ON LAGUERRE POLYNOMIALS, BESSEL FUNCTIONS, HANKEL TRANSFORM and a SERIES in the UNITARY DUAL of the SIMPLY-CONNECTED COVERING GROUP of Sl(2, R)

REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 4, Pages 181–224 (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)thereis a continuous family {πr}, −1 Hilbert space is L2((0, ∞)). First of all one exhibits a representation, Dr, of g = Lie G by second order diﬀerential operators on C∞((0, ∞)). For x ∈ (r) −x r (r) (r) (0, ∞), −1 Bessel function, then for any y (0, ), the function Jr,y (x)=Jr(2 xy) is a Whittaker vector. Other weight vectors are given and the highest weight vector is given by a limiting behavior at 0.

0. Introduction 0.1. Throughout this paper r is a real number where −1

Received by the editors December 2, 1999 and, in revised form, January 21, 2000. 2000 Mathematics Subject Classiﬁcation. Primary 22D10, 22E70, 33Cxx, 33C10, 33C45, 42C05, 43-xx, 43A65. Research supported in part by NSF grant DMS-9625941 and in part by the KG&G Foundation.

c 2000 American Mathematical Society 181 182 BERTRAM KOSTANT

Let g = Lie Sl(2, R)andletU(g) be the universal enveloping algebra over g with coefficients in C.Let{h, e, f}⊂g be the S-triple basis of g where 10 0 −1 00 h = ,e= ,f= . 0 −1 00 −10 Let G be a simply-connected Lie group with Lie algebra g. The center Z of G is infinite cyclic and one identifies Sl(2, R)withG/Z2. The metaplectic group, Mp(2, R), the double cover of Sl(2, R), is identified with G/Z4. Conjugation of g −10 by 01 defines an outer automorphism κ of G. Let L be a Hilbert space and assume that π : G → U(L) is an irreducible unitary representation of G.LetL∞ ⊂Lbe the dense subspace of infinitely differentiable vectors in L. From the general theory of unitary representations the “differential” of π induces a representation π∞ : U(g) → End L∞. The group G is not in the Harish-Chandra category because Z is infinite. In particular, even though the element e − f ∈ g is elliptic, the subgroup K of G corresponding to R(e − f) is not compact. Nevertheless, much of the Harish-Chandra theory is still valid. In particular, if h0 = −i(e − f), then the linear span, LHC,of the π∞(h0) eigenvectors in L∞ is a dense U(g)-submodule of L∞. We refer to LHC as the Harish-Chandra module associated to π. We will say that π is one-sided if the spectrum of π∞(h0)|LHC is either a set of positive numbers or a set of negative numbers. To distinguish between these two cases, and adopting terminology which is valid for the discrete representations of Sl(2, R), we will say that π is holomorphic (resp. anti-holomorphic) if the spectrum of π∞(h0)|LHC is a set of positive (resp. negative) numbers. It is immediate that π is holomorphic if and only if πκ is anti-holomorphic, where for g ∈ G, πκ(g)=π(κ(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-sided representations are exactly the members of the second series. We are concerned, in the present paper, with a construction, and resulting harmonic analysis, of a model, using Laguerre functions, for the members of this second series. Using κ it is enough to give the model for the members of the holomorphic subfamily. Due − to a sign difference the members of this subfamily are denoted, in [Pu], by D` where `>0. In the present paper these representations will appear as πr where, as stated in the first sentence of the introduction, −1

0.2. Let Diff C∞((0, ∞)) be the algebra of all differential operators on (0, ∞). Let x ∈ C∞((0, ∞)) be the natural coordinate function. Identifying elements of C∞((0, ∞)) with the corresponding multiplication operators, it is easy to see that ∞ there is a representation, Dr : U(g) → Diff C ((0, ∞)) so that d D (h)=2x +1, r dx Dr(e)=ix, d2 d r2 D (f)=i(x + − ). r dx2 dx 4x LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 183

HHC { 0 0 0} Next, one finds that r is stable under Dr(U(g)). In fact, h ,e,f is an S- 0 1 − 0 1 triple in the complexification, gC,ofg,wheree = 2 ( ih+e+f),f = 2 (ih+e+f) and, as above, h0 = −i(e − f). When applied to the Laguerre functions one finds (see Theorem 2.6), for n ∈ Z+,

0 (r) (r) Dr(h )ϕn =(2n + r +1)ϕn , 0 (r) (r) (0.1) Dr(e )ϕn = iϕn+1, 0 (r) 2 (r) Dr(f )ϕn = i(nr + n )ϕn−1,

(r) 0 |HHC where we put ϕ−1 = 0. In particular, the spectrum of Dr(h ) r is positive. (r) r −x Furthermore, ϕ0 (x)=x 2 e spans the eigenspace for the minimal eigenvalue, 0 |HHC 0 (r) r +1,ofDr(h ) r and Dr(f )ϕ0 =0. ∈ |HHC For u g the operator Dr(u) r is formally skew-symmetric. But “formal” isn’t good enough since the Laguerre functions do not vanish as x approaches 0 for r ≤ 0. In fact, they become unbounded for r<0. Nevertheless, these operators are skew-symmetric. But much more is true. The hypotheses for Nelson’s beautiful |HHC − Theorem 5 in [Ne] are satisﬁed so Dr r can be exponentiated for all 1

Theorem 0.1. There exists a unique irreducible unitary representation πr : G → H |HHC { } U( ) whose Harish-Chandra module is Dr r . Furthermore, the family πr , − ∞ − 1 0.

The representation πr descends to Sl(2, R)whenr is an integer. In such a case πr deﬁnes a holomorphic discrete series representation of Sl(2, R)whenr is positive and π0 is often referred to as a limit of such series. The representation πr descends to the metaplectic group Mp(2, R)whenr ∈ Z/2. We will say πr is special if 0 < |r| < 1. Obviously the special representations occur in pairs, {πr,π−r}.

Remark 0.2. What seems to be particularly striking about a special pair {π−r,πr}, and what seems to be illustrative of the power of Nelson’s theorem, is that, whereas clearly Dr = D−r, one has the inequivalence of πr and π−r. The only special pair which descends to the metaplectic group Mp(2, R)is{π− 1 ,π1 }.Asonemight 2 2 suspect these two representations are the two irreducible components (even and odd) of the holomorphic metaplectic (or oscillator) representation. See Theorem 3.13. In a number of ways the special pairs can be regarded as a generalization of the metaplectic representation. For example, if one defines an equivalence relation by putting πr ∼ πr0 in case πr and πr0 have the same infinitesimal character, then the equivalence classes are the special pairs and the singlets {πr} where πr is not special. There is, however, an important distinction between the set of special representations where r is positive and the set where r is negative. Pukanszky in [Pu] shows that the former has positive Plancherel measure whereas the latter has zero Plancherel measure. H∞ ∞ H 0.3. Let r denote the space of C vectors in with respect to πr.One establishes ∞ ∞ ⊂H∞ ⊂ ∞ ∞ Co ((0, )) r C ((0, )) 184 BERTRAM KOSTANT where subscript o denotes compact support. Furthermore, ∞ |H∞ (0.2) πr = Dr r . See Propositions 4.6 and 4.7. As is standard in representation theory the action of H∞ U(g) induces a Fréchet topology on r which we will refer to as the πr-Fréchet topology. The πr-Fréchet topology is in fact strictly finer than the topology induced H∞ ∞ ∞ ∞ ∞ E ∞ on r by its inclusion in C ((0, )) when C ((0, )) = ((0, )) is given the ∞ ∞ familiar Fréchet topology of distribution theory. In fact, Co ((0, )) is not dense H∞ in r with respect to the πr-Fréchet topology. See Theorem 5.26. ∞ The representation πr can be defined on the convolution algebra, Disto(G), ∈ ∞ of distributions of compact support on G and for any ν Disto(G), πr (ν)is continuous with respect to the πr-Fréchet topology. See e.g. [Ca]. We recall that Disto(G) contains both G and U(g). Contragrediently, there is a Disto(G)-module H−∞ −∞ r with respect to a representation πr and one has inclusions H∞ ⊂H⊂H−∞ r r { } ∈H∞ ∈H−∞ together with a sesquilinear form ϕ, ρ for ϕ r and ρ r such that H∞ 7→ { } every continuous linear functional on r is uniquely of the form ϕ ϕ, ρ for ∈H−∞ ∞ −∞|H∞ ρ r . In addition, πr = πr r . There is a natural embedding of the ∞ ∞ H−∞ space, Disto((0, )), of distributions of compact support on (0, )in r .See H−∞ Proposition 4.11. In addition, Va embeds into r where a>0andVa is the space of all Borel measurable functions, ψ,on(0, ∞)whichareL2 on (0,a)and O(xk)on[a, ∞), for some k ∈ N. See Proposition 4.13. ∈ ∈H−∞ ∈ C Let u gC. Thenanelementρ r is called a u-weight vector of weight λ −∞ in case πr (u)(ρ)=λρ.Incaseu = h, e or f, we prove that the corresponding weight space is at most 1-dimensional. For a more general statement see Theorem 5.15. An e-weight (resp. f-weight) vector of ρ of weight λ will be referred to as an e-Whittaker (resp. f-Whittaker) vector in case λ =6 0. It will be referred to as a highest (resp. lowest) weight vector if λ = 0. Here we are adopting terminology from finite dimensional representation theory emphasizing e (resp. f)behaviorand not extremal h-weights. In fact, h-weights are unbounded. Theorem 5.17 asserts that for any y ∈ (0, ∞) the Dirac measure δy at y is (up to scalar multiplication) the unique e-Whittaker vector of weight iy.NowletJr(z) be the Bessel function ∈ ∞ ∈ ⊂H−∞ of order r. For√ any y (0, )letJr,y V1 r be defined by putting ∈H∞ Jr,y(x)=Jr(2 yx)sothatforanyϕ r , Z ∞ √ (0.3) {ϕ, Jr,y} = Jr(2 yx)ϕ(x)dx. 0 Then one has (see Theorem 5.17)

Theorem 0.3. The function Jr,y(x) is, up to scalar multiplication, the unique f- Whittaker vector of weight −iy.

Remark 0.4. If 0 < |r| < 1, then the Bessel functions Jr(z)andJ−r(z) are a basis of the space of solutions of the corresponding Bessel (2nd order diﬀerential) equation. One sees then that for the special pair {π−r,πr} of unitary representations both solutions become involved in the determination of f-Whittaker vectors. ∈ π − Let ko K be deﬁned by putting ko = exp 2 (e f). Let −∞ (0.4) Ur = crπr (ko) LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 185

− r+1 πi where cr = e 2 so that Ur|H is a unitary operator. It is immediate from the ﬁrst equation in (0.1) that (r) − n (r) (0.5) Ur(ϕn )=( 1) ϕn HHC so that Ur is of order 2 and stabilizes the Harish-Chandra√ module r .Onthe other hand, the operator with kernel function Jr(2 yx) is classical and is known as the Hankel transform (modulo a slight change in parameters). Furthermore, it is also classical that Z ∞ √ − n (r) (r) (0.6) ( 1) ϕn (y)= Jr(2 yx)ϕn (x)dx, 0 a result that Carl Herz in [He] says is implicit in the 19th century work of Sonine. The result (0.6) is stated as Exercise 21, p. 371 in [Sz] where a reference to the paper [Ha] of G.H. Hardy is given. The result is important for us since it implies

Theorem 0.5. Ur|H is the Hankel transform. For completeness we include a proof of (0.6) given to us by John Stalker. Remark 0.6. The domain of the Fourier transform is normally regarded to be the space of tempered distributions. It is not generally appreciated but, since it stabilizes the Harish-Chandra module of Hermite functions, it operates contragrediently on the algebraic dual of this module. This makes its domain much larger than the space of tempered distributions. The same statement is true of the Hankel H−∞ transform. Nevertheless, we only require here that it operates (see (0.4)) on r .

Let G1 be a semisimple Lie group whose corresponding symmetric space X is Hermitian. The holomorphic discrete series of G1 is normally constructed on a space of square integrable holomorphic sections of a line bundle on X.IfX is of tube type, then Ding and Gross in [DG] have recognized that the natural generalization of the Hankel transform is given by the action of an element in G1 and have used this fact in [DG] to transfer the discrete series, constructed on X, to another model built on a symmetric cone. Applied to the case considered here this constructs our representations πn,n=1, 2, .... This method, however, would not yield πr for r ≤ 0. As pointed out to us by David Vogan, for such values of r, πr cannot be realized on a space of square integrable holomorphic sections of a line bundle on X arising from the natural action of G1 on X.Forr rational this follows from Harish-Chandra’s classiﬁcation of the discrete series for a group with ﬁnite center.

0.4. Note that Ad ko deﬁnes the non-trivial element of the Weyl group of G with respect to the pair (Rh, g). That is, Ad ko(h)=−h. It follows that the Hankel transform Ur carries h-weight vectors of weight λ to h-weight vectors of weight −λ. The following result is stated in Theorem 5.22. ∈ C λ−1 µ ∈ Theorem 0.7. Assume λ where Re λ > 0.Letµ = 2 so that x ⊂H−∞ µ V1 r . Then, up to scalar multiplication, x is the unique h-weight vector of µ weight λ and Ur(x ) is the unique h-weight vector of weight −λ. Remark 0.8. Note for Re λ > 0theh-weight vector is independent of r whereas for Re λ < 0 there is an apparent dependence on r. This dependence is made explicit in Theorem 5.25 for the case where λ = −(r + 1). The situation when λ is purely imaginary will be explored elsewhere. 186 BERTRAM KOSTANT

Now note that Ad(k0)(e)=−f so that the Hankel transform Ur carries e-weight vectors of weight λ to f-weight vectors of weight −λ.Infact,fory ∈ (0, ∞)one has, for Whittaker vectors,

Ur(δy)=Jr,y . See Theorem 5.17. More interesting is the question about highest and lowest weight H−∞ H−∞ vectors if they exist at all in r . In fact, they do exist in r . The highest (resp. lowest) weight vector, given below, in Theorem 0.8, as δr,0 (resp. Jr,0)isthe H−∞ unique element (up to scalar multiplication) in r which is simultaneously an h-weight vector and an e-weight (resp. f-weight) vector. See Theorem 5.25. ∈H∞ ∈H∞ Theorem 0.9. There exists a unique element δr,0 r such that for any ϕ r the limit in (0.7) below exists and one has − r (0.7) {ϕ, δr,0} = lim x 2 ϕ(x). x→0

Moreover, δr,0 is, up to scalar multiplication, the unique highest weight vector (i.e. ∈ −∞ − δr,0 Kerπr (e)). Furthermore, δr,0 is also an h-weight vector of weight (r+1). The h-weight vector, Jr,0, of weight r +1 given by Z ∞ 1 r (0.8) {ϕ, Jr,0} = ϕ(x)x 2 dx Γ(r +1) 0 ∈H∞ for any ϕ r , (see Theorem 0.7) is, up to scalar multiplication, the unique ∈ −∞ lowest weight vector (i.e. Jr,0 Kerπr (f)). In addition, with respect to the Hankel transform, one has

(0.9) Ur(δr,0)=Jr,0. ∈ ∞ ∞ Remark 0.10. Note that (0.7) vanishes if ϕ Co ((0, )) thereby establishing that ∞ ∞ H∞ Co ((0, )) is not dense in r . 0.5. Following this introduction there are ﬁve chapters:

1. The representation Dr and the Casimir element 2. Laguerre functions and Harish-Chandra modules 3. The series, {πr}, −1

1. The representation Dr and the Casimir element

1.1. Let gC = Lie Sl(2, C)andletg = Lie Sl(2, R)sothatg is a real form of gC. Let {h, e, f}⊂g be the S-triple given by 10 0 −1 00 h = ,e= ,f= . 0 −1 00 −10 Consider the algebra Diff C∞((0, ∞)) of differential operators on the open interval (0, ∞). Let x ∈ C∞((0, ∞)) be the natural coordinate function and let H be the Hilbert space L2((0, ∞)) with respect to the usual measure defined by dx. Unless stated otherwise we will always assume r ∈ R where r>−1. Let ∞ Dr : gC → Diff C ((0, ∞)) be the linear map, where identifying elements in C∞((0, ∞)) with the corresponding multiplication operators,

Dr(e)=ix, d D (h)=2x +1, (1.1) r dx d2 d r2 D (f)=i(x + − ). r dx2 dx 4x d2 − d Noting that [x, x dx2 ]= 2x dx one readily establishes that Dr is a Lie algebra homomorphism. Thus if U(g) is the universal enveloping algebra of gC,thenDr ∞ extends to an algebra homomorphism Dr : U(g) → Diff C ((0, ∞)). Let Cas ∈ U(g) be the quadratic Casimir element corresponding to the Killing form on gC. Clearly, 1 h2 (1.2) Cas = ( + ef + fe). 4 2 Computation yields

Proposition 1.1. The diﬀerential operator Dr(Cas) is the scalar operator given by 1 (1.3) D (Cas)= (r2 − 1). r 8 Proof. We ﬁrst note that

2 d 2 Dr(h /2) = (2x +1) /2 (1.4) dx d d =2(x2( )2 +2x +1/4). dx dx Next, 2 2 d 2 d r d 2 d r Dr(ef + fe)=ix(i(x( ) + − )) + i(x( ) + − )ix (1.5) dx dx 4x dx dx 4x d d r2 = −2x2( )2 − 4x − 1+ . dx dx 2 But then (1.3) follows from (1.1), (1.2), (1.4) and (1.5). QED 188 BERTRAM KOSTANT

Let G 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| < 1. Later in this paper the representation Dr will lead to an irreducible unitary representation πr of G. The representation πr will depend on more than just Dr. In fact, for 0 < |r| < 1 the representations πr and π−r will be inequivalent whereas Dr and D−r are clearly the same. It is convenient to think of the parameter r as an analogue of dimension. In fact, if r ∈ N and νr is the ﬁnite dimensional irreducible representation of U(g) 1 2 − having dimension r,then 8 (r 1) is the scalar value of νr(Cas). Indeed, with the usual deﬁnition of ρ the “strange formula” of Freudenthal-de Vries asserts that | |2 1 | |2 dim g ρ = 8 because the formula asserts ρ = 24 andinourcasedim g =3. However, (r − 1)ρ is the highest weight of νr. But then the scalar value of νr(Cas) | |2 −| |2 1 2 − is rρ ρ = 8 (r 1).

2. Laguerre functions and Harish-Chandra modules (r) 2.1. Recall −1 monomials 1,x,x2, ..., with respect to the ﬁnite measure xre−xdx on (0, ∞). See §2, Chapter X in [Ja]. But then upon multiplication by the square root of the weighting factor xre−x it follows that for n =0, 1, 2..., n r − x (r) n − r x d r+n −x (2.1) x 2 e 2 L (x)=(−1) x 2 e 2 (x e ) n dxn is an orthogonal family of functions in H. If we replace x by 2x in (2.1) and divide r by 2 2 , it then follows that r − (r) 2 x (r) ϕn (x)=x e Ln (2x) n r −x n −r 2x d r+n −2x = x 2 e (−1) x e (x e ) (2.2) dxn n n − r x d r+n −2x =(−1) x 2 e (x e ) dxn is again an orthogonal basis of functions in H. We will refer to the sequence of (r) functions ϕn (x),n=0, 1, ..., as Laguerre functions (of type r).

(r) Proposition 2.1. The set of Laguerre functions {ϕn (x)},n=0, 1, ..., is an orthogonal basis of H.Furthermore, Z ∞ | (r) |2 1 (2.3) ϕn (x) dx = r+1 n!Γ(n + r +1). 0 2 r − x (r) Proof. By (5.7.1) in Theorem 5.7.1, p. 104, in [Sz] the functions x 2 e 2 Ln (x),n= 0, 1, ..., span a dense subspace of H and hence, in the Hilbert space sense, are an orthogonal basis of H. But then the same statement is true after scaling and replacing x by 2x. Hence the Laguerre functions are also an orthogonal basis of H. LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 189

Let ψn be the function defined by the left side of (2.1). Now by the computation of dn on the bottom of p. 185 in [Ja], one has Z ∞ 2 (2.4) |ψn(x)| dx = n!Γ(n + r +1). 0 (r) − r But then (2.3) follows from (2.4) since ϕn (x)=2 2 ψn(2x). QED 0 0 0 2.2. We now consider a new S-triple {h ,e,f}⊂gC defined by putting 1 −i {h0,e0,f0} = A{h, e, f}A−1 where A = √1 . One readily has 2 −i 1 0 0 i h = (2.5) −i 0 = −i(e − f). Next, 0 1 −i −1 e = 2 −1 i (2.6) 1 = (−ih + e + f). 2 Also, 0 i −1 f = −1 −i (2.7) 1 = (ih + e + f). 2 The element e−f is elliptic in g and hence h = −i(e−f) ∈ gC is hyperbolic. We ∞ are looking for a g-submodule of C ((0, ∞)), with respect to Dr, which is spanned 0 0 d 2 d r2 by eigenvectors of Dr(h ). By (2.5) Dr(h )=−(x( ) + − − x). Hence if ∞ 0 dx dx 4x ϕ ∈ C ((0, ∞)) is an eigenvector for Dr(h ) with eigenvalue λ we have (recalling (1.1)) the differential equation d d r2 (2.8) (x( )2 + − − x + λ)ϕ =0. dx dx 4x Lemma 2.2. For n =0, 1, ..., one has 0 (r) (r) (2.9) Dr(h )ϕn =(2n + r +1)ϕn . Proof. By (3) on p. 186 in [Ja] (or (5.1.2) on p. 96 in [Sz]) one has d d (2.10) (x( )2 +(r +1− x) + n)L(r)(x)=0. dx dx n If A is the operator on C∞((0, ∞)) defined by putting Af(x)=f(2x), then conjugating the differential operator in (2.10) by A and then multiplying by 2 yields the differential equation d d (2.11) (x( )2 +(r +1− 2x) +2n)L(r)(2x)=0. dx dx n Conjugating the differential operator in (2.11) by the multiplication operator M = −x r e x 2 then establishes the differential equation d d (2.12) M(x( )2 +(r +1− 2x) +2n)M −1ϕ(r) =0. dx dx n 190 BERTRAM KOSTANT

However, d d r (2.13) M M −1 = +(− +1) dx dx 2x so that d d r M(r +1− 2x) M −1 =(r +1− 2x) +(r +1− 2x)(− +1) dx dx 2x (2.14) d r2 r =(r +1− 2x) +2r +1− 2x − − . dx 2x 2x But (2.13) also implies d d r Mx( )2M −1 = x( +(− +1))2 dx dx 2x (2.15) d 2 d r r2 = x +(−r +2x) + + − r + x. dx dx 2x 4x But then adding (2.14), (2.15) and 2n yields d d d d r2 M(x( )2 +(r +1− 2x) +2n)M −1 = x( )2 + − − x +(2n + r +1). dx dx dx dx 4x (r) But then (2.9) follows from (2.12) and (2.8) with λ =2n+r+1 and ϕ = ϕn .QED Remark 2.3. With our assumptions on r and n note that the eigenvalue 2n + r +1 is always positive. HHC { (r)} ∈ Z 2.3. Let r be the linear span of the ϕn ,n +. By Proposition 2.1 it HHC H HHC follows that r is dense in . We wish to show that r is stable under the HHC 0 action of Dr(gC). By Lemma 2.2 r is clearly stable under Dr(h ). But now by (1.1), (2.6) and (2.7) one has i d2 d r2 (2.16) D (e0)= (x +(1− 2x) − − 1+x) r 2 dx2 dx 4x and i d2 d r2 (2.17) D (f 0)= (x +(1+2x) − +1+x). r 2 dx2 dx 4x 0 0 Remark 2.4. Note that Dr(e )andDr(f ) diﬀer only in the sign of 2 terms. Recall (2.2). We make the following computations: 2 n d (r) n − r x r r −1 d r+n −2x x ϕ (x)=(−1) x 2 e ((( )( +1)x − r + x) (x e ) dx2 n 2 2 dxn dn+1 (2.18) +(−r +2x) (xr+ne−2x) dxn+1 dn+2 + x (xr+ne−2x)). dxn+2 Next,

d (r) n − r x r −1 (1 ∓ 2x) ϕ (x)=(−1) x 2 e ((− x dx n 2 (2.19) dn dn+1 +1∓ (−r) ∓ 2x) (xr+ne−2x)(1 ∓ 2x) (xr+ne−2x)) dxn dxn+1 LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 191 and ﬁnally, 2 2 n r (r) n − r x r −1 d r+n −2x (2.20) (− ∓ 1+x)ϕ (x)=(−1) x 2 e ((− x ∓ 1+x) (x e )). 4x n 4 dxn It follows from (2.16) upon adding (2.18), (2.19) and (2.20) choosing the top signs that

0 (r) i n − r x d d n+1 r+n −2x (2.21) D (e )ϕ = (−1) x 2 e ((1 − r)+x )( ) x e . r n 2 dx dx But d dn+1 dn+1 (2.22) [x , ]=−(n +1) . dx dxn+1 dxn+1 Thus, d dn+1 dn+1 d dn+1 x (xr+ne−2x)= x (xr+ne−2x) − (n +1) (xr+ne−2x) dx dxn+1 dxn+1 dx dxn+1 dn+1 = ((r + n)xr+n − 2xr+n+1 − (n +1)xr+n)e−2x dxn+1 dn+1 dn+1 =(r − 1) (xr+ne−2x) − 2 (xr+n+1e−2x). dxn+1 dxn+1 But then by (2.2) and (2.21),

n+1 0 (r) n+1 − r x d r+n+1 −2x δr(e )ϕ (x)=i(−1) x 2 e (x e ) (2.23) n dxn+1 (r) = iϕn+1. We have proved HHC 0 Lemma 2.5. r is stable under Dr(e ).Infact,

0 (r) (r) (2.24) Dr(e )ϕn = iϕn+1. Now from (2.17) and adding (2.18), (2.19) and (2.20) choosing the bottom signs and taking n = 0 one has

0 (r) i − r x d d 2 r −2x D (f )ϕ (x)= x 2 e ((−2r +2+4x)+(1− r +4x) + x( ) )x e . r 0 2 dx dx But it is straightforward to verify that xre−2x satisﬁes the diﬀerential equation d d (x( )2 +(1− r +4x) )+(−2r +2+4x))xre−2x =0 dx dx and hence,

0 (r) (2.25) Dr(f )ϕ0 =0. Let k = R(e−f)=iRh0 and let K ⊂ G be the subgroup corresponding to k.Since the center G is not ﬁnite, the group G is not in the Harish-Chandra class and K is not compact. Accordingly, we will modify the deﬁnition of a Harish-Chandra module to suit the case at hand. In this paper a module for U(g) will be called a Harish-Chandra module if it is spanned by 1-dimensional submodules under the action of k. Partly summarizing some of the results above one has 192 BERTRAM KOSTANT

− ∞ HHC H Theorem 2.6. As always 1

0 (r) r −x In particular, the spectrum of Dr(h ) is positive. Furthermore, ϕ0 (x)=x 2 e spans the “minimal k-type” and 0 (r) (2.27) Dr(f )ϕ0 =0. Moreover, for n>0 one has 0 (r) 2 (r) (2.28) Dr(f )ϕn = i(nr + n )ϕn−1. Proof. The equations (2.26) have been previously established. See Lemmas 2.2 and 2.5. Also (2.27) is just (2.25). We will prove (2.28) by induction on n.Theresult (r) has actually been established for n = 0 by putting ϕ−1 = 0. Inductively, assume (2.28) for n.Then (r) 0 (r) (2n + r +1)ϕn = Dr(h )ϕn 0 0 − 0 0 (r) =(Dr(e )Dr(f ) Dr(f )Dr(e ))ϕn 2 0 (r) − 0 (r) = i(nr + n )Dr(e )ϕn−1 iDr(f )ϕn+1 − 2 (r) − 0 (r) = (nr + n )ϕn iDr(f )ϕn+1. Thus, 0 (r) 2 (r) Dr(f )ϕn+1 = i((2n + r +1)+nr + n )ϕn 2 (r) = i((n +1)r +(n +1) ))ϕn . HHC This establishes (2.28) for all n.Thus r is stable under Dr(gC). It is then clearly 0 a Harish-Chandra module for g where all the eigenvalues of Dr(h )arepositiveand have multiplicity 1. But since nr + n2 = n(r + 1) is positive for all n =1, ..., it HHC (r) follows from (2.28) that any nonzero gC-submodule of r must contain ϕ0 and HHC hence, by the second equation in (2.26), must be equal to r . This establishes irreducibility. QED

3. The series, {πr}, −1

∞ ∞ ∞ L and L ⊂ Domπ(u) for any u ∈ g.Furthermore,ifπ (u)=πe(u)|L ,then L∞ is stable under π∞(u)andπ∞ deﬁnes a Lie algebra representation of g on L∞. In addition, π∞(u) is essentially skew-adjoint. See [Se]. The representation π∞ ∞ ∞ extends to a representation π : U(g) → EndL . It follows easily that if Lo ⊂L is a subspace such that Lo ⊂ Domπ(vi) for a basis {vi} of g and Lo is stable under ∞ ∞ all πe(vi), then Lo is a U(g)-submodule of L with respect to π .NowPukanszky, in [Pu], observes that there exists a character χ on K such that χπ|K descends to a representation of the circle group. Hence, if we let Lo be the span of the eigenvectors of πe(e − f), then Lo is dense in L. He then proves (see pp. 99 and 100 in [Pu]) that, for a basis {vi} of g, Lo ⊂ Domπ(vi)andLo is stable under πe(vi). ∞ HC Thus Lo is a U(g) submodule of L . Consequently, if we deﬁne L to be the ∞ 0 HC HC span of the eigenvectors of π (h ), then one has L = Lo.Inparticular,L is a Harish-Chandra module for U(g) with respect to π∞ and we refer to LHC as the Harish-Chandra module associated to π. Pukanszky lists three families of irreducible unitary representations. From the ∞ list one readily notes that π (Cas)=Cπ for a scalar Cπ and if we denote the ∞ 0 HC spectrum of π (h )|L by Sπ,theneachλ ∈ Sπ has multiplicity 1. Furthermore by inspection one has

Proposition 3.3. Let πi,i=1, 2, be two irreducible unitary representations of G. 1 2 Then π is equivalent to π if and only if there exists λi ∈ Sπi such that, as pairs,

(3.3) (Cπ1 ,λ1)=(Cπ2 ,λ2). In particular, any irreducuble unitary representation of G is determined by its Harish-Chandra module.

To align the notation of [Pu] with that used here one notes that if H0 and the Casimir eigenvalue q aredeﬁnedasin[Pu],then 1 (3.4) H |LHC = − π∞(h0)|LHC 0 2 and

(3.5) q = −2Cπ.

The members of the second family (denoted by II) of irreducible unitary representations are characterized by the property that the numbers in Sπ are either all positive or all negative. Correspondingly, this family, respectively, breaks up into 2 − + ∞ − series, D` and D` both for 0 <`< (note the sign change in (3.4)). For π = D` one has

Sπ = {2`, 2` +2, 2` +4, ...} and 1 (3.6) C = `(` − 1). π 2 Although it is an abuse of terminology, for suggestive reasons, we will refer to { −} ∞ the set of representations, D` ,for0<`< , as the holomorphic series of representations of G. LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 195

3.3. Let u1 = e − f,u2 = e + f and u3 = h so that {ui}, 1=1, 2, 3, is a basis of g. Moreover, the basis is an orthogonal basis with respect to the Killing form. − 2 2 2 2 − 0 2 Clearly, 8 Cas = u1 + u2 + u3.Butu1 = (h ) by (2.5) so that X3 − 0 2 2 (3.7) 8 Cas 2(h ) = ui . i=1 On the other hand, − 0 2 (r) 2 − − 2 (r) Dr(8 Cas 2(h ) )ϕn =((r 1) 2(2n + r +1) )ϕn (3.8) − (r) = (8n(n + r +1)+(r +3)(r +1))ϕn by Lemma 1.1 and (2.9). The power of Nelson’s theorem on analytic vectors enables |HHC H ∞ us to exponentiate Dr(g) r to a unitary representation of G on = L2((0, )). Theorem 3.4. For any −1

Proof. Let ui,i=1, 2, 3 be the basis of g deﬁned in (3.7). By (3.7)P and (3.8) the { (r)} H 3 2 elements of the orthogonal basis, ϕn ,of are eigenvectorsP of Dr( i=1 ui )and 3 2 |HHC the eigenvalues are real. It is immediate then that Dr( i=1 ui ) r is essentially |HHC ∈ self-adjoint. On the other hand, Dr(u) r is skew-symmetic for any u g by Lemma 3.2. It follows then from Theorem 5, p. 602 in [Ne], that there exists a H § HHC unitary representation, πr of G on such that, in the notation of 3.2, r is in the domain of πfr(u) for all u ∈ g and f |HHC |HHC (3.9) πr(u) r = Dr(u) r . 0 But now u1 = ih by (2.5) and hence by (2.9) and (3.9) the elements of the orthog- (r) onal basis, {ϕn },ofH are eigenvectors of πfr(u1) with pure imaginary eigenvalues. f |HHC HHC But then πr(u1) r is essentially skew-adjoint. Hence r is clearly the Harish- ∈ HHC Chandra module of πr and, by (3.9), the action of u g on r is given by |HHC Dr(u) r . 1 Let H be any nonzero closed subspace of H which is stable under πr(G). Using the spectral theorem for πfr(u1) it follows that there exists some n ∈ Z+ such (r) 1 (r) 1 that ϕn ∈H . But then using (3.9) it follows that Dr(U(g))(ϕn ) ⊂H.Thus HHC ⊂H1 H1 H r by Theorem 2.6. Hence = .Thusπr is irreducible. But now if we deﬁne ` by (3.10) r =2` − 1, ∞ { } then ` is arbitrary in the interval (0, ). Furthermore, Sπr = 2`, 2` +2, 2` +4, ... 1 2 − 1 2 − 1 − by (2.26). But Cπr = 8 (r 1) by (1.3). But 8 (r 1) = 2 `(` 1). Thus, − recalling (3.5), (3.6) and Proposition 3.3, πr is equivalent to D` in the notation of [Pu]. QED

Let Q be the set of representations {πr} where 0 < |r| < 1. A representation πr will be called special if πr ∈ Q. It is clear from (1.3) that πr is special if and only if the inﬁnitesimal character of πr takes a negative value on Cas.Itmaybenoted from Pukanszky’s list (see p. 102 in [Pu]) that this inﬁnitesimal character is shared 196 BERTRAM KOSTANT

τ τ only by elements in the family (Pukanszky’s notation) Eq . The family Eq is the analogue for G of the complementary series of Sl(2, R). Obviously, Q is a union of pairs [ (3.11) Q = {π−r,πr}. 0

Q = Q− ∪ Q+ { } − { −} 1 where Q− = πr , 1

Remark 3.6. The subset, Q−, of the unitary dual of G diﬀers in a fundamental way from the subset Q+. As established by Pukanszky, Q− has zero Plancherel measure whereas Q+ has positive Plancherel measure. See [Pu], formula (2.19), p. 117. See also the bottom of p. 102 in [Pu]. 3.4. Let exp : g → G be the exponential map for G.SinceG is simply-connected, one knows that (3.12) R → K, t 7→ exp t(e − f) is a diﬀeomorphism. Furthermore, if Z = CentG,thenZ ⊂ K.SinceZ is the 0 ∼ kernel of the adjoint representation and (see (2.5)) e−f = ih , it follows that Z = Z and (3.13) Z = {exp mπ(e − f) | m ∈ Z}.

Return to the representation πr. For any element exp t(e − f) ∈ K one has − (r) it(2n+r+1) (r) (3.14) πr(exp t(e f))ϕn = e ϕn by (2.26). The scalar value on H of any central element is then given by

imπ(r+1) (3.15) πr(exp mπ(e − f)) = e I. We have proved

Proposition 3.7. For any m ∈ Z one has exp mπ(e − f) ∈ Kerπr if and only if m(r +1)∈ 2Z. For k =1, 2, ..., clearly G/Zk is the k-fold covering of the adjoint group PSl(2, R). As an immediate consequence of Proposition 3.7 one has

Proposition 3.8. The unitary representation πr descends to a unitary represen- k r+1 ∈ Z tation of G/Z if and only if k( 2 ) . LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 197

Three cases stand out, namely, when k =1, 2, 4. We may identify PSl(2, R)=G/Z, (3.16) Sl(2, R)=G/Z2, Mp(2, R)=G/Z4, where Mp(2, R) is the 2-fold covering of Sl(2, R), the so-called metaplectic group. But then Proposition 3.8 implies

Proposition 3.9. The unitary representation πr descends to PSl(2, R) if and only if r is an odd integer. It descends to Sl(2, R) if and only if r is an integer. From the spectrum (see (2.26)), in the latter case, πr is a holomorphic discrete series representation if r =1, 2, ... and is the so-called limit of such series when r =0. The unitary representation πr descends to the metaplectic group Mp(2, R) if and ∈ 1 Z only if r 2 .

3.5. It is clear from Proposition 3.9 that {π− 1 ,π1 } is the only special pair for the 2 2 metaplectic group Mp(2, R). From the spectrum (2.26) it is clear that these representations are the two components of the “holomorphic” metaplectic representation µ of Mp(2, R)onL2(R). Recall µ = µeven ⊕ µodd where µeven (resp. µodd)isthe subrepresentation of µ defined by the even (resp. odd) square integrable functions ∼ on R. In this section we will be explicit about the equivalences π− 1 = µeven and ∼ 2 π 1 = µodd. 2 We recall some well-known properties of the metaplectic representation. Let w ∈ C∞(R) denote the linear coordinate function. Then ∞ Dµ : U(g) → Diff C (R) is a representaion of U(g) by differential operators where i D (e)= w2, µ 2 d 1 (3.17) D (h)=w + , µ dw 2 i d D (f)= ( )2. µ 2 dw

Let the Hermite polynomials Hm(w),m=0, 1, ..., on R be deﬁned as in [Sz], p. 101 (not [Ja]). Thus these polynomials are the Gram-Schmidt orthogonalization 2 of the monomials 1,w,w2, ..., with respect to the measure e−w dw on R.Itthen − w2 follows from (5.7.2) in Theorem 5.7.1, p. 104 in [Sz] that ψm(w)=e 2 Hm(w), for m =0, 1, ..., deﬁne an orthogonal basis of the Hilbert space L = L2(R) with respect to Lebesgue measure dw. But now by (5.5.2), p. 102 in [Sz] one has (recalling (2.5)) 1 (3.18) D (h0)ψ =(m + )ψ . µ m 2 m LHC LHC For m =0, 1, ..., let even be the span ψ2m and let odd be the span of ψ2m+1.Let LHC LHC ⊕LHC = even odd . Now by the second equation in (5.5.10) in [Sz], p. 102 one has d (3.19) (− +2w)H = H . dw m m+1 198 BERTRAM KOSTANT

But then this readily implies (“creation operator”) d (3.20) (− + w)ψ = ψ . dw m m+1 Squaring the operator on the left side of (3.20) implies d d (3.21) ( )2 − 2w + w2 − 1)ψ = ψ . dw dw m m+2 Lemma 3.10. For m =0, 1, ..., one has i (3.22) D (e0)ψ = ψ . µ m 4 m+2 Proof. It is immediate from (2.6) and (3.7) that the diﬀerential operator on the left 4 0 side of (3.21) equals i Dµ(e ). But then (3.22) follows from (3.21). QED Squaring the “annihilation operator” yields Lemma 3.11. One has 0 0=Dµ(f )ψ0 (3.23) 0 = Dµ(f )ψ1.

− w2 Proof. Obviously, ψ0(w)=e 2 .But

d − w2 − w2 (3.24) (e 2 )=−w(e 2 ). dw Hence, d (3.25) ( + w)ψ =0. dw 0 − w2 But (3.20) and (3.24) imply that ψ1(w)=2we 2 .Butthen d (3.26) ( + w)ψ =2ψ . dw 1 0 d 2 d 2 d 2 d 2 Hence, ( dw + w) annihilates ψ0 and ψ1.But(dw + w) =(dw ) +2w dw + w +1. d 2 2 0 But then by (2.7) one has ( dw + w) = i Dµ(f ). This implies (3.23). QED The argument establishing (2.28) in the proof of Theorem 2.6 when applied in the present case easily yields 0 (3.27) Dµ(f )ψm = im(m − 1)ψm−2 LHC LHC for m =2, 3, 4, ... . It follows then from (3.18), (3.22) and (3.27) that even and odd are both stable and irreducible under Dµ(U(g)). Next note that, by (3.17), Dµ(u) is manifestly skew-symmetric for any u ∈ g. In addition, one readily computes that 3 (3.28) D (Cas)=− . µ 32 P 3 2 |LHC Consequently, as in the proof of Theorem 3.4 one has that Dµ( i=1 ui ) is HC essentially self-adjoint. Furthermore, Dµ(u1)|L is essentially skew-adjoint by L L LHC LHC L (3.18). Let even (resp. odd) be the closure of even (resp. odd )sothat even (resp. Lodd) is the subspace of even (resp. odd) functions in L = L2(R)andof course

L = Leven ⊕Lodd. LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 199

As in the proof of Theorem 3.4, Nelson’s Theorem 5, p. 602 in [Ne], applies and now recovers (without needing the intervention of the Heisenberg group) the following well-known result about the metaplectic representation.

HC Proposition 3.12. The representation Dµ|L exponentiates to deﬁne a unitary representation µ of G on L = L2(R).Furthermore,µ descends to the metaplectic group Mp(2, R). Moreover, Leven and Lodd are stable subspaces and respectively deﬁne irreducible unitary representations µeven and µodd of Mp(2, R).Finally, LHC LHC even (resp. odd ) is the Harish-Chandra module of µeven (resp. µodd) where any ∈ |LHC |LHC u gC operates as Dµ(u) even (resp. Dµ(u) odd ). Proof. The only matter to be checked is the descent to Mp(2, R). Let g ∈ Z,using the notation of (3.13), so that g = exp kπ(e − f)fork ∈ Z.Butthen

ikπ(m+ 1 ) (3.29) µ(g)ψm = e 2 ψm by (3.18). It follows then that g ∈ Kerµ if and only if g ∈ Z4. But then the result follows from (3.16). QED

(r) Our deﬁnition of Laguerre polynomials Ln is taken from [Ja] and not from [Sz]. (r) Denoting the latter by Ln (Szego) it follows from a comparison of (1), p. 184 in [Ja] and (5.1.5), p. 97 in [Sz] that 1 (3.30) L(r)(Szego)=(−1)m L(r). m m! m But then equations (5.6.1), p. 102 in [Sz], writing Hermite polynomials in terms of Laguerre polynomials, simpliﬁes to − 1 2m ( 2 ) 2 H2m(w)=2 Lm (w ), 1 2m+1 ( 2 ) 2 H2m+1(w)=2 wLm (w ). But then

w2 − 1 2m − ( 2 ) 2 ψ2m(w)=2 e 2 Lm (w ), (3.31) w2 1 2m+1 − ( 2 ) 2 ψ2m+1(w)=2 we 2 Lm (w ). But on the other hand, the maps

ηeven : L2((0, ∞)) →Leven,

ηodd : L2((0, ∞)) →Lodd are manifestly unitary isomorphisms where, for ϕ ∈ L2((0, ∞)), 2 w 1 w (η ϕ)(w)=| | 2 ϕ( ), even 2 2 (3.32) 2 w 1 w (η ϕ)(w)=| | 2 ϕ( ) sign w. odd 2 2 But by deﬁnition

− 1 1 − 1 ( 2 ) − −x ( 2 ) ϕm (x)=x 4 e Lm (2x),

1 1 1 ( 2 ) −x ( 2 ) ϕm (x)=x 4 e Lm (2x). 200 BERTRAM KOSTANT

But then by (3.31) and (3.32)

1 2 1 (− ) w 1 1 − 1 − w (− ) 2 (η (ϕ 2 ))(w)=| | 2 2 4 |w| 2 e 2 L 2 (w ) even m r2 m 1 w2 − 1 (3.33) − − ( 2 ) 2 =2 4 e 2 Lm (w ) −2m− 1 =2 4 ψ2m(w). Again by (3.31) and (3.32)

1 2 1 ( ) w 1 − 1 1 − w ( ) 2 (η (ϕ 2 ))(w)=| | 2 2 4 |w| 2 sign w e 2 L 2 (w ) odd m 2 m 3 w2 1 (3.34) − − ( 2 ) 2 =2 4 we 2 Lm (w ) −2m− 7 =2 4 ψ2m+1(w).

Theorem 3.13. The unitary isomorphism ηeven : L2((0, ∞)) →Leven (see (3.32)) intertwines the irreducible unitary representations π− 1 and µeven of the metaplec- 2 tic group Mp(2, R). Also the unitary isomorphism ηodd : L2((0, ∞)) →Lodd intertwines the irreducible unitary irreducible representations π 1 and µodd of Mp(2, R). 2 Proof. It is clear from (3.33) and (3.34) that HHC LHC ηeven( − 1 )= even, (3.35) 2 HHC LHC ηodd( 1 )= odd . 2 Since all the unitary representations in question arise by application of Nelson’s the- ∈ |HHC |HHC orem, it suﬃces to show, for all u gC, (1) that ηeven − 1 intertwines D− 1 (u) − 1 2 2 2 |LHC |HHC |HHC |LHC and Dµ(u) and (2) that ηodd 1 intertwines D 1 (u) 1 and Dµ(u) . even 2 odd 0 2 2 But since e and e clearly generate the Lie algebra gC it suﬃces to prove (1) and (2) only for u = e and u = e0.Inthecaseu = e, (1) and (2) are obvious from (1.1), (3.17) and (3.32). Now assume u = e0. By (2.26), (3.22) it follows from (3.33) that − 1 − 1 0 ( 2 ) ( 2 ) ηeven(D− 1 (e )ϕm )=ηeven(iϕ ) 2 m+1 −2(m+1)− 1 = i2 4 ψ2(m+1)

−2m− 1 i =2 4 ( )ψ 4 2(m+1) 0 −2m− 1 = Dµ(e )2 4 ψ2m − 1 0 ( 2 ) = Dµ(e )ηeven(ϕm ). This establishes (1). The argument establishing (2), using (3.34) is identical. QED

Distribution theory on ∞ and the spaces H∞ and H−∞ 4. (0, ) r r

4.1. As usual −1

H∞ ∈ Also, r is stable under πr(g) for any g G. But now if Disto(G) denotes the convolution algebra of distributions of compact support on G, then we may regard G ⊂ Disto(G)whenG is identified with the group of Dirac measures. In addition, U(g) ⊂ Disto(G)whenU(g) is identified with the algebra of distributions having support at the identity. But one has a representation ∞ → H∞ (4.2) πr : Disto(G) End r ∞ where if ν ∈ Disto(G)andϕ ∈H ,then r Z ∞ (4.3) πr (ν)ϕ = πr(g)ϕν(g)dg. G ∞ The homomorphism πr extends the domain of (4.1) and in addition includes ∞ 1 |H ∈ k k { } 2 ∈H πr(g) r for any g G.If ϕ = ϕ, ϕ for any ϕ , one defines a Fréchet H∞ k k k k ∈ topology on r by taking as semi-norms ϕ v = πr(v)ϕ for v U(g)and ∈H∞ ϕ r . We will refer to this topology as the πr-Fréchet topology. The image of (4.2) are continuous operators and hence, by transpose, define operators on the con- H∞ ∗ tinuous dual to r . This may be put in a neater form. One defines a -operation ∗ ∗ on Disto(G), ν 7→ ν , by defining ν so that Z Z ψ(g)ν∗(g)dg = ψ(g−1)ν(g)dg G G ∈ ∞ ∞ ∞ for any ψ Co (G). Here Co (G)isthespaceofC functions of compact support on G. It is easy to see that this ∗-operation is an extension of the ∗-operation defined on gC considered in (3.1). Clearly, there now exists a vector space (πr-tempered H−∞ { } ∈H∞ ∈H−∞ distributions) r and a sesquilinear form ϕ, ρ for ϕ r and ρ r such 7→ { } H∞ that ϕ ϕ, ρ is a continuous linear functional on r and every continuous linear functional is uniquely of this form. See e.g. [Ca]. Clearly, one has a representation −∞ → H−∞ (4.4) πr : Disto End r ∈H∞ ∈H−∞ ∈ so that for ϕ r ,ρ r and ν Disto(G), { ∞ ∗ } { −∞ } πr (ν )ϕ, ρ = ϕ, πr (ν)ρ . 7→ { } ∈H∞ ∈H { } The map ϕ ϕ, ψ for ϕ r and ψ ,where ϕ, ψ is the ordinary H H∞ inner product in , is clearly a linear functional on r which is continuous with respect to the πr-Fréchet topology. In this way one has an embedding of L2((0, ∞)) H−∞ in r and hence inclusions H∞ ⊂ ∞ ⊂H−∞ (4.5) r L2((0, )) r . Furthermore, −∞ |H∞ ∞ (4.6) πr (ν) r = πr (ν) ∈ H∞ for any ν Disto(G). Thus r as a Disto(G)-module defined by (4.2) is just a H−∞ submodule of r (defined by (4.4)). In particular, both G and U(g)operateon H−∞ r and −∞ | ∞ πr (g) L2((0, )) = πr(g).

For any u ∈ g let Domr(u) be the domain of πfr(u). If u ∈ g,notethat f −∞ | (4.7) πr(u)=πr (u) Domr(u). 202 BERTRAM KOSTANT

∈H∞ ∈ Indeed, this is clear since if ϕ r and ψ Domr(u), one has {− ∞ } { f } (4.8) πr (u)ϕ, ψ = ϕ, πr(u)ψ . H∞ ⊂ The equality (4.8) follows from the obvious fact that r Domr(u)and ∞ f |H∞ (4.9) πr (u)=πr(u) r . For completeness we should note that, as a consequence of Theorem 3.4, one has HHC ⊂H∞ (4.10) r r and by (4.9) and (3.9) |HHC ∞ |HHC (4.11) Dr(u) r = πr (u) r .

A vector ϕ ∈Hwill be called πr-analytic if the map G →H,g7→ πr(g)ϕ is analytic. In such a case obviously ϕ ∈H∞ and one knows (Harish-Chandra, see Theorem 4.4.5.4, p. 278 in [Wa]) that there exists a neighborhood, U of 0 in g such that for any u ∈U the left side of (4.12), below, converges in L2 and ∞ X 1 (4.12) π∞(um)ϕ = π (exp u)ϕ. m! r r m=0 ∈HHC ∈ Proposition 4.1. Any ϕ r is πr-analytic. Furthermore, if u g,then |HHC f Dr(u) r is essentially skew-adjoint. That is, πr(u) is the operator closure of |HHC Dr(u) r . Proof. One has, restating (3.9), |HHC f |HHC (4.13) Dr(u) r = πr(u) r .

It is a theorem of I.E. Segal (see [Se]) that πfr(u) is the operator closure of the ∞ f restriction of πr (u) to the Garding space. But then πr(u) is the operator clo- |HHC sure of Dr(u) r by the last statement of Nelson’s Theorem 5 in [Ne]. That is, |HHC Dr(u) r is essentially skew-adjoint.P This and more also follows directly from 3 2 ∈HHC [Ne]. Indeed, put A = Dr( j=1 ui ), using the notation of (3.7). Let ϕ r . But ϕ is a ﬁnite sum of eigenvectors for A. Obviously, ϕ is then an analytic vector for A − 1, in the terminology of [Ne]. But then, by Lemma 6.2 in [Ne], ϕ is an ∈ |HHC analytic vector for Dr(u) for any u g.ThisimpliesthatDr(u) r is essentially ∈HHC skew-adjoint by Lemma 5.1 in [Ne]. Finally, any ϕ r is a πr-analytic vector by Goodman’s theorem. Stated as Theorem 4.4.6.1 in [Wa] one notes, in the notation of that theorem, ϕ is an analytic vector for B since it is, clearly, a ﬁnite sum of eigenvectors for B.QED

Remark 4.2. As a consequence of the phenomenon noted in Remark 3.5, one cannot −∞ have the equality of πr (u) with the differential operator Dr(u) on all infinitely differentiable functions in L2((0, ∞)) for all u ∈ g when 0 < |r| < 1. Indeed, for −∞ ∈ such values of r the equality of πr (u)andDr(u) for all u g must already HHC fail on −r . In fact, by Remark 3.5, (2.26), and using the notation of (3.7), the |HHC −∞ |H operator Dr(u1) −r has eigenvalues which cannot be eigenvalues of πr (u1) . −∞ If πr (u1) had such an eigenvalue, the corresponding nonzero eigenvector, by necessity, would have to be orthogonal to each element of the orthogonal basis { r } H ϕn of . This is of course a contradiction. LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 203

4.2. Recall that Dr(e)=ix. ∈H∞ k ∈H∞ ∈ Z Proposition 4.3. If φ r ,then(ix) φ(x) r for any k +.Furthermore, ∞ k k (4.14) πr (e )φ(x)=(ix) φ(x). Also, for any t ∈ R, itx (4.15) πr(exp te)ψ(x)=e ψ(x) for any ψ ∈ L2((0, ∞)). ∈H∞ |HHC Proof. Let φ r .Thenφ is in the domain of the closure of Dr(e) r by ∈HHC Proposition 4.1. Thus there is a sequence φn r such that φn converges to ∞ ∞ φ and ixφn converges to πr (e)(φ)inL2((0, )). But then, almost everywhere, ∞ φnj converges to φ and ixφnj converges to πr (e)(φ) for some subsequence φnj of ∞ ∈H∞ φn. But this implies πr (e)(φ)=ixφ.Inparticular,ixφ r . By iteration one obtains (4.14) and the first statement of Proposition 4.3. ∈HHC Let ψ r .Thensinceψ is a πr analytic vector by Proposition 4.1, there exists a neighborhood V of 0 in R such that for t ∈Vthe left side of (4.16) below converges in L2 and ∞ X 1 (4.16) tmD (em)ψ = π (exp te)ψ. m! r r m=0 In particular, a subsequence of the sequence of cut-off sums on the left side of (4.16) converges almost everywhere to the right side of (4.15). But the left side of (4.16) itx itx converges everywhere to e ψ(x). Thus πr(exp te)ψ(x)=e ψ(x). But then if A is the skew-adjoint infinitesimal generator (given by Stone’s theorem) of the 1- 7→ itx H HHC parameter group, t e , of unitary operators on , one has that r is in the |HHC |HHC domain of A and A r = Dr(e). But Dr(e) r is essentially skew-adjoint by Proposition 4.1 and hence A = πfr(e). This proves (4.15). QED 4.3. If X is a manifold, let M(X) denote the space of Borel measurable functions on X and if µ is a Borel measure on X,letL2(x, µ) be the Hilbert space of square integrable functions on X with respect to µ.Letα : M((0, ∞)) → M((0, ∞)) be the linear isomorphism defined by putting 1 (4.17) α(ϕ)(x)=x 2 ϕ(x). It is immediate that dx (4.18) α: H→L ((0, ∞), ) 2 x dx ∞ is a unitary isomorphism and one notes that x is Haar measure on (0, )asa multiplicative group. Next note that (4.19) α: C∞((0, ∞)) → C∞((0, ∞)) d ∞ is a linear isomorphism and 2x dx is a left- (or right)-invariant vector field on (0, ) d with respect to the multiplicative group structure. Recall that Dr(h)=2x dx +1. One easily has d (4.20) 2x ◦ α = α ◦ D (h) dx r on C∞((0, ∞)). Now ε : R → (0, ∞) is an isomorphism of Lie groups where ε(w)= ew.Letβ : M((0, ∞)) → M(R) be defined by putting β(ψ)=ψ ◦ ε. Then clearly 204 BERTRAM KOSTANT

∞ dx → R ∞ ∞ → β : L2((0, ), x ) L2( ,dw) is a unitary isomorphism and β : C ((0, )) ∞ R ◦ d d ◦ ∞ ∞ C ( ) is an algebra isomorphism. Furthermore, β 2x dx =2dw β on C ((0, )). Finally, let γ = β ◦ α. Then by composition

(4.21) γ : H→L2(R,dw) is a unitary isomorphism. Furthermore, (4.22) γ : C∞((0, R)) → C∞(R) is a linear isomorphism d (4.23) 2 ◦ γ = γ ◦ D (h) dw r ∞ R H∞ on C ((0, )). We are now ready to establish that the functions in r are (inﬁnitely) smooth. Proposition 4.4. One has H∞ ⊂ ∞ ∞ (4.24) r C ((0, )).

Also, the 1-parameter group of unitary operators on H, t 7→ πr(exp th), is given by t 2t (4.25) πr(exp th)(ϕ)(x)=e ϕ(e x) for any ϕ ∈H.

Proof. Let W be the space of all functions in L2(R,dw) which are absolutely continuous on every finite interval and whose first derivative is again in L2(R,dw). d Then it is classical that 2 dw operating on W defines a skew-adjoint operator B. HHC (See e.g. Example 3, p. 198 in [Yo].) Let V = γ( r ) so that by (4.21), (4.22) and (4.23) one has ∞ V ⊂ C (R) ∩ L2(R,dw) d ⊂ and that furthermore, V is stable under 2 dw . But clearly V W and d B|V =2 |V. dw d | But 2 dw V is essentially skew-adjoint by Proposition 4.1, and hence B must be the d | closure of 2 dw V . Also, one must have

W = γ(Domr(h)) (using the notation of Remark 4.1) and

(4.26) B ◦ γ = γ ◦ πfr(h) H∞ ⊂ on Domr(h). But since, obviously, r Domr(h), one has (4.27) V ⊂ Y ⊂ W H∞ where Y = γ( r )and ◦ ◦ ∞ (4.28) B γ = γ πr (h) H∞ H∞ ∞ on r .But r is stable under πr (h). Hence, Y is stable under B by (4.28). But ⊂ ∞ R H∞ ⊂ ∞ ∞ then, by the deﬁnition of B, one must have Y C ( ). Thus r C ((0, )) by inverting (4.23). It is classical and immediate that the 1-parameter group of unitary operators, tB tB e ,onL2(R,dw) generated by B, is the translation group given by e ϕ(w)= ϕ(w +2t) for any ϕ ∈ L2(R,dw). It follows then from (4.26) and the deﬁnition of LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 205

tB πfr(h)thate ◦ γ = γ ◦ πr(exp th). But if t 7→ π(t) is the 1-parameter group of unitary operators on H deﬁned by the right side of (4.25), then, by the deﬁnition of γ, clearly etB ◦ γ = γ ◦ π(t). This proves (4.25). QED ∞ ∞ ∞ ∞ 4.4. Let Co ((0, )), Disto((0, )) and Dist((0, )) be, respectively, the spaces, on (0, ∞), of all C∞ functions of compact support, all distributions of compact support and the space of all distributions.

Remark 4.5. As one easily sees, the diﬀerential operators, Dr(u), for u ∈ g (see (1.1)) are “formally skew-symmetric” in the sense that they are skew-symmetric d for pairings when dx operates as a skew-symmetric operator. This is certainly the ∈ ∞ ∞ ∈ ∞ ∞ ∈ case of a pairing of φ C ((0, )) with ψ Co ((0, )). That is, for any u g, Z ∞ Z ∞

(4.29) Dr(u)ψ(x)φ(x)dx = − ψ(x)Dr(u)φ(x)dx. 0 0 Another theorem of Nelson enables us to establish Proposition 4.6. One has ∞ ∞ ⊂H∞ (4.30) Co ((0, )) r and for any u ∈ g one has | ∞ ∞ ∞ | ∞ ∞ (4.31) Dr(u) Co ((0, )) = πr (u) Co ((0, )). Proof. By (4.29) one has

(4.32) {Dr(u)ψ,φ} = −{ψ,Dr(u)φ} P ∈HHC ∈ ∞ ∞ 3 2 for any φ r and ψ Co ((0, )). Let ∆ = i=1 ui using the notation of (3.7). Then (4.32) implies

(4.33) {Dr(∆)ψ,φ} = {ψ,Dr(∆)φ}. |HHC But, as established in the proof of Theorem 3.4, Dr(∆) r is essentially self- |HHC adjoint. Let A be the closure of Dr(∆) r . Then (4.33) implies ∞ ∞ ⊂ (4.34) Co ((0, )) Dom(A) and | ∞ ∞ | ∞ ∞ (4.35) A Co ((0, )) = Dr(∆) Co ((0, )). ∞ |HHC But since πr (∆) is clearly a symmetric extension of Dr(∆) r , it follows that ∞ ∞ ∞ A is the closure of πr (∆). But Co ((0, )) is stable under A by (4.35). Thus ∞ ∞ ⊂ k ∈ Z ∞ ∞ ⊂H∞ Co ((0, )) Dom(A ) for all k +.ButthenCo ((0, )) r by The- orem 4.4.4.5, p. 270 in [Wa] (another result of Nelson). But then (4.32) implies HHC H (4.31) since r is Hilbert space dense in .QED ∞ ∞ H∞ 4.5. We can now prove that (4.31) is true when we replace Co ((0, )) by r . H∞ ⊂ ∞ ∞ Recall that r C ((0, )) by (4.24). One should keep in mind the possible −∞ ∞ ∞ ∩H | | non-equality of Dr(u)withπr (u)onC ((0, )) when 0 < r < 1. See Remark 4.2. Proposition 4.7. For any u ∈ g one has ∞ |H∞ (4.36) πr (u)=Dr(u) r . 206 BERTRAM KOSTANT

See (4.24). Furthermore, the injection map H∞ → ∞ ∞ (4.37) r C ((0, )) H∞ § ∞ ∞ is continuous where r has the πr-Fréchet topology (see 4.1) and C ((0, )) (written as E((0, ∞)) in distribution theory) has the Fréchet toplogy defined as in distribution theory. That is, uniform convergence of all derivatives on compact sets. ∈ ∞ ∞ ∈H∞ ∈ { } Proof. Let ψ Co ((0, )),φ r and u g. Then, by (4.31), Dr(u)ψ,φ = −{ ∞ } ψ,πr (u)φ . But then, by (4.29), Z ∞ − ∞ ψ(x)(Dr(u)φ(x) πr (u)φ(x))dx =0. 0 ∈ ∞ ∞ ∞ Since ψ Co ((0, )) is arbitrary, this clearly implies Dr(u)φ(x)=πr (u)φ establishing (4.36). H∞ Now assume that a sequence φn converges to zero in r with respect to the πr-Fréchet topology. Then, recalling (1.1), for any k ∈ Z+, with the norm in H, k d k k (2x dx +1) φn(x) converges to zero. With the notation of (4.21) one has that ∈ ∞ R d k R ϕn C ( )and(2dw ) ϕn converges to zero in L2( ,dw). That is, using the notation of §8, p. 55 in [Sc], ϕn converges to 0 in DL2 . But then it is classical that ϕn converges to zero in E(R). (See the inclusion on p. 166 in [Sc]. One multiplies by a suitable function in D(R) and uses integration by parts.) But now since x is bounded away from 0 and also from above on any closed subinterval [a, b] ⊂ (0, ∞), d k it follows by induction, that ( dx ) φn converges to 0 uniformly on [a, b]. This proves the continuity of (4.37). QED ∈H−∞ 4.6. Let ρ r . Then, by (4.30), (4.38) ψ 7→ {ψ,ρ} ∞ ∞ defines a linear functional on Co ((0, )). D ∞ ∞ ∞ Lemma 4.8. Let ((0, )) denote Co ((0, )) when endowed with the LF- topology of distribution theory. Then (4.38) is continuous. That is, (4.38) defines an element in Dist((0, ∞)). ∞ ∞ Proof. We have only to show that if ψn is a sequence in Co ((0, )) with support ⊂ ∞ d k in some closed interval [a, b] (0, ) such that ( dx ) ψn converges uniformly to 0 in [a, b] for any k ∈ Z+,then

(4.39) {ψn,ρ} converges to 0.

But, recalling (4.31), the assumption on ψn clearly implies that if B is any differential operator on (0, ∞) with coeﬃcients in the ring C[x, x−1], then, with the H-norm, kB(ψn)k converges to zero. But then, recalling (1.1), kDr(v)(ψn)k converges to zero for any v ∈ U(g). But by (4.31) this implies ψn converges to 0 in the πr-Fr´echet topology. But then one has (4.39). QED 4.7. As a consequence of Lemma 4.8 one has a linear map H−∞ → ∞ (4.40) ζr : r Dist ((0, )) ∈H−∞ ∈ ∞ ∞ such that for any ρ r and ψ Co ((0, )), Z ∞

(4.41) {ψ,ρ} = ψ(x)ζr(ρ)(x)dx. 0 LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 207

∞ ∞ It will be established later, in Theorem 5.26, that Co ((0, )) is not dense in H∞ r with respect to the πr-Fréchet topology. In particular, using the Hahn-Banach theorem, the kernel of ζr is always non-trivial. For the case 0 < |r| < 1thiscanbe established now. | | ∞ ∞ H∞ Proposition 4.9. If 0 < r < 1,thenCo ((0, )) is not dense in r with respect to the Fréchet topology of §4.1. In particular, (via the Hahn-Banach theorem) the kernel of ζr is non-trivial. ∈H∞ ∈HHC ∈ ∞ ∞ Proof. Let ψ r ,ϕ −r and u g. Assume Co ((0, )) is dense in H∞ ∈ ∞ ∞ r . Then there exists a sequence ψn Co ((0, )) such that (see (4.31)) ψn ∞ H { } and Dr(u)ψn respectively converge to ψ and πr (u)ψ in .But Dr(u)ψn,ϕ = −{ } { ∞ } −{ } ψn,Dr(u)ϕ by (4.29). Taking the limit implies πr (u)ψ,ϕ = ψ,Dr(u)ϕ . { ∞ } −{ −∞ } But πr (u)ψ,ϕ = ψ,πr (u)ϕ . Hence, one has the equality Dr(u)ϕ = −∞ πr (u)ϕ. This contradicts the statement of Remark 4.2. QED 1 Remark 4.10. For the case (metaplectic case) where r = 2 one can already exhibit elements in Kerζr. Indeed, let η− 1 = ηeven and η 1 = ηodd using the notation of 2 2 (3.32). Then clearly, from (3.32), ηr(ϕ) vanishes in a neighborhood of 0 for any ∈ ∞ ∞ H∞ ϕ Co ((0, )).However,onemusthavethatη( r ) equals the space of even − 1 Schwartz functions or odd Schwartz functions according to whether r = 2 or 1 H−∞ H−∞ r = 2 . Upon extending the domain of ηr to r the image of r is, accordingly, −1 the space of even or odd tempered distributions. But then (η− 1 ) (δ0) ∈ Kerζ− 1 2 2 −1 0 0 and (η 1 ) (δ ) ∈ Kerζ1 where δ0 is the Dirac measure at the origin and δ is its 2 0 2 0 first derivative. If V ⊂ Dist ((0, ∞)) is a subspace, then a linear map →H−∞ ξV : V r will be called a cross-section of ζr in case ζr ◦ ξV is the identity on V .WhenξV is a specified cross-section and there is no possibility of ambiguity we will, when suitable, identify V with ξV (V ). We have already done so when V = L2((0, ∞)). WenowdosowhenV = Disto((0, ∞)). ∞ H−∞ Proposition 4.11. We may (and will) embed Disto((0, )) in r so that for ∈H∞ ∈ ∞ any ϕ r and ν Disto((0, )) one has Z ∞ (4.42) {ϕ, ν} = ϕ(x) ν(x)dx. 0 H∞ Proof. We have only to establish that the linear functional on r defined by the H∞ right side of (4.42) is continuous with respect to the πr-Fréchet topology on r . But this is immediate from the continuity of (4.37). Recall that Disto((0, ∞)) is the dual space to E((0, ∞)). QED Remark 4.12. A very important consequence, for us, of Proposition 4.11 is the now ∈H−∞ ∈ ∞ established fact that δy r for any y (0, )whereδy is the Dirac measure at y.

Next, let V1 ⊂ Dist(R) be the space of all Borel measurable functions ρ on (0, ∞) such that (A),

(4.43) ρ|(0, 1) ∈ L2((0, 1),dx) 208 BERTRAM KOSTANT and (B), there exists a positive constant C and k ∈ Z+ (with k and C dependent on ρ) such that, on [1, ∞), (4.44) |ρ(x)|

Proof. Since ϕ ∈Hone has ϕ|(0, 1) ∈ L2((0, 1),dx). But then

(4.46) ϕρ|(0, 1) ∈ L1((0, 1),dx) by (4.43). Let C and k be as in (4.44). Then, by (4.44),

−(k+1) (4.47) x ρ(x)|[1, ∞) ∈ L2([1, ∞),dx). ∞ k+1 ∈H∞ ∞ k+1 ∈H On the other hand, πr (e )(ϕ) r .Inparticular,πr (e )(ϕ) .But ∞ k+1 k+1 πr (e )(ϕ)(x)=(ix) ϕ(x) by (1.1) and (4.36). Thus k+1 (ix) ϕ(x)|[1, ∞) ∈ L2([1, ∞),dx). Multiplying by the conjugate of the function in (4.47), together with (4.46), proves that the integrand on the right side of (4.45) is absolutely integrable on (0, ∞). ∈H∞ Now assume that ϕn r converges to zero in the πr-Fr´echet topology. We have only to show that Z ∞

(4.48) ϕn(x)ρ(x)dx converges to 0. 0

But now, among other conditions, ϕn converges to 0 in H.Thusϕn|(0, 1) converges to zero in L2((0, 1),dx). Hence, by (4.43), Z 1 (4.49) ϕn(x)ρ(x)dx converges to 0. 0 k ∞ k+1 k k+1 | ∞ But also πr (e )ϕn converges to zero. Thus, in particular, x ϕn(x) [1, ) converges to 0 in L2([1, ∞),dx). But then, upon multiplication with the conjugate of the function in (4.47), Z ∞ ϕn(x)ρ(x)dx converges to 0. 1 But then (4.48) follows from (4.49). QED

(r) − 1 (r) (r) 1 2 { } 4.8. Let ψn =(2r+1 n!Γ(n + r +1)) ϕn so that, by (2.3), ψn ,n=0, 1, ..., is an orthonormal basis of L2((0, ∞),dx). By classical Hilbert space theory any ϕ ∈Hhas the expansion X∞ (r) (4.50) ϕ = anψn n=0 LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 209

{ (r)} where an = ϕ, ψn . We will referP to (4.50) as the Fourier-Laguerre expansion (of ∈ Z j (r) ∈HHC type r)ofϕ.Forj + let ϕj = n=0 anψn so that ϕj r and

(4.51) lim ϕj = ϕ j→∞ P H ∈H∞ 3 2 in . Now assume ϕ r . As in the proof of Proposition 4.6 let ∆ = i=1 ui . Then, for any m ∈ Z+, clearly { ∞ − m (r)} { ∞ − m (r)} πr ((∆ 1) )ϕ, ψn = ϕ, πr ((∆ 1) )ψn .

(r) ∞ − m But then since the ψn are eigenvectors for πr ((∆ 1) ), by (3.8), one has ∞ − m ∞ − m (4.52) πr ((∆ 1) )(ϕj )=(πr ((∆ 1) )(ϕ))j .

But by (4.51) this implies ∞ − m ∞ − m (4.53) lim πr ((∆ 1) )(ϕj)=πr ((∆ 1) )(ϕ) j→∞ in H. ∈ ∞ H Lemma 4.14. Let v U(g).Thenπr (v)(ϕj ) is a Cauchy sequence in .

Proof. Let UR(g) be the enveloping algebra of g, defined using only coefficients in R.SinceU(g)=UR(g)+iUR(g), it clearly suffices to assume v ∈ UR(g). But then v ∈ UR(g)2m for some m ∈ Z+, using the notation of the standard filtration of UR(g). But by (6.7), p. 588 in [Ne] there exists a constant k such that for any i, j ∈ Z+ with i

∞ ∞ In contradistinction to the non-density statement of Theorem 4.9 for Co ((0, )) one has

HHC H∞ Proposition 4.15. The subspace r is dense in r with respect to the πr- H∞ ∈HHC ∈HHC Fréchet topology on r .Infact,ifϕ r and ϕj r is defined as in H∞ (4.51), then ϕj converges to ϕ in the πr-Fréchet topology of r . ∈H∞ ∈ Proof. Let ϕ r and let v U(g). Then ϕ is contained in the domain of the ∞ |HHC ∞ closure of πr (v) r by (4.51) and Lemma 4.14. On the other hand, πr (v)isan ∞ |HHC ∞ extension of πr (v) r . But, using (4.9), it is clear that πr (v)itselfadmitsa H∞ closure (the dense subspace r is in the domain of its adjoint). Hence, one must have, in the notation of Lemma 4.14,

∞ ∞ (4.54) lim πr (v)(ϕj )=πr (v)ϕ. j→∞

That is kϕj − ϕkv converges to 0. QED 210 BERTRAM KOSTANT

H−HC 4.9. As a consequence of Proposition 4.15 we may introduce a vector space r where H−∞ ⊂H−HC (4.55) r r { } ∈HHC ∈H−HC and there is a sesquilinear form ψ,ρ for ψ r and ρ r which extends HHC H−∞ the sesquilinear pairing of r and r and is such that every (algebraic) linear HHC 7→ { } ∈H−HC functional on r is uniquely of the form ψ ψ,ρ for a unique ρ r . Furthermore, if (U(g),K) is the subalgebra of Disto(G) generated by U(g)andK one has a representation −HC → H−HC (4.56) πr :(U(g),K) End r ∈HHC ∈H−HC ∈ such that in the notation deﬁning (4.4), for ψ r ,ρ r and ν (U(g),K), { ∞ ∗ } { −HC } πr (ν )ψ,ρ = ψ,πr (ν)ρ . It is immediate that −∞ −HC |H−∞ (4.57) πr (ν)=πr (ν) r 5. Whittaker vectors, Bessel functions and the Hankel transform

5.1. We will consider the classical Bessel functions Jr(x) restricted to (0, ∞) where, as usual in this paper, r ∈ (−1, ∞). By deﬁnition Jr(x)isthesolution of the diﬀerential equation d d (5.1) (x2( )2 + x + x2 − r2)J(x)=0 dx dx which is given explicitly by the well-known power series (1.17.1), p. 14 in [Sz]. It has the property that one can write r ∗ (5.2) Jr(x)=x Jr (x) ∗ C ∞ ∗ where Jr is an entire function on which is real on (0, )andissuchthatJr (0) = 2−r.

Remark 5.1. For the special values of r where 0 < |r| < 1 the functions Jr(x)and J−r(x) are 2 linearly independent solutions of (5.1). Hence Jr(x)andJ−r(x)span the space of all solutions. See e.g. bottom paragraph “To sum up...” of p. 43 in [Wt]. One also knows that as, x −→ ∞ , − 1 (5.3) Jr(x)=O(x 2 ).

See e.g. (1.71.7), p. 15 or (1.71.11), p. 16 in [Sz]. Now for any y ∈ (0, ∞)letJr,y be the function on (0, ∞) deﬁned by putting √ (5.4) Jr,y (x)=Jr(2 yx). Proposition 5.2. Using the notation of Proposition 4.13 one has, for any y ∈ ∞ ∈ ∈H−∞ (0, ), Jr,y V1 so that Jr,y r . r Proof. Clearly, by (5.2), Jr,y (x)=O(x 2 )asx approaches 0. Since Jr,y is smooth, it follows that Jr,y |(0, 1) ∈ L2((0, 1),dx). On the other hand, by (5.3), there exists − 1 a positive constant C such that |Jr,y (x)|

(5.5) |Jr,y(x)|

∈H∞ ∞ 5.2. Now for any ϕ r let Tr(ϕ) be the function on (0, ) deﬁned so that for y ∈ (0, ∞),

(5.6) Tr(ϕ)(y)={ϕ, Jr,y}. 0 ∈ Dropping the√ factor 2 in Jr,y let Jr,y V1 be the function deﬁned by putting 0 0 1 { 0 } 0 Jr,y(x)=Jr( yx)andletTr(ϕ)(y)= 2 ϕ, Jr,y . Regard Tr and Tr as linear maps H∞ ∞ from r to the space Φ of all functions on (0, ). Let R be the operator on Φ deﬁned so that, for φ ∈ Φandy ∈ (0, ∞), Rφ(y)= √1 φ( y ). Note that, by (4.25), √ 2 2 if ao ∈ G is given by ao = exp toh for to = −log 2, then

(5.7) R|H = πr(ao). −1 H∞ In particular, R and R stabilize r and one notes that −1 0 (5.8) RTrR = Tr. (r) (r) r −x (r) Now recall that the Laguerre functions ϕn are given by ϕn (x)=x 2 e Ln (2x). But then − (r+1) (r) 2 [r] (5.9) Rϕn (x)=2 ϕn (x) [r] ∈H∞ wherewedeﬁneϕn r by putting r − x [r] 2 2 (r) (5.10) ϕn (x)=x e Ln (x). [HC] [r] Let Hr be the span of the functions {ϕn },n=0, 1,... . [r] Remark 5.3. Note that, by (5.7), the functions {ϕn },n=0, 1, ..., are again an [HC] orthogonal basis of H and that Hr ⊂His the Harish-Chandra module for πr 0 −1 when K is replaced by the conjugate group K = ao Kao . The following is a famous result with a long and interesting history. Theorem 5.4. For n =0, 1, ..., one has 0 [r] − n [r] (5.11) Trϕn =( 1) ϕn . Remark 5.5. Carl Herz in [He] says that the result is implicit in work of Sonine. He also cites an early 20th century reference but that reference deals only with the case r = 0. In [He], Herz establishes the result in great generality where (0, ∞)is replaced by a space of matrices. G.H. Hardy claims the result in [Ha]. However, he gives a proof only for a restricted set of values for r. The complete result is given as Exercise 21, p. 371 in [Sz] with some hints and a reference to [Ha]. For an understanding of Theorem 5.4 we will present a proof given to us by John Stalker. It begins with the following equality Z ∞ r − 2 a − a2 st r+1 4s (5.12) Jr(at)e t dt = r+1 e 0 (2s) where s ∈ C and Re s > 0. Equation (5.12) is established in [Wt], §13.3 (4), p.394wherewehavewrittens = p2. Watson remarks that (5.12) is the basis of investigations of Sonine in an 1880 paper. Writing t2 = x and a2 = y, (5.12) becomes the following Laplace transform equality Z ∞ r 1 √ r − y 2 − y 2 sx 4s (5.13) Jr( yx)x e dx = r+1 e . 2 0 (2s) 212 BERTRAM KOSTANT

Remark 5.6. Note that Theorem 5.4, for the case n = 0, is established by (5.13) by putting s = 1 . The proof of (5.12) in [Wt] is straightforward as soon as Watson 2 0 justiﬁes the term-by-term integration of the power series expansion of Jr,y (x). Proof of Theorem 5.4. (Stalker). Now it is well-known (see e.g. [Ja], §4 of Chapter X, p. 187) that ∞ X n − n (r) t −(r+1) xt (5.14) (−1) L (x) =(1− t) e 1−t n n! n=0 as an equality of two analytic functions in (x, t)wherex is arbitrary and |t| < 1. r − x Multiplying both sides of (5.14) by x 2 e 2 one has

X∞ n n [r] t −(r+1) r − x 1+t (5.15) (−1) ϕ (x) =(1− t) x 2 e 2 1−t . n n! n=0 1 1+t | | Now put s = 2 1−t and note that Re s > 0when t < 1. Substituting this expression for s in (5.13) one has Z ∞ 1 √ r x 1+t r 1 − t y 1−t − − r+1 − Jr( yx)x 2 e 2 1 t dx = y 2 ( ) e 2 1+t 2 0 1+t and hence Z ∞ 1 √ r x 1+t r y 1−t −(r+1) − − (r+1) − (5.16) Jr( yx)(1 − t) x 2 e 2 1 t dx =(1+t) y 2 e 2 1+t . 2 0 √ r But Jr( yx)x 2 , as a function of x, is integrable at 0 by (5.2) and is bounded by a constant multiple xk,fork sufficiently large, as x tends to ∞ by (5.3). Thus if one ∈ Z 1 differentiates (5.13) n-times for any n +, with respect to s, and then sets s = 2 , the computation on the left side can be carried out under the integral sign. But since s is an analytic function of t for |t| < 1, the same differentiation statement is true for (5.16) where t replaces s and we set t = 0. But then Theorem 5.4 follows from (5.15). QED

[r] Since the elements {ϕn } are an orthogonal basis of H (see Remark 5.3), it follows immediately from Theorem 5.4 that there exists a unique unitary operator 0 H 0 |H[HC] 0|H[HC] 0 Ur on such that Ur r = Tr r and that furthermore Ur has order 2. 0 The operator Ur is often referred to as the Hankel transform. However, it is more −1 convenient here to make a slight change (conjugating by πr(ao )) and refer to this −1 altered operator as the Hankel transform. That is, ﬁrst of all, by applying πr(ao ) to (5.11) it follows from (5.7), (5.8) and (5.9) that Theorem 5.4 can be rewritten as Theorem 5.7. For n =0, 1, ..., one has (r) − n (r) (5.17) Trϕn =( 1) ϕn .

We will let Ur be the unique unitary operator (necessarily of order 2) on H such that |HHC |HHC Ur r = Tr r . Clearly, −1 0 πr(ao)Urπr(ao) = Ur.

We refer to Ur as the Hankel transform. LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 213

5.3. If G1 is a linear semisimple Lie group and X = G1/K1 is a Hermitian symmetric space, where K1 is a maximal compact subgroup of G1, then the holomorphic discrete series of G1 is generally constructed on spaces of square integrable holomorphic sections of line bundles on X.IncaseX is of tube type, Ding and Gross in [DG] use this construction to create another model, for the holomorphic discrete series, which is built on the symmetric cone Ω associated to X.Incase G1 = Sl(2, R), then Ω = (0, ∞) and the new model is (essentially), πr,wherer is a positive integer. The main point in their construction of the new model is their recognition 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 positive integer, then for some cr ∈ C and k ∈ K1,

(5.18) Ur = crπr(k). Even if one went to the covering group, G, their construction does not lead to πr for −1

Ad ko(h)=−h,

(5.19) Ad ko(e)=−f,

Ad ko(f)=−e. 2 In particular, ko is central in G. See also (3.13). − r+1 πi Theorem 5.8. Let cr = e 2 .Then

(5.20) Ur = crπr(ko). Proof. By (2.5) and (2.26) one has π (r) 2 (2n+r+1)i πr(ko)ϕn = e . Hence, (r) nπp (r) crπr(ko)ϕn = e ϕn (5.21) − n (r) =( 1) ϕn . 214 BERTRAM KOSTANT

But then, since both crπr(ko)andUr are unitary operators on H, (5.20) follows from (5.21) and (5.17). QED

One immediate consequence of Theorem 5.8 is the following property of the Hankel transform Ur. This property is an analogue of the fact that the Fourier transform stabilizes the Schwartz space and is continuous with respect to its Fréchet topology. Theorem 5.9. For the Hankel transform one has H∞ →H∞ (5.22) Ur : r r H∞ and (5.22) is continuous with respect to the πr-Fréchet topology on r . −∞ |H Since πr(ko)=πr (ko) we can initially extend the domain of the Hankel H−∞ −∞ transform to r by putting Ur = crπr (ko). But in fact, recalling (4.57), one H−HC can go further and extend the domain of the Hankel transform Ur to r by putting −HC (5.23) Ur = crπr (ko). 2 OneofcoursestillhasUr = Identity. See Theorem 5.7. | | ∈H−∞ Remark 5.10. Note that if 0 < r < 1andρ r , the problem of computing Ur(ρ), by Proposition 4.9, seems to be complicated by the fact that ρ may vanish as a distribution on (0, ∞). ∈H Even if ϕ , as in the case of the Fourier transform, it is not transparent√ when Ur(ϕ) can be determined by integrating ϕ(x) against the kernel Jr(2 yx). ∈H∞ The following result says that if ϕ r everything is fine. ∈H∞ ∈H∞ Theorem 5.11. Let ϕ r so that (by Theorem 5.9) Ur(ϕ) r and hence ∞ Ur(ϕ)(y) is a C -function of y. Furthermore, recalling Proposition 5.2, Ur(ϕ)(y) is given by

(5.24) Ur(ϕ)(y)={ϕ, Jr,y } for any y ∈ (0, ∞). ∈HHC Proof. There exists, by Proposition 4.15, a sequence ψn r that converges to ϕ H∞ in the πr-Fr´echet topology of r .Butthenix ψn converges to ix ϕ in this topology since πr(e) is certainly continuous in the πr-Fr´echet topology. In particular, ψn and ix ψn converge respectively to ϕ and ix ϕ in H.Lety ∈ (0, ∞). Then since Jr,y(x) is in L2 at 0 one has Z Z 1 √ 1 √ (5.25) lim ψ (x)J (2 yx)dx = ϕ(x)J (2 yx)dx. →∞ n r r n 0 0 −1 On the other hand, ix Jr,y(x)|[1, ∞) ∈ L2([1, ∞),dx) by (5.3). But then Z ∞ Z ∞ −1 ψn(x)Jr,y (x)dx = ixψn(x)(ix) Jr,y(x)dx 1 1 converges to Z ∞ Z ∞ −1 ixϕ(x)(ix) Jr,y (x)dx = ϕ(x)Jr,y (x)dx 1 1 LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 215

Hence, by (5.25) Z ∞ Z ∞ (5.26) lim ψ (x)J (x)dx = ϕ(x)J (x)dx. →∞ n r,y r,y n 0 0

But the left side of (5.26) is just Ur(ψn)(y), by deﬁnition of Ur. However, by H∞ Theorem 5.9, Ur(ψn)convergestoUr(ϕ)intheπr-Fr´echet topology of r .But then, by Proposition 4.7, Ur(ψn) certainly converges pointwise to Ur(ϕ). But then Ur(ϕ)(y) is given by the right side of (5.26). QED

∈ ∈H−HC 5.4. Let u gC. A vector ρ r will be called a u-weight vector of weight ∈ C −HC λ if πr (u)ρ = λρ. The span of the set of all such vectors will be referred to as the corresponding u-weight space, or simply weight space if u is understood. If u = e (resp. u = f)andλ =6 0, then the e-weight (resp. f-weight) vector will also be 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- −HC −HC nomer since it refers, respectively to the kernels of πr (e)andπr (f) and not, as in ﬁnite dimensional representation theory, to h-weights. Note that, as a con- ∈H−HC sequence of (5.23), ρ r is a u-weight vector of weight λ if and only if its Hankel transform Ur(ρ)isanAd ko(u)-weight vector of weight λ. In particular, if u = h, then, by (5.19), ρ is an h-weight vector of weight λ if and only if its Hankel transform Ur(ρ)isanh-weight vector of weight −λ. Furthermore, again by (5.19), ρ is an e-weight vector of weight λ if and only if its Hankel transform Ur(ρ)isan f-weight vector of weight −λ. Now if u ∈ gC,wecanwrite u = α(u)h0 + β(u)e0 + γ(u)f 0

(∗) for some unique α(u),β(u),γ(u) ∈ C.LetgC = {u ∈ gC | γ(u) =06 }.

∈ (∗) ∈ C H−HC Theorem 5.13. Let u gC . Then for any λ the u-weight space, in r , (∗) of weight λ is 1-dimensional. Furthermore h, e, f ∈ gC so that in particular for H−HC u = h, e, f the u-weight space, in r , of weight λ 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 (e0)∗ = −f 0,(h0)∗ = h0 and (f 0)∗ = −e0. It then follows immediately that β(u∗)=−γ(u). Hence,

(∗) ∗ (5.27) v ∈ gC ⇐⇒ β(v ) =06 However, it is immediate from (2.5), (2.6) and (2.7) that h = i(e0 − f 0), 1 e = (ih0 + e0 + f 0), 2 1 f = (ih0 + e0 + f 0), 2 (∗) ∗ so that h, e, f ∈ gC .Letλ ∈ C and let v = u − λ.Toprovethetheoremit HHC clearly suﬃces to prove that (recalling (4.36)) if Y = Dr(v)( r ), then Y has 216 BERTRAM KOSTANT

HHC ∈ Z { (r)} codimension 1 in r .Form + let Xm be the span of ϕn ,n=0, 1, ..., m. But now it is immediate from (2.26), (2.27), (2.28) and (5.27) that |HHC (a) KerDr(v) r =0,

(5.28) (b) X0 6⊂ Y,

(c) Dr(v)(Xm) ⊂ Xm+1. 6 HHC It follows from (b) of (5.28) that Y = r .Butdim Xm = m + 1. Hence dim Dr(v)(Xm)=m + 1 by (a) of (5.28). But then

(5.29) Xm = X0 ⊕ Dr(v)(Xm−1) HHC ⊕ by (b) and (c) of (5.28). But then clearly r = X0 Y .QED Remark 5.14. AWeylgroupofG has order 2. Note that Theorem 5.13 is not a contradiction of Theorem 6.8.1 on p. 182 of [Ko] (which, in this case, asserts the existence of a 2-dimensional space of algebraic Whittaker vectors). Indeed the HHC hypothesis of this result in [Ko] requires that one must add, to r ,theHarish- Chandra module of the anti-holomorphic representation corresponding to πr.See (6.8.1) in [Ko]. ∈H−HC ∈H−∞ An element ρ r will be called πr-tempered in case ρ r .Wenow obtain explicit results on the “temperedness” of certain Whittaker vectors. For λ = −iy, y ∈ (0, ∞) the corresponding f-Whittaker vector is πr-tempered and is given in terms of the Bessel function Jr. ∈ ∞ ∈H−∞ Theorem 5.15. Let y (0, ) so that δy r where δy is the Dirac measure at y (see Remark 4.12). Then Rδy is the weight space of e-Whittaker vectors of weight iy. In particular, any such√ Whittaker vector is πr-tempered. H−∞ The function Jr,y(x)=Jr(2 yx) is in r by Proposition 5.2. Furthermore, RJr,y is the weight space of f-Whittaker vectors of weight −iy. In particular, any such Whittaker vector is πr-tempered. Furthermore,

(5.30) Ur(δy)=Jr,y where, we recall Ur is the Hankel transform. ∈H−∞ Proof. Let ϕ r . Then (see (4.36)) { −∞ } { } ϕ, πr (e)δy = ixϕ, δy = −iyϕ(y)

= {ϕ, iyδy}. −∞ This proves πr (e)δy = iyδy. Now, ∞ − (5.31) πr (f)Ur(δy)= iyUr(δy) by Remark 5.14. On the other hand, since Ur has order 2, it follows from (5.23) and (5.24) that

{ϕ, Ur(δy)} = {Ur(ϕ),δy}

= {ϕ, Jr,y }, This proves (5.30). The theorem then follows from (5.31). QED LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 217

5.5. Let ∆ ∈ U(g) be as in (4.52). Using the notation of (4.50) we may, by (3.8), define a quadradic polynomial q on R such that for all n ∈ Z, ∞ − (r) (r) (5.32) πr (∆ 1)(ψn )=q(n)ψn . ∈H∞ { } Now let ϕ r and let an be, as in (4.50), the Fourier-Laguerre coefficients of ∞ − m ϕ. Then, by (4.53), the Fourier-Laguerre expansion of πr ((∆ 1) )(ϕ)isgiven by X∞ ∞ − m m (r) (5.33) πr ((∆ 1) )(ϕ)= q(n) anψn n=0 for any m ∈ Z+. Since this is an expansion of an element in H,thesumofnorm squares of the coefficents is finite. In particular, the norm of the coefficents is bounded. On the other hand, the coefficents of q are positive, since r>−1. See (3.8). Thus for all m ∈ Z+ there exists a positive constant Cm such that for all n 2 −m (5.34) |an| 0, ϕfj converges to ϕe pointwise and uniformly on the closed interval [0,ω],usingthe notation of (4.51).

(r) Proof. In the proof of Proposition 3.12 we noted that if Ln (Szego) is the normal- (r) ization of Ln as given in (5.1.5), p. 97 of [Sz], then 1 (5.36) L(r)(Szego)=(−1)n L(r). n n! n − 1 It follows then from (7.6.11), p. 173 in [Sz] that there exists a> 2 and positive constants C and D such that for all x ∈ [0,ω]andn ∈ Z+, | (r) | a Ln (2x) n!. Hence 1 (n + r +1) 2 (5.38) n! d < 2r+1 . n Γ(r +2) 218 BERTRAM KOSTANT

(r) But recalling the deﬁnition of the orthonormal basis {ψn } (see (4.50)) one has

g(r) − x 2 (r) (5.39) ψn = dne Ln (2x).

Hence, for all n ∈ Z+ and x ∈ [0,ω] there exists positive constants E,F and k ∈ N such that

g(r) α |ψn (x)|

g(r) 2 −(k+1) k |anψn (x)|

This proves that ϕfj converges pointwise to a continuous function ψ on [0, ∞)and uniformly so on any closed interval [0,ω]. But, by (4.51), a subsequence ϕji con- ∞ f verges almost everywhere to ϕ on (0, ). Hence ϕji converges almost everywhere to ϕe on (0, ∞). Hence ϕe = ψ on (0, ∞). But then ϕe(0) is deﬁned and ϕe = ψ as functions on [0, ∞). QED

Now by (5.1.7), p. 97 in [Sz] and (5.45) one has n + r (5.41) L(r)(0) = (−1)nn! . n n But by standard properties of the Γ-function Γ(n + r +1) n + r (5.42) = Γ(r +1). n! n But using the notation of (5.39) one has, using (5.41) g n + r ψ(r)(0) = d (−1)nn! n n n (5.43) 1 1 2r+1 2 n + r 2 = (−1)n . Γ(r +1) n As an immediate consequence of (5.43) and Theorem 5.16 one has ∈H∞ Theorem 5.17. Let ϕ r and let (4.50) be its Fourier-Laguerre expansion. Then the sum on the right side of (5.44), below, absolutely converges and 1 X∞ 1 2r+1 2 n + r 2 (5.44) ϕe(0) = (−1)na . Γ(r +1) n n n=0 We can determine the highest weight vector (unique up to scalar multiplication ∈H−HC by Theorem 5.13) of πr and establish that it is πr-tempered. Let δr,0 r be { } e ∈HHC deﬁned so that ψ,δr,0 = ψ(0) for any ψ r . LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 219

Theorem 5.18. The linear functional δr,0 is a highest weight vector and spans the ∈H−∞ highest weight space. Furthermore, δr,0 is πr-tempered (i.e. δr,0 r )andfor ∈H∞ any ϕ r one has

{ϕ, δr,0} = ϕe(0) (5.45) − r = lim x 2 ϕ(x). x→0 (See Theorm 5.8.) In particular, if (4.50) is the Fourier-Laguerre expansion of ϕ, then {ϕ, δr,0} is given by the convergent sum (5.44). ∈HHC Proof. Let ψ r .Then { −HC } { ∞ } ψ,πr (e)(δr,0) = πr (e)(ψ),δr,0 { } (5.46) = ixψ, δr,0 − r = lim ixx 2 ψ(x). x→0 − r e But limx→0 x 2 ψ(x)=ψ(0) exists. Hence, the limit in (5.46) (with the extra x- −HC factor) must be zero. Thus πr (e)(δr,0)=0sothatδr,0 is a (clearly nonzero) 0 H∞ highest weight vector. Now let δr,0 be the linear functional on r defined so 0 e that δr,0(ϕ)=ϕ(0). To prove that δr,0 is πr-tempered and all the remaining 0 statements of the theorem, it clearly suffices only to prove that δr,0 is continuous H∞ ∈H−∞ in the πr-Fréchet topology on r . Indeed one would then have that δr,0 r { } 0 ∈H∞ and ϕ, δr,0 = δr,0(ϕ) for all ϕ r . Let k ∈ N be fixed such that k ≥ r.Then n + r ≤ n + k n n n + k = k (n + k)k ≤ . k! n+r ≤ k Thus, there exists positive constants A, B, such that n An + B and hence there exists positive constants C, D such that

1 2 n + r k (5.47) ≤ Cn 2 + D n ∈ N ≥ k Now recalling (5.33) let m be such that 2m 2 + 2. But since the coeﬃcients of the quadratic polynomial q are positive, there exist positive constants E,F such P k ∈ Z m ≥ 2m 2 2m that for all n +, q(n) En + F . But clearly n=0(Cn + D)/(En + F ) converges. Hence, by (5.47) one has a convergent sum X∞ 1 n + r 2 (5.48) /q(n)2m = M. n n=0 − m ∈ ∈H∞ Now in the notation of (5.33) let v =(∆ 1) U(g). Let ϕ r so that, by k ∞ k k k deﬁnition, πr (v)(ϕ) = ϕ v. Let (4.50) be the Fourier-Laguerre expansion of ϕ. m Then, by (5.33), |q(n) an|≤kϕkv for all n ∈ Z+.Thatis,foralln ∈ Z+, m (5.49) |an|≤kϕkv/q(n) . 220 BERTRAM KOSTANT

But then if N is the constant preceding the sum in (5.44), one has by (5.44), (5.48), and (5.49) X∞ 1 n + r 2 |δ0 (ϕ)|≤N |a | r,0 n n n=0 X∞ 1 (5.50) n + r 2 ≤kϕk N | /q(n)m| v n n=0 ≤ NMkϕkv. 0 k k But this proves that δr,0 is continuous with respect to the seminorm ϕ v and 0 H∞ hence δr,0 is a continuous linear functional on r with respect to the πr-Fr´echet topology. QED

5.6. We now turn to the lowest weight vector. Let Jr,0 = Ur(δr,0)sothat,see Remark 5.12, Jr,0 is a lowest weight vector. Furthermore, R Jr,0 is the lowest weight H∞ space and since Ur stabilizes r (see (5.23)), one has that Jr,0 is smooth. As an easy consequence of Theorem 5.18 one has ∈H∞ Theorem 5.19. Let ϕ r .Then Z ∞ − r √ (5.51) {ϕ, J } = lim y 2 J (2 yx)ϕ(x)dx r,0 → r y 0 0 noting, in particular, that the limit on the right side of (5.51) exists.

Proof. Since the Hankel transform, Ur, is of order 2, it follows from (5.23), (5.24), (5.30) and (5.45) that

{ϕ, Jr,0} = {Ur(ϕ),Ur(Jr,0)}

= {Ur(ϕ),δr,0} − r = lim y 2 Ur(ϕ)(y) y→0 − r = lim y 2 {ϕ, Jr,y }. y→0 But this is just the statement of (5.51). QED QED Now consider h-weight vectors. By Theorem 5.13 any weight is possible and all have multiplicity 1. But of course there is the question as to whether the weight vector is πr-tempered. ∈ C − 1 µ ∈ Theorem 5.20. Let µ be such that Re µ > 2 so that x V1,inthe § µ ∈H−∞ notation of 4.7, and hence, by Proposition 4.13, x r .Letλ =2µ +1 so that Re λ > 0.Thenxµ (uniquely up to a scalar multiple) is an h-weight vector of r weight λ.Inparticular,x 2 is a smooth h-weight vector of weight r +1. µ On the other hand, Ur(x ) is a smooth h-weight of weight −λ for Re λ < 0.In r particular, Ur(x 2 ) is a smooth h-weight vector of weight −(r +1). ∈H∞ Proof. Let ϕ r . By (4.36) d (5.52) {−(2x +1)(ϕ),xµ} = {ϕ, π−∞(h)(xµ)} dx r ∈ Z k ∈H∞ H∞ ∞ k But now, for any k , x ϕ(x) r ,since r is stable under πr (e )and ∞ k hence R →His a C map, where t 7→ πr(exp (−t)h)(x ϕ(x)). But, by (4.25), k −2tk k 2tk πr(exp (−t)h)(x ϕ(x)) = e x πr(exp (−t)h)(ϕ(x)). We may multiply by e LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 221

k and conclude that t 7→ x πr(exp (−t)h)(ϕ(x)) also defines a differentiable map from R to H. Furthermore, one must have d d (5.53) (xkπ (exp (−t)h)(ϕ)) = xk(−(2x + 1))(ϕ) dt r t=0 dx in H since the difference quotient whose limit in H defines the left side of (5.53) clearly converges pointwise to the right side of (5.53), by (4.25) and the smoothness of ϕ. But by a change of variables one has Z ∞ µ −2t µ −t {πr(exp (−t)h)(ϕ),x } = ϕ(e x)x e dt 0 = {ϕ, et(2µ+1)xµ}. Hence, d (5.54) ({π (exp (−t)h)(ϕ),xµ}) = {ϕ, (2µ +1)xµ}. dt r t=0 R b ∞ µi i µi For any ψ ∈H and i =1, 2, let {ψ,x }i = ψ(x)x dx where a1 =0,b1 = r ai ∞ − 1 ∈ C { µ} 1,a2 =1,b2 = ,Reµ1 > 2 and µ2 is arbitrary. Clearly, ψ,x = µ µ {ψ,x }1 + {ψ,x }2. But now d d (5.55) ({π (exp (−t)h)(ϕ),xµ} ) = {−(2x +1)(ϕ),xµ} dt r 1 t=0 dx 1 by (5.53) where we put k = 0. On the other hand, if we choose k so that xµ−k|[1, ∞) ∈ L2((1, ∞),dx), then by (5.53), d d ({π (exp (−t)h)(ϕ),xµ} ) = ({xkπ (exp (−t)h)(ϕ),xµ−k } ) dt r 2 t=0 dt r 2 t=0 d (5.56) = {−xk(2x +1)(ϕ),xµ−k} dx 2 d = {−(2x +1)(ϕ),xµ} . dx 2 But then by (5.54), (5.55) and (5.56) one has d {−(2x +1)(ϕ),xµ} = {ϕ, (2µ +1)xµ}. dx −∞ µ µ But then πr (h)(x )=(2µ +1)x by (5.52). The last statements follow from H−∞ Remark 5.12 and the fact that Ur stabilizes r . See (5.23). QED Remark 5.21. It is clear in Theorem 5.20 that the h-weight vectors for weights λ when Re λ is positive is independent of r. On the other hand, there is an apparent dependence on r when Re λ is negative. Since δr,0 has an obvious dependence on r the dependence of the h-weight vector of weight −(r +1),onr, will be verified in Theorem 5.23.

From the commutation relations in g it is obvious that highest weight vector δr,0 and the lowest weight vector Jr,0 must be h-weight vectors. The question is: what are the h-weights? This is determined in Theorem 5.23 below. Contrary to one’s experience for ﬁnite dimensional representations, it turns out that the h-weight of the highest weight vector (recall deﬁnitions in §5.4) is smaller than the h-weight of the lowest weight vector. In fact, the latter is positive and the former is just its negative. The lowest weight vector Jr,0 is given in Theorem 5.19 as a limit. In Theorem 5.23 it will be given as an integral. 222 BERTRAM KOSTANT

Remark 5.22. By the multiplicity 1 statement of Theorem 5.13 and the commutation relations in g note that (up to scalar multiplication) δr,0 is the only nonzero H−HC H−∞ element in r , and a fortiori in r , which is simultaneously an h-weight vector and an e-weight vector. The same is true of Jr,0 when f replaces e. Theorem 5.23. One has ∞ − (5.57) πr (h)(δr,0)= (r +1)(δr,0) and ∞ (5.58) πr (h)(Jr,0)=(r +1)(Jr,0).

Furthermore, Jr,0 given by (5.51), can also be given by Z ∞ 1 r (5.59) {ϕ, Jr,0} = ϕ(x)x 2 dx Γ(r +1) 0 ∈H∞ for any ϕ r so that

1 r (5.60) U (δ )= x 2 . r r,0 Γ(r +1) ∞ ∞ ∞ − ∞ Proof. From the commutation relations 2πr (e)=[πr (h),πr (e)] and 2πr (f)= ∞ ∞ [πr (h),πr (f)] together with the multiplicity 1 statement of Theorem 5.13 it is immediate that there exists scalars µ, ν ∈ C such that ∞ πr (h)(δr,0)=µδr,0, ∞ πr (h)(Jr,0)=νJr,0. But then (5.61) ν = −µ ∈H∞ by Remark 5.12. Thus, for any ϕ r one has d {−(2x +1)(ϕ),δ } = {ϕ, π∞(h)(δ )} dx r,0 r r,0 = µ{ϕ, δr,0}. Hence,

− r d − r (5.62) lim x 2 (−(2x +1)(ϕ))(x)=µ lim x 2 ϕ(x). x→0 dx x→0 (r) − r − r x 2 2 − d Now put ϕ = ϕ0 so that ϕ(x)=e x .Thenx ( (2x dx +1)(ϕ))(x)= −(r +1)e−x +2e−xx. Thus the limit on the right side of (5.62) is −(r +1).But − r −x x 2 ϕ(x)=e . Hence, the limit on the right side of (5.62) is µ.Thisproves µ = −(r +1). Butthen ν =(r + 1) by (5.61). But then by the multiplicity r 1 statement of Theorem 5.13 one must have Jr,0 = cx 2 for some constant c by Theorem 5.20. Thus Z ∞ − r r (5.63) lim y 2 {ϕ, J } = c ϕ(x)x 2 dx → r,y y 0 0 ∈H∞ (r) { (r) } (r) for any ϕ r . Now as above choose ϕ = ϕ0 .But ϕ0 ,Jr,y = ϕ0 (y)by (r) −y r Theorem 5.7. Since ϕ0 (y)=e y 2 one has Z ∞ (5.64) lim e−y = c e−xxrdx. → y 0 0 LAGUERRE POLYNOMIALS AND THE UNITARY DUAL OF Slf(2, R) 223

But the left side of (5.64) is 1 and, by definition of the Γ-function, the right side is 1 cΓ(r +1).Hencec = Γ(r+1) .QED H−∞ → ∞ Recall the map ζr : r Dist((0, )). See (4.40). ∞ ∞ H∞ Theorem 5.24. The subspace Co ((0, )) is not dense in r with respect to the πr-Fréchet topology. In fact, the highest weight vector, δr,0, and the Category-O −∞ ∞ ∞ module, πr (U(g))(δr,0), it generates, vanishes on Co ((0, )). That is, −∞ ⊂ (5.65) πr (U(g))(δr,0) Kerζr. H∞ → ∞ ∞ Furthermore, (see (4.37)) the injection map r C ((0, )) is continuous H∞ but is not a homeomorphism onto its image where r has the πr-Fréchet topology and C∞((0, ∞)) = E((0, ∞)) has the Fréchet topology of distribution theory.

Proof. It is obvious from (5.45) that δr,0 ∈ Kerζr. But this implies (5.65) since ∞ ∞ ∞ Co ((0, )) is stable under πr (U(g)) by (4.36). The continuity of (4.37) is established in Proposition 4.7. It is not a homeomorphism onto its image since, as one ∞ ∞ E ∞ knows, Co ((0, )) is dense in ((0, )) with respect to the distribution theory Fr´echet topology on E((0, ∞)). QED

Remark 5.25. In contrast to the highest weight vector, δr,0, note that, by (5.59), the lowest weight vector, Jr,0,isnotinthekernelofζr. See (4.40).

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 E-mail address: [email protected]