Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2,R)

Acta Applicandae Mathematicae 71: 261–277, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

261

Laguerre Polynomials, Restriction Principle, andHolomorphic Representations of SL(2,R)

MARK DAVIDSON1, GESTUR ÓLAFSSON1,� and GENKAI ZHANG2,��1Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.email: {davidson,olafsson}@math.lsu.edu2Department of Mathematics, Chalmers University of Technology and Göteborg University,S-412 96 Göteborg, Sweden. e-mail: [email protected]

(Received: 1 June 2001)

Abstract. The restriction principle is used to implement a realization of the holomorphic representa-tions of SL(2,R) on L2(R+, tα dt) by way of the standard upper half plane realization. The resultingunitary equivalence establishes a correspondence between functions that transform according to thecharacter θ �→ e−i(2n+α+1)θ under rotations and the Laguerre polynomials. The standard recursionrelations amongst Laguerre polynomials are derived from the action of the Lie algebra.

Mathematics Subject Classifications (2000): 22E46, 43A85, 33C45, 33C80.

Key words: Laguerre polynomials, representation theory, Lie groups, special functions, Segal–Barg-man transform, restriction principle, SL(2,R).

Introduction

It is a well-known fact that orthogonal polynomials and other special functionsare closely related to representations of Lie groups and harmonic analysis on ho-mogeneous spaces. The spherical harmonics are matrix coefficients of irreduciblerepresentations of SO(n) and hypergeometric functions are solutions to invariantdifferential equations on rank one symmetric spaces. These and other similar re-lations can then be used as a guideline to generalize classical special functions tomore general settings, i.e., higher dimensions or more general manifolds. A goodexample of this is the work of G. Heckmann and E. Opdam on hypergeometricfunctions, (see [14, 22, 23] and the references therein). Another good exampleare the Hermite polynomials. They and the natural relations that they satisfy areclosely related to the representation of the Heisenberg group on Fock spaces asis shown in [24]. One of the main tools in the paper by B. Ørsted and the thirdauthor was the restriction principle. The idea is to restrict holomorphic functionson C

n to the totally real submanifold Rn ⊂ C

n. It turns out that the unitary part ofthe polarization of the restriction map is the classical Segal–Bargmann transform

� Research by G. Olafsson supported by NSF grant DMS 0070607.�� Research by G. Zhang supported by Swedish Research Council (NFR).

262 MARK DAVIDSON ET AL.

between the Fock space F of holomorphic functions on Cn and L2(Rn). The same

concept was also used to get the quantization as a unitary map between Fock spaceand the Hilbert space of Hilbert–Schmidt operators on Fock space. This idea wastaken up in [21] where the general restriction principle was introduced as a wayto construct a natural intertwining unitary map. This principle works if one of theHilbert spaces is a reproducing Hilbert space of holomorphic functions on a com-plex manifold MC and the other one is an L2-space on a totally real submanifoldM ⊂ MC. In this generality the principle is independent of the representationtheory of Lie groups. Nevertheless, the main motivation was to understand classi-cal integral transforms and quantization and to construct new natural equivariantintegral transforms related to the restriction of unitary representations to symmetricsubgroups. It was shown in [21] (see also [9, 20] and the references therein), thatthe classical Segal–Bargmann transform and the Hall transform on compact Liegroups are both special cases of the restriction principle. Further natural spaces arethe symmetric real forms of Hermitian symmetric spaces [16]. In particular, theequivariant Berezin transform [2, 3, 5, 6, 19] can also been derived in this way.

In [21] the restriction of holomorphic functions on a bounded symmetric do-main was considered. Here the reproducing Hilbert spaces in question correspondedto unitary highest weight representations of the group of bi-holomorphic automor-phisms of the domain. Furthermore, a totally real symmetric subspace H/HK wasconsidered. It was then shown that the restriction map was densely defined witha dense image in L2(H/HK). It was finally shown that the unitary part of therestriction map, i.e., the generalized Segal–Bargmann transform, was a convolutionoperator. No explicit formula for the convolution kernel was derived, but in thecase of the unit disk a formula for the symbol of the operator was given. This wasgeneralized by G. Zhang in [31] where the symbol was worked out for generalbounded symmetric domains.

The aim of this paper is twofold. First, we discuss the restriction principleusing the simple example of the upper half plane and the group SL(2,R), forthe unbounded realization of Hermitian symmetric spaces of tube type. Second,we relate this to classical orthogonal polynomials by showing how the Laguerrepolynomials can be related in a natural way to the holomorphic representationsof SL(2,R) using the unitary part of the restriction map. We recall that there aretwo different ways to view the classical Segal–Bargmann transform. One way is toincorporate the Gaussian function into the measure and consider the correspondingL2-space on R

n or to include the Gaussian into the restriction map. In [21] thebounded realization and the second approach was used to construct the generalizedSegal–Bargmann transform. In this paper we use the more probabilistic view toinclude the density into the measure of the unbounded realization, which has theadvantage of being related to harmonic analysis on symmetric cones and henceEuclidean Jordan algebras. Then the restriction map simply becomes the restrictionof the function. It turns out that the unitary part of the polarization is then exactlythe Laplace transform from the L2-space on the positive reals to the space of holo-

LAGUERRE POLYNOMIALS AND REPRESENTATION THEORY 263

morphic functions [25]. The Laguerre polynomials are then defined as the imageof the inverse Segal–Bargmann transform of a ‘natural’ K-finite orthogonal basisof the realization of the representation in the space of holomorphic functions. Theclassical formula for the generating function and the recursion relations betweenthose polynomials are simple consequences of the representation theory as we willshow later in this note. Now most of the Riemannian symmetric spaces H/HK

can be realized as a totally real symmetric submanifold of a Hermitian symmetricspace. This idea allows us therefore to generalize the Laguerre polynomials – andpossibly other series of orthogonal polynomials – to almost all Riemannian sym-metric spaces. Furthermore the holomorphic representations have a natural familyof raising and lowering operators. Transforming those operators to L2(H/HK)

gives natural creation and annihilation operators. In particular there is a natural‘vacuum’ state in all of these spaces. Evaluating the relations between the ortho-normal basis using those raising and lowering operators gives then correspondingrelations for the generalized polynomials. In this paper we discuss only the caseof SL(2,R) acting on the upper half plane. Our aim is to explain the ideas forreaders that are not specialists in representation theory of semisimple Lie groups.The case of symmetric cones as real forms in tube type domains will be discussedin [4]. The other real forms has been treated in [33].

The Laguerre polynomials Lαn(x) can be defined in many different ways. One

of the oldest definitions is in terms of the generating function

(1 − w)−α−1 exp

(xw

w − 1

)=

∞∑n=0

Lαn(x)w

n, |w| < 1, −1 < α. (0.1)

Another way is to use the formula

Lαn(x) = exx−α

n!dn

dxn(e−xxn+α). (0.2)

The result is that

Lαn(x) =

n∑k=0

�(n + α + 1)

�(k + α + 1)

(−x)k

k!(n − k)! (0.3)

= �(n + α + 1)

�(n + 1)1F1(−n, α + 1; x). (0.4)

Finally the Laguerre polynomials are orthogonal on R+ with respect to the mea-

sure xαe−x dx. In particular, we have∫ ∞

0Lα

n(x)Lαm(x)x

αe−x dx = δn,m�(n + α + 1)

�(n + 1). (0.5)

One could try to use any one of these as a definition. Our point of view is clos-est to (0.5) as we use an orthogonal basis in the Hilbert space of holomorphic


functions and our generalized Segal–Bargmann transform to define an orthogonalbasis {�αn(x)} on the space L2(R+, dµα) where dµα(x) = xα dx. Using (0.2) wethen show that

�αn(x) = e−xLαn(2x). (0.6)

There are other generalizations of the Laguerre polynomials. S. Gindikin de-fined hypergeometric functions for cones in [11]. This gives, in terms of (0.4), a nat-ural generalization of the Laguerre polynomials. But the first systematic treatmentis in the book by J. Faraut and A. Koranyi [10], where generalized Laguerre poly-nomials on cones are defined by an explicit formula analogous to formula (0.3).From that the expression in terms of the Gindikin hypergeometric function wasderived. Finally the Laplace transform of the corresponding Laguerre functions�αn(x) = e−Tr(x)Lα

n(2x) was evaluated, generalizing Equation (0.6). As a corollarythe norm and the orthogonality of those polynomials followed. Their work usesthe theory of Jordan algebras and therefore does not show immediately how togeneralize this to the vector valued case or to symmetric spaces more generalthan the symmetric cones. This is where we think the relation to holomorphicrepresentations shows its advantages.

1. The Holomorphic Representations of SL(2,R)

Denote by G the special linear group SL(2,R) and by GC the complexificationSL(2,C). For

g =(a b

c d

)∈ GC and z ∈ C

let

g · z = az + b

cz + d

if cz + d �= 0. This defines a transitive action of G on the upper half planeH = {z ∈ C | Im(z) > 0}, and a transitive action of

SU(1, 1) ={(

a b

b a

)| |a|2 − |b|2 = 1

}= cGc−1 ⊂ GC

on the unit disk D = {z ∈ C | |z| < 1}. Here

c = 1√2

(1 i

i 1

).

Notice that c: D → H , z �→ c · z, is an isomorphism. This allows us to performcalculations in either the bounded picture using cGc−1 and D or the unboundedpicture using G and H . We notice also that the action of G on H can be lifted to theuniversal covering group G of G. We will denote this action also by (g, z) �→ g · z.


For each α > −1 let dµα be the measure on the positive real half-line givenby dµα(t) = tα dt , where dt is usual Lebesgue measure. Define the holomorphicfunction Kα: H → C by

Kα(z) = �(α + 1)

(−iz)α+1= L(µα)(−iz) =

∫ ∞

0eizt tα dt. (1.1)

As α will be fixed most of the time we simply write K = Kα . Then K(z,w) =K(z − w) is the reproducing kernel for a Hilbert space Hα ⊂ Hol(H ,C) of holo-morphic functions on H on which G acts unitary by a multiplier representation.More specifically, we define Jα(g, z) as the unique lifting of the map

(g, z) → (cz + d)α+1, where g =(a b

c d

),

such that Jα(e, z) = 1. Then the formula Tα(g)F (z) = Jα(g−1, z)−1F(g−1 · z)

defines a unitary action on Hα. By differentiation we get also a representation ofthe Lie algebra sl(2,R):

X · F(z) = d

dtTα(exp tX)F (z)

∣∣∣∣t=0

= d

dt((c(t)z + d(t))−(α+1)F

(a(t)z + b(t)

c(t)z + d(t)

)∣∣∣∣t=0

, (1.2)

where

exp(−tX) =(a(t) b(t)

c(t) d(t)

)and F ∈ H∞

α ,

the space of smooth vectors. Finally we extend the representation by complex lin-earity to sl(2,C). Since the functions in Hα are holomorphic (1.2) is valid for allX ∈ sl(2,C).

If α > 0 then the inner product on Hα is given by the integral

(F | G) = 2α

2π�(α)

∫H

F(z)G(z) yα−1 dx dy. (1.3)

Let Kw(z) = K(z,w). We have

||Kw||2 = (Kw | Kw) = K(w,w) = 2−(α+1)�(α + 1)

yα+1, w = x + iy.

The restriction of this action to K = SO(2) decomposes into one-dimensionalsubspaces generated by the functions

γ 0n,α(z) =

(z − i

z + i

)n

(z + i)−(α+1). (1.4)


Those functions are orthogonal with norm

∥∥γ 0n,α

∥∥2 = �(n + 1) 2−(α+1)

�(n + α + 1). (1.5)

Let

γn,α = iα+1 �(n + α + 1)

�(n + 1)γ 0n,α, n ∈ N0 (1.6)

and define

e+ = 1

2

(−i 11 i

), e− = 1

2

(i 11 −i

), e0 =

(0 −i

i 0

)∈ sl(2,C).

Then [e+, e−] = e0 and we have the following relations:

LEMMA 1.1. Let the notation be as above. Then the following holds:

(1) e+ · F(z) = (α + 1)( z+i2 )F (z) + (z+i)2

2 F ′(z),(2) e− · F(z) = (α + 1)( z−i

2 )F (z) + (z−i)2

2 F ′(z),(3) e0 · F(z) = i((α + 1)zF (z) + (1 + z2)F ′(z)).

In particular:

(1) e+ · γn,α = i(n + α)γn−1,α,(2) e− · γn,α = i(n + 1)γn+1,α ,(3) e0 · γn,α = −(2n + α + 1)γn,α .

2. The Restriction Principle

For a function F defined on the upper half plane let RF(t) = F(it), where t > 0.The map R is known as the restriction map, see [21]. Since the functions in Hα

are holomorphic it follows that R is injective on Hα. For a > 0 let ka = Kia ∈ Hα.Then

Rka(t) = �(α + 1)

(t + a)α+1.

LEMMA 2.1. The linear span of {ka | a > 0} is dense in Hα.Proof. Assume that f ∈ Hα is perpendicular to all ka , a > 0. Then f (ia) = 0

for all a > 0. As f is holomorphic it follows that f = 0. ✷Let L2(R+, dµα) be the space of measurable functions square integrable with

respect to the measure dµα .

LEMMA 2.2. With the notation as above we have Rka ∈ L2(R+, dµα), for allα > −1.


Proof. Let α > −1. Then

‖Rka‖2 =∥∥∥∥ �(α + 1)

(t + a)α+1

∥∥∥∥2

= �(α + 1)2∫ ∞

0

tα

(t + a)2(α+1)dt

= �(α + 1)2

aα+1

∫ ∞

0

tα

(t + 1)2(α+1)dt (t → at)

= �(α + 1)2

aα+1B(α + 1, α + 1),

where B is the Beta function. ✷Let L denote the Laplace transform. For f ∈ L2(R+, dµα), Lf is defined

because t �→ e−st ∈ L2(R+, dµα) for all α > −1. Furthermore,

|Lf (s)| �∥∥e−st

∥∥ ‖f ‖ � ks− α+12 .

From this it follows that sαLf (s) in turn has a Laplace transform.

LEMMA 2.3. The set {Rka | a > 0} is dense in L2(R+, dµα).Proof. Let f ∈ L2(R+, dµα) and suppose f is orthogonal to all ka, a > 0.

Then by (1.1)

0 = (f | ka) =∫ ∞

0f (t)

�(α + 1)

(t + a)α+1tα dt

=∫ ∞

0f (t)L(e−assα)(t)tα dt

=∫ ∞

0L(f (t)tα)(s)e−assα ds

= L(sαL(taf (t)))(a).

From this and the injectivity of the Laplace transform it follows that f = 0. ✷By the previous lemmas it follows that the restriction map R: Hα →L2(R+, dµα)

is densely defined and has dense range. It is easily seen to be closed. Hence, wecan polarize RR∗ to obtain a unitary map Uα: L2(R+, dµα) → Hα . As α will befixed most of the time we will simply write U instead of Uα. Let f ∈ L2(R+, dµα).Then

RR∗f (y) = R∗f (y)

= (R∗f | K(·, iy))Hα

= (f | K(·, iy))L2


=∫ ∞

0f (x)K(ix, iy) dµα(x)

= �(α + 1)∫ ∞

0f (x)

xα

(x + y)α+1dx

=∫ ∞

0f (x)L(tαe−ty)(x)xα dx

=∫ ∞

0tαe−tyL(xαf (x))(t) dt (because of the symmetry of L)

= L(tαL(xαf (x)))(y).

Define Pf (y) = L(xαf (x))(y). Then P 2 = RR∗. As (P (f ), f ) � 0 for f ∈L2(R+, dµα) it follows that

√RR∗ = P . Polarization now yields an operator

U : L2(R+, dµα) → Hα such that R∗ = UP .

THEOREM 2.4. The unitary map U : L2(R+, dµα) → Hα is given by

Uf (z) =∫ ∞

0eizt f (t) dµα(t).

Proof. We have by definition R∗ = UP . Hence, U ∗R∗ = P . Taking adjointswe get RU = P . Thus if z = iy with y > 0 it follows from this that the restrictionof U is P and

Uf (iy) =∫ ∞

0e−ytf (t)tα dt

=∫ ∞

0eiztf (t) dµα(t).

Since Uf is holomorphic the claim follows. ✷Remark 2.5. The map U is the well-known unitary intertwining operator first

obtained by Rossi and Vergne [25]. Hence, the restriction principle gives a newproof of the unitarity of this map.

COROLLARY 2.6. Assume that α > 0. Then U ∗: Hα → L2(R+, dµα) is givenby the integral operator

U ∗F(t) = 2α

2π�(α)

∫∫H

F(z)e−iztyα−1 dx dy.

Proof. Denote the right-hand side by G(t). We then have

(U ∗F | f ) = (F | Uf )Hα

= 2α

2π�(α)

∫∫H

F(z)Uf (z)yα−1 dx dy


= 2α

2π�(α)

∫∫H

F(z)

∫ ∞

0eizt f (t)tα−1dt yα−1 dx dy

= 2α

2π�(α)

∫∫∫F(z)e−iztyα−1 dx dy f (t)tα−1dt

= (G, f )L2(R+,dµα). ✷Recall that we started with a unitary representation Tα of G (or G) on Hα.

As U ∗: Hα → L2(R+, dµα) is an unitary isomorphism we can carry this represen-tation over to L2(R+, dµα) by tα(g) = UTα(g)U

∗.In the next proposition we calculate the sl(2,C)-action of the basis e0, e+, e−

on L2(R+, dµα).

PROPOSITION 2.7. Suppose f ∈ L2(R) is twice differentiable. Then

(1) e0f (t) = (tD2 + (α + 1)D − t)f (t),(2) e+f (t) = −i

2 (tD2 + (2t + (α + 1))D + (t + α + 1))f (t),(3) e−f (t) = −i

2 (tD2 − (2t − (α + 1))D + (t − (α + 1))f (t).

We first state and prove a useful lemma.

LEMMA 2.8. For f ∈ L2(R+, dµα) be differentiable. Then the following hold:

(1) izUf (z) = −U(f ′)(z) − αU(f

t)(z),

(2) i(Uf )′(z) = −U(tf )(z),(3) iz2U(f )′(z) = −z2U(tf )(z) = −iz(U(tf ′)(z) + (α + 1)U(f )(z)).

Proof. (1) We have

izUf (z) =∫ ∞

0f (t)iz eizt tα dt

=∫ ∞

0f (t)tα

d

dteizt dt

= f (t)tαeizt |∞0 −∫ ∞

0eizt

d

dt

(f (t)tα

)dt

= −∫ ∞

0eizt (f ′(t)tα + αf (t)tα−1) dt

= −U(f ′)(z) − αU

(f

t

)(z).

For (2) we calculate:

iU(f )′(z) = i

∫d

dzeizt f (t)tα dt

= −∫

eizt tf (t)tα dt

= −U(tf (t))(z).


(3) From (1) and (2) we get:

iz2U(f )′(z) = −z2U(tf )(z)

= (iz)(−U((tf )′)(z) − αU(f )(z))

= −iz(U(f )(z) + U(tf ′)(z) + αU(f )(z))

= −iz(U(tf ′)(z) + (α + 1)U(f )(z)). ✷Proof of Proposition 2.7. We have

e0Uf (z) = i((α + 1)zUf (z) + (1 + z2)U(f )′(z))= i(α + 1)zUf (z) + iU(f )′(z) − iz(U(tf ′)(z) + (α + 1)Uf (z))

= iU(f )′(z) − izU(tf ′)(z)= −U(tf )(z) + U((tf ′)′) + αU(f ′)= U(−tf + f ′ + tf ′′ + αf ′)(z)= U(tf ′′ + (α + 1)f ′ − tf )(z).

This proves (1). The proof for e+ is similar. First we calculate

(z + i)Uf (z) = zUf (z) + iUf (z) = iU

(f ′ + f + α

f

t

).

Also,

(z + i)2U(f )′(z) = (z + i)2iU(tf )(z)

= i(z + i)iU((tf )′ + tf + αf )

= −(z + i)U(tf ′ + tf + (α + 1)f ).

Using these formulae we obtain

e+Uf (z) = 12 ((α + 1)(z + i)Uf (z) + (z + i)2U(f )′(z))

= 12 ((α + 1)(z + i)Uf (z) − (z + i)U(tf ′ + tf + (α + 1)f ))

= − 12 (z + i)U(tf ′ + tf )

= − 12 iU((tf ′ + tf )′ + tf ′ + tf + α(f ′ + f ))

= −i

2U(f ′ + tf ′′ + f + tf ′ + tf ′ + tf + α(f ′ + f ))

= −i

2U(tf ′′ + (2t + (α + 1))f ′ + (t + (α + 1))f ).

This proves (2). Formula (3) is done similarly. ✷The reuse of notation should not cause confusion. Since these operators are

unitarily equivalent to those on Hα they form an sl(2,C) algebra of differentialoperators. Since e+ lowers the K-types and, in particular, is locally nilpotent on


the space of K-finite vectors, we can view it as an annihilation operator. Similarly,e− raises the K-types and we can view it as a creation operator. In the next sectionwe determine the K-types in L2(R+, dµα).

3. The Laguerre Polynomials and K-Finite Vectors

Let n � 0 and α > −1. Then we can define a series of orthogonal functions inL2(R+, dµα) by �αn(t) := U ∗(γn,α) where γn,α is as in (1.6).

THEOREM 3.1. Let α > −1 and n � 0. Then �αn(t) = e−tLαn(2t).

Proof. Recall the formula (0.2) from the introduction

Lαn(x) = exx−α

n!dn

dxn(e−xxn+α).

Hence

U(e−xLαn(2x))(z) =

∫ ∞

0eizxe−xxαLα

n(2x) dx

= 1

2α+1

∫ ∞

0exp

(iz − 1

2u

)uαLα

n(u) du

= 1

2α+1�(n + 1)

∫ ∞

0exp

(iz + 1

2u

)dn

dun

(e−uun+α

)du

= (−1)n(iz + 1)n

2α+n+1�(n + 1)

∫ ∞

0exp

(iz − 1

2u

)un+α du

= (−1)nin(z − i)n

�(n + 1)

∫ ∞

0e(iz−1)xxn+α dx

= (−1)nin�(n + α + 1)(z − i)n

�(n + 1)(−iz + 1)n+α+1

= iα+1 �(n + α + 1)

�(n + 1)

(z − i

z + i

)n

(z + i)−(α+1)

= γn,α. ✷We now derive the formula (0.1) for the generating function of the Laguerre

polynomials. We will reformulate the formula so that it becomes the Schwarz ker-nel for the unitary intertwining map of two different realizations of the holomorphicrepresentation Tα of G. For the related expansion of spherical functions in terms oforthogonal polynomials; see [32] and [33].

THEOREM 3.2. Let α > −1. The following formula holds

(1 − w)α+1e− 1+w1−w x =

∞∑n=0

wn�αn(x), |w| < 1.


Notice that multiplying both sides by ex gives (0.1). To clarify the signifi-cance of the above formula we consider the space Hα(D) of holomorphic func-tions on the unit disk D = {w ∈ C | |w| < 1} with the reproducing kernel2−(α+1)/�(α + 1)(1 − wz)−(α+1). The Cayley map c induces a unitary map fromHα(D) onto Hα , which we also denote by c

cf (z) = f

(z − i

z + i

)(z + i)−(α+1).

The polynomials wn, n = 0, 1, . . . , form an orthogonal basis of Hα(D) and wehave c(wn) = γ 0

n,α. Let

V = U ∗c: Hα(D) �→ L2(R+, dµα).

Thus V is a unitary opertor. Let en(w) = wn/‖wn‖ be the orthonormal basis ofHα(D). Thus {V en} is an orthonormal basis of L2(R+, dµα). Theorem 3.2 cannow be written as:

COROLLARY 3.3. Let α > −1. With the above notation we have

(1 − w)α+1e− 1+w1−w

x = 2−(α+1)i(α+1)∞∑n=0

en(w) ⊗ V en(x).

Thus, up to the constant 2−(α+1)i(α+1), the right-hand side gives the Schwarzkernel for the unitary operator V . We now prove Theorem 3.2, which at the sametime gives a proof of the corollary.

Proof. Let qm(x) be the coefficient in the expansion

(1 − w)α+1e− 1+w1−w x =

∞∑m=0

wmqm(x).

We prove that qm(x) = �αm(x). Notice that qm(x) can be obtained by

qm(x) = 1

m!(

d

dw

)m

(1 − w)α+1e− 1+w1−w x |w=0,

and that qm(x) is a product of e−x and a polynomial of x and thus is in L2(R+, dµα).Let further

pn(x) := U ∗(γ 0n,α)(x),

which is a constant multiple of �αn(t),

pn(x) = 2α+1i−(α+1)‖γ 0n,α‖2�αn(x).

Thus γ 0n,α = U(pn), and in view of Theorem 2.4 this reads as follows

γ 0n,α(z) =

∫ ∞

0eizxpn(x) x

α dx.


We perform the inverse Cayley transform, z = i(1 + w)/(1 − w). So that

2−(α+1)i−(α+1)wn(1 − w)α+1 =∫ ∞

0e− 1+w

1−wxpn(x) dµα(x),

or

2−(α+1)i−(α+1)wn =∫ ∞

0(1 − w)−(α+1)e− 1+w

1−wxpn(x) dµα(x).

Then we perform the differentiation 1/m!(d/dw)m on the above formula and eval-uate at w = 0. This gives

2−(α+1)i−(α+1)δn,m =∫ ∞

0qm(x)pn(x) dµα(x)

and the complex conjugate is

2−(α+1)i(α+1)δn,m =∫ ∞

0qm(x)pn(x) dµα(x),

since qm(x) is clearly a real-valued function. However, the functions {pn(x)} forman orthogonal basis of the space L2(R+, dµα), thus qm is a constant multiple of pm

qm = 2−(α+1)i(α+1) pm

‖pm‖2= 2−(α+1)i(α+1) pm

‖γ 0m,α‖2

.

Using the formula for the definition of γm,α and Theorem 3.1 we find thatqm = �αm. ✷

Lemma 1.1 of Section 1 and Proposition 2.7 now combine to give the action ofe0, e+, and e−, on the Laguerre functions. A further calculation gives the classicalrecursion relations for the Laguerre polynomials.

THEOREM 3.4. We have the following relations between the Laguerre func-tions �αn(x).

(1) (tD2 + (α + 1)D + (2n + α + 1 − t)�αn = 0,(2) (tD2 + (2t + (α + 1))D + (t + α + 1))�αn = −2(n + α)�αn−1,(3) (tD2 − (2t − (α + 1))D + (t − (α + 1)))�αn = −2(n + 1)�αn+1.

These formulas, in turn, imply the following recursion relations for the Laguerrepolynomials Lα

n .

(1) (tD2 + (α − t + 1)D + n)Lαn(t) = 0,

(2) tDLαn(t) = nLα

n(t) − (n + α)Lαn−1(t),

(3) tDLαn(t) = (n + 1)Lα

n+1(t) − (n + α + 1 − t)Lαn(t).


4. Other Recursion Relations

Various recursion relations occur among the Laguerre polynomials in the α para-meter by observing simple relations among the K-finite vectors in Hα. For example,notice that (z − i)γ ◦

n,α = γ ◦n+1,α−1 if α > 0. This observation allows us to de-

fine a dense operator M: Hα → Hα−1 given by Mf (z) = (z − i)f (z). Theunitarity of the operators Uα and Uα−1 then induces a corresponding operatorm: L2(R+, dµα) → L2(R+, dµα−1) such that the following diagram commutes:

HαM Hα−1

L2(R+, dµα)

Uα

m L2(R+, dµα−1)

Uα−1

LEMMA 4.1. Assume α > 0. The operator m is given by the formula m =i(tD + (α − t)).

Proof.

(z − i)Uαf =∫

zeiztf (t)tαdt − iUαf

= −i

∫d

dt(eizt )f (t)tαdt − i

∫eizt f (t)tαdt

= i

∫eizt (f ′(t)tα + αtα−1f (t))dt − i

∫eiztf (t)t tα−1dt

= iUα−1(tf′(t) + αf (t)) − iUα−1(f (t)t)

= i(Uα−1(tf′ + (α − t)f ).

PROPOSITION 4.2. Assume α > 0. For each n we obtain

(tD + (α − t))�αn = (n + 1)�α−1n+1 .

Proof. This follows from the lemma and the fact that Uα(�αn) = γn,α . ✷

In a similar way, we observe that (z + i)γ ◦n,α = γ ◦

n,α−1. The operator M: Hα →Hα−1 given by multiplication by (z + i) induces an operator m: L2(R+, dµα) →L2(R+, dµα−1). Reasoning as we did above, we obtain the following proposition:

PROPOSITION 4.3. Assume α > 0. For each n we obtain

(tD + (α + t))�αn = (n + α)�α−1n .

As a final example, we note that 2iγ ◦n,α = γ ◦

n,α−1 +γ ◦n+1,α−1. If M: Hα → Hα−1

is multiplication by 2i then we obtain


PROPOSITION 4.4. For each n and α we obtain

2t�αn = (n + α)�α−1n − (n + 1)�α−1

n+1 .

We now summarize these results in one theorem. We include the recursion rela-tions for the Laguerre polynomial Lα

n . They are implied by the recursion relationsfor the Laguerre functions �αn . We have used the recursion formulas given in 3.4 tofurther simplify the final formulas.

THEOREM 4.5. We have the following recursion relations among the Laguerrefunctions.

(1) (tD + (α − t))�αn = (n + 1)�α−1n+1 ,

(2) (tD + (α + t))�αn = (n + α)�α−1n ,

(3) 2t�αn = (n + α)�α−1n − (n + 1)�α−1

n+1 .

These, in turn, imply the following relations among the Laguerre polynomi-als Lα

n .

(1) tLαn = (n + α + 1)Lα−1

n − (n + 1)Lα−1n+1 ,

(2) tLαn = (n + α)Lα−1

n−1 − (n − t)Lα−1n ,

(3) Lα−1n = Lα

n − Lαn−1.

Final Remark 4.6. As one can see these recursion relations (Theorem 3.4 andTheorem 4.5) have been derived with a thorough knowledge of the representationtheory assumed. This point of view can be reversed, however. In the recent article[18] by Kostant (see also [30]) the recursion formulas for the Laguerre functionsare assumed to be known. From there the Lie algebra action is derived which,by a theorem of Nelson, integrates to a representation of the universal coveringof SL(2,R). Many of the basic formulas presented there are different from whatwe present due to the difference in the measure on L2(0,∞). In the article byJakobsen and Vergne [17], tensor products of highest-weight representations Hα

with themselves were studied. The resulting representations are decomposable intoa series of representations each of which are again a highest weight representation.This implies various relations among the K-finite vectors and, hence, recursionrelations among the Laguerre polynomials. We have not pursued this idea here.

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Documents

Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2,R)