Laguerre Bessel

7/23/2019 Laguerre Bessel

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REPRESENTATION THEORYAn Electronic Journal of the American Mathematical SocietyVolume 4, Pages 181224 (April 26, 2000)S 1088-4165(00)00096-0

ON LAGUERRE POLYNOMIALS, BESSEL FUNCTIONS,HANKEL TRANSFORM AND A SERIES

IN THE UNITARY DUAL OF THE SIMPLY-CONNECTED

COVERING GROUP OF Sl(2,R)

BERTRAM KOSTANT

Abstract. Analogous to the holomorphic discrete series of Sl(2,R) there isa continuous family{r},1 < r


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182 BERTRAM KOSTANT

Letg = LieSl(2,R) and letU(g) be the universal enveloping algebra overg withcoefficients in C. Let{h,e,f} g be the S-triple basis ofg where

h= 1 00 1 , e= 0 10 0 , f= 0 01 0 .

Let G be a simply-connected Lie group with Lie algebra g. The center Z of Gis infinite cyclic and one identifies Sl(2,R) with G/Z2. The metaplectic group,M p(2,R), the double cover ofSl(2,R), is identified with G/Z4. Conjugation ofgby1 0

0 1

defines an outer automorphism ofG.

LetLbe a Hilbert space and assume that : GU(L) is an irreducible unitaryrepresentation ofG. LetL Lbe the dense subspace of infinitely differentiablevectors inL. From the general theory of unitary representations the differentialof induces a representation

: U(g)End L.

The group G is not in the Harish-Chandra category because Z is infinite. Inparticular, even though the element e f g is elliptic, the subgroup KofG cor-responding to R(e f) is not compact. Nevertheless, much of the Harish-Chandratheory is still valid. In particular, ifh =i(e f), then the linear span,LHC, ofthe(h) eigenvectors inL is a denseU(g)-submodule ofL. We refer toLHCas the Harish-Chandra module associated to . We will say that is one-sided ifthe spectrum of(h)|LHC is either a set of positive numbers or a set of negativenumbers. To distinguish between these two cases, and adopting terminology whichis valid for the discrete representations ofS l(2,R), we will say that is holomor-phic (resp. anti-holomorphic) if the spectrum of(h)|LHC is a set of positive(resp. negative) numbers. It is immediate that is holomorphic if and only if

is anti-holomorphic, where for gG, (g) = ((g)).Lajos Pukanszky in [Pu]has determined, among other things, the unitary dual

of G. See p. 102 in [Pu]. There are 3 series of representations. The one-sidedrepresentations are exactly the members of the second series. We are concerned,in the present paper, with a construction, and resulting harmonic analysis, of amodel, using Laguerre functions, for the members of this second series. Using itis enough to give the model for the members of the holomorphic subfamily. Dueto a sign difference the members of this subfamily are denoted, in [Pu], by Dwhere > 0. In the present paper these representations will appear as r where,as stated in the first sentence of the introduction,1< r


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LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OFSl(2,R) 183Next, one finds thatHHCr is stable underDr(U(g)). In fact,{h, e, f} is an S-triple in the complexification,gC, ofg, wheree =

12 (ih+e +f), f= 12 (ih+ e+ f)

and, as above, h =

i(e

f). When applied to the Laguerre functions one finds

(see Theorem 2.6), for n Z+,Dr(h

)(r)n = (2n + r+ 1)(r)n ,

Dr(e)(r)n =i

(r)n+1,

Dr(f)(r)n =i(nr+ n

2)(r)n1,

(0.1)

where we put (r)1 = 0. In particular, the spectrum of Dr(h

)|HHCr is positive.Furthermore,

(r)0 (x) = x

r2 ex spans the eigenspace for the minimal eigenvalue,

r+ 1, ofDr(h)|HHCr and Dr(f)(r)0 = 0.For u g the operator Dr(u)|HHCr is formally skew-symmetric. But formal

isnt good enough since the Laguerre functions do not vanish asx approaches 0 for

r 0. In fact, they become unbounded forr < 0. Nevertheless, these operatorsare skew-symmetric. But much more is true. The hypotheses for Nelsons beautifulTheorem 5 in [Ne]are satisfied so Dr|HHCr can be exponentiated for all1< r 0. Let = 12 so that x V1 Hr . Then, up to scalar multiplication,x is the uniqueh-weight vector ofweight andUr(x) is the uniqueh-weight vector of weight.Remark 0.8. Note forRe >0 theh-weight vector is independent ofr whereas forRe


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186 BERTRAM KOSTANT

Now note thatAd(k0)(e) =fso that the Hankel transform Ur carriese-weightvectors of weight to f-weight vectors of weight. In fact, fory (0, ) onehas, for Whittaker vectors,

Ur(y) = Jr,y.

See Theorem 5.17. More interesting is the question about highest and lowest weightvectors if they exist at all inHr . In fact, they do exist inHr . The highest(resp. lowest) weight vector, given below, in Theorem 0.8, asr,0 (resp. Jr,0) is theunique element (up to scalar multiplication) inHr which is simultaneously anh-weight vector and an e-weight (resp. f-weight) vector. See Theorem 5.25.

Theorem 0.9. There exists a unique elementr,0 Hr such that for any Hrthe limit in (0.7) below exists and one has

{, r,0}= limx0

xr2 (x).(0.7)

Moreover,r,0 is, up to scalar multiplication, the unique highest weight vector (i.e.

r,0Ker r (e)). Furthermore,r,0 is also anh-weight vector of weight(r+1).Theh-weight vector,Jr,0, of weightr+ 1 given by

{, Jr,0}= 1(r+ 1)

0

(x)xr2 dx(0.8)

for any Hr , (see Theorem 0.7) is, up to scalar multiplication, the uniquelowest weight vector (i.e. Jr,0 Kerr (f)). In addition, with respect to theHankel transform, one has

Ur(r,0) =Jr,0.(0.9)

Remark 0.10. Note that (0.7) vanishes ifCo ((0, )) thereby establishing thatCo ((0, )) is not dense inHr .0.5. Following this introduction there are five chapters:

1. The representationDr and the Casimir element2. Laguerre functions and Harish-Chandra modules3. The series,{r},1< r


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LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OFSl(2,R) 187it is not transparent, particularly for1 < r < 0, it is one of the results of ourpaper (see Lemma 3.2) that Dr|HHCr is indeed a representation by skew-symmetricoperators. This is a key point since it sets the stage for the use of Nelsons theorem

to establish that Dr|HHCr integrates to a unitary representation ofG.1. The representation Dr and the Casimir element

1.1. Let gC = Lie Sl(2,C) and let g= Lie Sl(2,R) so that g is a real form ofgC.Let{h,e,f} g be the S-triple given by

h=

1 00 1

, e=

0 10 0

, f=

0 01 0

.

Consider the algebra DiffC((0, )) of differential operators on the open in-terval (0, ). Let x C((0, )) be the natural coordinate function and letH be the Hilbert space L2((0, )) with respect to the usual measure defined bydx. Unless stated otherwise we will always assume r R where r >1. LetDr : gC Diff C((0, )) be the linear map, where identifying elements inC((0, )) with the corresponding multiplication operators,

Dr(e) =ix,

Dr(h) = 2x d

dx+ 1,

Dr(f) =i(xd2

dx2+

d

dx r

2

4x).

(1.1)

Noting that [x, x d2

dx2] =2x d

dx one readily establishes that Dr is a Lie algebra

homomorphism. Thus ifU(g) is the universal enveloping algebra of gC, then Drextends to an algebra homomorphismDr :U(g)Diff C((0, )). Let C asU(g) be the quadratic Casimir element corresponding to the Killing form on gC.Clearly,

Cas =1

4(

h2

2 + ef+ f e).(1.2)

Computation yields

Proposition 1.1. The differential operator Dr(Cas) is the scalar operator givenby

Dr(Cas) =1

8(r2 1).(1.3)

Proof. We first note that

Dr(h2/2) = (2x

d

dx+ 1)2/2

= 2(x

2

(

d

dx )

2

+ 2x

d

dx + 1/4).

(1.4)

Next,

Dr(ef+ f e) = ix(i(x( d

dx)2 +

d

dx r

2

4x)) + i(x(

d

dx)2 +

d

dx r

2

4x)ix

=2x2( ddx

)2 4xddx

1 + r2

2.

(1.5)

But then (1.3) follows from (1.1), (1.2), (1.4) and (1.5). QED

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188 BERTRAM KOSTANT

LetG be a simply-connected covering group with Lie algebra g.

Remark 1.2. Note the scalar value of Dr(Cas) is negative if and only if|r| 0 when|t| < 1. Substituting this

expression fors in (5.13) one has1

2

0

Jr(

yx)xr2 e

x21+t1t dx= y

r2 (

1 t1 + t

)r+1ey21t1+t

and hence

1

2

0

Jr(

yx)(1 t)(r+1)x r2 e x2 1+t1t dx= (1 + t)(r+1)y r2 ey2 1t1+t .(5.16)

ButJr(

yx)xr2 , as a function ofx, is integrable at 0 by (5.2) and is bounded by a

constant multiplexk, fork sufficiently large, as x tends toby (5.3). Thus if onedifferentiates (5.13)n-times for anyn Z+, with respect tos, and then setss = 12 ,the computation on the left side can be carried out under the integral sign. Butsinces is an analytic function oft for|t|< 1, the same differentiation statement istrue for (5.16) where t replaces s and we set t = 0. But then Theorem 5.4 followsfrom (5.15). QED

Since the elements{[r]n} are an orthogonal basis ofH (see Remark 5.3), itfollows immediately from Theorem 5.4 that there exists a unique unitary operator

Ur onH such that Ur|H[HC]r = Tr|H[HC]r and that furthermore Ur has order 2.The operator Ur is often referred to as the Hankel transform. However, it is moreconvenient here to make a slight change (conjugating by r(a1o )) and refer to thisaltered operator as the Hankel transform. That is, first of all, by applying r(a

1o )

to (5.11) it follows from (5.7), (5.8) and (5.9) that Theorem 5.4 can be rewritten as

Theorem 5.7. Forn= 0, 1, ...,one has

Tr(r)n = (1)n(r)n .(5.17)

We will letUr be the unique unitary operator (necessarily of order 2) onHsuchthat

Ur|HHCr =Tr|HHCr .Clearly,

r(ao)Urr(ao)1 =Ur.

We refer to Ur as the Hankel transform.



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LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OFSl(2,R) 2135.3. If G1 is a linear semisimple Lie group and X=G1/K1 is a Hermitian sym-metric space, where K1 is a maximal compact subgroup ofG1, then the holomor-phic discrete series of G1 is generally constructed on spaces of square integrable

holomorphic sections of line bundles on X. In case X is of tube type, Ding andGross in [DG]use this construction to create another model, for the holomorphicdiscrete series, which is built on the symmetric cone associated to X. In caseG1 =S l(2,R), then = (0, ) and the new model is (essentially), r, where r isa positive integer. The main point in their construction of the new model is theirrecognition of the fact that the Hankel transform can be represented, up to a scalar,by a group element in K1. ForG1 = Sl(2,R) this means that, if r is a positiveinteger, then for somecr Cand kK1,

Ur = crr(k).(5.18)

Even if one went to the covering group, G, their construction does not lead tor for1 < r 0 since in this range, as pointed out to us by David Vogan, rcannot be represented on Xby spaces of square-integrable holomorphic sections

of line bundles. (Ifr is rational, this follows from Harish-Chandras classificationof discrete series represenations since, in that case, r descends to a group in theHarish-Chandra class.) Our construction ofr , using Nelsons theorem, and ourknowledge of the Harish-Chandra module, immediately (see Theorem 5.8 below)yields (5.18) forG and all1< r


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214 BERTRAM KOSTANT

But then, since both crr(ko) and Ur are unitary operators onH, (5.20) followsfrom (5.21) and (5.17). QED

One immediate consequence of Theorem 5.8 is the following property of theHankel transform Ur. This property is an analogue of the fact that the Fouriertransform stabilizes the Schwartz space and is continuous with respect to its Frechettopology.

Theorem 5.9. For the Hankel transform one has

Ur :Hr Hr(5.22)and (5.22) is continuous with respect to ther-Frechet topology onHr .

Since r(ko) = r (ko)|H we can initially extend the domain of the Hankel

transform toHr by puttingUr = crr (ko). But in fact, recalling (4.57), onecan go further and extend the domain of the Hankel transform Ur toHHCr byputting

Ur = crHCr (ko).(5.23)

One of course still has U2r = Identity. See Theorem 5.7.

Remark 5.10. Note that if 0


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LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OFSl(2,R) 215Hence, by (5.25)

limn

0

n(x)Jr,y(x)dx=

0

(x)Jr,y(x)dx.(5.26)

But the left side of (5.26) is just Ur(n)(y), by definition of Ur. However, byTheorem 5.9,Ur(n) converges to Ur() in the r-Frechet topology ofHr . Butthen, by Proposition 4.7, Ur(n) certainly converges pointwise to Ur(). But thenUr()(y) is given by the right side of (5.26). QED

5.4. Let u gC. A vector HHCr will be called a u-weight vector of weightCifHCr (u)= . The span of the set of all such vectors will be referred toas the corresponding u-weight space, or simply weight space ifu is understood. Ifu= e (resp. u= f) and = 0, then the e-weight (resp. f-weight) vector will alsobe referred to as an e-Whittaker (resp. f-Whittaker) vector. An e-weight (resp.f-weight) vector of weight 0 will be called a highest (resp. lowest) weight vector.

Remark 5.12. The term highest and lowest is a convenient misnomer. It is a mis-nomer since it refers, respectively to the kernels ofHCr (e) and

HCr (f) and not,

as in finite dimensional representation theory, to h-weights. Note that, as a con-sequence of (5.23), HHCr is a u-weight vector of weight if and only if itsHankel transform Ur() is an Ad ko(u)-weight vector of weight . In particular, ifu= h, then, by (5.19), is anh-weight vector of weight if and only if its HankeltransformUr() is anh-weight vector of weight. Furthermore, again by (5.19), is an e-weight vector of weight if and only if its Hankel transform Ur() is anf-weight vector of weight.

Now ifugC, we can writeu= (u)h+ (u)e+ (u)f

for some unique (u), (u), (u)C. Let g(

)C ={ugC|(u)= 0}.

Theorem 5.13. Letug()C

. Then for any C theu-weight space, inHHCr ,of weight is 1-dimensional. Furthermoreh,e,f g()

C so that in particular for

u= h,e,f theu-weight space, inHHCr , of weight is 1-dimensional.Proof. Recalling the definition of the-operation in gC (see (3.1)) it is clear from(2.5), (2.6) and (2.7) that (e) =f, (h) =h and (f) =e. It then followsimmediately that(u) =(u). Hence,

vg()C

(v)= 0(5.27)However, it is immediate from (2.5), (2.6) and (2.7) that

h= i(e f),e=

1

2(ih+ e+ f),

f=1

2(ih+ e+ f),

so that h,e,f g()C

. Let C and let v = u. To prove the theorem itclearly suffices to prove that (recalling (4.36)) if Y = Dr(v)(HHCr ), then Y has



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216 BERTRAM KOSTANT

codimension 1 inHHCr . For mZ+ letXm be the span of{(r)n}, n= 0, 1,...,m.But now it is immediate from (2.26), (2.27), (2.28) and (5.27) that

(a) Ker Dr(v)|HHCr = 0,

(b)X0Y,(c) Dr(v)(Xm) Xm+1.

(5.28)

It follows from (b) of (5.28) that Y =HHCr . But dimXm = m+ 1. HencedimDr(v)(Xm) = m + 1 by (a) of (5.28). But then

Xm = X0 Dr(v)(Xm1)(5.29)by (b) and (c) of (5.28). But then clearlyHHCr =X0 Y. QEDRemark 5.14. A Weyl group of G has order 2. Note that Theorem 5.13 is nota contradiction of Theorem 6.8.1 on p. 182 of [Ko] (which, in this case, assertsthe existence of a 2-dimensional space of algebraic Whittaker vectors). Indeed the

hypothesis of this result in [Ko]requires that one must add, toHHCr , the Harish-Chandra module of the anti-holomorphic representation corresponding to r. See(6.8.1) in [Ko].

An element HHCr will be called r-tempered in case Hr . We nowobtain explicit results on the temperedness of certain Whittaker vectors. For =iy, y (0, ) the corresponding f-Whittaker vector is r-tempered and isgiven in terms of the Bessel function Jr.

Theorem 5.15. Lety (0, ) so thaty Hr wherey is the Dirac measureat y (see Remark 4.12). Then Ry is the weight space of e-Whittaker vectors ofweight iy. In particular, any such Whittaker vector isr-tempered.

The functionJr,y(x) = Jr(2

yx) is inHr by Proposition 5.2. Furthermore,RJr,y is the weight space off-Whittaker vectors of weight

iy. In particular, any

such Whittaker vector isr-tempered. Furthermore,

Ur(y) = Jr,y(5.30)

where, we recallUr is the Hankel transform.

Proof. Let Hr . Then (see (4.36)){, r (e)y}={ix, y}

=iy(y)={,iyy}.

This provesr (e)y = iyy.Now,

r (f)Ur(y) =iyUr(y)(5.31)by Remark 5.14. On the other hand, since Ur has order 2, it follows from (5.23)and (5.24) that

{, Ur(y)}={Ur(), y}={, Jr,y},

This proves (5.30). The theorem then follows from (5.31). QED



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LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OFSl(2,R) 2175.5. Let U(g) be as in (4.52). Using the notation of (4.50) we may, by (3.8),define a quadradic polynomial qon Rsuch that for all nZ,

r ( 1)((r)

n ) = q(n)

(r)

n .(5.32)Now let Hr and let{an} be, as in (4.50), the Fourier-Laguerre coefficients of. Then, by (4.53), the Fourier-Laguerre expansion ofr (( 1)m)() is givenby

r (( 1)m)() =n=0

q(n)man(r)n(5.33)

for any m Z+. Since this is an expansion of an element inH, the sum of normsquares of the coefficents is finite. In particular, the norm of the coefficents isbounded. On the other hand, the coefficents ofqare positive, since r >1. See(3.8). Thus for all mZ+ there exists a positive constantCm such that for all n

|an|< Cm(n2 + 1)m.(5.34)Now for any function on (0, ) letbe the function on (0, ) defined by putting(x) = x r2 (x). In case the limit of(x) exists asx tends to 0, we will put(0) = lim

x0(x).(5.35)

In such a case if is continuous on (0, ), then of courseis continuous on [0, ).If HHCr , then is clearly the restriction to (0, ) of an entire function andhence(0) exists and is continuous on [0, ).Theorem 5.16. Let Hr . Then(0) exists. Furthermore, for any > 0,j converges to pointwise and uniformly on the closed interval [0, ], using thenotation of (4.51).

Proof. In the proof of Proposition 3.12 we noted that ifL(r)n (Szego) is the normal-

ization ofL(r)n as given in (5.1.5), p. 97 of [Sz], then

L(r)n (Szego) = (1)n1

n!L(r)n .(5.36)

It follows then from (7.6.11), p. 173 in [Sz]that there exists a >12 and positiveconstantsCand D such that for all x[0, ] andnZ+,

|L(r)n (2x)|< n! (Cna + D).But then certainly

|e x2 L(r)n (2x)|< n! (Cna + D)(5.37)for all n Z+ and x [0, ]. Let dn = ( 12r+1 n! (n+ r + 1))

12 . Thenn! dn =

(2r+1 n!(n+r+1) )12 . But

(n + r+ 1) = (n + r)(n+ r 1) (r+ 2)(r+ 2)

= (n + r+ 1)(n + r) (r+ 2)((r+ 2)n + r+ 1

).

But (n+ r+ 1)(n+ r) (r+ 2)> n!. Hence

n! dn 0. Thenx (uniquely up to a scalar multiple) is anh-weight vector ofweight. In particular, x

r2 is a smoothh-weight vector of weightr+ 1.

On the other hand, Ur(x) is a smoothh-weight of weight

forRe


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LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OFSl(2,R) 221and conclude that t xkr(exp (t)h)((x)) also defines a differentiable mapfrom RtoH. Furthermore, one must have

d

dt (xkr(exp (t)h)())t=0 = xk((2x

d

dx + 1))()(5.53)

inH since the difference quotient whose limit inH defines the left side of (5.53)clearly converges pointwise to the right side of (5.53), by (4.25) and the smoothnessof. But by a change of variables one has

{r(exp (t)h)(), x}=

0

(e2tx)xetdt

={, et(2+1)x}.Hence,

d

dt({r(exp (t)h)(), x})t=0={, (2 + 1)x}.(5.54)

For any Hr and i = 1, 2, let{, xi

}i = biai (x)xidx where a1 = 0, b1 =1, a2 = 1, b2 =, Re 1 > 12 and 2 C is arbitrary. Clearly,{, x} ={, x}1+ {, x}2. But now

d

dt({r(exp (t)h)(), x}1)t=0 ={(2xd

dx+ 1)(), x}1(5.55)

by (5.53) where we putk = 0. On the other hand, if we choosekso thatxk|[1, )L2((1, ), dx), then by (5.53),

d

dt({r(exp (t)h)(), x}2)t=0 = d

dt({xkr(exp (t)h)(), xk}2)t=0

={xk(2x ddx

+ 1)(), xk}2

={

(2xd

dx+ 1)(), x

}2.

(5.56)

But then by (5.54), (5.55) and (5.56) one has

{(2xddx

+ 1)(), x}={, (2 + 1)x}.But then r (h)(x

) = (2+ 1)x by (5.52). The last statements follow fromRemark 5.12 and the fact that Ur stabilizesHr . See (5.23). QEDRemark 5.21. It is clear in Theorem 5.20 that the h-weight vectors for weights whenRe is positive is independent ofr. On the other hand, there is an apparentdependence onr whenRe is negative. Sincer,0 has an obvious dependence on rthe dependence of the h-weight vector of weight(r+ 1), on r, will be verified inTheorem 5.23.

From the commutation relations in g it is obvious that highest weight vectorr,0and the lowest weight vectorJr,0 must beh-weight vectors. The question is: whatare the h-weights? This is determined in Theorem 5.23 below. Contrary to onesexperience for finite dimensional representations, it turns out that the h-weight ofthe highest weight vector (recall definitions in5.4) is smaller than the h-weightof the lowest weight vector. In fact, the latter is positive and the former is justits negative. The lowest weight vectorJr,0 is given in Theorem 5.19 as a limit. InTheorem 5.23 it will be given as an integral.



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222 BERTRAM KOSTANT

Remark 5.22. By the multiplicity 1 statement of Theorem 5.13 and the commuta-tion relations in g note that (up to scalar multiplication) r,0 is the only nonzeroelement in

HHCr , and a fortiori in

Hr , which is simultaneously an h-weight

vector and an e-weight vector. The same is true ofJr,0 when f replaces e.

Theorem 5.23. One has

r (h)(r,0) =(r+ 1)(r,0)(5.57)and

r (h)(Jr,0) = (r+ 1)(Jr,0).(5.58)

Furthermore,Jr,0 given by (5.51), can also be given by

{, Jr,0}= 1(r+ 1)

0

(x)xr2 dx(5.59)

for any Hr so that

Ur(r,0) = 1(r+ 1)

x r2 .(5.60)

Proof. From the commutation relations 2r (e) = [r (h),

r (e)] and2r (f) =

[r (h), r (f)] together with the multiplicity 1 statement of Theorem 5.13 it is

immediate that there exists scalars, Csuch thatr (h)(r,0) = r,0,

r (h)(Jr,0) = Jr,0.

But then

=(5.61)by Remark 5.12. Thus, for any Hr one has

{(2xddx

+ 1)(), r,0}={, r (h)(r,0)}={, r,0}.

Hence,

limx0

xr2 ((2x d

dx+ 1)())(x) = lim

x0x

r2 (x).(5.62)

Now put = (r)0 so that (x) = e

xxr2 . Then x

r2 ((2x d

dx+ 1)())(x) =

(r+ 1)ex + 2exx. Thus the limit on the right side of (5.62) is(r+ 1). Butx

r2 (x) = ex. Hence, the limit on the right side of (5.62) is . This proves

=(r + 1). But then = (r + 1) by (5.61). But then by the multiplicity1 statement of Theorem 5.13 one must have Jr,0 = cx

r2 for some constant c by

Theorem 5.20. Thus

limy0

yr2 {, Jr,y}= c

0

(x)xr2 dx(5.63)

for any Hr . Now as above choose = (r)0 . But{(r)0 , Jr,y} = (r)0 (y) byTheorem 5.7. Since

(r)0 (y) = e

yyr2 one has

limy0

ey =c

0

exxrdx.(5.64)



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LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OFSl(2,R) 223But the left side of (5.64) is 1 and, by definition of the -function, the right side isc(r+ 1). Hence c = 1(r+1) . QED

Recall the mapr :Hr Dist((0, )). See (4.40).Theorem 5.24. The subspaceCo ((0, )) is not dense inHr with respect to ther-Frechet topology. In fact, the highest weight vector, r,0, and the Category-Omodule, r (U(g))(r,0), it generates, vanishes onC

o ((0, )). That is,

r (U(g))(r,0) Kerr.(5.65)Furthermore, (see (4.37)) the injection mapHr C((0, )) is continuous

but is not a homeomorphism onto its image whereHr has ther-Frechet topologyandC((0, )) =E((0, ))has the Frechet topology of distribution theory.Proof. It is obvious from (5.45) that r,0 Ker r. But this implies (5.65) sinceCo ((0, )) is stable under r (U(g)) by (4.36). The continuity of (4.37) is estab-lished in Proposition 4.7. It is not a homeomorphism onto its image since, as oneknows, Co ((0, )) is dense inE((0, )) with respect to the distribution theoryFrechet topology onE((0, )). QEDRemark 5.25. In contrast to the highest weight vector, r,0, note that, by (5.59),the lowest weight vector, Jr,0, is not in the kernel ofr. See (4.40).

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224 BERTRAM KOSTANT

[Wt] G. Watson,Theory of Bessel Functions, Cambridge Univ. Press, 1966.

[Yo] K. Yosida, Functional Analysis, Grundlehren, 123, Springer-Verlag, 1971.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139

E-mail address: [email protected]

Laguerre Bessel

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