C OMPOSITIO M ATHEMATICA
D AVID K EYS
Principal series representations of special unitary groups over local fields
Compositio Mathematica, tome 51, n
o1 (1984), p. 115-130
<http://www.numdam.org/item?id=CM_1984__51_1_115_0>
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115
PRINCIPAL SERIES REPRESENTATIONS OF SPECIAL UNITARY GROUPS OVER LOCAL FIELDS
David
Keys~
© 1984 Martinus Nijhoff Publishers, The Hague. Printed in the Netherlands
Intertwining operators
are used with amultiplicity-one
result tocompute
certain c-functions onSU(3)
over a non-Archimedean local field. Reduci-bility
of theunitary
andnon-unitary principal
seriesrepresentations
isdetermined. The c-functions are used to determine the Plancherel mea- sure for the
unitary principal
series ofSU(2n
+1).
TheR-groups
forSU(2n
+1)
areexplicitly
classified.1. Definitions and
preliminaries
Let F be a non-Archimedean local field and E a
separable quadratic
extension of F with Galois
automorphism
x ~ x.Define G =
SU(2n
+1)
to be the groupG is the
algebraic
grouppreserving
the hermitian formh(x, y) = xny-n+···+x0y0+···+x-nyn
onE2n+l
.Let A be the torus
{diag(dn,···,1,···,d-n)|dl ~ F, did-l
= 1 for alli}, M=ZG(A)~(E*)n,
and N the group of uppertriangular unipotent
matrices of G. P = MN is a minimal
parabolic subgroup
of G.Extend a character À of M
trivially
across N and define theprincipal
series
representations IndGP03BB using
left translation andunitary
induction.Let
K 0
= K =G(OE)
be the standard maximalcompact subgroup
ofG,
and setKm = {g~G|g ~ I mod pmE} for m 1.
More
generally,
if H is anysubgroup
ofG,
defineHm= {h~H|h~I mod pmE} for m 1.
:j: Research partially supported by NSF grant MCS-78-01826.
If V = tN is the lower
unipotent subgroup
ofG,
thenKm
has Iwahori factorizationVmMmNm, m
1.Let a =
Hom(X(A), Z) ~ R
be the real Liealgebra
ofA,
with duala* =
X(A)
0 R.Canonically,
a =Hom(X(M), 7l )
0 R and a* =X( M )
0R,
as in§5.3.1
of[9].
An element s Ea *c
determines aquasi-character
Xs of M as follows. Let H: M - a be themapping
whichsatisfies |~(m)| =
q(x,H(m»
for allm ~ M, X E X(M).
Then defineXs(m) = ql(s,H(m»
form E M. For À E
M",
we set03BBs(m) = 03BB(m)~s(m).
If n =1, 03BB ~ (E*)",
weset Xs(x) = 03BB(x)|x|sE
for SEC. Also defineRe 03BBs
to be Re s.Each element of
X(A) corresponds
to aunique
element ofa *,
called the associatedweight.
A non-trivial rational character a of A whichoccurs in the
adjoint representation
of A on the Liealgebra
Y of G iscalled a root character.
Let £
be theeigenspace
in 5 associated to a. Theweights
of the root characters form a rootsystem 4Y
in a *. A root a issaid to be reduced if ta OE 4Y with t E Q
implies
t ~ Z.In Section 2 we prove a
multiplicity
one result based on ideas of R.Howe.
Intertwining operators
are studied and used to define certainmeromorphic
functions in s in Section 3.Reducibility,
of therepresentation IndGP03BBs
is related to the zeros andpoles
of these functions in Section 4. The c-functions arecomputed
forthe rank 1 groups
SL2
andSU(3)
in Section 5. Theexplicit
formulas for the c-functions and Plancherel measure are used to constructcomplemen- tary
series in Section 6 and to determinereducibility
in Sections 7 and 8.We note that Casselman has
independently
determined the reducibleunitary principal
series ofSU(3) by
a differentanalysis
based onJacquet
modules.
2. A
multiplicity-one
resultWe first define a character
"4/;m
ofKm
foreach m
0. Let03C80
~ 1 onKo.
For a
positive,
let£
be thesubgroup
of V whose Liealgebra
is5-a +5-2a.
The groupVa/V2a
isisomorphic
to the additive group of E.We fix a
character 03C8
of E with conductor thering
ofintegers (9E,
anddefine a character
"4/;a,m
ofV03B1/V203B1 by 03C803B1,m(x)=03C8(03C0-2mx),
x EE,
where03C0 is a
prime
inmE.
We will assume"4/;a,m
factorsthrough
the trace from Eto F. Then
03A003C803B1,m ( a simple)
determines a character ofV/03A0V03B2 (03B2
non-simple),
hence of V. Let"4/;m
be its restriction toVm.
Now define
03C8m
onKm = VmMmNm by 03C8m(03C5ln) = 03C8m(03C5)
for03C5 ~ Vm, l ~ Mm, n ~ Nm.
We want to show
THEOREM 1:
Suppose
03C0 is an irreduciblerepresentation of
G. Then themultiplicity of 03C8m
in03C0|Km
is 0 or 1.Define an
algebra
of( Km, 03C8m)-spherical functions, m > 0, by 9’m = ( f
e
C~c(G)|f(k1gk2)=03C8m(k1k2)f(g)
forkl, k2~Km, g~G}.
Theorem 1 will follow from
THEOREM 2:
Sm is a
commutativealgebra
under convolution.The case m = 0 is
well-known, i.e.,
thetheory
of K-dass-1representa-
tions. Theproof
forgeneral
m is based onarguments
due toRoger
Howe.With restrictions on F and G a
Chevalley
group, themultiplicity
oneresult is
equivalent
touniqueness
of Whittaker models. SeeCorollary
2 ofthe main theorem of
[7].
For G =
SU(2n
+1),
char F =0,
andE/F unramified,
one may usearguments
similar to theproof
of Lemma 4 of[7]
to prove the commuta-tivity
ofJm
and themultiplicity
one result. Thesearguments
wereoriginally developed by
Howe in the case ofGLn .
Theorem 2 holds for
S’L2 ( E )
andSU(3)
with no restrictions onE,
F.The
proof
is based on a standardgeneral argument.
Let a be anantiautomorphism
of G which preservesKm
and"4/;m.
Then a acts on9’m.
Further,
A double coset
Km gKm supports
a non-zerospherical
functionin Jm
ifand
only
ifSuppose
that any double cosetKm gKm
contains an element fixedby
cl, up to anélément
of ker"4/;m.
Then a acts as theidentity
onY’m’
soY’m
iscommutative
by (2.1).
For G =
SU(3),
we setThen a
long
calculation shows that if g satisfies(2.2),
then the double cosetKm gKm
contains an element fixedby
a, up to an element of ker"4/;m.
In Section 3 we will
apply
themultiplicity
one result for,SL2
andSU(3)
to theprincipal
seriesrepresentations,
in which case it is easy togive
a directproof.
3.
Intertwining operators
and Plancherel measureFix a coset
representative w
for w E W and define the usualintertwining operators
betweenIndGP03BB
andIndGPw03BB (À
aquasi-character) by
It is well-known that these
operators satisfy
thecocycle
relationprovided
l(w1w2) = l(w1) + l(w2).
Choose a coset
representative
for each w ~ W as in[3]
and writeA ( w, À)
insteadof A(w, 03BB).
We will define normalizedoperators U(w, 03BB)
which
satisfy
thecocycle
relationwith no condition on the
lengths of W1
and W2.We define a function on
VP,
which is open and dense onG, by
where
03BB~M^, s ~ U*C, h = conductor 03BB, and m max{1, h}.
Then fm,03BB,s
is inIndGP03BBs
and transforms as03C8m
underKm.
Recallthat, by
BruhatTheory,
therepresentations IndGP03BBs
aregenerically
irreducible.Consider first the action of
A(waÀs) on fm,03BB,s
for w = w«, a asimple
root.
Suppose Re s03B1 >
0. ThenA(w03B1,03BBs)fm,03BB,s
converges, is inIndGPw03B103BBs
and transforms as
03C8m
underKm. By
themultiplicity
one result forMa
=ZG (ker a)
=SL2
orSU(3),
it must be a scalartimes fm,w03BB,ws, i.e., A(w,03BBs)fm,03BB,s(g)=03B3m,03B1(03BB,s)fm,w03BB,ws(g)
for allg E G, (3.4)
w = wa a
simple
reflection.Now,
for any w ~W,
consider theproduct decomposition of A(w, 03BBs)
into rank-one
operators corresponding
to the reducedexpression w
=walwa2 ... wa, as a product
ofsimple
reflections.(3.1)
and(3.4) give
for all g E
G,
wheres)
isgiven by
aproduct
formula in terms of the03B3m,03B1(03BB, s).
The formula(3.4’)
is similar to the case m = h =0;
see forexample, [2].
Besides the
product formula,
the 03B3m,w alsosatisfy
In Section
5,
weexplicitly
calculate03B3m(03BB, s)
for G =SL2
andSU(3) by setting
g = 1 in(3.4)
andevaluating
theintegrals
The c-functions
Ym(À, s )
are defined for all sby analytic
continuation.Then
But,
from Harish-Chandra’stheory
of theintertwining operators,
where
y( G/P )
is a constant ands)
is Plancheral measure. See[8],
or works of
Silberger.
Set 03BC03B1 = 03BCw if w = Wa’ with a reduced. If both aand 2 a are roots,
formally
set 03BC203B1 = 03BC03B1.Thus for the rank 1 group
SL2
andSU(3),
We may determine the Plancherel measure for the
unitary principal
seriesin
general by
aproduct
formulaThis is a
product
of rank 1 factorswhere
a ranges over the set(D,
ofpositive
reduced roots.Here,
c is apositive
constant. SeeCorollary
5.4.3.3 of
[9].
REMARK 1:
By
ageneral
theorem ofSilberger [11],
for reductive groups the Plancherel measure is aproduct
of Euler factorsUj(03C9, z), j = 0,
1.Our results in Section 5
determine,
forSU(3),
whether either factor isidentically
one and the values ofSilberger’s
constants03BBJ, j
=0,
1. Wenote also that
Silberger
has shown that one may associatecomplementary
series
representations
to any zero of the Plancherel measure.REMARK 2: The functions
03B3m,w(03BB, s)
are related to the local coefficientsC~(03B3,
qr,0, w )
studiesby
Shahidi in[8].
Section 3.2 of[8]
containsformulas for
C~(03BB,
qr,B, w )
for the real groupsSL2 ( F ),
F = R orC,
andSU(2,1).
4. The
reducibility
group We fix an m h and define the normalizedoperators
Then U(w-1, w03BB)U(w,03BB)=I,
first for W=Wa asimple
reflection.Then one may show as in
Chapter 1,
Section 2 of[3]
that thecocycle
relation
(3.2)
holds for normalizedoperators
with no conditions on thelengths of wl
and w2.Thus if
W03BB = {w~W|w03BB=03BB},
thenw ~ U(w, 03BB)
is a homomor-phism
onW03BB
and notmerely
aprojective homomorphism.
We define the
p-adic analogue
of theKnapp-Stein R-group.
For À
unitary,
set 0394’ ={03B1|03BC03B1(03BB, 0)
=0}
and let W’ be thesubgroup
ofW
generated by
the reflectionswa, a E
A’. ThenW’={w~W03BB|U(w, 03BB)
is
scalar) by
results of[10].
Define R =
{w~ W03BB|03B1 ~ 0394’
and a > 0imply
wa >0}.
ThenW.
is asemi-direct
product W03BB
= R oc W’.THEOREM 1
(Harish-Chandra [9]): {U(w,03BB)|w~ W03BB}
spans the commut-ing algebra W(X) of
theunitary principal
seriesIndGP03BB.
THEOREM 2
(Silberger [10]):
The dimensionof
thecommuting algebra L(03BB) is |W03BB/W’|.
A
priori,
themap w ~ U(w, 03BB)
fromWx
into thecommuting algebra
is
only
aprojective homomorphism.
Thus R abelian does not im-mediately imply
thatmultiplicity
one holds. But with our normalization and choice of cosetrepresentatives
forWeyl
groupelements,
thecocycle
relation
(3.2)
holds with no condition on thelengths
of theWeyl
group elements. Then the above map isactually
ahomomorphism.
Since the
self-intertwining operators corresponding
to elements of W’are
scalar,
Theorem1,
thecocyle
relation(3.2),
and thedecomposition Wx = R OE
W’imply
that{U(w, 03BB)|w ~ R}
spans thecommuting alge-
bra.
Then, by
Theorem2,
theseoperators
form a linear basis for thecommuting algebra.
Sincer ~ U(r, 03BB), r E R,
is ahomomorphism,
weget
thefollowing.
THEOREM 3: The
commuting algebra b(03BB) of IndGP03BB
isisomorphic
to thegroup
algebra C[R].
The
R-groups
forp-adic Chevalley
groups are classified in[3].
Weshow in Section 8 that
R ~ Z2
... XZ2,
with the number of factors of7L2
boundedby n
= rankG,
for G =SU(2n
+1).
In
fact,
weexplicitly
construct a list of characters À of M with non-trivialR03BB,
such that any character with non-trivial R group isconjugate
to one on the list.Note that the zeros of the
03B3m(03BB, s)
are related to thereducibility
of thenon-unitary principal
seriesby
Theorem 6.6.2 of[1].
5.
Computation
of some c-functionsSee
[4]
for the case m = h =0,
withapplications
toreducibility. 03B30(03BB, s)
isthe Harish-Chandra c-function.
We first indicate the calculation of
03B3m(03BB, s)
forSL2(F),
mmax{1,h}.
Fix an additivecharacter 03C8
of F with conductor thering
ofintegers (9F
and set03C8m(03C5)
=03C8(03C0-2m03C5).
For 03BB ~ (Fx)^,
defineThen for Re s >
0,
where
0393(03BBs)
is the gamma function of[6], i.e.,
one of Tate’s local factors03C1(03BBs).
Thus on
SL2(F),
we have the well-known resultA(w-1, w03BB)A(w, 03BB)=0393(w03BBs)0393(03BBs)I.
Now,
suppose G =SU(3),
mmax{1,h}.
Then for Re s >0,
03B3m(03BB, s) = Nfm,03BB,s(n03C9)dn
The
difficulty
is that theintegral
is over those x, y ~ Esatisfying y + y + xx = 0.
We use the formulas
03BBs(x) = 03BB(x)|x|sE, IxlE = IxxlF’
and03B4(y) = |y|E|yy|F = |y|2E.
For
E/F unramified,
qF = q, qE =q 2.
ForE/F ramified,
qE = qF = q.Suppose
for now char F =1= 2 and write x, ysatisfying y + y
+ xx = 0 asx=u+vr, Y= -1xx+zr,
where u, v,z ~ F, E = F[03C4],
03C4 = -03C4 andmE = (flF
~TmF.
Let
dEx
= dx be Haar measure on E withvol(mE)
= 1 anddFz
= dzbe Haar measure on F with
vol(OF)
= 1. Haar measure on N isgiven by dEx dFz.
It is convenient to assume m
ord E 2.
The case char F = 2 may be handled
similarly.
Making
thechange
of variables z H -1 2xxz
in the lastintegral
for03B3m(03BB, s ),
thenletting x ~ 2x(1
+z03C4)-1
andfinally taking x H x-1,
weget 03B3m(03BB, s) =
where B(x)=F~(xx03C4)-1pm+ord2E and B(n) = F~03C4-1pm-2n+ord2E.
Now, if degree 03BB = h 1,
letB(n, h) = B(n)~03C4-1p-h+1E,
and writeThe last
intégral,
over FnT-1 P-h+1E,
is similar to a Beta function and is theonly tricky part
of03B3m(03BB, s)
to evaluate. One way is to use Fourieranalysis
as follows: For notationalsimplicity,
assumeE/F is
unramified.Define a function
B(u)
on Eby
we want the value
B(1). B(u)~L1(E)
if Re s > 0. Define the Fourier transformB(w)
=EB(u)03C8(uw)du
where we let u H u - z T and then u ~
w-1u.
If w = a + b T and we
choose 03C8(w)=03C8(a) to
have conductorOE,
then(w) = (a + b03C4)
using
results of[6].
If Re s >
1, th en Ê E L1(E),
andby
the Fourier inversionformula, B(u)
=E(w)03C8(uw)dw.
But it is then easy to calculateB(1)
in terms ofthe gamma functions of
[6].
This line of
analysis
leads to thefollowing
result.THEOREM: Let m
max {1,h}.
(1)
For G =SL2(F), ym(À, s)= À(’7T)2mq-2msf(Às).
(2) for
G= SU(3),
char F =1=2,
the03B3m(03BB, s)
aregiven
asfollows:
It isconvenient to
change
s toassume 03BB(03C0E)
= 1.First assume
E/F
is unramified.(a)
If À isunramified, 03BBs(x) = |x|sE,
then(b)
If À is ramified ofdegree
h 1and À 1 FX
=1,
then(c)
If À is ramified ofdegree h 1, 03BB|Fx
ramified ofdegree
h’1,
thenw03BBs ~ 03BBs
forany s and 03B3m(03BB, s )
=Now assume
F/F
is ramified.(d) If 03BB(x) = |x|sE
isunramified,
then(e)
If À is ramified ofdegree h >
1 andÀIFX
=1,
then03B3m(03BB, s) = |2|1-2sFq-4ms03BB(03C4)qsq(h/2~2s-1).
(f)
If À is ramified ofdegree h 1,
andÀ 1 (2 F X
has order2,
then(g)
If À is ramified ofdegree h
1and x ~ 03BB(xx)
is ramified on EX ofdegree h’,
thenym(À, s) = |2|1-2sF03BB-1(s)q-4ms03BB(03C003C0)2m.
REMARK: The case char F = 2 may be handled
similarly.
Write xand y
with
y + y + xx = 0 as y = z + xx(03C4 + 03C4)-103C4,
wherez E F, x E E,
and(9E = OF~03C4OF.
Note that trace r = T + T is nonzero sinceE/F is
assumedseparable.
Then03B3m(03BB, s)
=where
B(n) = {z
EFlz
+ T E(T
+03C4)pm-2nE}.
6.
Complementary
seriesComplementary
seriesrepresentations
forSU(3)
may be constructed in the standard manner.Note that the
contragredient
ofIndGP03BBs
isnaturally isomorphic
toIndGP(03BBs)-1
via thenon-degenerate
G-invariant bilinear formg ~ IndGP03BBs.
For Às
withwÀs
=(03BBs)-1,
define the hermitian formIf
03BC(03BB: 0)
=0,
thena(w, 03BB)
acts as theidentity
onIndGP03BB,
and( .1. ) is
the usual inner
product.
But the restrictionof f E IndGP03BBs
to Kdepends only
onÀ,
not on s. Thusby continuity, (·|·) will
bepositive
definite of snear
0, i.e.,
untila ( w, 03BBs)
has apole, i.e.,
until03B3m(03BB, s )
has a zero.Thus
IndGP03BBv
is in thecomplementary
seriesif 03BB~(Ex)^, wÀ=À,
v E
R,
and if03B3m(03BB, s )
has apole
at s = 0 and is nonzero for 0 s v.7. Reducible
principal
séries ofSU(3)
We determine the
reducibility
of theunitary
andnon-unitary principal
series of
SU(3).
Reducibility
of theunitary principal
seriesIndGP03BB
is determinedby
thetheory
of theR-group
outlined in Section 3 andknowledge
of thePlancherel measure;
IndGP03BB
is reducible if andonly
if wÀ = À and03B3m(w03BB, -s)03B3m(03BB, s) is holomorphic
at s = 0. In this case,IndGP03BB splits
asthe sum of two
inequivalent
irreduciblesubrepresentations,
each occur-ring
withmultiplicity
1.Let 03BB be a
quasicharacter.
By
Theorem 6.6.2. of[1 ],
thenon-unitary principal
seriesIndGP03BB
isreducible if and
only
if03B3m(w03BB, -s)03B3m(03BB, s)
is zero at s = 0. We mayassume Re À > 0. Then the kernel of
A(w, 03BB)
is an irreducible invariantsubspace
ofIndGP03BB,
which transforms as aspecial representation
of G.THEOREM: Let G =
SU(3), 03BB
E( E")"
= MA. Note the wÀ = Àif and only if 03BB(xx) = 1.
(1)
Theunitary principal
seriesIndGP03BB
is reducible if andonly
if 03BB ~1, w03BB = 03BB,
and03BB|Fx ~
1.(2) Suppose À E (Ex)^
and Re s > 0. The reduciblenon-unitary prin- cipal
seriesIndGP03BBs
are thefollowing:
Assume
E/F
unramified.(a)
The unramified03BBs(x) = |x|sE
for s = 1 or s= 1
+03C0i(2
Inq)-’.
(b) Àramified
ofdegree
h1, 03BB|Fx
=1,
and s= 1
+03C0i(2
Inq)-1.
Now assume
E/F
ramified.(c) Unramified 03BBs(x) = IxlE
f or s = 1.(d) À
ramified ofdegree h, 03BB|OFx
of order2,
and s= 1.
8.
R-groups
f orSU(2n
+1)
We
classify
theR-groups
which occur for thequasi-split special unitary
groups in odd dimension associated to
separable quadratic
extensionsE/F.
We realize the non-reduced root
system 03A6 as {±el±ej,±ei,± 2el|1
i ~ j n} with {el - el+1, en|1
in}
as a basis for thepositive
roots.The
Weyl group Sn
ocZn2
acts aspermutations
andsign changes
on the el . Let cI denote thesign change
on el. Then cn is thesimple
reflectioncorresponding
to the root en .Extend the dual roots aU : Fx ~ A to aU : Ex ~ M.
For À E MA,
definea
character 03BB03B1
of EXby 03BB03B1(x) = 03BB(03B103C5(x)).
We will be concerned with the03BB03B1 corresponding
to03B1~{±el±ej,±2ej}.
Let x ~ Ex. The elementsaU ( x )
of M aregiven by
thefollowing diagonal
matrices in G.For a =
2ei, aU(x)
=diag(1, ··· ,1,
x, ...,1, xx-1, 1,
...x-1,1,
···,1),
with an x in thei th place.
For 03B1 = el-ej, 03B103C5(x) = diag(1,···,x,···,x-1,···,1,···,x,···,
x-1,··· ,1 )
with x andX-1
in thei th and j th places.
For 03B1 = el + ej, 03B103C5(x)=diag(1,
···" x ··· x,...,1,
..., x-1,
...,x
There is a well-defined action of the
Weyl
group W *N(A)/Z(A)
oncharacters of
M = Z(A), given by w-103BB(m)=03BB(wmw-1).
Thus W actson the characters
Àa
ofE’, by w-103BB03B1(x) = 03BB(w03B103C5(x)w-1) = Àwa(x).
A character À of M is determined
by
the ncharacters 03BB03B1
of Excorresponding
to the roots a = el - e2, e2 - e3, ... ,en-1 -en,2en,
sinceM ~ (Ex)n by
the ncorresponding one-parameter subgroups
a’. Notethat wX = À if and
only
ifwÀa = 03BB03B1
for each a = el - e2,...,en-1-en,2en.
From the definitions of the
a",
weget
the relationsand
These relations hold in
general.
Additional relations will follow from the conditionsw03BB03B1=03BB03B1.
Recall that R
= (w
EW03BB|03B1
~ ~’ and a > 0imply
that wa >01.
It isenough
to consider roots a ~ ~’ such that 2 a is not a root, since a ispositive
if andonly
if 2 a ispositive. Here,
~’ ={03B1|U(w03B1, 03BB)
isscalar}.
For a ei ±
ej,
a E ~’if,
andonly if 03BB03B1 ~ 1,
a well-knownSL2
result.For a
= 2el, by
our Theorems forSU(3),
a E ~’if,
andonly if 03BB03B1(xx)
= 1for all x E
Ex,
andeither Àa
~ 1or 03BB03B1|Fx ~
1.THEOREM: For G =
SU(2n
+1),
R ~Z2 ···
XI2
with the numberof factors of
the 2-element groupll 2
boundedby
n = rank G.Explicitly, any À
with non-trivialR-group
isconjugate
to a character Àdefined by specifying
distinctcharacters of Ex, for
a =2el,
wherek
i nfor
somek, satisfying 03BB03B1(xx)
=1, 03BB03B1 ~
1and ÀalFx
~ 1. ThenR x
lS
ck,ck+1,
..., cn> ~ Zn-k+12
.The
commuting algebra of Ind GX
isgiven
as the groupalgebra C[R]:
thus
Ind’À decomposes into IR
2n irreducibleinequivalent
components.The
proof
of the Theorem is similar to the one in[3]
for groups oftype Cn .
The difference is that the reflection ensends 03BB2en(x) to 03BB2en(x)-1
instead of
À2e (X)-l.
Note also thatÀe +e (x) = Àe -e (x)À2e (xx):
We may
replace À by
aconjugate
wÀ with no loss ofgenerality.
ThenWÀ
and W’ arereplaced by conjugates wW03BB
and "’W’. ButRWÀ =1= wRÀ
ingeneral, although
these groups areisomorphic. However,
we prove thatcertain w are not in any
R03BB
in an invariant manner. Consider thesublattice L=L03BB=Z[03B103C5|03BB03B1 ~ 1 ]
inZ[03A603C5].
Then w03BB = 03BBif,
andonly
ifw03B103C5 - aU is in L for all a.
Suppose
there is a03B203C5
= w03B103C5 - aU in L~03A603C5,
so03BB03B2
~ 1 and thus03B2
E ~’.Then, summing
overi, 0
iord(w), 03A3wl03B203C5
=03A3wi(w03B103C5 - aU )
= 0. Thisimplies
there is a13
E~’, 13
>0,
withwlf3
0 forsome i. So wl~R and then w ~ R. This
argument
is invariant underconjugation,
so noconjugate
of w may be in anyR-group.
LEMMA 1:
Suppose
w = sc ER, with s E Sn
and c EZn2.
Then s = 1.PROOF:
Suppose s
has a non-trivialcycle. By
the aboveremark,
we mayconjugate by
apermutation
to assume that one suchcycle
is( k
k + 1 ...n). Then, conjugating by
asign change,
we may assume that cchanges
the
sign
of at most one el,k 1
n.First,
supposec( el)
= el forall k
i n. Then wÀ = Àimplies 03BB2e
=w03BB2en = 03BBw-1(2en) = 03BB2en-1.
Thisimplies Àen-l-en
=03BB2en-103BB-12en~1,
soen-1
- en ~ ~’.
But thenen-1 - en > 0
andw(en-1 - en) = en - ek 0,
con-tradicting
w E R.Now suppose
c(el) = el
fork 1 n
andc(en) = -en.
Then w03BB = 03BBimplies 03BBel-el+1 = w03BBel-el+1 + 03BBel-1-el
for k i n.Hence 03BBel-el+1
=Àen-l-en
fork
i n.Also, À2en
=wÀ2en = À2en-l implies Àen-l-en
= 1.Multiplying
the trivialcharacters 03BB03B1 together,
for a = ek - ek+1,...,en-1 - en, weget 03BBek-en ~ 1. Also, 03BB2ek(x) = w03BB2ek(x) = c-1n03BB2en(x) = 03BB2en(x)-1. 50 we get 03BB2en(xx) = 03BBek-en(x)03BB2en(xx) =
03BB2ek(x)03BB2en(x)-103BB2en(xx)=03BB2ek(x)03BB2en(x) ~
1.Finally, 03BBen-1+en(x) = 03BBen-1-en(x)03BB2en(xx) ~
1implies
en -1+ en E
0’.However,
en-1+ en
> 0 andw(en-1 + en)
0 contradict w E R.Thus s cannot contain a non-trivial
cycle,
and R is contained in thegroup Zn2
ofsign changes.
LEMMA 2 :
I f
ckck+1 ···en ER,
then each CIER, k i
n.PROOF : Since w03BB = 03BB
if,
andonly
ifw03BB03B1 = 03BB03B1
for all a = e1 - e2,..., en-1- en,
2en, w
= ck ’ ’ ’ en fixes Àif,
andonly
if03BB2en(xx) = 03BBel-el+1(xx)
= 1for all
k
i n. But then eachproduct cj···
en fixesÀ,
kj
n, so infact each
Cj
fixesÀ,
kj
n. Note that this result is true for anyproduct
of
sign changes.
Since for anya > 0, ckck+1...cn03B1>0 implies
thatcj03B1
> 0 for each kj
n, weget
that ckck+1 ···cn E R implies
that eachcj ~ R, k j n.
We may now
explicitly classify
all À E M" with non-trivialRÀ.
Weknow that
R-group
is contained in the groupll2
ofsign changes
in W.Consider the
longest product
ofsign changes
in R. Each factor will then be inR,
and R isgenerated by
thesesign changes.
We may in factreplace
À
by
aconjugate
to assume that ckck+1 ··· cn is thelongest product
ofsign changes
in R. ThenR ~ ck, ··· , cn>.
To
complete
theclassification,
we show that characters À existhaving
these
R-groups by listing
all of them. A character À of M will be fixedby
each
cj generating R,
kj
n,if,
andonly if 03BB2en(xx) = 03BBel-el+1(xx) ~
1for all
k
i n.Since Àe -e = 03BB2el03BB-12el+1,
we mayequivalently require
03BB2el (xx)
= 1 fork
i n. Note that this conditionimplies X,@ = 03BBel-el.
Then,
to haveR03BB = ck,···,cn>,
we need a OE 0’ for each a =el ± ej and
2 em ,
wherek
ij n and k m
n. So we mustrequire À 2 e
1 andÀ2e IFx == 1, k
m n, and also03BBel-ej ~ 1
fork i j n.
Equivalently,
we may define the character Àby specifying
distinctcharacters Àa
ofEX, a
=2 el, k 1
n,satisfying 03BB03B1(xx)
=1, 03BB03B1 ~ 1
and03BB03B1|Fx~1.
We note that the conditions for
reducibility,
as well as theexplicit
classification of the
R-groups,
are the same as these for theanalogous
real group, F =
R,
E = C.REMARK: The above theorem determines the
reducibility
of theunitary principal
series of G. But note that for R =ck,··· cn>,
the charactersÀe,-e1+1’ 1
ik,
arearbitrary.
If wereplace
these charactersby quasi- characters,
thenon-unitary principal
seriesIndGP03BB
will also be reducible.One may use the
intertwining operators {A(w, 03BB)|w ~ R}
to construct|R|
nonzeroprojections
onto invariantsubspaces,
thusshowing operator reducibility.
References
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Reductive Groups, to appear.
[2] W. CASSELMAN: The unramified principal series of p-adic groups, I. The spherical
function. Comp. Math. 40 (3) (1980) 387-406.
[3] D. KEYS: On the decomposition of reducible principal series representations of p-adic Chevalley groups. Pacific J. Math. 101 (1982) 351-388.
[4] D. KEYS: Reducibility of unramified unitary principal series representations of p-adic
groups and class-1 representations. Math. Annalen 260 (1982) 397-402.
[5] A.W. KNAPP and E.M. STEIN: Intertwining operators for semi-simple groups, II.
Invent. Math. 60 (1980) 9-84.
[6] P.J. SALLY and M.H. TAIBLESON: Special functions on locally compact fields. Acta Math. 116 (1966) 279-309.
[7] F. RODIER: Models de Whittaker et characteres de representations. In: Lecture Notes in Math. 466, Non-Commutative Harmonic Analysis, Springer-Verlag (1975), pp.
151-171.
[8] F. SHAHIDI: On certain L-functions. Amer. J. Math. 103 (2) (1981) 297-355.
[9] A. SILBERGER: Introduction to harmonic analysis on reductive p-adic groups (based on
lectures by Harish-Chandra at IAS). Math Notes no. 23,Princeton University Press, N.J. (1979).
[10] A. SILBERGER: The Knapp-Stein dimension theorem for p-adic groups. Proc. Amer.
Math. Soc. 68 (1978) 243-246, and "Correction", 76 (1979) 169-170.
[11] A. SILBERGER: Special representations reductive p-adic groups are not integrable. Ann.
Math. 111 (1980) 571-587.
(Oblatum 22-X-1981 & 8-VI-1982 & 10-1-1983) Department of Mathematics
University of Utah
Salt Lake City, UT 84112
USA
Current address:
Rutgers, The State University Department of Mathematics, NCAS Newark, NJ 07102
USA