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Seperate and joint Gevrey vectors for representations of LiegroupsCitation for published version (APA):Elst, ter, A. F. M. (1989). Seperate and joint Gevrey vectors for representations of Lie groups. (RANA : reportson applied and numerical analysis; Vol. 8922). Eindhoven: Technische Universiteit Eindhoven.
Document status and date:Published: 01/01/1989
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Download date: 15. Jul. 2020
Eindhoven University of Technology Department of Mathematics and Computing Science
RANA89-22 October 1989
SEPARATE AND JOINT GEVREY VECTORS FOR REPRESENTATIONS
OF LIE GROUPS by
AF.M. ter Eist
Reports on Applied and Numerical Analysis Department of Mathematics and Computing Science Eindhoven University of Technology
P.O. Box 513 5600 MB Eindhoven The Netherlands
Separate and joint Gevrey vectors for representations of Lie groups
by
A.F.M. ter Elst
Abstract
In the present paper we prove that for every Lie group there exists a
basis in its Lie algebra such that for each A ~ 1 and each of its unitary
representations, a vector in the Hilbert space which is a Gevrey vector
of order A in each direction of the basis elements is in fact a (joint)
Gevrey vector of order A. As a corollary we obtain that a vector
is a Gevrey vector if and only if the corresponding positive definite
function is a Gevrey function.
AMS 1980 Subject Classification: 22E45.
1
1 Introd uction and notations
Let G be a d-dimensional (real) Lie group, let Y1, ••• ,Yd be analytic vector fields on G which
are linearly independent at each point of G, let H be a Hilbert space and let>. ;::: 1. A
function f from G into H is called a Gevrey function of order>' if and only if f is infinitely
differentiable and for each compact subset K of G there exist c, t > 0 such that for all n E INo
sup sup 1I(¥i1 0 ••• 0 ¥inf)(g)1I :S ctnn!).. il •... ,inE{l, ... ,d}gEK
It follows from [Nel, Theorem 2] and [GW, Theorem 1.1] that this definition does not depend
on the choice of Y1 , ••. ,Yd. The Gevrey functions of order 1 are just the real analytic functions
from G into H. Now let 71' be a representation of G into H. For u E H define u : G -lo H by
u(g) := 7rg U (g E G). A vector U E H is said to be infinitely differentiable, analytic respectively
a Gevrey vector of order>' for 71' if and only if the map u is infinitely differentiable, (real)
analytic respectively a Gevrey function of order>. from G into H. (Cf. [ Gar], [N ell and [GW]
respectively.) Let HOO(7r), HW(7r) and H).(7r) denote the space of all infinitely differentiable
vectors, of all analytic vectors and of all Gevrey vectors of order>. for 71', respectively. Note
that HW(7r) = H1(7r). We only consider continuous unitary representations. For each X in
the Lie algebra {} of G let d7r(X) be the infinitesimal generator of the one-parameter unitary
group t ..... 7rexp tx. Let 01!'(X) be the restriction of dl!'(X) to HOO(7r). The map X ..... 01!'(X)
extends uniquely to an associative algebra homomorphism, denoted by 011' also, from the
complex universal enveloping algebra U(g) of 9 into the set of all linear operators from HOO( 11') into HOO( 71').
There exist infinitesimal characterizations for the spaces HOO(7r), HW(7r) and H),(7r), Le.
characterizations in terms ofthe infinitesimal generators. To this end we introduce the concept
of multi-index. Let V be a non-empty finite set. We define the set M(V) of multi-indices over
V by kI(V) := U~o V n • Here VO is the set with one element, called the empty sequence,
which is denoted by ( ). For a E M(V) define the length of a by 110'11 = n, where n E INo is
the unique number such that a E vn and the reverse a r of a by
(y .- ()
N ow let V := {1, ... , d} and let AI,.'" Ad be operators in H. For a E M (V) define the
operator Aat by
AO .- I,
A(il •... Jn) .- Aj1o ... oAjn (nEIN,jl, ... ,jnEV).
Define the joint Coo-domain Doo (A 1, •.• , Ad) of the operators AI, ... ,Ad by
DOO(A}, ... ,Ad):= n DCA.). Q'EM(V)
2
Here D(T) denotes the domain of the operator T. Let"\ ~ 1. Define the Gevrey space
S>. (AI, ... ,Ad) of order ,.\ relative to {AI, ... ,Ad} by
S>.(AI, .• • , Ad) := {u E DOO(Al,' .. ,Ad) : 3c,t>o'v' o-EM(V) [IIAo-ul! =5 ctllalillall!>']}.
(ef. [GW, Section 1].) We have the following infinitesimal description of the spaces HOO(7f)
and H>.(7f).
Theorem 1 Let 7f be a representation of a Lie group G in a Hilbert space H. Let Xl,'" ,Xd
be any basis in the Lie algebra g of G. Let"\ ~ 1. Then
d
HOO(1r) = DOO(d1r(XI)"" ,d1r(Xd» = n DOO (d1r(Xk)) k=I
and
Proof. See [Go02, Proposition 1.1] and [Goo1, Theorem 1.1] for the space HOO(1r) and [GW,
Proposition 1.5] for the space H>.(7f-). o
So a vector in H is infinitely differentiable for 1r if and only if it is infinitely differentiable
for each 1rIG" separately, where Gk is the one dimensional group {exptXk : t E IR}.
Let ,.\ ~ 1. Then clearly
d
S>.(d1r(XI ), ... , d1r(Xd» C n S>.(d1r(Xk» (1) k=I
for any basis Xl, •. "Xd in g. The space on the right hand side of (1) is much easier to
describe than the space on the left hand side. In the present paper we shall show that there
exists a basis X I, ..• ,Xd in g such that
d
H>.(1r) = S>.(d1r(X1), ... ,d1r(Xd» = n S>.(d1r(Xk)). (2) k=l
As a corollary of (2) we obtain that for every ,.\ ~ 1 a vector u E H is a Gevrey vector of
order ,.\ for 1r if and only if the corresponding positive definite function 9 ....... (1r 9 U, u) from G
into C is a Gevrey function of order "\.
In fact, (2) generalizes a theorem of Flato and Simon ([FS, Theorem 3]), which states that
there exists a basis Xl, ... ,Xd in g such that HW(1r) = nt=l Sl(d1r(Xk». Taking ,.\ = 1, this
paper yields an easier proof for the theorem of Flato and Simon.
2 Gevrey vectors for direct sums
In this section we prove that in case the operators AI, ... ,Ad span a Lie algebra, we can always
write the corresponding Gevrey spaces as an intersection of two Gevrey spaces relative to a
reduced number of operators.
3
Theorem 2 Let 9 be a Lie algebra of skew-Hermitian operators in a Hilbert space defined on
a common invariant domain. Let d E IN, d ~ 2 and let AI, ... , Ad E g. Suppose
9 = span( {AI, ... , Ad}).
Let A ~ 1 and let dl E {I, ... ,d - I}. Then
SA(AI, ..• , Ad) = SA(A}, ... , AdJ n SA (Ad1 +1, ... , Ad).
Proof. Let V1 := {I, ... ,dl }, V2:= {dl + I, ... ,d} and V:= {I, ... ,d}. By assumption,
for ali i,j E V there exist cl,j, ... ,c1.i E ill. such that [Ai,Aj] = Et=ICf.;Ak. Let M:=
1 + max{lc7,il : i,j, k E V}.
Let U E SA(Al, ... , Ad1 ) n SA(Adl+1,'" ,Ad)' Then there exist c > 0 and t ~ Md such
that IIAau11 ::; ctllalillctlllA and IIApuil ::; ctIlPIIII,sll!A for ali ct E M(Vd and ,s E MCV2). For
N E INo the hypothesis peN) states:
I (A')'Aau, u)1 ::; c22Ihlltllall+lhll(IIctil + II,II}!A for ali a E M(Vt) and, E VN.
Clearly hypothesis P(O) holds. Let N E IN and suppose hypothesis peN - 1) holds. Let
, E VN and let a E M(V1 ). Let iI, ... ,jN E V such that, = (j}, ... ,iN)' We consider two
cases.
Case I. Suppose ji E V2 for ali i E {I, ... , N}.
Then, E M(V2)' so by Schwarz' inequality we obtain that
I(A')'Aau, u)1 = I(Aau, A-yru)1 ::; IIAauil lIA"rUIl ::; ctllalillall!Actlhllll,II!A
< c2tllall+lhll(IIail + II,II)!A.
Case II. Suppose there exists i E {I, .. . ,N} with ji E VI.
Let k := ij. Pushing the operator Ak in A" to the ultimate right hand side and taking into
account ali commutators, we obtain that there exist 6 E VN-I, Cll"" C(N-l)d E ill. and
(h, ... , ()(N-I)d E VN-l such that
(N-I)d A" = AsAk + L cpAop
p=1
and [Cpl ::; M for ali p. Now the induction hypothesis peN - 1) and the inequality dM ::; t
yield the estimates
(N-l)d
< I(As(AkAa)U, u)1 + L Icpll(AopAau, u)1 p=1
< c22Ihli-ltllali+lhIlCliali + II,II)!" + (11111- I)dMc22Ihll-ltllall+lhll-I(lIall + 11,11- I)!'\
< c22Ihll-ltllall+lhll(lIall + II,II)!" (1 + 11111- 1 ) (liall + 11111)"
< c22Ihlltllall+lhll(llall + '"11)!"· This proves hypothesis peN).
By induction, for ali, E M(V) and a E M(Vd we obtain that
4
Now let IE M(V). Then
1IA.-yuIl2 = 1(A.-yrA')'u,u)1 ~ c2(2t?lhll(21h'ID!" ~ [C(2t2")lhlllh'II!>.f
and the theorem follows o
For other theorems in this direction, we refer to the thesis [tEJ.
The third equality in the following corollary has been firstly proved by Flato and Simon.
(See [FS, Theorem 2].)
Corollary 3 Let G be a real Lie group with Lie algebra 9. Let 91 and 92 be subalgebras of 9
such that 9 = 91 + 92- (Not necessarily a direct sum.) Let G1 and G2 be subgroups of G which
have Lie algebras 91 and 92 respectively. Let 7l' be a representation of G in a Hilbert space H,
and let 7l'1 and 7l'2 be the restrictions of 7l' to G1 and G2 respectively. Let Zl,' .. ,Zd be a basis
in.9:, let X b •• - ,Xdl be a basis in 91 and let Y1, ..• , Ydz be a basis in 92' Let ,\ ~ 1. Then
In particular,
and
Proof. By the previous theorem we obtain that S>.(87l'(X1 ), ••• ,87l'(Xdl)' 87l'(Yd, ... ,87l'(Yd2 » = S,,(87l'(X1 ), ... ,87l'(XdJ)nS>.(87l'(Y1 ), ••• , 87r(Yd2»' By an elementary counting argument,
it follows that S>.( 87r (Xt) , ... , 87l'(XdJ, 87r(Y1), ... ,87r(Yd2» = S>.( 87r(ZI), ... ,87r(Zd». Since
o
Corollary 4 Let 7l' be a representation of a solvable real Lie group G. Let XI, ... , X d be a
basis in the Lie algebra 9 of G such that Ci := span( {XI, ... ,Xi}) is a subalgebra of 9 and
Ci is an ideal in CHI for all i E {I, ... , d}. Let ,\ ~ 1. Then
d
S,,(d7l'(X1 ), ••• , d7l'(Xd» = n S,,(d7r(Xk». k=1
3 Separate and joint Gevrey vectors
The following lemma in case ,\ = I can be found in [FS, Lemma 1], but there the proof is
based on different arguments.
Lemma 5 Let A be a Hermitian or skew-Hermitian operator in a Hilbert space which has an
invariant domain D. Let'\ ~ 1. Let c, t > O. Then
5
{U ED: 'v'melNo [IIA2m ull ~ ct2m
(2m)!'\]} C S,\(A)
Proof. Let s:== 4'\t. Let U E D and suppose that IIA2m
u II ~ ct2"'(2m )!,\ for all m E lNo. For
n E IN hypothesis P( n) states
IIAku11 ~ cskk!'\ for all k E {1, ... , n}.
Clearly hypothesises P(l) and P(2) are valid. Let n E IN, n ~ 2 and suppose pen - 1) is
valid. If 2 log n E IN then hypothesis P( n) holds. Suppose 2 log n ¢ IN. There exist unique
m, k E lNo such that n == 2m + k and 1 ~ k < 2m. Then 2k < 2m + k == n, hence 2k ~ n - 1.
So by assumption and hypothesis P( n - 1) we obtain:
IIAnuII 2 == I(A2m+lu,A2ku)1 ~ IIA2m+1ullllA2kull ~ c2t2m+ls2k(2m+1)!"{2k)!'\
< c2t2m+1 s2k2,\2m+122,\k(2m)!2'\k!2,\ ~ c2t2m+l s2k(2'\?(2m+k)(2m + kW,\
~ [csnn!AF·
By induction, the lemma follows. o
For the case>. = 1 the following result has been stated in [FS, Theorem 1].
Theorem 6 Let G be a compact Lie group with Lie algebra g. Let >. ~ 1 and let 1r be a
representation of G in a Hilbert space H. Let XI, ... ,Xd be any basis in g. Then
d
S,\(d1r(Xd,··· ,d1r(Xd»== n S,\(d1r(Xk». k=l
Proof. The compactness of G insures that there exists a positive definite invariant real
symmetric bilinear form f3 on 9 x g. (See [Hoc, Theorem XIII.1.l].) Let Yt, ... , Yd be a basis
in 9 such that f3(Yi, Yj) = iii,j for all i,j E {I, ... ,d}. By [Nel, Lemma 6.1] there exists a
constant Ml ~ 1 such that for all i,j E {I, ... , d} and all u E Hoo(1r) we have
1l 81r (XiX j)ull ~ M l Il81r(I - Ax)ull,
where ll.x := r:i=l X~ E U(g). So there exists a constant M ~ 1 such that for all u E HOO(1r):
M 1181r(I - ll.y )ull ~ d + 11l81r(I - Ax )ull,
where ll.y := r:~=l Yf E U(g).
Now let u E nt:l SA(d1r(Xk)). Then u E ni=l Doo(d1r(Xk» == H OO (1r). (See Theorem 1.) Since G is compact, there exists an index set f and for all a E f there exists a 1r-invariant
subspace H 0: of H such that 1r 0: := 1rIHo is irreducible and H == ffiaEI H 0:' Let Uo: E H 0: be
the projection of u on Ho:. Note that Ho: C Hoo(1r).
Let a E 1. By [Bou, Chapter I §3.7 Proposition Il], Ay belongs to the center of
U(g). Since 1ra is irreducible, it follows by Schur's lemma that there exists ca E C such
that 81r 0:( ll.y) = -oexf. Because the operator 01r( Ay) is negative, we obtain that Cex ~ O.
Then (1 + liex)lIuexll == 1181ra(I - Ay)uall == 1181r(I - Ay )uall ~ ltlIl81r(f - Ax )uexll :$;
l:l r:t=o 1I81r(Xk)2uall, where Xo := fEe C U(g).
For all m E IN 0 let hypothesis P( m) state:
6
d 2m 1 2m,\:"""" 2m+1 (1 + 6a ) lIual! $ diM L..,lltJ7r(Xk) uall·
+ k=O
We have already proved hypothesis P(O). Let mE INo and suppose P(m) holds. Then by
HOlders inequality:
(1 + oa?m+lllualllluall = [(1 + ooymlluallf
< 1 M2m+1 [~lltJ (X )2m+l 11]2 (d + 1)2 L- 7r k ua
k=O d
$ d: 1 M2m+l L IItJ7r(Xk)2m
+1
ua ll 2
k=O
d
= -d 1 M2m+l L 1 (tJ7r(Xk»2ffl
+2
Ua ! ua)1 + 1 k=O
d
< -d 1 M 2m+1 L IItJlI'(Xk)2
m+
2 ualilluall· + 1 k=O
So P( m) is valid for all m E INo.
Since U E ni=l S" (d7r( X k», there exist c, t > 0 such that for all k E {1, ... , d} and all
n E INo: IId7r(xk)nu ll $ ctnn!". Let mE INo. Then
So
IltJ7r(I - ,6,y)2m
uII2 = L lIo7r(I - ,6,y)2m ua I12 = L [(1 + oa)2m ll ua ll]
2
a€I a€I
< L [_1_M2m t lIo7r (Xk f n+
1 Uall]
2
a€I d + 1 k=O
1 d 2 < - L L [M2mlltJ7r(Xk)2m+lUall]
d + 1 a€I k=O
d = _1_ L [M2fflIlO7r(Xk)2ffl+lulI]2 $ [c(Mt2)2m(2m+1)!Af
d+1 k=o
$ [c (22"Mt2rm (2m )!2>.f.
2m
lIo7r(I - ,6,y)2m
ull $ C (22)'Mt2) (2m)!2>'
for all mE INo. Hence U E S2>.(07r(I - ,6,y» by Lemma 5. Since A 2:: 1 we can deduce that
u E H>.(7r) from [GW] Example following Theorem 1.7.
We arrive at the main theorem of this section.
Theorem 7 Let G be a Lie group with Lie algebra g. Then there exists a basis Xl! ... ,Xd
in 9 such that for all oX 2:: 1 and all representations 7r of G we have
d
S>.( d7r(X1 ), ... ,d7r(Xd)) = n S>.( d7r(Xk)). k=l
7
Proof. We prove the theorem by induction to dim 9. If dim 9 = 1 then nothing has to be
proved. For d E IN let hypothesis P( d) state:
For any Lie group G with Lie algebra 9 and d1 := dim 9 $; d there exists a basis
Xl, ... ,Xdl in 9 such that for all ), 2 1 and all representations 1r of G we have
dl
SA(d1r(XI ), ... ,d1r(Xd1 )) = n SA(d1r(Xk)). k=l
Let d E IN, d 2 2 and suppose hypothesis PC d - 1) is valid. Let G be a Lie group with
Lie algebra 9. Suppose dim 9 = d. We shall prove:
Assertion 1: There exists a basis Xl, ... ,Xd in 9 such that for all ), 2 1 and all repre
sentations 1r of G we have
d
SA(d1r(X1), ••• ,d1r(Xd) = n SAC d1r(Xk)). k=l
First we prove the following assertion:
Assertion 2: Let 9}192 be subalgebras of 9 such that 9 is the direct sum of 91 and 92'
Suppose dim 91 2 1 and dim 92 2 1. Then Assertion 1 holds.
Proof of Assertion 2. Let G1 and G2 be subgroups of G which have Lie algebras 91 and 92
respectively. By induction hypothesis P(d - 1) there exist a basis Xl. ... ,Xdl in 91 and a
basis Y1 , ..• , Yd2 in 92 such that for every representation 1rl of GI and for every representation
1r2 of G2 and all ), 2 1 we have
d1
SAC d1rl (X d,·· . ,d1r1 (Xdl)) = n SAC d7r1(Xk» k=1
and
d2 S.\(d1r2(Yd,···, d1r2(Yd2)) = n S.\(d1r2(Yk».
k=l
Then Xl, ... ,Xd1 , YI, ... ,Yd2 is a basis in 9.
Now let 1r be a representation of G in a Hilbert space H and let), 2 1. Let 1rl and 1r2 be
the restrictions of 1r to G1 and G2 respectively. Then d1r(Xk) = d1rl(Xk) for all k E {l, ... ,d}
and d1r(Yk) = d1r2(Yk) for all k E {l, ... , d2}. So by Corollary 3 we obtain that
SA(d1r(Xd, .• · ,d1r(Yd2» =
= S.\(d1rl (XI), ... ,d1rl(Xdl»n SA(d1r2(Y1 ), ... , d1r2(Yd2» dl d2
= n SA (d7r1 (XI.)) n n S.\(d1r2(Yk» 1.=1 k=l d1 d2
= n SA(d1r(Xk» n n S.\(d7r(Yk». 1.=1 k=1
8
This proves Assertion 2.
Now we prove Assertion 1. If 9 is solvable, then Assertion 1 follows by Corollary 4. So
we may assume that 9 is not solvable. Let q be the radical of g. By [VarI, Theorem 3.14.1]
there exists a semisimple subalgebra m of 9 such that 9 is the direct sum of q and m. (This
is a Levi decomposition of g.) If dim q ~ 1, then Assertion 1 follows by Assertion 2.
So we may assume that dim q = O. Then 9 = m is semisimple. Let 9 = t + (l + n be an
Iwasawa decomposition of g. (See [ReI, Theorem VI.3.4].) Let S := (l + n. Then -t and s are
subalgebras of g, S is solvable and 9 is the direct sum of ~ and s. Since 9 is semisimple, always
dim £ ~ 1. In case dim S ~ 1, Assertion 1 follows again by Assertion 2.
So we may assume that dim S = O. Then 9 = t So the Lie algebra 9 is compact.
But the Lie group G need not be compact and we cannot immediately apply Theorem 6.
Corresponding to the Lie algebra 9 there exists a connected simply connected Lie group G1
with Lie algebra g. (See [VarI, Theorem 3.15.1].) Then G1 is compact by [Wal, Theorem
3.6.6]. Let Xt, ..• ,Xd be any basis in g. Let A ~ 1 and let tr be a representation of G
in a Rilbert space H. Now X l-+ 8tr(X) is a representation of the Lie algebra 9 by skew
symmetric operators in H and the operator 8tr(XI)2 + ... +8tr(Xd? is essentially self-adjoint.
(See [Nel, Theorem 3).) So by [Nel, Corollary 9.1] there exists a representation u of G1 such
that du(X) = dtr(X) for all X E g. Therefore we obtain by Theorem 6 that
S>.(dtr(Xt}, ... ,dtr(Xd» = S>.(du(Xt}, ... ,du(Xd» d d
= n SX(dU(Xk» = n S>.(dtr(Xk». k=l k=l
This proves the theorem. 0
The following corollary is a kind of Rartogs' theorem.
Corollary 8 Let tr be a representation of a Lie group G. Let 9 be the Lie algebm of G and
let A ~ 1. Then
H>.(tr) = n H>.(trIG1 ) = n S>.(dtr(X». Gl subgroup of G X eg
dimGl=l
4 Gevrey vectors and positive definite functions
Let tr be a representation of a Lie group G in a Rilbert space H. For U E H define Pu : G -+ C
by Pu(g) := (trgU, u) (g E G). Now we present a description of the infinitely differentiable
vectors for tr and the Gevrey vectors of order A for tr in terms of the positive definite functions
Ptl, U E H. More precisely, we prove that
HOC( tr) = {u E H : the function Pu is infinitely differentiable on G}
and
H>. (tr) = {u E H: the function Pu is a Gevrey function of order A on G}
for all ). 2:: 1. We need a well-known lemma.
Lemma 9 Let A be a self-adjoint opemtor in a Hilbert space H. Let u E H. Define F : IR -+
C by
F(t) := (eitAu, u) (t E IR).
Let V be an open neighborhood of O. Suppose the restriction Flv is infinitely differentiable.
Then u E DOO(A). Moreover, for all n E INo we have F(2n)(0) = (_I)nIlAnuIl 2•
Proof. We may assume that A is the multiplication operator by the function h in the Hilbert
space H = L2(Y,m) for some measure space (Y,B,m). Let f: IR -+ IR be defined by f(t):=
ReF(t) = J cos(th)luI2dm, t E IR. Since f is an even function and infinitely differentiable on
V, we obtain that 1'(0) = O. Then by Fatou's lemma:
J h2 1ul 2dm = 2 J li~~f n2 (1- cos(~h») lul2dm
< 2liminf J n2 (1- cos( 1.h») lul2dm n-+oo n
::: -2lim inf n2 (f( 1.) - f(O) - 1. f'(O») n ..... oo n n
= - fl/(O) < 00.
So u E DCA) and 1"(0) = -IlAuIl2. Hence by Lebesque's theorem on dominated convergence
we obtain that F is twice differentiable on V and F"(t) = _(eitAAu, Au) for all t E V. By
induction, the lemma follows. o
Theorem 10 Let 1r be a representation of a Lie group G in a Hilbert space H. Let). 2:: 1.
Then
and
H>. ( 1r) = {u E H: the function Pu is a Gevrey function of order). on G}
In particular,
HW ( 1r) = {u E H : the function Pu. from G into C is real analytic}.
Proof. Let u E H. Suppose Pu E GOO(G). Let Xl, ... ,Xd be a basis in the Lie algebra 9
of G. Let k E {l, ... ,d}. Then the function t 1-+ (etd1r(Xk )u,u) = pu(exptXk) from IR into
C is infinitely differentiable, so by Lemma 9 we obtain that u E DOO(d1r(Xk». Therefore,
u E n%=l DOO(d1r(Xk)) = HOO(1r) by Theorem 1.
Now let). 2:: 1, let u E H and suppose the function Pu is a Gevrey function of order). on G.
Then Pu E GOD( G), so by the previous part we obtain that u E HODe 1r). Let K be a compact
neighborhood of the identity e in G. Let X E g. Then by definition there exist c, t > 0
such that for all n E INo and all g E I( we have I[Xnpu](g)1 ~ Gtnn!>., where X denotes the
10
corresponding left invariant vector field on G. Define F : IR - C by F(t) := (e td1l"(X)u, u),
tEnt. Then by Lemma 9 we obtain for all n E INo:
III d,(X)]'ull' = [F(2')(O)[ = \ (~t)" pur exp tX) ['=0\
= I[X 2npu)( e)1 $ Ct2n(2n )!,\ :5 C(2t?nn!2'\.
Hence u E S,\(d1l"(X». Therefore u E nxeg S,x(d1l"(X» = H,\(1I") by Corollary 8. o
A vector u E H is called weakly analytic if for all v E H the function 9 ..... (11" 9 U, v) is an
analytic function from G into C.
Corollary 11 Let 11" be a representation of Lie group in a Hilbert space H. Let u E H. Then
u is weakly analytic if and only if u is analytic.
This corollary has been proved before in (Var2, page 303], by using the BOOre category
theorem.
References
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mathematics, Hermann, Paris, 1975.
[tE] ELST 1 A.F.M. TER, "Gevrey spaces related to Lie algebras of operators," Thesis,
Eindhoven University of Technology, Eindhoven, The Netherlands, November 1989,
to appear.
[FS] FLATO, M. AND J. SIMON, Separate and joint analyticity in Lie groups represen
tations. J. Funct. Anal. 13 (1973), 268-276.
IG.h] GARDING, L., Note on continuous representations of Lie groups. Proc. Nat. Aead.
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[Hel] HELGASON, S., "Differential geometry, Lie groups, and symmetric spaces," Aca
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[Nel] NELSON, E. Analytic vectors. Ann. Math. 70 (1959),572-615.
[Vad] VARADARAJ AN, V.S., "Lie groups, Lie algebras, and their representations," Grad
uate Texts in Mathematics 102, Springer-Verlag, New York etc., 1984.
[Var2] VARADARAJAN, V.S., "Harmonic analysis on real reductive groups," Lect. Notes
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[\Val] WALLACH, N.R., "Harmonic analysis on homogeneous spaces," Pure and appl.
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