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.

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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

[Bou] BOURBAKI, N., "Lie groups and Lie algebras," Part I: Chapters 1-3. Elements of

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.

Se. U.S.A. 33 (1947),331-332.

(Gool] GOODMAN, R., Analytic and entire vectors for representations of Lie groups. Trans.

Amer. Math. Soc. 143 (1969),55-76.

[Goo2] GOODMAN, R., One-parameter groups generated by operators in an enveloping

algebra. J. Funct. Anal. 6 (1970),218-236.

[GW] GOODMAN, R. AND N.R. WALLACH, Whittaker vectors and conical vectors. J.

Funct. Anal. 39 (1980), 199-279.

[Hel] HELGASON, S., "Differential geometry, Lie groups, and symmetric spaces," Aca­

demic Press, New York etc., 1978.

[Hoc] HOCHSCHILD, G., "The structure of Lie groups," Holden-Day, San Francisco etc.,

1965.

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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

in Math. 576, Springer-Verlag, Berlin etc., 1977.

[\Val] WALLACH, N.R., "Harmonic analysis on homogeneous spaces," Pure and appl.

math. 19, Marcel Dekker, New York, 1973.

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