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ContemporaryMathematics 3 1 Special Functions in Minimal Representations 0 2 Toshiyuki Kobayashi n a J Dedicated to Igor Frenkel on the occasion of his 60th birthday with great admire 3 2 Abstract. MinimalrepresentationsofarealreductivegroupGarethe‘small- ] est’irreducibleunitaryrepresentationsofG. Wediscussspecialfunctionsthat T ariseintheanalysisofL2-modelofminimalrepresentations. R . h t a 1. Introduction m An irreducible unitary representation of a real reductive Lie group G is called [ minimal if its infinitesimal representation is annihilated by the Joseph ideal [11] 1 in the enveloping algebra. Loosely speaking, minimal representations of G are the v ‘smallest’ infinite dimensional unitary representations. 5 0 TheWeilrepresentation,knownforaprominentroleinnumbertheory(e.g.the 5 theta correspondence), provides minimal representations of the metaplectic group 5 Mp(n,R). The minimal representation of a conformal group SO(4,2) appears in . 1 mathematical physics, e.g., as the bound states of the hydrogen atom, and inci- 0 dentally as the quantum Kepler problem. In these classical examples the repre- 3 sentationsarehighestweightmodules, however,for moregeneralreductive groups, 1 minimal representations (if exist) may not be highest weight modules, see a pio- : v neering work of Kostant [21] for SO(4,4). Xi In the last decade I have been developing a geometric and analytic theory of minimalrepresentationswithmycollaborators,S.BenSa¨ıd,J.Hilgert,G.Mano,J. r a Mo¨llers,B.Ørsted,andM.Pevzner,see[1, 6, 7, 8, 9, 12, 13, 15, 16, 17, 18, 19, 20]. Amongall,inthispaper,wefocuson‘specialfunctions’thatarisenaturallyin the L2-model of minimal representations. Needless to say, the interaction between specialfunctionsandgrouprepresentationshasalonghistoryandthereisextensive literatureonthis subject. Anew feature inour setting forminimalrepresentations isthatthe representationofthe groupis realizedonthe HilbertspaceL2(Ξ) where the dimension of a manifold Ξ (see (2.1), or more generally Section 2.3 for the definition of Ξ) is strictly smaller than the dimension of any nontrivial G-space in most cases. This means that G cannot act geometrically on Ξ but there is 2010 Mathematics Subject Classification. Primary22E30;Secondary22E46, 33C45. Theauthor waspartiallysupportedbyGrant-in-AidforScientificResearch(B)(22340026), JapanSocietyforthePromotionofSciences. (cid:13)c0000(copyrightholder) 1 2 TOSHIYUKIKOBAYASHI a natural action of G on L2(Ξ). As a result, the Casimir element of a compact subgroup acts as a fourth-order differential operator. The indefinite orthogonal groupG=O(p+1,q+1) is the most interesting for this purpose in the sense that thegroupGitselfcontainstwoparameterspandq,andweshallhighlightthiscase by giving occasionally some perspectives to other reductive groups. Acknowledgements: I first met Igor Frenkel when I visited Yale University in 2009to give a colloquium talk. It was then a surprising pleasure that Igor told me his recent theory on quaternionic analysis [3, 4, 5] with Libine uses some of my work [17, 19] on geometric analysis of minimal representations, which encourages me to develop further the analytic theory of minimal representations. IwouldliketothanktheorganizersP.Etingof,M.Khovanov,A.KirillovJr.,A. Lachowska,A.Licata,A.SavageandG.Zuckermanfortheirhospitalityduringthe stimulating conference “Perspectives in Representation Theory” in honor of Prof. Igor Frenkel’s 60th birthday at Yale University, 12–16 May 2012. Thanks are also due to an anonymous referee for his/her careful comments. 2. A generalization of the Fourier transform 2.1. Algebraic characterization of Fourier transforms. We begin with analgebraiccharacterizationoftheEuclideanFouriertransform Rn. LetQj :=xj F be the multiplication operators by coordinates, and P := 1 ∂ . Then we have: j √ 1∂xj − Proposition 2.1. Any continuous operator A on L2(Rn) satisfying A Q =P A, A P = Q A on (Rn) (1 j n) j j j j ◦ ◦ ◦ − ◦ S ≤ ≤ isascalar multipleoftheEuclidean Fourier transform Rn. Inparticular, anysuch F continuous operator A is unitary up to scaling. In place of Rn, let us consider the isotropic cone (2.1) Ξ:= x Rp+q 0 :x2+ +x2 x2 x2 =0 , { ∈ \{ } 1 ··· p− p+1−···− p+q } equipped with a measure dµ= 1rp+q 3drdωdη in the bipolar coordinates: 2 − (2.2) R+ Sp−1 Sq−1 ∼ Ξ, (r,ω,η) (rω,rη). × × → 7→ Remark 2.2. This cone Ξis a specialcase ofthe Lagrangiansubmanifold ofa minimal real coadjoint orbit, denoted by the same letter Ξ, given in Theorem 2.10 where we deal with more general reductive groups. Recallfrom[17]thatthe fundamental differential operators R (1 j p+q) j ≤ ≤ on Ξ are mutually commuting operators which are obtained as the restriction of the tangential differential operators ∂ ε x (cid:3) (2E+p+q 2) j j − − ∂x j to Ξ, where ε = 1 (1 j p); = 1 (p+1 j p+q), (cid:3) := p+qε ∂2 (the j ≤ ≤ − ≤ ≤ a=1 a∂x2a Laplacian on Rp,q) and E := p+qx ∂ (the Euler operator). Then we have a=1 a∂xa P Theorem 2.3 ([17, TheoPrem 1.2.3]). Suppose p+q is even, 4. Then there ≥ exists a unitary operator on L2(Ξ) satisfying the following relation for A: Ξ F (2.3) A◦Qj =Rj ◦A, A◦Rj =Qj ◦A on Cc∞(Ξ) (1≤j ≤p+q). SPECIAL FUNCTIONS IN MINIMAL REPRESENTATIONS 3 Conversely, any continuous operator A satisfying (2.3) is a scalar multiple of the unitary operator . Ξ F 2.2. Unitary inversion operator Ξ. The similar nature of Rn and Ξ F F F indicatedinProposition2.1andTheorem2.3isderivedfromthecommonfactthat they arise as the unitary inversion operators in the L2-model (Schrödinger model) of minimal representations of real reductive groups Mp(n,R) and O(p+1,q+1), respectively. To see some more details of Theorem 2.3, let n, n, and l be the Lie algebras generatedbytheoperatorsQ ,R ,and[Q ,R ](1 i,j p+q). Theng:=n+l+n i j i j is isomorphic to o(p+1,q+1), and p=l+n (o≤(p,q)≤+R)⋉Rp+q is a maximal ≃ parabolic subalgebra of g. For p+q even, 4, we proved in [19] that there exists an irreducible unitary ≥ representation π of the group G := O(p+1,q+1) on the Hilbert space L2(Ξ) of which the infinitesimal representationis given by Q (the action of n) and R (the j j action of n), see also [17, Chapter 1]. We set I O (2.4) w := p G. O Iq ∈ (cid:18) − (cid:19) Geometrically,p is the Lie algebraofthe conformaltransformationgroup(O(p,q) R )⋉Rp,q oftheflatpseudo-RiemannianEuclideanspaceRp,q,andw inducesthe· >0 conformal inversion of Rp,q by the Mo¨bius transform. The unitary operators π(g) are of simple form if g G belong to the maximal ∈ parabolic subgroup P with Lie algebra p, namely, they are given by the multiplication of certain elementary functions on Ξ and the translations coming from the geometricactionoftheLevisubgroupofP onΞ(see[17,Chap.2,Sect.3]). Inview of the Bruhat decomposition G=P PwP, ∐ it is enough to find an explicit formula of the unitary operator π(w) in order to giveaglobalformulaofthe G-actiononL2(Ξ). We callπ(w) the unitary inversion operator, and set (2.5) :=π(w). Ξ F We initiated in a series of papers [15, 16, 17] the following: Program 2.4 ([17, Program 1.2.5]). Use the unitary inversion operator Ξ F for minimal representations as an analog of the Euclidean Fourier transform Rn, F and develop a theory of ‘Fourier analysis’ on Ξ. In the classical Schrödinger model of the Weil representation of the metaplectic group Mp(n,R) on L2(Rn), the unitary inversion operator is nothing but the Euclidean Fourier transform Rn (up to scalar of modulus one), see Example 3.1. We note that Mp(n,R) and OF(p+1,q+1) with p+q even are simple Lie groups of type C and D, respectively. The first stage of Program 2.4 is to establish a framework of the L2-model (Schrödinger model) of minimal representations, and to introduce the unitary in- versionoperator with analgebraiccharacterizationsuchasTheorem2.3. In [8] Ξ F we gavesucha model anddefined by using Jordanalgebras,see Section2.3. In Ξ F this case Ξ is a Lagrangian subvariety of a minimal nilpotent coadjoint orbit and the resulting representations on L2(Ξ) include a slightly wider family of unitary 4 TOSHIYUKIKOBAYASHI representations than minimal representations (e.g. the full complementary series representations of O(n,1)). The second stage is to solve the following: Problem 2.5. Find an explicit formula of the integral kernel of . Ξ F We will discuss Problem 2.5 in Section 3. It is noteworthy that I. Frenkel and M. Libine have developed their original theory on quaterionic analysis in a series of papers [3, 4, 5] from the viewpoint of representation theory of the conformal group SL(2,HC) SL(4,C) and its real forms, and have demonstrated a close ≃ connection between minimal representations of various O(p,q)’s and quaternionic analysis. Forinstance, the explicitformulaof forO(3,3)whichwasobtainedin Ξ Kobayashi–Mano[17] is used in [5] for the stuFdy of the key operator (Pl in their R notation) in the analysis of the space HR of split quaternions. The third stage and beyond will be based on the algebraic property (Theorem 2.3) and analytic property (Problem 2.5) of the unitary inversion operator . Ξ F Among various, potentially interesting directions of the ‘Fourier analysis’ on Ξ, here are some few topics: AtheoryofholomorphicsemigroupswasgiveninHowe[10]forMp(n,R) • and in Kobayashi–Mano[16] for SO(n+1,2). A deformationtheory of the Euclidean Fourier transform Rn [1], e.g. an • F interpolation between Rn and the unitary inversion operator of SO(n+ F 1,2). A generalization of the classical Bargmann–Segal transform. See [9] in • the case G/K is of tube type. Stage 1 already includes a solution for the Plancherel-type theorem of . A Ξ F natural but open question would be a Paley–Wiener type theorem of : Ξ F Question 2.6. Find an explicit characterization of (C (Ξ)). FΞ c∞ Another important space of functions is an analog of Schwartz functions. For this we may consider: Definition 2.7 (Schwartz space on Ξ). Let (Ξ) be the Fréchet space of S smooth vectors of the unitary representation of G on L2(Ξ). This definition makes sense in a more general setting (see Theorem 2.10). By the general theory of unitary representations, we have: Proposition 2.8. induces automorphisms of the Hilbert space L2(Ξ) and Ξ F the Fréchet space (Ξ). S Ξ :L2(Ξ) ∼ L2(Ξ) (Plancherel type theorem), F → ∪ ∪ (Ξ) ∼ (Ξ) (Paley–Wiener type theorem). S → S The following question is also open: Question 2.9. Find an explicit characterization of (Ξ). S SPECIAL FUNCTIONS IN MINIMAL REPRESENTATIONS 5 2.3. Schro¨dinger model of minimal representations. Suppose that V is a real simple Jordan algebra. Let G and L be the identity components of the conformal group and the structure group of the Jordan algebra V, respectively. Then the Lie algebra g is a real simple Lie algebra and has a Gelfand–Naimark decomposition g = n+l+n, where n V is regarded as an Abelian Lie algebra, ≃ l str(V) the structure algebra, and n acts on V by quadratic vector fields. ≃ Let OGR ( g ) be a (real) minimal nilpotent coadjointorbit. By identifying g min ⊂ ∗ with the dual g , we consider the intersection ∗ Ξ:=V OGR , ∩ min which is a Lagrangiansubmanifold of the symplectic manifold OGR endowed with min the Kirillov–Kostant–Souriau symplectic form. There is a natural L-invariant RadonmeasureonΞ,andwewriteL2(Ξ)fortheHilbertspaceconsistingofsquare integrable functions on Ξ. Theorem 2.10 (Schrödinger model [8]). Suppose V is a real simple Jordan algebra such that its maximal Euclidean Jordan algebra is also simple. Among all such Jordan algebras V, we exclude the case where V Rp,q with p+q odd (see ≃ Examples 2.13 and 2.14). 1) For an appropriate finite covering group G of G there exists a natural unitary representation of G on L2(Ξ). It is irreducible if and only if Ξ is connected. 2) The Gelfand–Kirillov dimension of π attaeins its minimum among all infinite dimensional represeentations of G, i.e. DIM(π)= 1dimOGR . 2 min 3) The annihilator of the differential representation dπ is the Joseph ideal in the enveloping algebra U(gC) if V isesplit and gC is not of type A. The simple Lie algebras g that appear in Theorem 2.10 are categorized into four cases as below: (2.6) sl(2k,R),so(2k,2k),so(p+1,q+1),e , 7(7) (2.7) sp(k,R),su(k,k),so (4k),so(2,k),e , ∗ 7( 25) − (2.8) sp(k,C),sl(2k,C),so(4k,C),so(k+2,C),e (C), 7 (2.9) sp(k,k),su (4k),so(k,1). ∗ Example2.11. IfV isaEuclideanJordanalgebra,thenGistheautomorphism group of a Hermitian symmetric space of tube type and the corresponding Lie algebra g is listed in (2.7). In this case Ξ consists of two connected components, andtheresultingrepresentationπisthedirectsumofanirreducibleunitaryhighest weight module and its dual. Remark 2.12. Inthecase(2.9)thecomplexminimalnilpotentorbitOGC does min not meet the real form g, and there does not exist an admissible representation of any Lie group with Lie algebra g. In particular, the representation π in Theorem 2.10 is not a minimal representation but still one of the ‘smallest’ infinite dimensional representations in the sense that the Gelfand–Kirillov dimension attains its minimum. Example 2.13. Let V =Sym(n,R). Then g=sp(n,R) and (2.10) Ξ= X M(n,R):X =tX,rankX =1 . { ∈ } 6 TOSHIYUKIKOBAYASHI Let Ξ := X V OGR : TraceX > 0 . Via the double covering map (folding + { ∈ ∩ min } map) Rn 0 Ξ , v vtv + \{ }→ 7→ wecanidentifytherepresentationonL2(Ξ )withtheevenpartoftheSchrödinger + model on L2(Rn) of the metaplectic group Mp(n,R) [2, 10]. See [8] for the re- alization of the odd part of the Weil representation in the space of sections for a certain line bundle over Ξ . + Example 2.14. We define a multiplication on Rp+q =R Rp+q 1 by − ⊕ p p+q (x ,x) (y ,y ):=(x y x y + x y ,x y +y x). 1 ′ 1 ′ 1 1 i i i i 1 ′ 1 ′ · − i=2 i=p+1 X X TheresultingJordanalgebraisdenotedbyRp,q (byalittleabuseofnotation). Itis asemisimpleJordanalgebraofranktwo,anditsconformalalgebraiso(p+1,q+1). SupposenowV =Rp,q withp+q even. ThenΞinTheorem2.10coincideswiththe isotropic cone given in (2.1). For q = 1, V is an Euclidean Jordan algebra, and Ξ consists of two connected components according to the sign of the first coordinate x , i.e.the pastandfuture cones. Forp,q 2, V is non-Euclidean,Ξ is connected, 1 ≥ and our representation π on L2(Ξ) is the same as the Schrödinger model of the minimal representation of O(p+1,q +1) constructed in [19, Part III], which is neither a highest nor a lowest weight module. Remark 2.15. There is no minimal representation for any group with Lie algebra o(p+1,q+1) with p+q odd, p, q 3 (see [23, Theorem 2.13]). ≥ 3. Unitary inversion operator Ξ F BytheSchwartzkerneltheorem,theunitaryinversionoperator canbegiven Ξ F by a distribution kernel K(x,y) (Ξ Ξ), namely, ′ ∈D × u(x)= K(x,y)u(y)dµ(y) for all u C (Ξ). FΞ ∈ c∞ ZΞ Problem 2.5 asks for an explicit formula of K(x,y). In the setting of Theorem 2.10, we can generalize the definition (2.5) of by taking w to be a lift of the Ξ F conformalinversionon V, see [9]. So far, Problem 2.5 has been solvedfor minimal representations in the following two cases: Case A. G is the biholomorphic transformation group of a Hermitian symmetric space of tube type ([9]). Case B. G=O(p+1,q+1) (Theorem 3.3). Case A includes the following earlier results: Example 3.1. 1) G=Mp(n,R), Ξ=Rn, π = the Weil representation. c K(x,y)= (2π)n2 e−√−1hx,yi In this case Ξ is the Euclidean Fourier transform Rn up to a phase F F factor c with c =1, see [2]. | | 2) G = SO(p+1,2), Ξ is the light cone (q = 1 in (2.1)), π = the highest weight representation of the smallest Gelfand–Kirillov dimension and of the smallest K-type. K(x,y)=cJp−3(2 2 x,y ) 2 h i p e SPECIAL FUNCTIONS IN MINIMAL REPRESENTATIONS 7 where J (t) := (t) λJ (t) is a renormalization of the J-Bessel function λ 2 − λ ([15]). e 3.1. Mellin–Barnes type integral expression. In[17]webroughtanidea of the Radon transform in the analysis of the unitary inversion operator for Ξ F minimal representations. Recall that the Euclidean Fourier transform Rn can be F writtenasthecompositionoftheone-dimensionalFouriertransformandtheRadon transform (plane wave decomposition). We can generalize this decomposition to the unitary inversion operator for the minimal representations and some small Ξ F representationsonL2(Ξ)givenbyTheorem2.10,namely,thereexistsadistribution Φ(t)ofonevariablesuchthatthedistributionkernelK(x,y)of isofthefollowing Ξ F form: (3.1) K(x,y)=Φ( x,y ), h i where , is some (natural) bilinear form of the ambient space V. Thus Problem h i 2.5 reduces to find a formula of Φ(t). Example 3.2. In Example 3.1 (2) we have seen Φ(t) = cJp−3(2√2t) when 2 G=O(p+1,2). Therefore, reduces to the Hankel transform composed with a Ξ F ‘Radon transform’ on Ξ in this case. e TheformulaofΦ(t)ismoreinvolvedforG=O(p+1,q+1)withp,q 2asthe ≥ corresponding minimal representation is not a highest weight module and Φ(t) is not always locally integrable (see Question 3.4 below). An explicit formula of Φ(t) inthiscasecanbegivenintermsof‘Besseldistributions’[17,Theorem5.1.1]. Here we give an alternative expression of Φ(t), namely, by using a distribution-valued Mellin–Barnes integral. We define a distribution of t with meromorphic parameter λ by Γ( λ) b(λ,t):= − (2t)λ. Γ(λ+ p+q 1) + 2 − HeretheRiesz distribution(2t)λ isdefinedasalocallyintegrablefunctiononR by + (2t)λ t>0 (2t)λ = + (0 t 0 ≤ for Reλ> 1, and is extended as a distribution by the meromorphic continuation on λ C. L−et m:= 1(p+q 4), and L be a contour starting at γ i , passes ∈ 2 − m − ∞ the real axis between ( m 1, m) and ends at γ+i when γ > 1. We define − − − ∞ − distributions Φp,q(t) by a distribution-valued Mellin–Barnes integral: b(λ,t)dλ (Case A-1), L0 Φp,q(t):= b(λ,t)dλ (Case B-1), RLm RLm(tba(nλ,πtλ) + bs(iλn,−πλt))dλ (Case B-2), according to the followingRthree cases: Case A-1. p=1 or q =1, Case B-1. p,q >1 and both odd, Case B-2. p,q >1 and both even. ThenΦp,q(t)isindependentofthechoiceofthe contourandγ undertheabove mentioned constraints. Let , be the (positive definite) inner product on Rp,q. Then we have h i 8 TOSHIYUKIKOBAYASHI Theorem 3.3 ([17, 6.2]). For G = O(p+1,q+1) with p+q even, 4, the § ≥ kernel K(x,y) of the unitary inversion is given by Ξ F K(x,y)=c Φp,q( x,y ) p,q h i for some constant c . p,q 3.2. Local integrability of the kernel. ThekerneloftheEuclideanFourier transform Rn isgivenbye−ihx,ξi,whichislocallyintegrable. Wemayasktowhich F extent this analytic feature remains to hold. To be more precise, let Φ(t) be the distribution on R as in (3.1). We ask Question 3.4. When is Φ(t) locally integrable? ForaEuclideanJordanalgebraV weprovedin[9]thatΦ(t)islocallyintegrable. See (2.7) for the list of the corresponding conformal Lie algebras g. For G = O(p+1,q+1) with p+q even > 2, the Mellin–Barnes type integral formula(Theorem3.3)leadstothe followingproposition(see[17,Theorem6.2.1]): Proposition 3.5. We have the identities modulo L1 (R,rp+q 3dr). loc − 0 (Case A-1), Φp,q(t)≡c1 ml=−01 2l(m(−1l)l1)!δ(l)(t) (Case B-1), c2Pml=−01 2l(m−l!l−1)!t−l−1 (Case B-2), −− for some nonzero constantsPc , c . Here m= 1(p+q 4). 1 2 2 − Thus we have a complete answer to Question 3.4 in this case: Corollary 3.6. Φ(t) is locally integrable if and only if g=o(p+1,2),o(2,q+ 1) or o(3,3) sl(4,R). ≃ We note that the minimal representations for g=o(p+1,2) or o(2,q+1) are highest (or lowest) weight modules, whereas minimal representations do not make sense for g = o(3,3) which is isomorphic to sl(4,R). (Recall that the Joseph ideal is defined when gC is not of type A.) ThedelicateanswerindicatedinCorollary3.6iscloselyrelatedtotheregularity of the ‘Radon transform’ on Ξ. To be more precise, the Radon transform on Ξ is defined as the integral over the codimension-one submanifold x,y =t in Ξ. h i which collapses when t=0. Accordingly, the Radon transform ( u)(x,t)= u(y)δ( x,y t)dy R h i− ZΞ hasabetterregularityasttendsto0. (Ontheotherhand,theasymptoticbehavior as t is similar to the Euclidean case.) The singular part of Φp,q(t) in | | → ∞ Proposition 3.5 fits well with the behavior of u(x,t) as t tends to 0. R Question3.4isopenforminimalrepresentationswithouthighestweightsexcept for the case G=O(p+1,q+1). SPECIAL FUNCTIONS IN MINIMAL REPRESENTATIONS 9 4. Fourth order differential equations 4.1. Gaussian kernel and minimal K-type. The EuclideanFouriertrans- form Rn is of order four, and therefore its eigenvalues in L2(Rn) are among F 1, √ 1 . Animportanteigenfunctionwitheigenvalue1isthe Gaussiankernel {e−±12kx±k2,−nam}ely, Rn(e−12kxk2)=e−21kxk2. F Thus the Gaussian kernel e−21kxk2 is a square integrable function on Rn satisfying the following property: (4.1) Rnf =f and f is O(n)-invariant. F Next let p q 1, p+q even, and we consider the isotropic cone Ξ in Rp,q ≥ ≥ as in (2.1). The unitary inversion operator of the minimal representation π of Ξ F O(p+1,q+1) is of order two because = π(w) and w2 = I (see (2.4)), and Ξ p+q F therefore its eigenvalues in L2(Ξ) are either 1 or 1. An important eigenfunction − of is K (2 x ) where K (t):=(t) λK (t) is a renormalization of the K- FΞ 21(q−2) k k λ 2 − λ Bessel function. This is a square integrable function on Ξ satisfying the following property:e e (4.2) f = f and f is O(p) O(q)-invariant. Ξ F ± × To be more precise, f if p q 0 mod 4, f = − ≡ Ξ F ( f if p q 2 mod 4. − − ≡ Example 4.1. For p=3 and q =1, we have Ξ(e−2kxk)= e−2kxk, F − because √π (4.3) K (t)= e t. 1 − −2 2 The function e−2kxk arises aes the wave function for the hydrogen atom with the lowest energy. From the view point of representation theory, the Gaussian kernel e−21kxk2 generatesthe minimal K-type of the Weil representationof Mp(n,R), whereasthe functionK (2 x )generatesthatoftheminimalrepresentationofO(p+1,q+ 21(q−2) k k 1) realized in L2(Ξ). e 4.2. The Mano polynomial. We recall a classical fact that the Hermite polynomials form an orthogonalbasis for the radial part of the Schrödinger model L2(Rn) of the Weil representation e.g. [2, 10], whereas the Laguerre polynomials arise in the minimal representation of the conformal group SO(4,2). The bottom of the series correspond to what we have discussed in Section 4.1. We notice that these two minimal representations are quite special, namely, they are highest weight modules. However, for more general reductive groups, minimal representations are not always highest weight modules, and we need new ‘orthogonalpolynomials’and‘specialfunctions’to describeanaturalbasisoffunc- tions satisfying (4.2) or alike. 10 TOSHIYUKIKOBAYASHI For µ ∈C\{−1,−2,−3,···} and ℓ ∈N, the Mano polynomials {Mjµ,ℓ(x)}j∈N are defined by Γ(j+µ+1) ∂j (4.4) Mµ,ℓ(x):= Gµ,ℓ(t,x), j j!2µΓ(j+ µ+1) ∂tj 2 (cid:12)t=0 (cid:12) where the generating function Gµ,l(t,x) is given b(cid:12)y (cid:12) x 2ℓ+1ex2 tx x (4.5) Gµ,ℓ(t,x):= ((cid:0)12−(cid:1)t)ℓ+µ+23Iµ2 (cid:18)2(1−t)(cid:19)Kℓ+21 (cid:18)2(1−t)(cid:19). Here I (z) := (z) αI (z) and K (z)e= (z) αK (ze) denote the renormalized I- α 2 − α α 2 − α andK-Besselfunctions. With this normalization,the polynomialMµ,ℓ(x) isofthe j followeing top term: e ( 1)j Mµ,ℓ(x)= − xj+ℓ+lower order terms. j j! Example 4.2 (Special values of the Mano polynomial). (1) Thebottomoftheserieswithj =0isrelatedwiththeK-Besselfunctions with half-integer parameter: ℓ (2ℓ k)! M0µ,ℓ(x)=π−12z2l+1ezKℓ+21(z)(= k!(ℓ− k)!xk). k=0 − X (2) The polynomials Mµ,ℓ(x) foreℓ=0 reduce to the Laguerre polynomials j Γ(n+ν+1) n n xk Mµ,0(x)=Lµ(x)(= ( 1)k ). j j n! − k Γ(k+ν+1) k=0 (cid:18) (cid:19) X (3) ThefunctionMµ,ℓ(x)isnotapolynomialwhenℓ N,butitisconvenient j 6∈ to include the negative integer case. In particular, for ℓ = 1, it follows from [6, Corollary 5.3] and [7, Lemma 3.2] that Mjµ,−1(x)−is essentially the Laguerre polynomial: xMjµ,−1(x)=Lµj(x) (j ∈N). Many of the classical orthogonal polynomials are obtained as eigenfunctions of self-adjoint differential operators of second-order, but the Mano polynomials Mµ,ℓ(x) are obtained as those of fourth-order. Indeed this is a requirement from j representationtheory because the Casimir operator (for a compact subgroup) acts as a fourth-order differential operator on Ξ. To see this we may recall that the Lie algebra n acts as a second order differential operator (e.g. the fundamental differential operator R in (2.3)). j We begin with a second order differential operator on R d x d x x :=(x +µ 2ℓ 1 )(x +µ ) ( )2, µ,ℓ R dx − − − 2 dx − 2 − 2 and introduce a fourth order differential operator 1 := . Pµ,ℓ x2Rµ,ℓR0,ℓ