribution attached to a distin-guished " # (and names it “relative”), and not our Whittaker–Period one,which is attached to a generic distinguished " #.

As in [F6] our work depends on a comparison of Fourier summationformulae on U!2! 1$!" and GL!2!!E". However these formulae simplify inour case as we take periods with respect to the anisotropic groups U!2$!"and D!!", instead of U!1! 1$!" and GL!2!!" as in [F6]. The comparisonis based on a transfer of Fourier orbital integrals of general (and spherical)functions between U!2! 1" and GL!2!E", and this was carried out in boththe split and non-split cases in [F6] (and [F7]). The required analysis in theremaining finite number of places v in % where U!2" and D ramify is carriedout here. It is easier than the analysis of [F6], since we deal with anisotropicgroups. We also explain the transfer from U!2!Fv" to U!1! 1$Fv", and fromDv to GL!2!Fv"; these local transfers can be used in the correspondingglobal comparisons (the one with Dv and GL!2!Fv" can replace C1 of[FH], but only when V ## of [FH] is empty, that with U!2" and U!1! 1" canbe used to give an alternative proof of our Theorems 0.1 and 0.2, as notedabove).

Our usage of a “Fourier” summation formula, which involves Fourier co-efficients of cusp forms, limits our discussion to the case of the genericrepresentations, those with a Whittaker model. It would be interestingto study the notion of cyclicity for degenerate U!2! 1$Fv" and U!2! 1$!"-modules (where a packet of representations is expected to contain only onegeneric element), and perhaps a bi-period summation formula for U!2! 1"—analogous to that of [FH] in the case of GL!2!E" and period GL!2!F"—would be of use. But we have not done that.

As is well known, some of our results can be obtained by the theta cor-respondence, but our interest is in the intrinsic approach of the summationformula.

1. THE GROUPS

We shall now define the groups which are studied in this paper. Let E/Fbe a quadratic extension of local or global fields of characteristic other than2. Denote by an overbar the action of the non-trivial element of Gal!E/F",and write g = !gij" for g = !gij" in GL!n!E". The unitary group in threevariables of interest to us here is

G = U!2! 1$E/F" =

!"

#g & GL!3!E"$ gJtg = J =

$

%0 1

'11 0

&

'

()

* #


The quasi-split unitary group U!1! 1"E/F# in two variables which is consid-ered in [F6] is related to the centralizer

C = U!1# $U!1! 1# = ZG!J0# =!g % G" gJ0 = J0g

"

of

J0 = diag!1!&1! 1#"

The anisotropic unitary group U!2!E/F# considered in this paper is relatedto

C# = U!1# $U!2# = ZG!J##! J# =

#

$0 1/2#

12# 0

%

&

In the local case we take # % F &NE/FE; here N = NE/F denotes the normmap from E to F . In the global case we take # % F such that # '% NEv/Fv

Ev

for all v % (, and # % NEv/FvEv for all v '% (. Then C# is anisotropic pre-

cisely at the places in (, a finite set of finite places with even cardinality.Up to isomorphism over F , the group C# depends only on #NE$; indeedJ#$$ is conjugate to J# by diag !$! 1! 1/$# % G !$ % E$#.

In [F6, Proposition 2], it was noted that J1 = g&10 J0g0, and

G = BC ) Bg0C = BC1 ) Bg&10 C1! g0 =

#

''$

1 &1 12

1 0 & 12

12

12

14

%

((& "

Here B denotes the upper triangular subgroup of G, and the index 1 in J1and C1 indicates # = 1. When # '% NE$ the group C# is anisotropic, andwe have a simpler decomposition.

1. Proposition. If # % F &NE, then G = BC#.

Proof. The 3$ 3 matrices which commute with J# have the form

g =

#

$a d c/4#2

b e b/2#c 2#d a

%

& "

Then

JtgJ =

#

$a &b/2# c/4#2

&2#d e &dc &b a

%

& !

and g % C# = ZG!J## if gJtgJ = I. Hence g % ZGL!3!E#!J## lies in C#

when there is % % E1 (thus %% = 1) with d = %b/2#! e = %!a + c/2##,


with (1) !a" c/2!#!a" c/2!# = 1 and with (2) !a+ c/2!#!a+ c/2!# =bb/! + 1 (note that any two of (1), (2), and (3) ac + ac = bb, imply thethird).

Let Y be the subvariety of x = !x1"x2"x3# in the projective 2-space overE with xJtx = 0. Then G acts transitively on Y by g$ x %& xg"1. Thestabilizer of x0 = !0" 0" 1# is B = stabG x0. Given x = !c""b" a# in Y wehave (3), hence a '= c/2! (since ! ( F "NE#, and dividing its componentsby a" c/2! we may assume that x satisfies (1), whence (2). For any choiceof # ( E)" ## = 1, define d = #b/2!" e = #!a + c/2!#. We then defineg ( C! with x = x0g

"1, and the proposition follows.

Corollary. When ! ( F " NE, the group C! consists of h!"! =diag!1" #" 1#" # ( E1, and

h =

!

"a b/2! c/4!2

b a+ c/2! b/2!c b a

#

$ "

with a" b" c ( E satisfying (1) !a " c/2!#!a " c/2!# = 1, and (2) !a +c/2!#!a+ c/2!# = bb/!+ 1.

Remark. (1) Note that g0 satisfies a = c/2! with ! = 1, where!c""b" a# = x = x0g0 = ! 12 "

12 "

14#, hence g"1

0 '( BC1.

(2) We have deth = a" c/2! ( E1$Any character %$ C! & ") of the group C! is of the form %!h!# =

%1!deth#%2!##, where %1" %2 are characters of E1. Recall that the principalseries representation I!µ&s# of G with central character '$ E1 & "1, whereµ$ E) & ") and &!x# = *x*E !x ( E)#, is defined on the space of smoothfunctions ( on G which satisfy (for any upper triangular unipotent u)

(%diag!)"*" )"1#ug

&= µ!)#

''

µ

(!*#*)*s+1

E (!g#%g ( G" ) ( E)" * ( E1&$

2. Proposition. We have HomC!!I!µ&s#" %# '= 0 precisely when µ =

%1%2 and ' = %1%22 on E1.

Proof. (a) When ! ( F " NE the group C! is compact, and we claimthat there is a non-zero linear form L on + = I!µ&s# which transformsunder C! by %, precisely when there is a non-zero vector u in + whichtransforms under C! by %. Indeed, given L there is w in + with L!w# '= 0,and u =

)C!

+!h#w · %!h#"1 dh satisfies +!t#u = %!t#u !t ( C!# and L!u# '=0, hence u '= 0. In the opposite direction, let l be a linear form on + with


l!u" #= 0, and define L by L!w" =!C!

l!"!h"w"#!h"$1 dh. Then L!u" #= 0,hence L has the required properties.

(b) If $ % " = I!µ%s" satisfies "!h"$ = #!h"$ !h % C!", since G =BC!, $ is determined by its values on B & C! = 'diag!a& 'a& a"; '' = 1&aa = 1(. There

µ!a"!(/µ"!a'"$!e" = $"diag!a& 'a& a"

#= #1!a"#2!'"$!e")

If $ #= 0, then #2 = (/µ and #1 = µ2/( = µ/#2, so µ = #1#2 and ( = #1#22

on E1. Conversely, if µ = #1#2 and ( = #1#22 on E1, then I!µ%s" contains

a one-dimensional space of $0 with "!h"$0 = #!h"$0 !h % C!", and thelinear form L!$" =

!C!

$!h"#!h"$1dh is non-zero on $0 and transformsunder C! via #.

(c) When ! = 1, the proposition is proven [F6, Proposition 29(a)] for#1 = 1 = #2. The extension to arbitrary #i is immediate.

Notations. Denote by C1! the subgroup of h! in C! with deth = a $

c/2! equals 1. Any h! % C! can be written in the form zh)!z/z withz = a$ c/2!, z = diag(1, z, 1), deth) = 1. Hence for " with central char-acter ( = #1#

22 , we have HomC!

!"& #" = HomC1!!"& #". On C1

! we have#!h!" = #2!'") Now J1 = g$1

0 J0g0, and g0C1g$10 = C. Matrix multiplica-

tion shows that the (2,2) entry of g0!g$10 is 1, while that of g0hg

$10 is deth.

The character # on C = 'g = !gij" % G* gij = 0 if i + j is odd( takes thenthe value #2!g11g33 $ g13g31"#1!g22" at g. Note that g0C

11g

$10 = C1 is 'g %

G* g22 = 1(, and the restriction of # to C1 is #!g" = #2!det g". If #1 = (/#22,for " with central character ( we have HomC!"& #" = HomC1!"& #2". As in[F6] to simplify the notations we shall often work with linear forms whichtransform under the subgroup C1

! or C1 by the character #2 + det. This iswhat we did in the Introduction (e.g., Theorem 0.4), where the cycle istaken to be U!2" or U!1& 1", denoted C (instead of C1

! as here), and thecharacter is denoted by # (instead of by #2 as here).

In view of these definitions, we restate Proposition 2 as follows.

2). Proposition. The space HomC1!!I!µ%s"& #2" is non-zero precisely

when µ = (/#2.

According to Keys [Ke] (as recorded in [F1, (3.1(3)), p. 558]), when ( =*3 and µ!z" = *!z/z"!z % E," for a character * of E1, the induced I!µ%"has a square-integrable “Steinberg” subrepresentation denoted St!µ%", anda one-dimensional quotient "!µ%"- g ./ *!det g". The space of St!µ%"consists of the $ in I!µ%" with

!B\G $!g"*!det g"$1 dg = 0, since I!µ%" is

of length 2.

Corollary. We have HomC!!St!µ%"& #" = '0()


Proof. Suppose that ! ! F " NE. By (a) above we need to deter-mine the " in I#µ#$ with (1) "#gh$ = $2#h$"#g$ #h ! C1

!$ and (2)!C1!"#h$%#deth$"1 dh = 0. It was shown in (b) that up to a scalar there is

a (unique) " satisfying (1), when $2 = &/µ on E1. But St#µ#$ exists onlywhen & = %3 on E1 and #µ = %/% on E% hence) µ = %2 on E1, namely$2 = % on E1. Consequently (2) implies that " = 0, since "#h$ = $2#deth$on C1

! .The case of ! = 1 is proven in [F6, Proposition 29(c)].

Given ! ! F "NE, where E/F is a quadratic extension of local fields withcharF &= 2, the anisotropic quaternion (division of rank 2) algebra over Fcan be realized as an algebra of 2 % 2 matrices over E, with multiplicativegroup

D! ="#

a b!b a

$' a' b ! E' aa" bb! &= 0

%( GL#2'E$(

This D! is an inner form of GL#2'F$: these two groups become isomorphicover E. Put B) = *

#a b0 d

$; a' d ! E%' b ! E+. The group GL#2'E$ acts

transitively by g, x -. xg"1 on the projective line over E, the stabilizer ofx0 = #0' 1$ is B). Hence

GL#2'E$ = B)D!'

and

GL#2'E$ = B)GL#2'F$ / B))1GL#2'F$ = B)D1 / B))D1(

Here )1 =#"1 i1 i

$' ) =

#"i i1 1

$, where E = F#i$ and i2 ! F . These decom-

positions hold also when E/F are global fields. Then D! splits precisely atthe v where ! ! NE%

v . It is anisotropic at the finite even set 0 of F-placeswhere ! ! Fv "NEv.

In the local case these decompositions are used in [F4, page 169], and[F6, Proposition 28], to show the following. Put µ#a$ = µ#a$ #a ! D%$ fora character on E%.

3. Proposition. When ! ! F " NE we have HomD!#I#µ1'µ2$'!$ &=

*0+ precisely when µ2 = µ"11 , and HomD!

#sp#µ$'!$ &= *0+ when the re-striction of µ to F% is non-trivial, but µ 1 NE% = 1. Further, HomGL#2'F$#I#µ1'µ2$'!$ &= *0+ precisely when µ2 = µ"1

1 or µi 1 F% = 1 #i = 1' 2$, andHomGL#2'F$ #sp#µ$'!$ &= *0+ when µ 1 F% &= 1' µ 1 NE% = 1.

Here sp#µ$ is the unique (square-integrable) subrepresentation ofthe induced I#µ#1/2' µ#"1/2$' ##x$ = 1x1 #x ! E%$, and I#µ1'µ2$is the GL#2'E$-module normalizedly induced from the character#a b0 d

$-. µ1#a$µ2#d$ of B).


2. FOURIER SUMMATION FORMULAE

The global tool used in our comparisons is a Fourier summation formula.Such formulae were developed and used already in [J1, J2,F4,F6], with re-spect to the quasi-split cycles GL!2!F" in GL!2!E" and U!1! 1" in U!2! 1".Here we shall have a simpler variant of these formulae, with respect to theanisotropic cycles D" and C1

" = U!2", where " # F $ NE. Fix characters#! $2 of !1

E/E1, and as usual, put #%!z" = #!z/z" !z # !&

E ".We first describe the Fourier summation formula on "% = D!!E"! D =

GL!2", for a test function f % =!

f %v on "% such that f %

v is smooth andcompactly supported on D%

v = D!Ev" modulo the center Z%v ' E&

v , trans-forming under Z%

v via #%v/$

%2v, with f %

v = f %v0 for almost all v. Here f %

v0 is the

unit element of the convolution algebra #%v of K%

v = D!REv"-biinvariant f %

v;a choice of a Haar measure is implicit. Let L!D%" be the space of smoothfunctions %( "% ) $ with %!&zh" = !#%/$%2"!z"%!h" !h # "%, z # %% ' !&

E !& # D% = D!E"" and

"%%D%\"% *%!h"*2 dh < +. The convolution operator

!r!f %"%"!g" =#

"%/%%f %!h"%!gh"dh =

#

%%D%\"%Kf % !g!h"%!h"dh

is an integral operator with kernel Kf % !g!h" =$

&#D%/Z% f %!g$1&h".The space L!D%" decomposes as the direct sum of three mutually or-

thogonal invariant subspaces: the space L0!D%" of cusp forms with centralcharacter #%/$%2, the space L1!D%" of functions %!g" = '!det g", where' is a character of !1

E/E1 with '2 = #%/$%2, and the continuous spectrum

Lc!D%". Denote the corresponding kernels by K0!K1!Kc . The Fourier sum-mation formula is the equality obtained on integrating K!n!h"&!n" onh # %D"\"" !D" = D"!F"! "" = D"!!"! % ' !&" and on n =

%1 x0 1

&

in N %\'% (i.e., on x in !E/E". Here & is a fixed non-trivial character of!/F , and (!n" = &!x+ x".

Using the disjoint decomposition D% = 'N %

%b 00 1

&D", union over b #

E&/E1, we note that the “geometric ” expression for the double integral""Kf % !n!h"(!n"dndh is

(

b#E&/E1

)!b! f %! ("!

)!b! f %! (" =#

'%

#

%\""

f %%n

%b 00 1

&h

&(!n"dndh*

Since f % is compactly supported modulo the center, the Bruhat decompo-

sition (and an application of the map g ,) g)%

0 "1 0

&g%0 "1 0

&$1*$1"! shows

that for a given f % the sum is finite, and the double integral ranges over a


compact in !! " "\#!. In the isotropic case, where ! = 1, there is anotherterm, "#0# f !# $$, in the geometric side; see [F4].

The integral of K1#n#h$$#n$ is zero since $ is non-trivial. The cuspidalkernel takes the form

K0#g#h$ =!

% !

!

&%% !

"% !#f !$&

##g$&#h$'

Here % ! ranges over the set of cuspidal representations of #! with centralcharacter (!/)!2, and & over an orthonormal basis of smooth functions in% !. The integral of K0#n#h$$#h$ is equal to

!

% !#W$P!$% ! #f !$#

where

#W$P!$% ! #f !$ =!

&%% !W$#% !#f !$&$P!#&$

is independent of the choice of the basis &&' of % !. Here W$#&$ =$N !\!! &#n$$#n$dn and P!#&$ =

$"D!\#!

&#h$dh.Next we record Kc#n#h$. Let µ1# µ2 be unitary characters of $"

E /E"

with µ1µ2 = (!/)!2. For any s % % consider the Hilbert space H !#µ1#µ2# s$of &( #! ) % with

&

%%a *0 b

&g

&=

''a/b''s+1/2E

µ1#a$µ2#b$&#g$"a# b % $"

E + g % #!#

and$&! ,&#k$,2 dk < -. Here &! = *K!

v#K!v = standard maximal compact

subgroup in D!v = GL#2#Ev$. The restriction to &! map & .) &,&! de-

fines an isomorphism from H !#µ1#µ2# s$ to H !#µ1#µ2$ = H !#µ1#µ2# 0$.Identify H !#µ1#µ2# s$ with H !#µ1#µ2$. Denote by &#µ1#µ2# s$ the ele-ment of H !#µ1#µ2# s$ corresponding to &#µ1#µ2$ in H !#µ1#µ2$. Denoteby I !#µ1#µ2# s$ the representation of #! on H !#µ1#µ2# s$ by right transla-tion, and

E#h# &#µ1#µ2# s$ =!

+%B!\D!&#+h#µ1#µ2# s$

"& = &#µ1#µ2$ % H !#µ1#µ2$

#

the associated Eisenstein series. The kernel on the continuous spectrum isgiven by

Kc#g#h$ =14%

!!

&

( -

/-E"g# I !#µ1#µ2# it+ f !$&#µ2#µ2# it

#

" E#h# &#µ1#µ2# it$dt'


Here ! ranges over an orthonormal basis of !!-finite functions inH !"µ1"µ2#. The first sum ranges over a set of representatives of theclasses of pairs "µ1"µ2# of unitary characters on "$

E /E$ under the equiva-

lence relation "µ!1"µ

!2# % "µ1"µ2# if "µ!

1"µ!2# = "µ1#

it"µ2#&it# "t ' ##.

Since N !\$! and %D$\&$ are compact, the integrals

E%"!"µ1"µ2" s# =!

N !\$!E"n" !"µ1"µ2" s#%"n#dn

and

E$"!"µ1"µ2" s# =!

%D$\&$

E"h" !"µ1"µ2" s#dh

converge, and we conclude that""

Kc"n"h#%"n#dndh is equal to

14&

#

("µ1"µ2#)

#

!

$! *

&*E%

$I !"µ1"µ2" it+ f !#!"µ1"µ2" it

%E$"!"µ1"µ2" it#dt' (4.1)

For a given f ! both sums are finite. In the quasi-split case, where $ = 1,the computation of

""Kc% is more involved; see [F4]. In summary, the

Fourier summation formula is the equality of the following.

4. Proposition. For every test function f ! we have#

b'E$/E1

("b" f !" %# =#

&,L0"D!#"W%P$#& ! "f !#+ "4'1#'

Next we develop the analogous formula on G = U"2" 1# and C$ = U"2#.Let f =

&fv be a smooth compactly supported function on ' = G""#

modulo % (= center of '" - "1E#, which transforms under % via )&1, with

fv = f 0v for almost all v. Here f 0

v is the unit element in the convolutionalgebra (v of the Kv = G"Rv#-biinvariant fv; Rv is the ring of integersin Fv; a choice of a Haar measure is implicit. Let L"G# be the space ofsmooth functions !. ' / ) with !"*zg# = )"z#!"g# "g ' '" z ' %"* ' G = G"F## and

"%G\' 0!"g#02 dg < *. The convolution operator

"r"f #!#"g# =!

'/%f "h#!"gh#dh =

!

%G\'Kf "g"h#!"h#dh

is an integral operator with kernel Kf "g"h# ='

*'G/Z f "g&1*h#'The space L"G# decomposes as the direct sum of three mutually orthog-

onal invariant spaces; the space L0"G# of cusp forms with central character), the space L1"G# of discrete-series non-cuspidal (necessarily non-generic)


representations, including the functions !!g" = "!det g", where " is a char-acter of !1

E/E1 with "3 = #, and the continuous spectrum Lc!G". Cor-

respondingly, Kf = K0 +K1 +Kc . The Fourier summation formula is theequality obtained on integrating K!n$h"%!n"&!h" on h # "C'\#' $ C1

'\#1'

!#' and #1' denote the groups of adele points on the algebraic groups C'

and C1' naturally defined to have C' and C1

' as their groups of F-points)and on

n =

!

"1 x y0 1 x0 0 1

#

$

in N\$ !$ = N!!"$N = N!F"$ N is the upper unipotent subgroup inB % G, thus x ranges over !E/E, and y over !E/E with y + y = xx". Alsowe put %!n" = %!x+ x".

The “geometric” expression for the double integral%%

Kf & !n$h"%!n" ·&2!h" over n # N\$ and h # C1

'\#1' is

&

b#E'/E1

(!b$ f$ %"$

where

(!b$ f$ %" ='

$

'

#1'

f(n diag!b$ 1$ b(1"h

)%!n"&2!h"dndh$

by virtue of the disjoint decomposition*

b N diag!b$ 1$ b(1"C' = *b NZ

diag!b$ 1$ b(1"C1' of G. Since f is compactly supported modulo the center,

applying the map g )* gJ'g(1 and using the Bruhat decomposition, we

conclude that for a given f the sum is finite, and the double integral rangesover a compact in $ ' #1

'. In the isotropic case, where ' = 1, one moreterm: (!0$ f$ %" of [F6], turns up.

The integral of K1!n$h"%!n"&2!h" over n is zero since % is non-trivial,and the forms ! contributing to K1 are non-generic:

%!!n"%!n"dn = 0

(see [F2]). We have'

N\$

'

C1'\#1

'

K0!n$h"%!n"&2!h"dndh =&

)

(W%P'$ &

)!f "$

where

W%!!" ='

N\$!!n"%!n"dn$ P'$ &!!" =

'

C1'\#1

'

!!h"&2!h"dh$

and(W%P'$ &

))!f " =

&

!#)W%!)!f "!"P'$ &!!"


is independent of the choice of the orthonormal basis !!" of ". The sumover " ranges over all cuspidal cyclic #P#$ %#!$ %= 0 for some ! & ") generic#W&#!$ %= 0 for some ! & "$ representations of !.

Finally we record Kc#n$h$. Let µ range over a set of representatives !µ"of the classes #µ' ( µ if µ' = µ'it #t & "$$ of unitary characters of #)

E /E).

For any s & $ consider the Hilbert space H#µ$ s$ of !* ! + $ with (forany upper triangular unipotent u)

!!diag#a$ b$ a,1$g

"= -a-s+1

E µ#a$#(/µ$#b$!#g$!a & #)

E $ b & #1E$ g & !

"

and#% -!#k$-2 dk < .. Here % = )Kv$Kv being the standard maximal

compact subgroup in Gv. As usual we use the map ! /+ !-% to iden-tify H#µ$ s$ with H#µ$ = H#µ$ 0$, and denote by !#µ$ s$ the element ofH#µ$ s$ corresponding to !#µ$ in H#µ$. The action of ! on H#µ$ s$ byright translation is denoted by I#µ$ s$, and the associated Eisenstein se-ries is

E#g$ !$µ$ s$ =$

*&B\G!#*g$µ$ s$ #! = !#µ$ & H#µ$$+

Then

Kc#g$h$ =14"

$

!µ"

$

!

% .

,.E!g$ I#µ$ it0 f $!$µ$ it

"E#h$ !$µ$ it$dt+

Here ! ranges over an orthonormal basis of %-finite functions in H#µ$.Since N\& and C1

#\$1# are compact, the integrals

E&#!$µ$ s$ =%

N\&E#n$ !$µ$ s$&#n$dn$

E#$ %#!$µ$ s$ =%

C1#\$1

#

E#h$ !$µ$ s$%2#h$,1 dh

are convergent, and we conclude that##

Kc#n$h$&#n$%2#h$dndh is equalto

14"

$

!µ"

$

!

% .

,.E&

!I#µ$ it0 f $!$µ$ it

"E#$ %#!$µ$ it$dt+ #5+1$

For a given f , both sums are finite. In the quasi-split case where # = 1 andC1# = U#1$ 1$, the computation of

##Kc&%2 is more involved; see [F6]. The

Fourier summation formula for the pair #G$C#$ is the equality in

5. Proposition. For every test function f we have$

b&E)/E1

,#b$ f$ &$ =$

"1L0#G$m#"$

!W&P#$ %

""#f $+ #5+1$0

here m#"$ denotes the multiplicity of " in the space of cusp forms on G\!0namely m#"$ = Hom!#"$L0#G$$+


3. MATCHING FUNCTIONS

For f =!

fv, the global integral !!b" f" #"" b = !bv" # !$E /!

1E , is the

product over v of

!!bv" fv" #v" ="

Nv

"

C1$" v

fv#diag!bv" 1" b

%1v "h

$#v!n"%2v!h"dndh&

#v is the component of # at v" %2v is that of %2, and bv ranges over E$v /E

1v ,

in the local case. Similarly, for f ' =!

f 'v on "' = GL!2"!E" and b =

!bv" # !$E /!

1E , the integral !!b" f '" #" is the product, over all places v of

F , of

!!bv" f 'v" #v" =

"

N 'v

"

D$" v

f 'v

#diag!bv" 1"h

$#v!n"dndh&

Definition. (1) We write !$!bv" f 'v" #v" for !!bv" f '

v" #v", and!$!bv" fv" #v" for !!bv" fv" #v", when the dependence on $ needs tobe made explicit.

(2) Denote by Cv = C(c !Gv"'

%1v " the space of complex valued

smooth functions fv on Gv which transform via '%1v on Zv and are

compactly supported modulo Zv.

(3) Denote by C 'v = C(

c !D'v" %

'2v/'

'v" the space of complex valued

smooth functions f 'v on D'

v which transform via %'2v/''v on Z'

v and are com-pactly supported modulo Z'

v.

(4) The functions fv # Cv and f 'v # C '

v are called matching if for everyb in E$

v /E1v we have !!b" fv" #v" = )b)1/2v !!b" f '

v" #v".

To relate the Fourier summation formulae we need to show that thereare sufficiently many matching functions.

6. Proposition. For every fv # Cv there is a matching f 'v # C '

v, and forevery f '

v # C 'v there is a matching fv # Cv.

Proof. Consider first the case of v such that D$" v and C$" v areanisotropic. Fixing such v we pass to local notations (i.e., omit v). Thedecomposition D' = N 'A'D$ implies that !!b" f '" #" is locally constantand compactly supported on E$/E1, and that given any locally con-stant and compactly supported function !'!b" on E$/E1 there is suchf ' with !!b" f '" #" = !'!b" for all b # E$. Similarly the decomposi-tion G = NAC1

$ implies that !!b" f" #" is locally constant and compactlysupported on b # E$/E1, and any compactly supported locally constantfunction !!b" on E$/E1 is so obtained. Since b *+ )b)1/2 is locally constant,the proposition follows for such v.


For all other v the groups D!" v and C!" v are isotropic, and the proposi-tion coincides with Proposition 7 of [F6]. Two cases are considered there,depending on whether v splits in E, or not. In both cases it is shown thatthere exists a function #$v

!b", and for each fv there is a complex num-ber %!0" fv" $v" (for each f #

v there is %!0" f #v" $v"", such that %!b" fv" $v"

is locally constant on E$v /E

1v , it is 0 for sufficiently small %b%v, and equal to

%b%v%!0" fv" $v"#$v!b" for all %b%v & B!fv". Moreover, all locally constant

functions %!b" on E$v /E

1v which vanish if %b%v is small and are equal to

%b%v%!0"#$v!b" for all %b%v & B!> 0", are of the form %!b" fv" $v" for some

such fv (see [F6, Lemmas 8 and 10]). Also %!b" f #v" $v" is locally constant on

E$v /E

1v , it is 0 if %b%v is small enough, and equal to %b%1/2v %!0" f #

v" $v"#$v!b"

for all %b%v & B!f #v". Moreover, all locally constant functions %#!b" on

E$v /E

1v which are zero if %b%v is small and are equal to %b%1/2v %#!0"#$v

!b" forall %b%v & B# !> 0", are of the form %!b" f #

v" $v" for some such f #v (see [F6,

Lemmas 9 and 11]). These characterizations imply Proposition 7 of [F6],which is our proposition, when v is a place of F such that ! ' NEv/Fv

E$v (in

particular, if v splits in E).

The characterizations described in the proof of Proposition 6 permit re-lating Fourier orbital integrals on Gv, with respect to different C!" v, andthose on D#

v, with respect to different D!" v.

Corollary. Let Ev/Fv be a quadratic extension of local fields, charFv (=2. Fix ! ' Fv )NEv.

(1) For every f !v ' C*

c !Gv"&)1v " there is f 1

v ' C*c !Gv"&

)1v "; and for

every f 1v ' C*

c !Gv"&)1v " such that %1!b" f 1

v " $v" is compactly supported onE$v /E

1v , namely it is 0 for all %b%v & B!f 1

v ", there is f !v ' C*

c !Gv"&)1v "; such

that %!!b" f !v " $v" = %1!b" f 1

v " $v" for all b ' E$v /E

1v .

(2) For every f #v! ' C*

c !D#v" '

#2v/&

#v" there is f #

v1 ' C*

c !D#v" '

#2v/&

#v",

and for every f #v1 ' C*

c !D#v" '

#2v/&

#v" such that %1!b" f #

v1" $v" is com-

pactly supported on E$v /E

1v there is f #

v! ' C*

c !D#v" '

#2v/&

#v" such that

%!!b" f #v!" $v" = %1!b" f #

v1" $v" for all b ' E$

v /E1v .

In other words, for ! ' Fv ) NEv, the %!!b" fv" $v" and %!!b" f #v" $v"

are the compactly supported functions amongst the %1!b" fv" $v" and%1!b" f #

v" $v"; the latter functions—in general—will have a specific type ofasymptotic behaviour as %b%v + *.

The global test functions f and f #, for which we need to relate the ge-ometric sides of the Fourier summation formulae, have local componentswhich are the unit elements in the respective Hecke algebras of sphericalfunctions, for almost all v. For almost all v the groups C! and D! are quasi-split over Fv since !!' F )NE/FE" lies in NEv (for almost all v). Moreover


Ev/Fv is unramified, v is finite, and !v is unramified (the maximal sub-ring of Fv on which !v is 1 is the ring Rv of integers), for almost all v.Propositions 14 and 16 of [F6] assert that these unit elements !f 0

v " Cv andf #v0 " C #

v$ are matching. More generally, the correspondence of unramifiedlocal representations stated in the introduction defines a homomorphismof the convolution Hecke algebras "v % Cv and "#

v % C #v of spherical

!Kv = G!Rv$- and K#v = GL!2!R#

v$-biinvariant) functions. The isolationargument used to derive the representation theoretic applications from theequality of the Fourier summation formulae is based on the fact (againproven in [F6, Propositions 14 and 16]) that such corresponding sphericalfunctions are matching. Since these results are used here, we briefly recalltheir statement.

In the case where v stays prime in E, and Ev/Fv! "v! #v! $v are unram-ified, we have #v = $v = 1, and the correspondence relates the unrami-fied D#

v-module I #!µ$!= I #!µ!µ&1$$ with the unramified Gv-module I!µ$,where µ is an unramified character of E'

v . The dual map D( "v ) "#v of

Hecke algebras is defined by f #v = D!fv$ if tr I #!µ! f #

v$ = tr I!µ! fv$ for allunramified characters µ of E'

v . The theory of the Satake transform impliesthat the function fv " "v (resp. f #

v " "#v$ is uniquely determined by the val-

ues of the traces tr I!µ! fv$ (resp. tr I #!µ! f #v$$, where µ runs through the

variety of unramified characters of E'v . Proposition 14 of [F6] asserts that

the corresponding fv " "v and f #v = D!fv$ " "#

v are matching.In the case where v is a finite place which splits in E, we have Ev =

E *F Fv = Fv + Fv, and E1v = ,!z! z&1$- z " F'

v ., since a = !a2! a1$ if a =!a1! a2$ " Ev. Put #0v!z$ = #v!!z! z&1$$ and $0v!z$ = $2v!!z! z&1$$. A char-acter µ of E'

v is a pair !µ1!µ2$ of characters µi of F'v . The correspondence

relates the unramified GL!2!Fv$-module % #v = I #!µ$ = I #!µ1!µ2$ (more

precisely the D#v-module % #

v ' % #v) with µ1µ2 = #0v/$0v, to the unrami-

fied GL!2!Fv$ = Gv-module %v = I!µ1!µ2! $0v$, whose central characteris #0v. Beware: not all principal series unramified Gv-modules with centralcharacter #0v are obtained by the correspondence, since one of the three in-ducing characters is taken to be $0v. The dual map D( "v ) "#

v of Heckealgebras is defined by f #

v = D!fv$ if tr I #!µ1!µ2- f #v$ = tr I!µ1!µ2! $0v- fv$

for all unramified characters µ1!µ2 of F'v with µ1µ2 = #0v/$0v. The the-

ory of the Satake transform implies that the function f #v " "#

v is uniquelydetermined by the values of the traces tr I #!µ1!µ2- f #

v$ for all unramifiedµ1!µ2 with µ1µ2 = #0v/$0v. But the map D( "v ) "#

v is not injectivesince fv will be uniquely determined by the traces tr I!µ1!µ2!µ3- fv$ or alltriples of characters µi of F'

v , but not by the subset where µ3 is limited to#0v/$0v. Proposition 16 of [F6] asserts the following.

7. Proposition. Corresponding fv " "v and f #v = D!fv$ " "#

v are match-ing.


In fact the proof of [F6, Proposition 16] is carried out (1) only with!0v = "0v = 1, but this restriction was made there merely to simplify thenotations and is easily removable; and (2) only for fv ! !v with value zeroat diag"#v$ 1$ 1# and diag"#2

v $ #v$ 1#, but this restriction does not limit theapplicability of our formula, since all f $

v ! !$v are nevertheless obtained via

D from this set of fv’s.

4. GLOBAL CYCLICITY

We begin with the following separation argument.

8. Proposition. Let V be a finite set of F-places containing all v whichare ramified in E, those with % ! Fv %NEv, and those where !$ " or & areramified. For each v in V let fv ! Cv and f $

v ! C $v be matching functions.

Put f = "!

v!V fv# & "!

v '!V f 0v # and f $ = "

!v!V f $

v# & "!

v '!V f $v0#' At each

v /! V fix corresponding unramified D$v- and Gv-modules # $

v and #v. Then"

# $(L0"D$#

#W&P%

$# $ "f $# =

"

#(L0"G#m"##

#W&P%$ "

$#"f #' "8'1#

Here # $ ranges over all cuspidal representations of "$ whose component atv '! V is # $

v, while # ranges over a set of representatives for the equivalenceclasses of cuspidal representations of # whose component at v '! V is #v.

Proof. Consider f V =!

v '!V fv, where fv ! !v for all v '! V , and fv = f 0v

for almost all v '! V , and f $V =!

v '!V f $v, where f $

v = D"fv# ! !$v. Note

that f $v0 = D"f 0

v #, and that fv and f $v are matching by Proposition 7. Note

that fv ) f 0v = fv and f $

v ) f $v0 = f $

v for all v '! V . Since ("b$ f ) f V $ &# =("b$ f $ ) f $V $ &# for all b ! E*/E1 (since +b+ = 1 by the product formulaon a global field), Propositions 4 and 5 imply that

"

# $(L0"D$#

#W&P%

$# $

#f $ ) f $V $+ "4'1#

="

#(L0"G#m"##

#W&P%$ "

$#"f ) f V #+ "5'1#

For each v '! V , the operator # $v"f $

v# acts on ) ! # $ "( L0"D$## as zerounless ) is K$

v = GL"2$R$v#-right invariant, and then it acts as the scalar

tr# $v"f $

v#. Note that if # $ contributes to the sum, then P%")# '= 0 for some), namely # $ is distinguished. Hence so is each component of # $, and atthe split places we have that # $

v = # $$v * # $$

v , and tr# $v"f $

v# = tr# $$v "f1v )

f )2v#, if f $

v""g1$ g2## = f1v"g1#f2v"g2#, and f )2v"g# = f2v"g%1#. To alleviate the

notations we take f2v = f 0v and write # $

v = # $v * # $

v and f $v = "f $

v$ f0v #, so


that ! !v denotes also the underlying GL"2"Fv#-module, and f !

v denotes theunderlying function on GL"2"Fv#. It then follows that

!W#P$

"! ! "f ! $ f !V # = tr! !V "f !V # ·

!W#P$

"! ! "f !#"

where tr! !V "f !V # = #v %&V tr! !

v"f !v#. This is zero unless ! !

v is unramified"v %& V #.

Similarly we have!W#P$" %

"!"f $ f V # = tr!V "f V # ·

!W#P$" %

"!"f #"

and this is zero unless !v is unramified for all v %& V . At a place v %& Vwhich stays prime in E, the component !v is the unramified constituent ofI"µv#, for some µv' E(

v ) !(, and tr!v"fv# = tr! !v"D"fv## where ! !

v isthe unramified constituent of I !"µv#.

Each ! which contributes to the sum is cyclic (since P$" %"&# %= 0 forsome & in !), hence so is each of its components. It is also generic (sinceW!"&# %= 0#, and each component !v of an automorphic representation isunitarizable. Consequently * by Proposition 0 of [F5]—if v splits in E thenthe generic unitarizable cyclic unramified GL"3"Fv#-module !v is of theform I"µ1"µ2" %0v#" µi' F(

v ) !( (unramified with µ1µ2%0v = '0v#, andtr!v"fv# = tr! !

v"D"fv## where ! !v = I !"µ1"µ2#.

The difference of the sums over ! and ! ! can be written then as$

! !+L0"D!#

!W#P$

"! ! "f !# · tr! !V "f !V #

=$

!+L0"G#m"!#

!W#P$" %

"!"f # · tr! !V "f !V #,

the difference (5.1)–(4.1) of integrals can similarly be expressed as an in-tegral over s & "/"log qv1#

*1# involving tr I !"µv1(isv1" f

!v1# for some v1 %& V

which stays prime in E. A standard argument of “generalized linear inde-pendence of characters” (on GL"2#; see, e.g., Theorem 2 of [FK]), based onthe absolute convergence of the two sums and two integrals here, the unitar-ity of all automorphic representations present, and the Stone–Weierstrasstheorem, implies (that "4)1# = "5)1# and) the proposition as stated.

Remark. By the rigidity and the multiplicity one theorems for GL"2#,the sum over ! ! consists of at most one term.

We need to relate the global distributions with products of local ones.This is done next.

If !v is irreducible then the dimension of each of the complex spacesHomNv

"!v" #v# and HomC$" v"!v" %v# is at most one (see the Remarks at

the end of the Appendix to [F6]). If the first space is non-zero we choose


a generator W!vand say that "v is generic. If the second space is non-

zero we choose a generator P#$ %vand say that "v is C#$ v-cyclic. If "v is

unramified then its space contains a Kv-fixed vector &0v. When !v and %v

are unramified the linear forms W!vand P#$ %v

are non-zero at &0v, and

they can be normalized to take the value 1 there. This &0v is used in the

presentation of a cuspidal " as a product!

"v; its space is spanned bylocal products

!&v, with &v ! "v for all v and &v = &0

v for almost all v.The linear forms W! and P#$ % on " are scalar multiples of

!W!v

and!P#$ %v

.Let "& = &#"v$% be a Kv-finite orthonormal basis of the space of "v.

Then"W!v

P#$ %v

#"v#fv$ =

$

&

W!v#"v#fv$&$P#$ %v

#&$

defines a linear form on Cv = C&c #Gv$'

'1v $. This functional is independent

of the choice of the basis "&%, it is zero unless "v is generic and cyclic, andfor inequivalent "vi #1 ( i ( k$ the #W!v

P#$ %v$"vi

are linearly independent.It transforms under left translations of fv by Nv via !v, and under righttranslations of fv by C#$ v via %v.

For every cuspidal representation ", non-trivial character !, element # !F)$ and character %, there is a complex number c#"$!$ %$ #$ such that

"W!P#$ %

#"

%&fv'= c#"$!$ %$ #$

(

v

"W!v

P#$ %v

#"v#fv$(

Both sides are zero unless all "v are generic and cyclic. The cuspidal " is(automorphically) cyclic precisely when all "v are cyclic, and c#"$!$ %$ #$ *=0. This constant depends on the various normalizations involved, but notethat when %v$ !v$ and "v are unramified, and "v is generic and cyclic, wehave that #W!v

P#$ %v$"v

#fv$ = tr"v#fv$ for spherical fv ! !v.

9. Proposition. Let " + be a "#-distinguished cuspidal representation of"+ = GL#2$#E$ with central character '+/%+2. Then " + corresponds to a $#-cyclic generic cuspidal representations " of % = U#2$ 1,#$ with central char-acter '.

Proof. We choose V and " +v = " +

v for all v *! V so that " + parametrizesthe only term on the left of (8.1). We need to show for each v ! V that thereis f +

v ! C +v = Cv#D+

v$ %+2v/'

+v$ (it matches some fv ! Cv by Proposition 6)

with #W!vP#$ v$" +

v#f +

v$ *= 0, where this last distribution is defined in closeanalogy to the one on Gv. Indeed, having shown this we would concludethat the right side of (8.1) is not identically zero, and any " occurringnon-trivially on the right would be cuspidal generic $#-cyclic with centralcharacter ', corresponding to " +.


So we fix v ! V , let !1 be a smooth vector in " "v with P#$ v#!1$ %= 0, and

!2 a smooth vector in " "v with W%v

#!2$ %= 0. We may assume that either!1 = !2 or that !1 is orthogonal to !2. Each of !1$ !2 can be multiplied bya scalar to have length 1, and we extend &!1$ !2' to an orthonormal basis&!i' of " "

v. Since " "v is irreducible and admissible, the set &" "

v#f "v$( f "

v ! C "v'

spans the algebra of endomorphisms of " "v, and we may choose f "

v such that" "v#f "

v$!i = &i$ 1!2. Then!W%v

P#$ v

"" "v#f "

v$ = W%v

!" "v#f "

v$!1

"P#$ v#!1$ = W%v

#!2$P#$ v#!1$ %= 0$

as required.

In the opposite direction, we have

10. Proposition. Let " be a !#-cyclic cuspidal generic representation of" = U#2$ 1(#$ with central character '. Then " corresponds to a unique$#-distinguished cuspidal representation " " of $" = GL#2$#E$ with centralcharacter '"/("2.

Proof. We apply (8.1) with a suitable set V and with "v equals "v foreach v %! V . To distinguish " of the proposition from the other representa-tions which index contributions to the right side of (8.1), we denote it by ".As noted in the proof of Proposition 8, each component "v of " is generic,unitarizable, and cyclic, hence when v %! V it is unramified and it corre-sponds to a generic unitarizable distinguished unramified representation " "

vof D"

v = GL#2$Ev$. We need to show that for some choice of fv#v ! V $,the right side of (8.1) is non-zero. Recall that #W%P#$ ($"#f $ is the productof a constant c#"$%$ ($ #$, which is non-zero when " is " since " is cyclicand generic, and

#v!V #W%v

P#$ (v$"v

#fv$, since #W%vP#$ (v

$"v#f 0

v $ = 1 for allv %! V .

Consider a place v ! V such that "v is supercuspidal. Such v is finiteand it stays prime in E (since a supercuspidal GL#3$Fv$-module cannot becyclic, by Proposition 0 of [F5], whose proof relies heavily on Bernstein–Zelevinsky [BZ2] and Gelfand–Kazhdan [GK]). Since "v is generic andcyclic its space contains vectors !1 and !2 of length 1 with P#$ (v

#!1$ %= 0and W%v

#!2$ %= 0. We may assume that !1 = !2 or that !2 is orthog-onal to !1. Extend &!1$ !2' to an orthonormal basis of "v. The ma-trix coefficient fv#x$ = #!2$ ˇ"#x$!1$ is a supercusp form which satisfies"v#fv$! = 0 for all ! orthogonal to !1, and "v#fv$!1 = !2 (up to a non-zero multiple). Consequently #W%v

P#$ (v$"v

#fv$ = 0 for all "v inequivalentto "v, and #W%v

P#$ (v$"v

#fv$ %= 0. Using such fv at each place v ! V where"v is supercuspidal, we conclude that the sum on the right of (8.1) ex-tends only over the " whose components at these v ! V are the super-cuspidal "v.


Next we consider a place v ! V such that !v is not supercuspidal. Then!v is the unique generic constituent in the composition series of an inducedrepresentation I"µv# if v stays prime, or Iv $ Iv" Iv = I"#2v $ $0v/%0v#and #2v is a generic unitarizable irreducible representation of GL"2"Fv#with central character %0v (if v splits in E). As usual, we choose a ba-sis &1" &2" ' ' ' for !v, and fv ! C%

c "Gv# with !v"fv#&i = (i" 1&2 and"W)v

P*" %v#!v

"fv# &= 0' Applying Bernstein’s decomposition theorem (whichis based on Bernstein’s analysis of the Bernstein center, see [F4, pp. 165–166]), we may replace here fv by its component fv"+ ! Cv , where +is the connected component +"!v# of the infinitesimal character ,"!v#of !v. Then !v"fv"+# acts as 0 on any !v with ,"!v# &! +"!v#, and!v"fv"+# acts as !v"fv# on !v. Using this fv"+ for fv in (8.1) we con-clude that the sum over ! on the right of (8.1) ranges precisely over all! whose components at the v &! V , or at the v ! V where !v is super-cuspidal, are the same as that of !; but at the remaining finite set ofplaces where !v is the generic constituent of the full induced I"µv# (ifv stays prime), or I"µ1v"µ2v"$0v/%0v# or I"#2v"$0v/%0v#" #2v supercus-pidal (if v splits), we only know that !v is a constituent of I"µv-

sv# or

I"µ1v-sv"µ2v" -

'sv "$0v/%0v# or I",v ( #2v"$0v/%0v# (,v unramified with

,2v = 1) for some s ! ! (as usual -v"x# = )x)v#. So far it appears that

the sum over ! on the right of (8.1) may range over a set larger than! alone, and cancellations may cause this sum on the right of (8.1) tovanish.

At this stage we use the rigidity theorem for automorphic representationsof " = U"2" 1#"## from [F2] and [F3], which asserts, in particular, that:there exists at most one (equivalence class of) cuspidal representation of "whose components are specified at almost all places, and such that at theremaining finite set of places the components are the generic constituents offully induced Gv-modules. Note that this is a weak form only, of the rigiditytheorems of [F2,F3]. We conclude that with our choice of fv"v ! V # thereis only one non-zero term in the sum of the right side of (8.1), it is indexedby our !, and so the left side of (8.1) is non-zero. By the rigidity theoremfor GL"2# the cuspidal (hence generic) distinguished ! * which parametrizesthe single term on the left of (8.1), is unique. It corresponds to our !, andthe proposition follows.

Propositions 9 and 10 imply that the correspondence is a bijection be-tween the set of $*-distinguished cuspidal representations ! * of GL"2"#E#with central character $*/%*2, and the set of packets of generic cuspidal !1

*-cyclic representations ! of " with central character $.

Theorem 0.2 of [FH] (which is quoted as Theorem 0.3 in the Intro-duction) asserts that a cuspidal representation ! * of GL"2"#E# is $*-distinguished precisely when it is $1 + GL"2"##-distinguished, and its


component ! !v at each v in the set " (of places where " # Fv $ NEv% is

not of the form I !&µ1#µ2% with µi trivial on F'v .

The Main Global Theorem of [F4], which is the quasi-split (" = 1%analogue of our Propositions 9 and 10, asserts that the correspondenceestablishes a bijection from the set of equivalence classes of automor-phic representations of GL&2#!E% with central character $!/%!2 whichare either cuspidal and GL&2#!%-distinguished, or induced of the formI !&µ!

1#µ!2%# µi( !1

E/E1 ) "'#µ1 *= µ2#µ

!i&z% = µi&z/z% &z # !'

E %, to theset of packets of generic cuspidal "1

1 = U&1# 1+!%-cyclic representations of# = U&2# 1+!% with central character $. Consequently we have:

Corollary. The packets of the "1"-cyclic generic cuspidal #-modules are

precisely the packets of the "11-cyclic generic cuspidal #-modules which cor-

respond to the cuspidal $1 , GL&2#!%-distinguished GL&2#!E%-modules,whose components at the v where " # Fv $NEv are not of the form I !&µ!

1#µ!2%

with µ!i trivial on F'

v .

This can also be stated in terms of the endoscopic &-lifting of [F2] fromU&1# 1%. Here &( !'

E /E'N!'

E ) "' is a character whose restriction to!'/F'N!'

E is nontrivial. The Global Theorem of [F6] asserts that thepackets of the generic "1

1-cyclic cuspidal representations of # with centralcharacter $ are the image under the &-endoscopic lifting of the packetsof the cuspidal representation ' of "1

1 = U&1# 1+!% with central character$/%2&. The packets of the generic ""-cyclic cuspidal representations of #with central character $ are the &-endo-lifts of those cuspidal represen-tations !0 of "1

1 = U&1# 1+!% with central character $/%2& which are notof the form !0&µ1#µ2%# µi( !'

E /E'!' ) "'# µ1 *= µ2 (those which base

change via the &-unstable base change lifting to I !&µ!1#µ

!2% on GL&2#!E%%,

and whose components at the v where " # Fv $ NEv is not of the form!0&µ1v#µ2v%# µi( E1

v ) "'&i = 1# 2% (those which base-change (via the &v-unstable base-change lifting) to I !&µ!

1v#µ!2v% on GL&2#Ev%; see [F2, Sects.

3.7, 3.8, p. 49]).

5. LOCAL CYCLICITY

As in [F6], where the case of " = 1 is considered, we define also a localcorrespondence, by means of an identity of Whittaker–Period distributionson the two groups in question. The case of " = 1 considered in [F6], weare mainly concerned here with " # F $NE, and v with " # Fv $NEv.

11. Proposition. For every component !v of a generic "1"-cyclic cuspidal

representation ! of # with central character $, there exists a unique D"# v-distinguished generic representation ! !

v of GL&2#Ev% with central character


!v/"2v, which is a component of a cuspidal !#-distinguished representation$ ! of GL"2%"E# with central character !/"2; and for each such component$ !v there exists a unique finite set $$v% of generic C1

#% v-cyclic representations ofGv, and constants c"$v% &v% "v% ##, such that the $v lie in one packet (see [F2])uniquely determined by $ !

v and are components of #1#-cyclic generic cuspidal

representations $ of $; such that for all matching f !v & C !

v and fv & Cv we have!W&v

P#% v

"$ !v"f !

v# =#

$v&$$v%c!$v% &v% "v% #

"!W&v

P#% "v

"$v"fv#' "11'1#

Proof. Given such global $ or the corresponding $ !, we set up the iden-tity (8.1) such that $ ! parametrizes the only term on the left, and $ occurson the right. At each v1 '= v in V we choose fv2 as in the proof of Propo-sition 10 to have that "W&v1

P#% "v1#$v1

"fv1# '= 0, and that the $ which occuron the right of (8.1) will have the component $v1

(at each v1 & V% v1 '= v#.As in the proof of Proposition 10 we use here the rigidity theorem forU"2% 1(E/F# of [F2]. Starting from $ !

v we proceed as in Proposition 9. Inany case we obtain (11.1) for all matching f !

v and fv, where the sum onthe right ranges over a subset of the packet of $v by virtue of the rigid-ity theorem for U"2% 1(E/F# of [F2]. This subset consists only of genericC1#% v-cyclic representations. The $$v% and $ !

v are uniquely determined byeach other since the packet of $$v% is determined by $ !

v via base-changeand endoscopic liftings and the Whittaker–Period distributions are linearlyindependent.

Definition. A D#% v-distinguished generic representation $ !v of

GL"2%Ev# and a C1#% v-cyclic generic representation $v of Gv = U"2% 1(Fv#

are said to correspond if they satisfy the relation (11.1) for all matching f !v

and fv.

We shall use this definition of correspondence only for representationsfor which the identity of the Whittaker–Period distributions is established,namely this is done below only for square-integrable $ !

v and $v. It is perhapsbest at this stage to define the local correspondence as the composition ofthe (inverse of the) (v-unstable base change from D#% v = U"2%Fv# to D!

v =GL"2%Ev#, and the (v-endoscopic lifting from D#% v to Gv = U"2% 1(Fv#.

Next we shall list the generic C1#% v-cyclic Gv-modules, and relate them to

the generic D#% v-distinguished D!v-modules. This has already been done in

[F6] when # = 1, hence we assume here that # & Fv ) NEv, in particularthat Ev = E *F Fv is a field. We also compare the notion of being C1

#% v-cyclic with being C1

1% v-cyclic, for a generic representation of Gv. To simplifythe notations, we use local notations (drop v) in the following.

Let E/F be a quadratic extension of non-archimedean local fields withcharF '= 2, and # & F )NE.


12. Proposition. (a) The correspondence is a bijection relating thegeneric D!-distinguished irreducible admissible representations " ! of GL"2#E#with the packets of the generic C1

!-cyclic irreducible admissible representations" of U"2# 1$F#.

(b) The packet of a generic C1!-cyclic " contains a C1

1 -cyclic generic "1.The packet of a generic C1

1 -cyclic "1 contains a generic C1!-cyclic " precisely

when "1 does not correspond to any " ! = I !"µ!1#µ

!2## µ!

i"z# = µi"z/z#"z %E&## µi' E1 ( !&# µ1 )= µ2, namely "1 is not the $-endoscopic lift of any"0"µ1#µ2## µ1 )= µ2, on U"2#F#.

Remark. Here " = "1 when "1 is not supercuspidal, and perhaps alsofor supercuspidal "1, but this is not shown here.

Proof. We shall go through the list of induced representations, theirconstituents, and supercuspidals, to verify our claims.

(1) If " ! is induced, then HomD!"" !# 1# )= *0+ when " ! = I !"µ#µ,1#,

by Proposition 3. When µ = %/&2 on E1 (equivalently µ! = %!/&!2 on E&#," ! corresponds to " = I"µ#. All induced " with HomC""# &2# )= 0 areof the form " = I"µ## µ = %/&2 on E1, if C = C1

! and if C = C11 , by

Proposition 2!. Both (a) and (b) follow in this case.

(2) If " is the Steinberg St"µ'#, then HomC"St"µ'## &2# = *0+ bythe corollary to Proposition 2, where C = C1

! or C = C11 .

(3) The reducible induced " = I"µ# are listed in [F1, (3.2), page 558].There are three cases of reducibility. The third in that list is the Steinberg,disposed of in (2) above. If µ is unitary, reducibility occurs precisely whenµ-F& = 1 and µ3 )= %!. Since I"µ# is tempered, it is the direct sum ofits two irreducible constituents, denoted in [F1, (3.2(1))], by "+ and ",.Then (1) above asserts that the packet *"+# ",+, which corresponds toI !"µ#µ,1##µ = %/&2 on E1# µ )= &!2 on E&, contains a generic Ci-cyclicrepresentation, with i = 1 and with i = !. Since only one of "+# ", isgeneric, namely "+, it is both C1

!-cyclic and C11 -cyclic.

(4) The square-integrable (“special” or “Steinberg”) subrepresenta-tion sp"µ$# of I !"µ$'1/2# = I !"µ$'1/2#µ$',1/2#, where µ' E& ( !& and$' E&/NE& ( !& has $-F& )= 1, is D!-distinguished (and GL"2#F#-distinguished) precisely when µ-F& = 1, by Proposition 3. It is the $-base-change of a special representation of U"1# 1$F#, which corresponds to aone-dimensional representation of the anisotropic inner form U"2$F#. Us-ing the trace formula on an anisotropic group U"2$"# over a global fieldwhose component at some place is our local U"2$F#, we can view this spe-cial representation as a component of a global cuspidal representation, thenlift the global representation via the $-base-change map as in [F4] to a cus-pidal representation of #! which is #!-distinguished, whose component at


our local place is our sp!µ!". Proposition 11, together with the usage ofthe rigidity theorem on U!2" 1#!" of [F2] explained in the proof of Propo-sition 10, implies that sp!µ!" corresponds to the square-integrable subrep-resentation #+

µ of the induced I!µ!$1/2" of [F1, (3.2(2))]; this I!µ!$1/2"is reducible precisely when µ$F% = 1, and its quotient is non-temperedand non-generic, denoted by #%

µ in [F1, (3.2(2))]. Moreover, this #+µ is the

only term in the sum on the right of (11.1), and it is generic and C1%-cyclic

(and also C11 -cyclic). This completes the proof of (a) and (b) for the non-

supercuspidal # (and # &). Note that the packet of sp!µ!" contains also asupercuspidal which we expect to be neither generic nor cyclic but havenot shown this as yet. This supercuspidal is classified here according to itspacket, hence it does not appear in (5) below.

(5) Given a supercuspidal D%-distinguished representation # & ofGL!2"E" it is easy to construct a global cuspidal "%-distinguished repre-sentation of "& whose component at some place (where the global % isnot a norm) is our local one. Applying Proposition 11 we conclude thatthe packet which corresponds to # & contains a generic C1

%-cyclic #; thepacket of # consists of supercuspidals by [F2] (since our correspondenceis !-base-change composed with !-endoscopic lifting).

Conversely, given a generic supercuspidal C%-cyclic # (which is not in thepacket of sp!µ!") we may construct a global cuspidal #%-cyclic represen-tation whose component (at a place where the global % is not a norm) isour local #, as in [F4, Proposition 14] (as corrected in [FH, after Propo-sition B17]). Applying Proposition 11, we conclude that # corresponds toa generic D%-cyclic representations of GL!2"E", which is supercuspidal bythe results of [F2] on the !-endo-lifting.

If # is a generic supercuspidal C11 -cyclic representation of G =

U!2" 1", then by [F6] it corresponds either to a supercuspidal GL!2"F"-distinguished representation of GL!2"E", or to an induced I &!µ&

1"µ&2""

µ&i' E%/F% ( #1" µ&

1 )= µ&2. Theorem 0.1 of [FH] establishes that a su-

percuspidal representation of GL!2"E" is GL!2"F"-distinguished if andonly if it is D%-distinguished. This result was proven independently andby purely local means by D. Prasad [P]. Hence the packets of the genericsupercuspidal C1

%-cyclic # are the packets of the generic supercuspidalC11 -cyclic # which correspond to the supercuspidal distinguished # &, but

not to the induced I &!µ&1"µ

&2"" µ&

1 )= µ&2" µ

&i' E%/F% ( #1, as asserted.

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Automorphic Forms with Anisotropic Periods on a Unitary Group · 2014-08-28 · automorphic forms...

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