Classification of K-type formulas for the Heisenberg ultrahyperbolic operator \square_s for \widetilde{SL}(3,\mathbb{R}) and tridiagonal determinants for local Heun functions
aa r X i v : . [ m a t h . R T ] J a n CLASSIFICATION OF K -TYPE FORMULAS FOR THE HEISENBERGULTRAHYPERBOLIC OPERATOR (cid:3) s FOR f SL (3 , R ) AND TRIDIAGONALDETERMINANTS FOR LOCAL HEUN FUNCTIONS
TOSHIHISA KUBO AND BENT ØRSTED
Abstract.
The K -type formulas of the space of K -finite solutions to the Heisenberg ultrahyper-bolic equation (cid:3) s f = 0 for the non-linear group f SL (3 , R ) are classified. This completes a previousstudy of Kable for the linear group SL ( m, R ) in the case of m = 3, as well as generalizes ourearlier results on a certain second order differential operator. As a by-product we also show sev-eral properties of certain sequences { P j ( x ; y ) } ∞ j =0 and { Q j ( x ; y ) } ∞ j =0 of tridiagonal determinants,whose generating functions are given by local Heun functions. In particular, it is shown thatthese sequences satisfy a certain arithmetic-combinatorial property, which we refer to as a palin-dromic property . We further show that classical sequences of Cayley continuants { Cay j ( x ; y ) } ∞ j =0 and Krawtchouk polynomials {K j ( x ; y ) } ∞ j =0 also admit this property. In the end a new proof ofSylvester’s formula for certain tridiagonal determinant Sylv( x ; n ) is provided from a representationtheory point of view. Contents
1. Introduction 12. Peter–Weyl theorem for the space of K -finite solutions 163. Specialization to ( f SL (3 , R ) , B ) 194. Heisenberg ultrahyperbolic operator for f SL (3 , R ) 235. Hypergeometric model dπ II n ( D ♭s ) f ( t ) = 0 306. Heun model dπ I n ( D ♭s ) f ( t ) = 0 427. Sequences { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 of tridiagonal determinants 468. Cayley continuants { Cay k ( x ; y ) } ∞ k =0 and Krawtchouk polynomials {K k ( x ; y ) } ∞ k =0 Introduction
The representation theory of a reductive Lie group G is intimately tied to the analysis on cor-responding flag manifolds G/P , where P is a parabolic subgroup. For a representation of P , oneconsiders the corresponding homogeneous vector bundle over the flag manifold G/P and the space of
Date : January 19, 2021.2020
Mathematics Subject Classification.
Key words and phrases. intertwining differential operator, Heisenberg ultrahyperbolic operator, Peter–Weyl theo-rem for solution spaces, K -type solution, polynomial solution, hypergeometric differential equation, Heun’s differentialequation, tridiagonal determinant, Sylvester determinant, Cayley continuant, palindromic property. sections either in the smooth sense or in some square integrable sense. Understanding the structureof this space of sections is crucial, for example, whether there are interesting subspaces invariantunder the left regular action of G . This might for example happen if there is given a G -invariantdifferential equation on the space of sections, so that its solutions form an invariant subspace. Inorder to analyze such a space of solutions, it is convenient to use the analogue of Fourier series,namely, via the maximal compact subgroup K in G ; here G = KP , so the flag manifold G/P isalso a homogeneous space for K , and sections can be described according to how they transformunder the action of K .In the case that we consider in this paper, there will be a one-parameter family (cid:3) s of naturalinvariant differential operators and explicit spaces of solutions. It turns out that these will berelated to classical function theory, namely, hypergeometric functions and local Heun functions,and some of the relevant identities analogous to classical tridiagonal determinants of Sylvester typeand Cayley type. Further, the representations obtained from the solution space to the differentialequation (cid:3) s f = 0 include ones, which are to be thought of as minimal representations (in somesense). We provide a new aspect on a connection between the representation theory of reductivegroups, ordinary differential equations in the complex domains, and sequences of polynomials.The aim of this paper is threefold. The first is the classification of K -type formulas of the spaceof K -finite solutions to the Heisenberg ultrahyperbolic equation (cid:3) s f = 0 for f SL (3 , R ). The secondis a study of certain sequences { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 of tridiagonal determinants arisingfrom the study of the K -finite solutions to (cid:3) s f = 0. The third is an application of the argumentsfor { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 to classical sequences of Cayley continuants { Cay k ( x ; y ) } ∞ k =0 and Krowtchouk polynomials {K k ( x ; y ) } ∞ k =0 . We describe these three topics in detail now.1.1. Heisenberg ultrahyperbolic operator (cid:3) s . For a moment, let g = sl ( m, C ) with m ≥ g of g such that there exists a parabolic subalgebra p = m ⊕ a ⊕ n withHeisenberg nilpotent radical n , namely, dim R [ n , n ] = 1. The Heisenberg condition on n forces g to be either sl ( m, R ) or su ( p, q ) with p + q = m . We write p = m ⊕ a ⊕ n for the complexificationof p = m ⊕ a ⊕ n . We write U ( g ) for the universal enveloping algebra of the complexified Liealgebra g = g ⊗ R C .In [18], under the framework of g and p as above, Kable introduced a one-parameter family (cid:3) s of differential operators with s ∈ C as an example of conformally invariant systems ([2, 3]). Theoperator (cid:3) s is referred to as the Heisenberg ultrahyperbolic operator ([18]). For instance, for m = 3,it is defined as (cid:3) s = R (( XY + Y X ) + s [ X, Y ]) , (1.1)where R denotes the infinitesimal right translation and X, Y are certain nilpotent elements in sl (3 , R ) (see (3.1)). The differential operator (cid:3) s for m = 3 in (1.1) in particular recovers the secondorder differential operator studied in [27] as the case of s = 0. Some algebraic and analytic propertiesof the Heisenberg ultrahyperbolic operator (cid:3) s as well as its generalizations are investigated in[17, 18, 19, 20] for a linear group SL ( m, R ).From a viewpoint of intertwining operators, the Heisenberg ultrahyperbolic operator (cid:3) s is anintertwining differential operator between parabolically induced representations for G ⊃ P , where G and P are Lie groups with Lie algebras g and p , respectively. Thus the space S ol ( (cid:3) s ) of smoothsolutions to the equation (cid:3) s f = 0 in the induced representation is a subrepresentation of G , andalso the space S ol ( (cid:3) s ) K of K -finite solutions is a ( g , K )-module. We then consider the followingproblem. Problem 1.2.
Classify the K -type formulas for S ol ( (cid:3) s ) K . In [18], an attempt for this direction was made for G = SL ( m, R ). In the paper, Kable introduceda notion of H -modules and developed an algebraic theory for them. Although the theory is powerfulto compute the dimension of the k ∩ m -invariant subspace of the space of K -type solutions for each K -type in S ol ( (cid:3) s ) K , the determination of the explicit K -type formulas was not achieved.In this paper, we consider G = f SL (3 , R ), the universal covering group of SL (3 , R ). By makinguse of a technique from our earlier paper [27], we successfully classified the K -type formulas of S ol ( (cid:3) s ) K for f SL (3 , R ). Our method is different from the algebraic theory used in [18]; we ratherutilize differential equations. In order to describe our results in more detail we next introduce somenotation.1.2. K -type formulas. For the rest of this introduction let G = f SL (3 , R ) with Lie algebra g . Fixa minimal parabolic subgroup B of G with Langlands decomposition B = M AN . Here the subgroup M is isomorphic to the quaternion group Q of order 8. Let K be a maximal compact subgroup of G so that G = KAN is an Iwasawa decomposition of G . We have K ≃ SU (2) ≃ Spin (3).Let Irr( M ) and Irr( K ) denote the set of equivalence classes of irreducible representations of M and K , respectively. As M ≃ Q , the set Irr( M ) may be given asIrr( M ) = { (+,+) , (+, − ) , ( − ,+) , ( − , − ) , H } , where ( ± , ± ) are some characters (see Section 3.4 for the definition) and H stands for the uniquetwo-dimensional genuine representation of M . The character (+,+) is, for instance, the trivialcharacter. Let ( n/
2) denote the irreducible finite-dimensional representation of K ≃ Spin (3) withdimension n + 1. Then we have Irr( K ) = { ( n/
2) : n ∈ Z ≥ } . For σ ∈ Irr( M ) and a character λ of A , we write I ( σ, λ ) = Ind GB ( σ ⊗ ( λ + ρ ) ⊗ )for the representation of G induced from the representation σ ⊗ ( λ + ρ ) ⊗ of B = M AN , where ρ is half the sum of the positive roots corresponding to B . We realize the induced representation I ( σ, λ ) on the space of smooth sections for a G -equivariant homogeneous vector bundle over G/B .Let Diff G ( I ( σ , λ ) , I ( σ , λ )) denote the space of intertwining differential operators from I ( σ , λ )to I ( σ , λ ).Let g denote the complexified Lie algebra of g = sl (3 , R ). Since g = sl (3 , C ), the Heisenbergultrahyperbolic operator (cid:3) s is given as in (1.1). We then set D s := ( XY + Y X ) + s [ X, Y ] ∈ U ( g ) , (1.3) TOSHIHISA KUBO AND BENT ØRSTED so that (cid:3) s = R ( D s ). It follows from Proposition 4.3 that we have R ( D s ) ∈ Diff G ( I ( (+,+) , − e ρ ( s )) , I ( ( − , − ) , e ρ ( − s ))) , where e ρ ( s ) is a certain weight determined by e ρ := ρ/ σ ∈ Irr( M ), R ( D s ) ⊗ id σ ∈ Diff G ( I ( σ, − e ρ ( s )) , I ( ( − , − ) ⊗ σ, e ρ ( − s ))) , where id σ denotes the identity map on σ .For notational convenience we consider R ( D ¯ s ) in place of R ( D s ), where ¯ s denotes the complexconjugate of s ∈ C (see Section 4.1 for the details). We define S ol ( s ; σ ) := the space of smooth solutions to ( R ( D ¯ s ) ⊗ id σ ) f = 0 , S ol ( s ; σ ) K := the space of K -finite solutions to ( R ( D ¯ s ) ⊗ id σ ) f = 0 . (1.4)It follows from a Peter–Weyl theorem for the solution space (Theorem 2.11) that the ( g , K )-module S ol ( s ; σ ) K decomposes as S ol ( s ; σ ) K ≃ M n ∈ Z ≥ ( n/ ⊗ Hom M (Sol( s ; n ) , σ ) , (1.5)where Sol( s ; n ) is the space of K -type solutions to D s (without complex conjugation on s ) on the K -type ( n/
2) = ( δ, V ( n/ ) ∈ Irr( K ), that is,Sol( s ; n ) = { v ∈ V ( n/ : dδ ( D ♭s ) v = 0 } . (1.6)(see (4.7) and (4.6)). Here dδ is the differential of δ and D ♭s denotes the compact model of D s (seeDefinition 2.5). Then the K -type decomposition S ol ( s ; σ ) K for ( s, σ ) ∈ C × Irr( M ) is explicitlygiven as follows. Theorem 1.7.
The following conditions on ( σ, s ) ∈ Irr( M ) × C are equivalent. (i) S ol ( s ; σ ) = { } . (ii) One of the following conditions holds. • σ = (+,+) : s ∈ C . • σ = ( − , − ) : s ∈ C . • σ = ( − ,+) : s ∈ Z . • σ = (+, − ) : s ∈ Z . • σ = H : s ∈ Z .Further, the K -type formulas for S ol ( s ; σ ) K are given as follows. (1) σ = (+,+) : S ol ( s ; (+,+) ) K ≃ M n ∈ Z ≥ (2 n ) for all s ∈ C .(2) σ = ( − , − ) : S ol ( s ; ( − , − ) ) K ≃ M n ∈ Z ≥ (1 + 2 n ) for all s ∈ C . (3) σ = ( − ,+) : S ol ( s ; ( − ,+) ) K ≃ M n ∈ Z ≥ (( | s | + 1) / n ) for s ∈ Z . (4) σ = (+, − ) : S ol ( s ; (+, − ) ) K ≃ M n ∈ Z ≥ (( | s | + 1) / n ) for s ∈ Z . (5) σ = H : S ol ( s ; H ) K ≃ M n ∈ Z ≥ (( | s | + 1) / n ) for s ∈ Z . We shall deduce Theorem 1.7 from Theorem 5.36 at the end of Section 5. The proof is notcase-by-case analysis on σ ∈ Irr( M ); each M -representation σ is treated uniformly via a recipe forthe K -type decomposition of S ol ( s ; σ ) K . See Section 4.4 for the details of the recipe. In regard toTheorem 1.7, we shall also classify the space Sol( s ; n ) of K -type solutions for all n ∈ Z ≥ . This isdone in Theorems 5.16 and 6.7. Recall from (1.4) that S ol ( s ; σ ) K concerns R ( D ¯ s ). Theorem 1.7shows that the K -type decompositions are in fact independent of taking the complex conjugate onthe parameter s ∈ C .There are several remarks on the space Sol( s ; n ) of K -type solutions and the K -type decompo-sitions of S ol ( s ; σ ) K . First, as mentioned above, the dimensions of k ∩ m -invariant subspaces ofthe spaces of K -type solutions are determined in [18] for SL ( m, R ) with arbitrary rank m ≥
3. Inparticular, as k ∩ m = { } for m = 3, the dimensions dim C Sol( s ; n ) for n ∈ Z ≥ are obtained inthe cited paper ([18, Thm. 5.13]). We note that our normalization on s ∈ C is different from one for z ∈ C in [18] by s = − z . In the paper, factorization formulas of certain tridiagonal determinantsare essential to compute dim C Sol( s ; n ). For further details on the factorization formulas, see theremark after (1.22) below. By making use of differential equations, we determine the dimensionsdim C Sol( s ; n ) for all n ∈ Z ≥ independently to the results of [18].Table 1 summarizes the results of [18] and this paper, concerning the K -type decompositions of S ol ( s ; σ ) K for g = sl (3 , R ). Table 1.
Comparison between [18] and this paper for g = sl (3 , R ) g = sl (3 , R ) dim C Sol( s ; n ) K -type decomposition of S ol ( s ; σ ) K [18] Done for SL (3 , R ) Not obtainedthis paper Done for f SL (3 , R ) Done for f SL (3 , R )Secondly, Tamori recently investigates in [36] the representations on the space of smooth solutionsto the same differential equation as the Heisenberg ultrahyperbolic equation R ( D s ) f = 0, as partof his thorough study on minimal representations. In particular, the K -type formula S ol ( s ; H ) K forthe case σ = H is determined in [36, Prop. 5.4.6]. The representations on S ol ( s ; H ) for s ∈ Z aregenuine representations and not unitarizable unless s = 0 ([36, Rem. 5.4.5]). We remark that, bothin [36] and in this paper, the K -type formula for S ol ( s ; H ) K is obtained by realizing the K -type TOSHIHISA KUBO AND BENT ØRSTED solutions in Sol( s ; n ) as hypergeometric polynomials; nevertheless, the methods are rather different.For instance, some combinatorial computations are carried out in [36], whereas we simply solve thehypergeometric equation. Further, we also give the K -type solutions by Heun polynomials. Weillustrate our methods in detail in Section 1.3.Lastly, the K -type formulas S ol (0; σ ) K for the case of s = 0 are previously classified by ourselvesin [27, Thm. 1.6]. In this case only three M -representations σ = (+,+) , ( − , − ) , H contribute to S ol (0; σ ) = { } with explicit K -type formulas S ol (0; (+,+) ) K ≃ M n ∈ Z ≥ (2 n ) , S ol (0; ( − , − ) ) K ≃ M n ∈ Z ≥ (1 + 2 n ) , S ol (0; H ) K ≃ M n ∈ Z ≥ ((1 /
2) + 2 n ) . It was quite mysterious that only the series of (3 / Z ≥ does not appear in the K -type formulas.Theorem 1.7 now gives an answer for this question.The representations realized on S ol (0; σ ) for σ = (+,+) , ( − , − ) , H are known to be unitarizableand the resulting representations are the ones attached to the minimal nilpotent orbit ([33]). Forinstance, the representation realized on S ol (0; H ) is the genuine representation so-called Torasso’srepresentation ([39]). Here are two remarks on a recent progress on the unitarity of the represen-tations on S ol (0; σ ). First, Dahl recently constructed in [9] the unitary structures for the threeunitarizable representations on S ol (0; σ ) by using the Knapp–Stein intertwining operator and theFourier transform on the Heisenberg group. Furthermore, Frahm also recently gives the L -modelsof these unitary representations in [11], as part of his intensive study on the L -realizations of theminimal representations realized on the solution space of conformally invariant systems.1.3. Hypergeometric and Heun’s differential equations.
The main idea for accomplishingTheorem 1.7 is use of the decomposition formula (1.5) and a refinement of the method applied forthe case s = 0 in [27]. The central problem is classifying the space Sol( s ; n ) of K -type solutions.In order to do so, one needs to proceed the following two steps.Step 1: Classify ( s, n ) ∈ C × Z ≥ so that Sol( s ; n ) = { } .Step 2: Classify the M -representations on Sol( s ; n ) = { } .In [27] (the case s = 0), via the polynomial realizationIrr( K ) ≃ { ( π n , Pol n [ t ]) : n ∈ Z ≥ } (1.8)of Irr( K ) (see (3.14)), we carried out the two steps by realizing Sol(0; n ) in (1.6) asSol(0; n ) = { p ( t ) ∈ Pol n [ t ] : dπ I n ( D ♭ ) p ( t ) = 0 } , the space of polynomial solutions to an ordinary differential equation dπ I n ( D ♭ ) p ( t ) = 0 via a certainidentification Ω I : k ∼ → sl (2 , C ) (see Sections 3.2 and 4.2). It turned out that the differential equation dπ I n ( D ♭ ) p ( t ) = 0 is a hypergeometric differential equation. Steps 1 and 2 were then easily carriedout by observing the Gauss hypergeometric functions arising from dπ I n ( D ♭ ) p ( t ) = 0.In this paper we apply the same idea for the general case s ∈ C . Nonetheless, in the generalcase, Steps 1 and 2 do not become as simple as the case of s = 0, as the differential equation dπ I n ( D ♭s ) p ( t ) = 0 is turned out to be Heun’s differential equation D H ( − , − ns − n , − n − , , − n − s t ) p ( t ) = 0 (1.9)with the P -symbol P − ∞ − n t − ns
412 1+ n + s n − s − n − . (See Section 9.1 for the definition of D H ( a, q ; α, β, γ, δ ; z ) and the notation of the P -symbol.) Assuch, one needs to deal with local Heun functions at 0 ([38]); these are not as easy to handle ashypergeometric functions for our purpose. For instance, in Step 1, one needs to classify ( s, n ) ∈ C × Z ≥ for which the local Heun functions in consideration are polynomials; however, as opposedto hypergeometric functions, classifying such parameters is not an easy problem at all, since onlyknown are necessary conditions in which local Heun functions are reduced to polynomials. To resolvethis problem, inspired by a work [36] of Tamori, we use a different identification Ω II : k ∼ → sl (2 , C ) sothat the differential equation dπ II n ( D ♭s ) p ( t ) = 0 becomes again a hypergeometric equation, namely, D F ( − n , − n + s − , − n + s t ) p ( t ) = 0 . (1.10)(See (4.39) for the definition of D F ( a, b, c ; z ).) For the detail, see Sections 3.2 and 4.2. We accom-plished the K -type formulas of S ol ( s ; σ ) K by using the hypergeometric model dπ II n ( D ♭s ) p ( t ) = 0(see Theorem 5.36).The Heun model dπ I n ( D ♭s ) p ( t ) = 0 and the hypergeometric model dπ II n ( D ♭s ) p ( t ) = 0 are relatedby a Cayley transform π n ( k ) given by an element k of K ((4.18)). Namely, we have dπ II n ( D ♭s ) p ( t ) = 0 ⇐⇒ dπ I n ( D ♭s ) π n ( k ) p ( t ) = 0(see Proposition 4.19). For instance, put u [ s ; n ] ( t ) := Hl ( − , − ns − n , − n − , , − n − s t ) (1.11)and a [ s ; n ] ( t ) := F ( − n , − n + s − , − n + s t ) , where Hl ( a, q ; α, β, γ, δ ; z ) denotes the local Heun function at z = 0 and F ( a, b, c ; z ) ≡ F ( a, b, c ; z )is the Gauss hypergeometric function. Then, for D [ s ; n ] H := D H ( − , − ns − n , − n − , , − n − s t )in (1.9) and D [ s ; n ] F := D F ( − n , − n + s − , − n + s t ) TOSHIHISA KUBO AND BENT ØRSTED in (1.10), we have D [ s ; n ] H u [ s ; n ] ( t ) = 0 and D [ s ; n ] F a [ s ; n ] ( t ) = 0 . (1.12)It will be shown in Lemma 6.3 that, for appropriate ( s, n ) ∈ C × Z ≥ at which u [ s ; n ] ( t ) and a [ s ; n ] ( t )are both polynomials, the two functions u [ s ; n ] ( t ) and a [ s ; n ] ( t ) are related as π n ( k ) a [ s ; n ] ( t ) ∈ C u [ s ; n ] ( t ) , (1.13)equivalently, (1 − √− t ) n a [ s ; n ] (cid:18) √− t − √− t · √− (cid:19) ∈ C u [ s ; n ] ( t ) . For the case of s = 0, the Gauss-to-Heun transformation by π n ( k ) from a [ s ; n ] ( t ) to u [ s ; n ] ( t ) can bereduced to a Gauss-to-Gauss transformation, as the local Heun function u [ s ; n ] ( t ) can be reduced toa hypergeometric function for s = 0. (See Remarks 4.37 and 6.4.)The proportional constant for (1.13) may be given by a ratio of shifted factorials. Indeed, let I − and J be certain subsets of Z ≥ (see (5.3)). Then, for n ≡ s ∈ C \ ( I − ∪ J ), wehave π n ( k ) a [ s ; n ] ( t ) = (cid:0) − n , n (cid:1)(cid:0) − n + s , n (cid:1) u [ s ; n ] ( t )with ( ℓ, m ) := Γ( ℓ + m )Γ( ℓ ) . (As the proportional constants do not play any role for our main purposes,we shall not discuss them in this paper.)We remark that a Cayley transform, which is slightly different from ours π n ( k ), is used alsoin [12] under the name of the MacWilliams transform, to construct the generating function of thediscrete Chebyshev polynomials from Jacobi polynomials via the Heun equation. Further, Gauss-to-Heun transformations are studied in, for instance, [29, 40, 41]. In the cited papers, consideredare the cases in which local Heun functions are pulled back from a hypergeometric function. Asopposed to the references, this paper concerns some cases that, for suitable ( s ; n ) ∈ C × Z ≥ , acertain linear combination of a [ s ; n ] ( t ) and the second solution b [ s ; n ] ( t ) to (1.10) is transformed to u [ s ; n ] ( t ). Namely, put b [ s ; n ] ( t ) := t n − s F ( − n + s − , − s − , n − s t ) . Then, for appropriate ( s, n ) ∈ C × Z ≥ , we have π n ( k ) (cid:0) a [ s ; n ] ( t ) ± C ( s ; n ) b [ s ; n ] ( t ) (cid:1) ∈ C u [ s ; n ] ( t ) , where C ( s ; n ) is some constant, to be defined in (5.14). For more details, see Section 6.1. Via thetransformation π n ( k ), we also give in detail the space Sol( s ; n ) of K -type solutions for the Heunmodel dπ I n ( D ♭s ) p ( t ) = 0 (Section 6).1.4. Palindromic property of a sequence { p k ( x ; y ) } ∞ k =0 of polynomials. In the last half of thispaper, we shall discuss certain arithmetic-combinatorial properties of four sequences of polynomialsof two variables, namely. { P k ( x ; y ) } ∞ k =0 , { Q k ( x ; y ) } ∞ k =0 , { Cay k ( x ; y ) } ∞ k =0 , and {K k ( x ; y ) } ∞ k =0 , (1.14) where { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 are obtained from the study of the Heun model dπ I n ( D ♭ ) p ( t ) =0. Since we could not find the property in consideration, we refer to it as a palindromic property .As such, we shall digress for a moment from the main results of this paper.To state the general definition, let { p k ( x ; y ) } ∞ k =0 be a sequence of polynomials of two variableswith p ( x ; y ) = 1, and { a k } ∞ k =0 a sequence of non-zero numbers. Let d : Z ≥ → R be a given map.For n ∈ Z ≥ , we put S ol k ( p ; n ) := { s ∈ C : p k ( s ; n ) = 0 } . Definition . A pair ( { p k ( x ; y ) } ∞ k =0 , { a k } ∞ k =0 ) is said to be a palindromicpair with degree d ( n ) if, for each n ∈ d − ( Z ≥ ), there exists a map θ ( p ; n ) : S ol d ( n )+1 ( p ; n ) → {± } such that, for all s ∈ S ol d ( n )+1 ( p ; n ), we have p k ( s ; n ) = 0 for k ≥ d ( n ) + 1 and p k ( s ; n ) a k = θ ( p ; n ) ( s ) p d ( n ) − k ( s ; n ) a d ( n ) − k for k ≤ d ( n ) . (1.16)We call the identity (1.16) the palindromic identity and θ ( p ; n ) ( s ) its sign factor . Further, for agiven palindromic pair ( { p k ( x ; y ) } ∞ k =0 , { a k } ∞ k =0 ), the sequence { a k } ∞ k =0 is said to be an associatedsequence of { p k ( x ; y ) } ∞ k =0 . If there exits a sequence { a k } ∞ k =0 such that ( { p k ( x ; y ) } ∞ k =0 , { a k } ∞ k =0 ) is apalindromic pair, then we say that { p k ( x ; y ) } ∞ k =0 admits a palindromic property .We note that our notion of palindromic property does not concern polynomials themselves butsequences of polynomials; for instance, each p k ( x ; n ) is not necessarily a palindromic polynomial. Example . Here are some simple examples.(a) If p ( x ; y ) = 1 and p k ( x ; y ) = 0 for all k ≥
1, then, for any sequence { a k } ∞ k =0 of non-zeronumbers, the pair ( { p k ( x ; y ) } ∞ k =0 , { a k } ∞ k =0 ) is a palindromic pair with degree 0.(b) If p ( x ; y ) = p ( x ; y ) = 1 and p k ( x ; y ) = 0 for all k ≥
2, then, for any sequence { a k } ∞ k =0 ofnon-zero numbers with a = ± a , the pair ( { p k ( x ; y ) } ∞ k =0 , { a k } ∞ k =0 ) is a palindromic pair withdegree 1.(c) If p k ( x ; y ) = x k , then, for any sequence { a k } ∞ k =0 of non-zero numbers, the pair ( { x k } ∞ k =0 , { a k } ∞ k =0 )is a palindromic pair with degree 0.We shall provide four examples for which the degree d ( n ) is not constant. (See Sections 1.7, 1.8,and 1.9.)Now suppose that ( { p k ( x ; y ) ∞ k =0 , { a k } ∞ k =0 ) is a palindromic pair with degree d ( n ), where theassociated sequence { a k } ∞ k =0 is of the form a k = ( c k )! for some sequence { c k } ∞ k =1 of non-negativeintegers with c = 1. Then, as p ( x ; y ) = 1 by definition, the palindromity of ( { p k ( x ; y ) ∞ k =0 , { a k } ∞ k =0 )implies that, for n ∈ d − ( Z ≥ ), the d ( n )th term p d ( n ) ( x ; y ) satisfies the identity p d ( n ) ( s ; n ) = θ ( p ; n ) ( s ) (cid:0) c d ( n ) (cid:1) ! for s ∈ S ol d ( n )+1 ( p ; n ) . (1.18) We refer to the identity (1.18) as the factorial identity of p d ( n ) ( x ; n ) on S ol d ( n )+1 ( p ; n ). Further, wemean by the refinement of a factorial (resp. palindromic) identity a factorial (resp. palindromic)identity for each explicit value of s ∈ S ol d ( n )+1 ( p ; n ).In order to show the refinements of a palindromic property and factorial identity for { p k ( x ; y ) } ∞ k =0 ,it is important to understand the zero set S ol d ( n )+1 ( p ; n ) of p d ( n )+1 ( x ; n ). For the purpose we alsoconsider the factorization formula of the polynomial p d ( n )+1 ( x ; n ).In summary, we shall investigate the following properties for the sequences of polynomials { p k ( x ; y ) } ∞ k =0 in (1.14): • Factorization formula of p d ( n )+1 ( x ; n ); • Palindromic property of { p k ( x ; y ) } ∞ k =0 ; • Factorial identity of p d ( n ) ( x ; n ); • Refinement of a factorial identity of p d ( n ) ( x ; n ).It is remarked that factorization formulas for { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 will be given in amore general situation. We are going to describe these topics in detail now.1.5. Sequences { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 of tridiagonal determinants. Recall from(1.12) that the local Heun function u [ s ; n ] ( t ) in (1.11) is a local solution at t = 0 to the Heundifferential equation (1.9). Let v [ s ; n ] ( t ) be the second solution (see (4.35)). It will be shown inProposition 7.16 that u [ s ; n ] ( t ) and v [ s ; n ] ( t ) may be given in power series representations as u [ s ; n ] ( t ) = ∞ X k =0 P k ( s ; n ) t k (2 k )! and v [ s ; n ] ( t ) = ∞ X k =0 Q k ( s ; n ) t k +1 (2 k + 1)! , where P k ( x ; y ) and Q k ( x ; y ) are certain k × k tridiagonal determinants (see Section 7.1). Forinstance, the n × n tridiagonal determinant P n ( x ; n ) for y = n ∈ Z ≥ is given as P ( x ; 0) = 1, P ( x ; 2) = 2 x , and for n ∈ Z ≥ , P n ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nx · − ( n − n ( n − x · − ( n − n −
2) ( n − x · . . . . . . . . . − · − ( n − x ( n − n − − · − ( n − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (1.19)Similarly, the n − × n − tridiagonal determinant Q n − ( x ; n ) for y = n ∈ Z ≥ ) is given as Q ( x ; 2) = 1, Q ( x ; 4) = 2 x , and for n ∈ Z ≥ , Q n − ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( n − x · − ( n − n −
1) ( n − x · − ( n − n −
3) ( n − x · . . . . . . . . . − · − ( n − x ( n − n − − · − ( n − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (1.20) Factorization formulas of P [( n +2) / ( x ; n ) and Q [( n +1) / ( x ; n ) . In 1854, Sylvester observedthat an ( n + 1) × ( n + 1) centrosymmetric tridiagonal determinantSylv( x ; n ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x n x n − x . . . . . . . . . x n − x n x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1.21)satisfies the following formula ([35]).Sylv( x ; n ) = ( ( x − )( x − ) · · · ( x − ( n − )( x − n ) if n is odd ,x ( x − )( x − ) · · · ( x − ( n − )( x − n ) if n is even . (1.22)By utilizing some results for the Heun model dπ I n ( D ♭ ) p ( t ) = 0, we show that P [( n +2) / ( x ; n ) and Q [( n +1) / ( x ; n ) enjoy similar but more involved factorization formulas. See Theorems 7.18 and 7.22for the details. Based on the factorization formula, we also express P ( n +1) / ( x ; n ) for n odd in termsof the Sylvester determinant Sylv( x ; n ) (Corollary 7.27).Here are some remarks on the factorization formulas in order. First, the factorization formulas P ( n +2) / ( x ; n ) and Q n/ ( x ; n ) for n ∈ Z also follow from a more general formula in [18, Prop. 5.11],which is obtained via some clever trick based on a change of variables and some general formulaon tridiagonal determinants (see, for instance, [5, p. 52] and [18, Lem. 5.10]). As mentioned in theremark after Theorem 1.7 above, the factorization formulas for P ( n +2) / ( x ; n ) and Q n/ ( x ; n ) playan essential role in [18] to determine dim C Sol( s ; n ). In this paper we somewhat do this processbackwards including the case of n odd. Schematically, the difference is described as follows. • [18]: “factorization → dim C Sol( s ; n )” ( n ∈ Z ≥ ) • this paper: “Sol( s ; n ) → Heun → factorization” ( n ∈ Z ≥ )(It is recalled that, as the linear group SL ( m, R ) is considered in [18], only even n ∈ Z ≥ appearfor the space Sol( s ; n ) of K -type solutions for SO (3) ⊂ SL (3 , R ) in the case of m = 3, whereas wehandle all n ∈ Z ≥ in this paper, as Spin (3) ⊂ f SL (3 , R ) is in consideration.)The techniques used in [18] can also be applied for the case of n odd. Nevertheless, it requiressome involved computations; for instance, the trick used in the cited paper does not make thesituation as simple as for the case of n even, and further, the general formula ([5, p. 52] and[18, Lem. 5.10]) cannot be applied either. By simply applying the results for the Heun model dπ I n ( D ♭ ) p ( t ) = 0, we successfully avoid such computations.1.7. Palindromic properties for { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 . We next describe the palin-dromic properties and factorial identities for { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 . The palindromicproperties are given as follows. Theorem 1.23 (Theorems 7.29 and 7.41) . The pair ( { P k ( x ; y ) } ∞ k =0 , { (2 k )! } ∞ k =0 ) is a palindromicpair with degree n . Similarly, ( { Q k ( x ; y ) } ∞ k =0 , { (2 k + 1)! } ∞ k =0 ) is a palindromic pair with degree n − . For the case for n odd, see Propositions 7.39 and 7.47. The key idea of proofs of Theorem 1.23 isto use a symmetry of the generating functions u [ s ; n ] ( t ) and v [ s ; n ] ( t ) with respect to the non-trivialWeyl group element m I2 ((3.11)) of SU (2) in the form of the inversion. See Section 7.4 for thedetails.It follows from Theorem 1.23 that P n ( x ; y ) ((1.19)) and Q n − ( x ; y ) ((1.20)) satisfy factorialidentities (see Corollaries 7.37 and 7.45). Corollaries 1.24 and 1.26 below are the refinements of theidentities via the factorization formulas (Theorem 7.18) of P n +22 ( x ; n ) and Q n ( x ; n ). Corollary 1.24 (Refinement of the factorial identity of P n ( x ; n )) . Let n ∈ Z ≥ . Then thefollowing hold. (1) n ≡ The values of P n ( s ; n ) are P n ( s ; n ) = n ! for all s ∈ C . (2) n ≡ The values of P n ( s ; n ) for s = ± , ± , . . . , ± ( n − are given as P n ( s ; n ) = ( n ! if s = 1 , , . . . , n − , − n ! if s = − , − , − . . . , − ( n − . (1.25) Corollary 1.26 (Refinement of the factorial identity of Q n − ( x ; n )) . Let n ∈ Z ≥ ) . Thenthe following hold. (1) n ≡ The values of Q n − ( s ; n ) for s = ± , ± , . . . , ± ( n − are given as Q n − ( s ; n ) = ( ( n − s = 3 , , , . . . , n − , − ( n − s = − , − , − , . . . , − ( n − . (1.27)(2) n ≡ The values of Q n − ( s ; n ) are Q n − ( s ; n ) = ( n − s ∈ C .We shall give proofs of Corollaries 1.24 and 1.26 in Sections 7.4.1 and 7.4.2.1.8. Sequence { Cay k ( x ; y ) } ∞ k =0 of Cayley continuants. In 1858, four years after the observationof Sylvester on (1.22), Cayley considered in [4] a sequence { Cay k ( x ; y ) } ∞ k =0 of k × k tridiagonal deter-minants Cay k ( x ; y ) and expressed each Cay k ( x ; n ) in terms of the Sylvester determinant Sylv( x ; n )(see, for instance, [4, 31] and [30, p. 429])). The first few terms are given asCay ( x ; y ) = 1 , Cay ( x ; y ) = x, Cay ( x ; y ) = (cid:12)(cid:12)(cid:12)(cid:12) x y x (cid:12)(cid:12)(cid:12)(cid:12) , Cay ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x y x y − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , . . . . The ( n +1)th term Cay n +1 ( x ; n ) with y = n is nothing but the Sylvester determinant Cay n +1 ( x ; n ) =Sylv( x ; n ). Further, the n th term Cay n ( x ; n ) may be thought of as the “almost” Sylvester deter-minant, as it is given as Cay n ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x n x n − x . . . . . . . . . x n − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1.28) (compare (1.28) with (1.21)). Following [31], we refer to each Cay k ( x ; y ) as a Cayley continuant.As part of the third aim of this paper, we shall show the palindromic property of the Cayleycontinuants { Cay k ( x ; y ) } ∞ k =0 as follows. Theorem 1.29 (Theorem 8.8) . The pair ( { Cay k ( x ; y ) } ∞ k =0 , { k ! } ∞ k =0 ) is a palindromic pair withdegree n . For the palindromic identity, see Theorem 8.8. As for the cases of { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 ,we prove Theorem 1.29 by observing a symmetry of the generating function of the Cayley contin-uants { Cay k ( x ; y ) } ∞ k =0 with respect to the inversion (see Section 8.2).Theorem 1.29 implies that the almost Sylvester determinant Cay n ( x ; y ) ((1.28)) also satisfiesa factorial identity. Corollary 1.30 below is the refinement of the factorial identity (see Theorem8.8) of Cay n ( x ; n ) on S ol n +1 (Cay; n ) via Sylvester’s factorization formula (1.22) of Sylv( x ; n ) =Cay n +1 ( x ; n ). Corollary 1.30 (Refinement of the factorial identity of Cay n ( x ; n )) . For n even, the values of thealmost Sylvester determinant Cay n ( s ; n ) for s = 0 , ± , . . . , ± ( n − , ± n are given as follows. • n ≡ n ( s ; n ) = ( n ! if s = 0 , ± , . . . , ± ( n − , ± n, − n ! if s = ± , ± , . . . , ± ( n − . • n ≡ n ( s ; n ) = ( − n ! if s = 0 , ± , . . . , ± ( n − , ± n,n ! if s = ± , ± , . . . , ± ( n − . Similarly, for n odd, the values of Cay n ( s ; n ) for s = ± , ± , . . . , ± ( n − , ± n are given as follows. • n ≡ n ( s ; n ) = ( n ! if s = 1 , − , , . . . , − ( n − , n, − n ! if s = − , , − , . . . , n − , − n. • n ≡ n ( s ; n ) = ( − n ! if s = 1 , − , , . . . , n − , − n,n ! if s = − , , − , . . . , − ( n − , n. We shall give a proof of Corollary 1.30 at the end of Section 8.2.It has been more than 160 years since Cayley continuants { Cay k ( x ; y ) } ∞ k =0 were introduced in[4]. We therefore suppose that Theorem 1.29 and Corollary 1.30 are already in the literature. Yet,in contrast to a large variety of a proof for Sylvester’s formula (1.22) (see the remark after Theorem7.22), we could not find them at all. It will be quite surprising if the classical identities in Corollary1.30 have not been found over a century.It is also remarked that Sylvester’s formula (1.22) readily follows from the representation theoryof sl (2 , C ) without any computation. Nonetheless, it seems not in the literature either. We shall then provide a proof of (1.22) from a representation theory point of view (see Section 10). Theidea of the proof is in principle the same as the one discussed in Section 1.3 for the classification of K -type formulas. Furthermore, by applying the idea for determining the generating functions of { P ( x ; y ) } ∞ k =0 and { Q ( x ; y ) } ∞ k =0 , we also give in Proposition 8.3 another proof for Sylvester’s formula(1.22) as well as the generating function of the Cayley continuants { Cay k ( x ; y ) } ∞ k =0 . (Thus we givetwo different proofs of Sylvester’s formula (1.22) in this paper.)1.9. Sequence {K k ( x ; y ) } ∞ k =0 of Krawtchouk polynomials. For k ∈ Z ≥ , let K k ( x ; y ) be a poly-nomial of homogeneous degree k such that K k ( x ; n ) for y = n ∈ Z ≥ is a Krawtchouk polynomialin the sense of [28, p. 137], namely, K k ( x ; y ) = k X j =0 ( − j (cid:18) xj (cid:19)(cid:18) y − xk − j (cid:19) (1.31)(see Definition 8.11). In this paper we also call the polynomials K k ( x ; y ) of two variables Krawtchoukpolynomials. From the observation of the generating functions of Cay k ( x ; y ) and K k ( x ; y ), it is im-mediate that K k ( x ; y ) = Cay k ( y − x ; y ) k ! (1.32)(see (8.4) and (8.13)). In the last part of this paper, via the identity (1.32), we deduce the palin-dromic property of the Krawtchouk polynomials {K k ( x ; y ) } ∞ k =0 from that of the Cayley continuants { Cay k ( x ; y ) } ∞ k =0 as follows. Theorem 1.33 (Theorem 8.19) . The pair ( {K k ( x ; y ) } ∞ k =0 , { } ∞ k =0 ) is a palindromic pair with degree n . See Theorem 8.19 for the palindromic identity. As the associated sequence { } ∞ k =0 for {K k ( x ; y ) } ∞ k =0 is of the form 1 = 1! for all k , the n th term K n ( x ; y ) satisfies a factorial identity. Corollary 1.34below is the refinement of the factorial identity of K n ( x ; n ) on S ol n +1 ( K , n ) (see Theorem 8.19). Corollary 1.34 (Refinement of the factorial identity of K n ( x ; n )) . Given n ∈ Z ≥ , the values of K n ( s ; n ) for s = 0 , , , . . . , n are given as follows. K n ( s ; n ) = ( s = 0 , , , . . . , n even , − s = 1 , , , . . . , n odd , (1.35) where n odd and n even are defined as n odd := ( n if n is odd ,n − n is even and n even := ( n − n is odd ,n if n is even . We shall give a proof of Corollary 1.34 at the end of Section 8.3 as a corollary of Theorem 8.19.We remark that the identities (1.35) can also be easily shown directly from the definition (1.31) byan elementary observation.The Krawtchouk polynomials {K k ( x ; y ) } ∞ k =0 are also a classical object at least for the case of y = n ∈ Z ≥ . Therefore Theorem 1.33 may be a known as, for instance, an exercise problem;nonetheless, we could not find it in the literature. Summary of the data for palindromic properties.
To close this introduction, we sum-marize some data on the palindromic property of the sequence { p k ( x ; y ) } ∞ k =0 of polynomials in(1.14). For the associated sequence { a k } ∞ k =0 of { p k ( x ; y ) } ∞ k =0 , we write g [ x ; y ] ( t ) := ∞ X k =0 p k ( x ; y ) t b k a k , where { b k } ∞ k =0 is some sequence of non-negative integers. Table 2 exhibits the generating function g [ x ; y ] ( t ), the associated sequence { a k } ∞ k =0 , the sequence { b k } ∞ k =0 of exponents, the degree d ( n ), andthe sign factor θ ( p ; n ) ( s ) for { p k ( x ; y ) } ∞ k =0 in (1.14). (For the sign factors θ ( P ; n ) ( s ) and θ ( Q ; n ) ( s ), see(7.28) and (7.40), respectively.) Table 2.
Summary on palindromic properties p k ( x ; y ) g [ x ; y ] ( t ) a k b k d ( n ) θ ( p ; n ) ( s ) K k ( x ; y ) (1 + t ) y − x (1 − t ) x k n ( − s Cay k ( x ; y ) (1 + t ) y + x (1 − t ) y − x k ! k n ( − n − s P k ( x ; y ) u [ x ; y ] ( t ) (4.34) (2 k )! 2 k n (7.28) Q k ( x ; y ) v [ x ; y ] ( t ) (4.35) (2 k + 1)! 2 k + 1 n − (7.40)1.11. Organization.
We now outline the rest of this paper. This paper consists of ten sectionsincluding this introduction. In Section 2, we overview a general framework established in [27] for thePeter–Weyl theorem (1.5) for the space of K -finite solutions to intertwining differential operators.In Section 3, we collect necessary notation and normalizations for f SL (3 , R ). Two identificationsΩ J : k ∼ → sl (2 , C ) for J = I , II are discussed in this section.The purpose of Section 4 is to recall from [18] the Heisenberg ultrahyperebolic operator (cid:3) s = R ( D s ) for f SL (3 , R ) and to study the associated differential equation dπ Jn ( D ♭s ) p ( t ) = 0. In thissection we identify the equation dπ Jn ( D ♭s ) p ( t ) = 0 with Heun’s differential equation ( J = I) and thehypergeometric differential equation ( J = II) via the identifications Ω J : k ∼ → sl (2 , C ). At the endof this section we give a recipe to classify the space Sol( s ; n ) of K -type solutions to (cid:3) s = R ( D s ).In accordance with the recipe given in Section 4, we show the K -type decompositions of S ol ( s ; σ ) K in the hypergeometric model dπ II n ( D ♭s ) p ( t ) = 0 in Section 5. This is accomplished in Theorem 5.36.We then convert the results in the hypergeometric model dπ II n ( D ♭s ) p ( t ) = 0 to the Heun model dπ I n ( D ♭s ) p ( t ) = 0 by a Cayley transform π n ( k ) in Section 6.Sections 7 and 8 are devoted to applications to sequences of polynomials. In Section 7, by utiliz-ing the results in the Heun model dπ I n ( D ♭s ) p ( t ) = 0, we investigate two sequences { P k ( x ; n ) } ∞ k =0 and { Q k ( x ; n ) } ∞ k =0 of tridiagonal determinants. The main results of this sections are factorization formu-las (Theorems 7.18 and 7.22) and palindromic properties (Theorems 7.29 and 7.41). We also showan expression of P n +12 ( x ; n ) for n odd in terms of the Sylvester determinant Sylv( x ; n ) (Corollary7.27). In Section 8, we show the palindromic properties for Cayley continuants { Cay k ( x ; y ) } ∞ k =0 andKrawtchouk polynomials {K k ( x ; y ) } ∞ k =0 . These are achieved in Theorems 8.8 and 8.19, respectively. The last two sections are appendices. In order to study { P k ( x ; n ) } ∞ k =0 and { Q k ( x ; n ) } ∞ k =0 , weuse some general facts on the coefficients of a power series expression of local Heun functions. Wecollect those facts in Section 9. In Section 10, for future possible convenience, we give a proof ofSylvester’s formula (1.22) from an sl (2 , C ) point of view.2. Peter–Weyl theorem for the space of K -finite solutions The aim of this section is to recall from [27, Sect. 2] a general framework established in [27].In particular, we give a Peter–Weyl theorem for the space of K -finite solutions to intertwiningdifferential operators between parabolically induced representations. This is done in Theorem 2.11.2.1. General framework.
Let G be a reductive Lie group with Lie algebra g . Choose a Cartaninvolution θ : g → g and write g = k ⊕ s for the Cartan decomposition of g with respectto θ . Here k and s stand for the +1 and − θ , respectively. We take maximalabelian subspaces a min0 ⊂ s and t min0 ⊂ m min0 := Z k ( a min0 ) such that h := a min0 ⊕ t min0 is a Cartansubalgebra of g .For a real Lie algebra y , we denote by y the complexification of y . For instance, the complex-ifications of g , h , a min0 , and m min0 are denoted by g , h , a , and m min , respectively. We write U ( y )for the universal enveloping algebra of a Lie algebra y .Let ∆ ≡ ∆( g , h ) be the set of roots with respect to the Cartan subalgebra h and Σ ≡ Σ( g , a )denote the set of restricted roots with respect to a . We choose a positive system ∆ + and Σ + insuch a way that ∆ + and Σ + are compatible. We denote by ρ for half the sum of the positive roots.Let n min0 be the nilpotent subalgebra of g corresponding to Σ + , so that p min0 := m min0 ⊕ a min0 ⊕ n min0 is a Langlands decomposition of a minimal parabolic subalgebra p min0 of g . Fix a standard parabolicsubalgebra p ⊃ p min0 with Langlands decomposition p = m ⊕ a ⊕ n . Let P be a parabolicsubalgebra of G with Lie algebra p . We write P = M AN for the Langlands decomposition of P corresponding to p = m ⊕ a ⊕ n .For µ ∈ a ∗ ≃ Hom R ( a , C ), we define a one-dimensional representation C µ of A as a e µ ( a ) := e µ (log a ) for a ∈ A . Then, for a finite-dimensional representation W σ = ( σ, W ) of M and weight λ ∈ a ∗ , we define an M A -representation W σ,λ as W σ,λ := W σ ⊗ C λ . (We note that the definition of W σ,λ was slightly different from the one in [27].) As usual, by letting N act on W σ,λ trivially, we regard W σ,λ as a representation of P . We identify the Fr´echet space C ∞ ( G/P, W σ,λ ) of smooth sections for the G -equivariant homogeneous vector bundle W σ,λ := G × P W σ,λ → G/P as C ∞ ( G/P, W σ,λ ) ≃ ( C ∞ ( G ) ⊗ W σ,λ ) P , that is, C ∞ ( G/P, W σ,λ ) ≃ n f ∈ C ∞ ( G ) ⊗ W σ,λ : f ( gman ) = σ ( m ) − e − λ ( a ) f ( g ) for all man ∈ M AN o , where G acts by left translation. Then we realize a parabolically induced representation I P ( σ, λ ) := Ind GP ( σ ⊗ ( λ + ρ ) ⊗ ) (2.1)of G on C ∞ ( G/P, W σ,λ + ρ ). For pairs ( σ, λ ) , ( η, ν ) of finite-dimensional representations σ, η of M and weights λ, ν ∈ a ∗ , wewrite Hom G ( I P ( σ, λ ) , I P ( η, ν )) for the space of intertwining operators from I P ( σ, λ ) to I P ( η, ν ).Then we setDiff G ( I P ( σ, λ ) , I P ( η, ν )) := Diff( I P ( σ, λ ) , I P ( η, ν )) ∩ Hom G ( I P ( σ, λ ) , I P ( η, ν )) , where Diff( I P ( σ, λ ) , I P ( η, ν )) is the space of differential operators from I P ( σ, λ ) to I P ( η, ν ).Let Irr( M ) fin be the set of equivalence classes of irreduicible finite-dimensional representationsof M . As M is not connected in general, we write M for the identity component of M . LetIrr( M/M ) denote the set of irreducible representations of the component group M/M . Viathe surjection M ։ M/M , we regard Irr( M/M ) as a subset of Irr( M ) fin . Let id ξ denote theidentity map on ξ ∈ Irr(
M/M ). Lemma 2.2 below shows that the tensored operator D ⊗ id ξ to D ∈
Diff G ( I P ( σ, λ ) , I P ( η, ν ) is also an intertwining differential operator. Lemma 2.2 ([27, Lems. 2.17, 2.21]) . Let
D ∈
Diff G ( I P ( σ, λ ) , I P ( η, ν )) . For any ξ ∈ Irr(
M/M ) ,we have D ⊗ id ξ ∈ Diff G ( I P ( σ ⊗ ξ, λ ) , I P ( η ⊗ ξ, ν )) . If the M -representation σ is the trivial character σ = χ triv , then Lemma 2.2 in particular showsthat differential operator D ∈
Diff G ( I P ( χ triv , λ ) , I P ( η, ν )) yields D ⊗ id ξ ∈ Diff G ( I P ( ξ, λ ) , I P ( η ⊗ ξ, ν )) for ξ ∈ Irr(
M/M ) . (2.3)2.2. Peter–Weyl theorem for S ol ( u ; λ ) ( ξ ) K . For a later purpose, we specialize the representations σ, η of M to be σ = χ triv and η = χ , where χ is some character of M . In this situation, it followsfrom (2.3) that, for D ∈
Diff G ( I P ( χ triv , λ ) , I P ( χ, ν )) and ξ ∈ Irr(
M/M ), we have D ⊗ id ξ ∈ Diff G ( I P ( ξ, λ ) , I P ( χ ⊗ ξ, ν )) . By the duality theorem between intertwining differential operators and homomorphisms betweengeneralized Verma modules (see, for instance, [6, 23, 25]), any intertwining differential operator
D ∈
Diff G ( I P ( χ triv , λ ) , I P ( χ, ν )) is of the form D = R ( u ) for some u ∈ U (¯ n ), where R denotes theinfinitesimal right translation of U ( g ) and ¯ n is the opposite nilpotent radical to n , namely,Diff G ( I P ( χ triv , λ ) , I P ( χ, ν )) = { R ( u ) for some u ∈ U (¯ n ) } . In particular, intertwining differential operators D are determined by the complex Lie algebra g and independent of its real forms.It follows from [15, Lem. 2.1] and [27, Lem. 2.24] that there exists a ( U ( k ∩ m ) , K ∩ M )-isomorphism U ( k ) ⊗ U ( k ∩ m ) C χ triv ∼ −→ U ( g ) ⊗ U ( p ) C χ triv , − ( λ + ρ ) , u ⊗ χ triv u ⊗ ( χ triv ⊗ − ( λ + ρ ) ) , (2.4)where K ∩ M acts on U ( k ) ⊗ U ( k ∩ m ) C χ triv and U ( g ) ⊗ U ( p ) C χ triv , − ( λ + ρ ) diagonally. Via the isomorphism(2.4), we define a compact model of u ♭ ∈ U ( k ) of u ∈ U (¯ n ) as follows. Definition . For R ( u ) ∈ Diff G ( I P ( χ triv , λ ) , I P ( χ, ν )) with u ∈ U (¯ n ), we denote by u ♭ ∈ U ( k ) anelement of U ( k ) such that the identity u ♭ ⊗ ( χ triv ⊗ − ( λ + ρ ) ) = u ⊗ ( χ triv ⊗ − ( λ + ρ ) ) (2.6) holds in U ( g ) ⊗ U ( p ) C χ triv , − ( λ + ρ ) . Remark . We remark that compact model u ♭ is not unique; any choice of u ♭ satisfying theidentity (2.6) is acceptable.Let Irr( K ) be the set of equivalence classes of irreducible representations of the maximal compactsubgroup K with Lie algebra k . For V δ := ( δ, V ) ∈ Irr( K ) and u ∈ U (¯ n ), we define a subspaceSol ( u ) ( δ ) of V δ by Sol ( u ) ( δ ) := { v ∈ V δ : dδ ( τ ( u ♭ )) v = 0 } . (2.8)Here dδ denotes the differential of δ and τ denotes the conjugation τ : g → g with respect to thereal form g , that is, τ ( X + √− X ) = X − √− X for X , X ∈ g . Lemma 2.9 ([27, Lem. 2.41]) . The space
Sol ( u ) ( δ ) is a K ∩ M -representation. We set Sol k ∩ m ( u ) ( δ ) := Sol ( u ) ( δ ) ∩ V k ∩ m δ , where V k ∩ m δ is the subspace of k ∩ m -invariant vectors of V δ . Clearly Sol k ∩ m ( u ) ( δ ) is a K ∩ M -subrepresentation of Sol ( u ) ( δ ). Further, since k ∩ m acts on ξ ∈ Irr(
M/M ) trivially, we haveHom K ∩ M (cid:0) Sol ( u ) ( δ ) , ξ (cid:1) = { } if and only if Hom K ∩ M (cid:16) Sol k ∩ m ( u ) ( δ ) , ξ (cid:17) = { } (2.10)via the composition of maps K ∩ M ֒ → M ։ M/M .Let I P ( χ triv , λ ) K denote the ( g , K )-module consisting of the K -finite vectors of I P ( χ triv , λ ).Then, given R ( u ) ∈ Diff G ( I P ( χ triv , λ ) , I P ( χ, ν )) and ξ ∈ Irr(
M/M ), we set S ol ( u ; λ ) ( ξ ) K := { f ∈ I P ( ξ, λ ) K : ( R ( u ) ⊗ id ξ ) f = 0 } . For the sake of future convenience, we now give a slight modification of a Peter–Weyl theorem[27, Thm. 1.2] for S ol ( u ; λ ) ( ξ ) K , although such a modification is not necessary for the main objectiveof this paper. We remark that this modified version is a more direct generalization of the argumentgiven after the proof of [18, Thm. 2.6] than [27, Thm. 1.2]. Theorem 2.11 (Peter–Weyl theorem for S ol ( u ; λ ) ( ξ ) K ) . Let R ( u ) ∈ Diff G ( I P ( χ triv , λ ) , I P ( χ, ν )) and ξ ∈ Irr(
M/M ) . Then the ( g , K ) -module S ol ( u ; λ ) ( ξ ) K can be decomposed as a K -representation as S ol ( u ; λ ) ( ξ ) K ≃ M δ ∈ Irr( K ) V δ ⊗ Hom K ∩ M (cid:16) Sol k ∩ m ( u ) ( δ ) , ξ (cid:17) . (2.12) Proof.
It follows from [27, Thm. 1.2] that the space S ol ( u ; λ ) ( ξ ) K can be decomposed as S ol ( u ; λ ) ( ξ ) K ≃ M δ ∈ Irr( K ) V δ ⊗ Hom K ∩ M (cid:0) Sol ( u ) ( δ ) , ξ (cid:1) . Now the equivalence (2.10) concludes the theorem. (cid:3)
When G is a split real group and P = M AN is a minimal parabolic subgroup of G , we have K ∩ M = M , M/M = M , and k ∩ m = { } . Thus in this case Theorem 2.11 is simplified as follows. Corollary 2.13.
Suppose that G is split real and P = M AN is minimal parabolic subgroup of G .Let R ( u ) ∈ Diff G ( I P ( χ triv , λ ) , I P ( χ, ν )) and σ ∈ Irr( M ) . Then the ( g , K ) -module S ol ( u ; λ ) ( σ ) K canbe decomposed as S ol ( u ; λ ) ( σ ) K ≃ M δ ∈ Irr( K ) V δ ⊗ Hom M (cid:0) Sol ( u ) ( δ ) , σ (cid:1) (2.14)3. Specialization to ( f SL (3 , R ) , B )The purpose of this section is to specialize the general theory discussed in Section 2 to a pair( f SL (3 , R ) , B ), where B is a minimal parabolic subgroup of f SL (3 , R ). In this section we in particulargive a recipe for computing K -type formulas of the space S ol ( u ; λ ) ( σ ) K of K -finite solutions for R ( u ) ∈ Diff G ( I B ( χ triv , λ ) , I B ( χ, ν )). It is described in Section 3.7.3.1. Notation and normalizations.
We begin by recalling from [27, Sect. 4.1] the notationand normalizations for f SL (3 , R ). Let G = f SL (3 , R ) with Lie algebra g = sl (3 , R ). Fix a Cartaninvolution θ : g → g such that θ ( U ) = − U t . We write k and s for the +1 and − θ ,respectively, so that g = k ⊕ s is a Cartan decomposition of g . We put a := span R { E ii − E i +1 ,i +1 : i = 1 , } and n := span R { E , E , E } , where E ij denote matrix units. Then b := a ⊕ n is aminimal parabolic subalgebra of g .Let K , A , and N be the analytic subgroups of G with Lie algebras k , a , and n , respectively.Then G = KAN is an Iwasawa decomposition of G . We write M = Z K ( a ), so that B := M AN is a minimal parabolic subgroup of G with Lie algebra b .We denote by g the complexification of the Lie algebra g of G . A similar convention is employedalso for subgroups of G ; for instance, b = a ⊕ n is a Borel subalgebra of g = sl (3 , C ). We write ¯ n for the nilpotent radical opposite to n .Let ∆ ≡ ∆( g , a ) denote the set of roots of g with respect to a . We denote by ∆ + and Πthe positive system corresponding to b and the set of simple roots of ∆ + , respectively. We haveΠ = { ε − ε , ε − ε } , where ε j are the dual basis of E jj for j = 1 , ,
3. The root spaces g ε − ε and g ε − ε are then given as g ε − ε = C E and g ε − ε = C E . We write ρ for half the sum of thepositive roots, namely, ρ = ε − ε .We define X, Y ∈ g as X = ! and Y = ! . (3.1)Then X and Y are root vectors for − ( ε − ε ) and − ( ε − ε ), respectively. The opposite nilpotentradical ¯ n is thus given as ¯ n = span { X, Y, [ X, Y ] } .3.2. Two identifications for so (3 , C ) ≃ sl (2 , C ) . As k = so (3) ≃ su (2), we have k = so (3 , C ) ≃ sl (2 , C ). For later applications we consider two identifications of so (3 , C ) with sl (2 , C ):Ω J : so (3 , C ) ∼ −→ sl (2 , C ) for J = I , II . Loosely speaking, these identifications are described as follows. (1) Identification Ω I : so (3 , C ) ∼ → sl (2 , C ): This is an identification via a Lie algebra isomorphismΩ I0 : so (3) ∼ → u (2). Schematically, we have so (3 , C ) ∼ Ω I / / sl (2 , C ) so (3) ⊗ R C O O ∼ Ω I0 / / su (2) ⊗ R C O O (2) Identification Ω II : so (3 , C ) ∼ → sl (2 , C ): This is an identification independent of su (2).Schematically, we have so (3 , C ) ∼ Ω II / / sl (2 , C ) so (3) ⊗ R C O O su (2)Let E + , E − , and E be the elements of sl (2 , C ) defined as E + := (cid:18) (cid:19) , E − := (cid:18) (cid:19) , E := (cid:18) − (cid:19) . (3.2)We now describe the identifications Ω I and Ω II in detail separately.3.2.1. Identification k ≃ sl (2 , C ) via Ω I . We start with the identification Ω I : k ∼ → sl (2 , C ). Firstobserve that k = so (3) is spanned by the three matrices B := −
10 0 01 0 0 ! , B := −
10 1 0 ! , B := − ! with commutation relations[ B , B ] = B , [ B , B ] = − B , and [ B , B ] = B . (3.3)On the other hand, the Lie algebra su (2) is spanned by A := (cid:18) √− −√− (cid:19) , A := (cid:18) − (cid:19) , A := (cid:18) √− √− (cid:19) with commutation relations[ A , A ] = 2 A , [ A , A ] = − A , and [ A , A ] = 2 A . Then one may identify k with su (2) via the linear mapΩ I0 : k ∼ −→ su (2) , B j A j for j = 1 , , . (3.4)Let Z + , Z − , Z be the elements of k = so (3 , C ) defined as Z + := B − √− B , Z − := − ( B + √− B ) , Z := [ Z + , Z − ] = − √− B . (3.5)Then we have k = span { Z + , Z − , Z } . (3.6) As A − √− A = 2 E + and − ( A + √− A ) = 2 E − , the isomorphism (3.4) yields a Lie algebraisomorphism Ω I : k ∼ −→ sl (2 , C ) , Z j E j for j = + , − , . (3.7)3.2.2. Identification k ≃ sl (2 , C ) via Ω II . We next discuss the identification Ω II : k ∼ −→ sl (2 , C ). Let W + , W − , and W be the elements of so (3 , C ) defined by W + := B + √− B , W − := − B + √− B , W := [ W + , W − ] = − √− B . (3.8)Then we have k = span { W + , W − , W } . (3.9)It follows from the commutation relations (3.3) that the triple { W + , W − , W } forms an sl (2)-triple,namely, [ W + , W − ] = W , [ W .W + ] = 2 W + , and [ W , W − ] = − W − . Therefore, k may be identifiedwith sl (2 , C ) via the Lie algebra isomorphismΩ II : k ∼ −→ sl (2 , C ) , W j E j for j = + , − , . (3.10)3.3. A realization of the subgroup M = Z K ( a ) . As K is isomorphic to SU (2), we realize M = Z K ( a ) as a subgroup of SU (2) via the isomorphism (3.7). To do so, first observe that theadjoint action Ad of SU (2) on su (2) yields a two-to-one covering map SU (2) ։ Ad( SU (2)) ≃ SO (3)on SO (3). We realize Ad( SU (2)) as a matrix group with respect to the ordered basis { A , A , A } of su (2) in such a way that the elements ± m I j ∈ SU (2) with m I0 := (cid:18) (cid:19) , m I1 := (cid:18) √− −√− (cid:19) , m I2 := (cid:18) − (cid:19) , m I3 := (cid:18) √− √− (cid:19) (3.11)are mapped to ± m I j m j ∈ SO (3) for j = 0 , , ,
3, where m = , m = − − , m = − − , m = − − . One can easily check that the map ± m I j m j respects the Lie algebra isomorphism Ω I : k ∼ → sl (2 , C ) in (3.7) ([27, Lem. 4.6]). Namely, for Z ∈ k , we haveΩ I (Ad( m j ) Z ) = Ad( m I j )Ω I ( Z ) for j = 0 , , , . As Z SO (3) ( a ) = { m , m , m , m } , we then realize M as a subgroup of SU (2) as M = (cid:8) ± m I0 , ± m I1 , ± m I2 , ± m I3 (cid:9) . The subgroup M is isomorphic to the quaternion group Q , a non-commutative group of order 8.3.4. Irreducible representations
Irr( M ) of M . In order to compute a K -type formula via (2.14),we next discuss the sets Irr( M ) and Irr( K ) of equivalence classes of irreducible representations of M and K , respectively. We first consider Irr( M ) via the isomorphism (3.7). As M is isomorphic to thequaternion group Q , the set Irr( M ) consists of four characters and one 2-dimensional irreducible Table 3.
Character table for ( ε, ε ′ ) for m I j ± m I0 ± m I1 ± m I2 ± m I3 (+,+) (+, − ) − − ( − ,+) − − ( − , − ) − − ε, ε ′ ∈ {±} , we define a character χ SO (3)( ε,ε ′ ) : Z SO (3) ( a ) → {± } of Z SO (3) ( a ) as χ SO (3)( ε,ε ′ ) (diag( a , a , a )) := | a | ε | a | ε ′ , where | a | + := | a | and | a | − := a . Via the character χ SO (3)( ε,ε ′ ) of Z SO (3) ( a ), we define a character χ ( ε,ε ′ ) : M → {± } of M as χ ( ε,ε ′ ) ( ± m I j ) := χ ( ε,ε ′ ) ( m j ) for j = 0 , , , . (3.12)We often abbreviate χ ( ε,ε ′ ) as ( ε, ε ′ ). The character (+,+) , for instance, is the trivial character of M . Table 3 illustrates the character table for ( ε, ε ′ ) = χ ( ε,ε ′ ) . The set Irr( M ) may then be describedas follows: Irr( M ) = { (+,+) , (+, − ) , ( − ,+) , ( − , − ) , H } , (3.13)where H is the unique genuine 2-dimensional representation of M ≃ Q .3.5. Irreducible representations
Irr( K ) of K . We next consider a polynomial realization ofIrr( K ). Let Pol[ t ] be the space of polynomials of one variable t with complex coefficients and setPol n [ t ] := { p ( t ) ∈ Pol[ t ] : deg p ( t ) ≤ n } . Then we realize the set Irr( K ) of equivalence classes of irreducible representations of K ≃ SU (2)as Irr( K ) ≃ { ( π n , Pol n [ t ]) : n ∈ Z ≥ } , (3.14)where the representation π n of SU (2) on Pol n [ t ] is defined as( π n ( g ) p ) ( t ) := ( ct + d ) n p (cid:18) at + bct + d (cid:19) for g = (cid:18) a bc d (cid:19) − . (3.15)It follows from (3.15) that the elements m I j defined in (3.11) act on Pol n [ t ] via π n as m I1 : p ( t ) ( √− n p ( − t ); m I2 : p ( t ) t n p (cid:18) − t (cid:19) ; m I3 : p ( t ) ( −√− t ) n p (cid:18) t (cid:19) . (3.16)Let dπ n be the differential of the representation π n . As usual we extend dπ n complex-linearlyto sl (2 , C ) and also naturally to the universal enveloping algebra U ( sl (2 , C )). It follows from (3.2)and (3.15) that E + , E − , and E act on Pol n [ t ] via dπ n as dπ n ( E + ) = − ddt , dπ n ( E − ) = − nt + t ddt , and dπ n ( E ) = − t ddt + n. (3.17) Peter–Weyl theorem for S ol ( u ; λ ) ( σ ) K with the polynomial realization of Irr( K ) . LetΩ : k ∼ → sl (2 , C ) be an identification of k with sl (2 , C ) such as (3.7) and (3.10). Then elements F ∈ U ( k ) can act on Pol n [ t ] via dπ n as dπ n (Ω( F )). To simplify the notation we write dπ Ω n ( F ) = dπ n (Ω( F )) for F ∈ U ( k ) . (3.18)Then, for R ( u ) ∈ Diff G ( I ( (+,+) , λ ) , I ( χ, ν )) with (+,+) the trivial character of M , we setSol ( u ) ( n ) := { p ( t ) ∈ Pol n [ t ] : dπ Ω n (cid:0) τ ( u ♭ ) (cid:1) p ( t ) = 0 } , (3.19)where u ♭ ∈ U ( k ) is a compact model of u ∈ U (¯ n ) (see Definition 2.5). Then the K -type decomposi-tion of S ol ( u ; λ ) ( σ ) K in (2.14) with the polynomial realization (3.14) of Irr( K ) becomes S ol ( u ; λ ) ( σ ) K ≃ M n ≥ Pol n [ t ] ⊗ Hom M (cid:0) Sol ( u ) ( n ) , σ (cid:1) . (3.20)We remark that since K ∩ M = M and m = { } , the ( U ( k ∩ m ) , K ∩ M )-isomorphism (2.4)reduces to an M -isomorphism U ( k ) ⊗ C (+,+) ∼ −→ U ( g ) ⊗ U ( b ) C (+,+) , − λ . In particular a compact model u ♭ ∈ U ( k ) of u ∈ U (¯ n ) is indeed unique in the present situation (see Remark 2.7).3.7. Recipe for determining the K -type formula for S ol ( u ; λ ) ( σ ) K . For later convenience wesummarize a recipe for computing the K -type formula for S ol ( u ; λ ) ( σ ) K via (3.20). Let R ( u ) ∈ Diff G ( I ( (+,+) , λ ) , I ( χ, ν )). Step 0:
Choose an identification Ω : k ≃ sl (2 , C ). Step 1:
Find the compact model u ♭ of u (see Definition 2.5). Step 2:
Find the explicit formula of the differential operator dπ Ω n (cid:0) τ ( u ♭ ) (cid:1) . Step 3:
Solve the differential equation dπ Ω n (cid:0) τ ( u ♭ ) (cid:1) p ( t ) = 0 and classify n ∈ Z ≥ such that Sol ( u ) ( n ) = { } . Step 4:
For n ∈ Z ≥ with Sol ( u ) ( n ) = { } , classify the M -representations on Sol ( u ) ( n ). Step 5:
Given σ ∈ Irr( M ), classify n ∈ Z ≥ with Sol ( u ) ( n ) = { } such thatHom M (cid:0) Sol ( u ) ( n ) , σ (cid:1) = { } . Heisenberg ultrahyperbolic operator for f SL (3 , R )The aim of this section is to discuss a certain second order differential operator called the Heisen-berg ultrahyperbolic operator R ( D s ). In this section we apply to R ( D s ) the theory described inSection 3 for arbitrary intertwining differential operators R ( u ) for f SL (3 , R ). We continue the no-tation and normalizations from Section 3.4.1. Heisenberg ultrahyperbolic operator R ( D s ) . We start with the definition of the Heisen-berg ultrahyperbolic operator R ( D s ) for f SL (3 , R ). As R ( D s ) being an intertwining differentialoperator (see Proposition 4.3 below), it is defined for the complex Lie algebra g = sl (3 , C ). Let X and Y be the elements of ¯ n defined in (3.1). For s ∈ C , we define D s as D s := ( XY + Y X ) + s [ X, Y ] ∈ U ( g ) . Then the second order differential operator R ( D s ) = R (cid:0) ( XY + Y X ) + s [ X, Y ] (cid:1) is called the Heisenberg ultrahyperbolic operator for g = sl (3 , C ). For the general definition for sl ( m, C ) with arbitrary rank m ≥
3, see [18, Sect. 3].Recall from Section 3.1 that the simple roots α, β ∈ Π are realized as α = ε − ε and β = ε − ε .Then, for s ∈ C , we set e ρ ( s ) := e ρ − s e ρ ⊥ (4.1)with e ρ := 12 ( ε − ε ) and e ρ ⊥ := 12 ( ε + ε ) . Remark that as e ρ ⊥ / ∈ a ∗ , we have e ρ ( s ) / ∈ a ∗ unless s = 0. For σ ∈ Irr( M ), we write I ( σ, e ρ ( s )) = I B ( σ, e ρ ( s ) | a ∗ ) (4.2)for the parabolically induced representation I B ( σ, e ρ ( s ) | a ∗ ) defined as in (2.1). Proposition 4.3 belowshows that the operator R ( D s ) is indeed an intertwining differential operator between parabolicallyinduced representations. Proposition 4.3.
We have R ( D s ) ∈ Diff G ( I ( (+,+) , − e ρ ( s )) , I ( ( − , − ) , e ρ ( − s )) . Consequently, for σ ∈ Irr( M ) , we have R ( D s ) ⊗ id σ ∈ Diff G ( I ( σ, − e ρ ( s )) , I ( ( − , − ) ⊗ σ, e ρ ( − s )) . Proof.
The first assertion readily follows from [18, Lem. 3.2] and [27, Lem. 6.4]. Lemma 2.2 thenconcludes the second. (cid:3)
Our aim is to compute the branching law of the space S ol ( D s ; − e ρ ( s )) ( σ ) K of K -finite solutions to( R ( D s ) ⊗ id σ ) f = 0. The K -type formula in (4.4) with ( u ; λ ) = ( D s ; − e ρ ( s )) gives S ol ( D s ; − e ρ ( s )) ( σ ) K ≃ M n ≥ Pol n [ t ] ⊗ Hom M (cid:0) Sol ( D s ) ( n ) , σ (cid:1) (4.4)with Sol ( D s ) ( n ) = { p ( t ) ∈ Pol n [ t ] : dπ Ω n (cid:0) τ ( D ♭s ) (cid:1) p ( t ) = 0 } . Observe that, for X and Y in (3.1), we have X, Y ∈ g = sl (3 , R ). Thus τ ( D ♭s ) for D s =( XY + Y X ) + s [ X, Y ] is simply given as τ ( D ♭s ) = τ ( D s ) ♭ = D ♭ ¯ s , (4.5)where ¯ s denotes the complex conjugate of s ∈ C . Then we writeSol( s ; n ) := Sol ( D ¯ s ) ( n ) so that Sol( s ; n ) = { p ( t ) ∈ Pol n [ t ] : dπ Ω n (cid:0) τ ( D ♭ ¯ s ) (cid:1) p ( t ) = 0 } = { p ( t ) ∈ Pol n [ t ] : dπ Ω n (cid:0) D ♭s (cid:1) p ( t ) = 0 } . (4.6)Similarly, we put S ol ( s ; σ ) K := S ol ( D ¯ s ; − e ρ (¯ s )) ( σ ) K . Then the K -type formula (4.4) yields S ol ( s ; σ ) K ≃ M n ≥ Pol n [ t ] ⊗ Hom M (Sol( s ; n ) , σ ) . (4.7)Hereafter we consider the branching law of S ol ( s ; σ ) K instead of S ol ( D s ; − e ρ ( s )) ( σ ) K .In order to solve the equation dπ Ω n (cid:0) D ♭s (cid:1) p ( t ) = 0 in (4.6), we next find the compact model D ♭s of D s , which is an element of U ( k ) such that D ♭s ⊗ ( (+,+) ⊗ e ρ ( s ) − ρ ) = D s ⊗ ( (+,+) ⊗ e ρ ( s ) − ρ )in U ( g ) ⊗ U ( b ) C (+,+) , e ρ ( s ) − ρ . For X, Y ∈ ¯ n in (3.1), define X ♭ , Y ♭ ∈ k as X ♭ := X + θ ( X ) and Y ♭ := Y + θ ( Y ) , where θ is the Cartan involution defined as θ ( U ) = − U t . Lemma 4.8.
We have D ♭s = X ♭ Y ♭ + Y ♭ X ♭ + s [ X ♭ , Y ♭ ] ∈ U ( k ) . (4.9) Proof.
One can easily verify that( X ♭ Y ♭ + Y ♭ X ♭ + s [ X ♭ , Y ♭ ]) ⊗ ( (+,+) ⊗ e ρ ( s ) − ρ ) = D s ⊗ ( (+,+) ⊗ e ρ ( s ) − ρ )in U ( g ) ⊗ U ( b ) C (+,+) , e ρ ( s ) − ρ . (cid:3) Relationship between dπ I n ( D ♭s ) and dπ II n ( D ♭s ) . In Section 3.2, we discussed two identifica-tions of k with sl (2 , C ), namely,Ω I : k ∼ → sl (2 , C ) and Ω II : k ∼ → sl (2 , C ) . (4.10)Thus compact model D ♭s may act on Pol n [ t ] via dπ n as dπ n (Ω I ( D ♭s )) and dπ n (Ω II ( D ♭s )) . (4.11)As for the notation dπ Ω n ( F ) = dπ n (Ω( F )) in (3.18), we abbreviate (4.11) as dπ Jn ( D ♭s ) = dπ n (Ω J ( D ♭s )) for J ∈ { I , II } . We next discuss a relationship between dπ I n ( D ♭s ) and dπ II n ( D ♭s ).We begin with the expression of Ω J ( D ♭s ) in terms of the sl (2)-triple { E + , E − , E } in (3.2). First,recall from (3.6) and (3.9) that k can be given as k = span { Z + , Z − , Z } = span { W + , W − , W } , where { Z + , Z − , Z } and { W + , W − , W } are the sl (2)-triples of k defined in (3.5) and (3.8), respec-tively. Lemma 4.12.
The compact model D ♭s = X ♭ Y ♭ + Y ♭ X ♭ + s [ X ♭ , Y ♭ ] is expressed as D ♭s = √−
12 (( Z + + Z − )( Z + − Z − ) − ( s − Z ) (4.13)= 12 (( W + + W − ) W − ( s − W + − W − )) . (4.14) Proof.
A direct computation shows that X ♭ and Y ♭ are expressed in terms of { Z + , Z − , Z } and { W + , W − , W } as X ♭ = √−
12 ( Z + + Z − ) = − √−
12 ( W + + W − ) ,Y ♭ = 12 ( Z + − Z − ) = √− W . By substituting these expressions into D ♭s , one obtains the lemma. (cid:3) Lemma 4.15.
The elements Ω I ( D ♭s ) and Ω II ( D ♭s ) in U ( sl (2 , C )) are given as Ω I ( D ♭s ) = √−
12 (( E + + E − )( E + − E − ) − ( s − E ) , (4.16)Ω II ( D ♭s ) = 12 (( E + + E − ) E − ( s − E + − E − )) . (4.17) Proof.
This follows from (3.7), (3.10), and Lemma 4.12. (cid:3)
We set k := 1 √ (cid:18) √− − −√− (cid:19) ∈ SU (2) . (4.18) Proposition 4.19.
We have Ω I ( D ♭s ) = Ad( k )Ω II ( D ♭s ) . (4.20) Consequently, for p ( t ) ∈ Pol n [ t ] , the following are equivalent: (i) dπ II n ( D ♭s ) p ( t ) = 0 ; (ii) dπ I n ( D ♭s ) π n ( k ) p ( t ) = 0 .Proof. One can easily check that E + + E − = − Ad( k )( E + + E − ) ,E + − E − = √− k ) E . Now the identity (4.20) follows from Lemma 4.15. Since dπ Jn ( D ♭s ) = dπ n (Ω J ( D ♭s )) for J ∈ { I , II } ,the equivalence between (i) and (ii) readily follows from (4.20). (cid:3) For J ∈ { I , II } , we writeSol J ( s ; n ) = { p ( t ) ∈ Pol n [ t ] : dπ Jn ( D ♭s ) p ( t ) = 0 } . (4.21) Table 4.
Character table for ( ε, ε ′ ) for m II j ± m II0 ± m II1 ± m II2 ± m II3 (+,+) (+, − ) − − ( − ,+) − − ( − , − ) − − s ; n ) with Sol J ( s ; n ), the K -type decomposition (4.7) becomes S ol ( s ; σ ) K ≃ M n ≥ Pol n [ t ] ⊗ Hom M (Sol J ( s ; n ) , σ ) . (4.22)It follows from Proposition 4.19 that π n ( k ) yields a linear isomorphism π n ( k ) : Sol II ( s ; n ) ∼ −→ Sol I ( s ; n ) . (4.23)For m I j ∈ M for j = 0 , , , m II j ∈ M as m II j := k − m I j k (4.24)so that the following diagram commutes.Sol II ( s ; n ) π n ( k ) ∼ / / π n ( m II j ) (cid:15) (cid:15) Sol I ( s ; n ) π n ( m I j ) (cid:15) (cid:15) Sol II ( s ; n ) π n ( k ) ∼ / / Sol I ( s ; n )A direct computation shows that m II j are given as follows. m II0 = (cid:18) (cid:19) , m II1 = (cid:18) − (cid:19) , m II2 = (cid:18) √− −√− (cid:19) , m II3 = − (cid:18) √− √− (cid:19) . (4.25)By (3.11) and (4.25), we have m II0 = m I0 , m II1 = m I2 , m II2 = m I1 , m II3 = − m I3 . (4.26)It then follows from (3.16) that m II j act on Pol n [ t ] via dπ n as m II1 : p ( t ) t n p (cid:18) − t (cid:19) ; m II2 : p ( t ) ( √− n p ( − t ); m II3 : p ( t ) ( √− t ) n p (cid:18) t (cid:19) . (4.27)Table 4 is the character table for ( ε, ε ′ ) with m II j .4.3. Differential equation dπ Jn ( D ♭s ) f ( t ) = 0 for J ∈ { I , II } . As indicated in the recipe in Section3.7, to determine the K -type decomposition of S ol ( s ; n ) K , it is crucial to determine the spaceSol J ( s ; n ) of K -type solutions, for which one needs to find polynomial solutions p ( t ) ∈ Pol n [ t ] todifferential equations dπ Jn ( D ♭s ) f ( t ) = 0 for J ∈ { I , II } . For this purpose we next investigate thesedifferential equations. Differential equation dπ I n ( D ♭s ) f ( t ) = 0 . We start with the differential equation dπ I n ( D ♭s ) f ( t ) =0. The explicit formula of dπ I n ( D ♭s ) is given as follows. Lemma 4.28.
We have − √− dπ I n ( D ♭s ) = (1 − t ) d dt + 2 (cid:0) ( n − t + s (cid:1) t ddt − n (cid:0) ( n − t + s (cid:1) . (4.29) Proof.
Recall from (4.16) that we have − √− I ( D ♭s ) = ( E + + E − )( E + − E − ) − ( s − E . Now the proposed identity follows from a direct computation with (3.17). (cid:3)
In order to study the differential equation dπ I n ( D ♭s ) f ( t ) = 0, we set D H ( a, q ; α, β, γ, δ ; z ) := d dz + (cid:18) γz + δz − εz − a (cid:19) ddz + αβz − qz ( z − z − a ) , (4.30)where a, q, α, β, γ, δ, ε are complex parameters with a = 0 , γ + δ + ε = α + β + 1. Then thedifferential equation D H ( a, q ; α, β, γ, δ ; z ) f ( z ) = 0 (4.31)is called Heun’s differential equation . For a brief account of the equation (4.31), see Section 9.1.The differential equation dπ I n ( D ♭s ) f ( t ) = 0 can be identified with Heun’s differential equation as inLemma 4.32 below. (We are grateful to Hiroyuki Ochiai for pointing it out.) Lemma 4.32.
We have − dπ I n ( D ♭s )4 = D H ( − , − ns − n , − n − , , − n − s t ) . Proof.
This follows from a direct computation with a change of variables z = t on (4.29). (cid:3) By Lemma 4.32, to determine the space Sol I ( s ; n ) of K -type solutions to dπ I n ( D ♭s ) f ( t ) = 0, itsuffices to find polynomial solutions to the Heun equation D H ( − , − ns − n , − n − , , − n − s t ) f ( t ) = 0 . (4.33)Let Hl ( a, q ; α, β, γ, δ ; z ) denote the power series solution Hl ( a, q ; α, β, γ, δ ; z ) = P ∞ r =0 c r z r with c = 1 to (4.31) at z = 0 ([38]). We set u [ s ; n ] ( t ) := Hl ( − , − ns − n , − n − , , − n − s t ) , (4.34) v [ s ; n ] ( t ) := tHl ( − , − ( n − s − n − , − n − , , − n − s t ) . (4.35)It follows from (9.1) in Section 9 that u [ s ; n ] ( t ) and v [ s ; n ] ( t ) are linearly independent solutions at t = 0 to the Heun equation (4.33). Therefore we haveSol I ( s ; n ) ⊂ C u [ s ; n ] ( t ) ⊕ C v [ s ; n ] ( t ) . (4.36)We shall classify the parameters ( s, n ) ∈ C × Z ≥ such that u [ s ; n ] ( t ) , v [ s ; n ] ( t ) ∈ Sol I ( s ; n ) in Section6 (see Propositions 6.5 and 6.6). Remark . Let F ( a, b, c ; z ) ≡ F ( a, b, c ; z ) denote the Gauss hypergeometric function. In [29,(3.8)], Maier found the following Heun-to-Gauss reduction formula: Hl ( − , α, β, γ, α + β − γ + 12 ; t ) = F ( α , β , γ + 12 ; t ) . (4.38)We remark that the case of s = 0 for dπ I n ( D ♭ ) f ( t ) = 0 gives special cases of this identity. Indeed,it follows from [27, Lem. 7.3] that a change of variables z = t yields the identity √− t dπ n ( D ♭ ) = D F ( − n , − n − ,
34 ; t ) , where D F ( a, b, c ; z ) denotes the differential operator D F ( a, b, c ; z ) = z (1 − z ) d dz + ( c − ( a + b + 1) z ) ddz − ab (4.39)such that D F ( a, b, c ; z ) f ( z ) = 0is the hypergeometric differential equation. Thus the equation dπ I n ( D ♭ ) f ( t ) = 0 is also equivalentto the hypergeometric equation D F ( − n , − n − ,
34 ; t ) f ( t ) = 0 . (4.40)Then (4.33) and (4.40) yield the following Heun-to-Gauss reductions: Hl ( − , − n , − n − , , − n −
12 ; t ) = F ( − n , − n − ,
34 ; t ); (4.41) Hl ( − , − n − , − n − , , − n −
12 ; t ) = F ( − n − , − n − ,
54 ; t ) . (4.42)4.3.2. Differential equation dπ II n ( D ♭s ) f ( t ) = 0 . We next consider dπ II n ( D ♭s ) f ( t ) = 0. The equation dπ II n ( D ♭s ) f ( t ) = 0 can be identified with the hypergeometric equation D F ( a, b, c ; z ) f ( z ) = 0 asfollows. Lemma 4.43.
We have dπ II n ( D ♭s )16 t = D F ( − n , − n + s − , − n + s t ) . Proof.
First, it follows from (4.17) and (3.17) that dπ II n ( D ♭s ) is given as2 dπ II n ( D ♭s ) = 2(1 − t ) t d dt + (( s + 3 n − t + ( s − n + 1)) ddt − n ( s + n − t. (4.44)Then a direct computation with a change of variable z = t concludes the lemma. (cid:3) Similar to the equation dπ I n ( D ♭s ) f ( t ) = 0, Lemma 4.43 shows that, to determine the spaceSol II ( s ; n ) of K -type solutions to dπ II n ( D ♭s ) f ( t ) = 0, it suffices to find polynomial solutions to thehypergeometric equation D F ( − n , − n + s − , − n + s t ) f ( t ) = 0 . (4.45) We set a [ s ; n ] ( t ) := F ( − n , − n + s − , − n + s t ) ,b [ s ; n ] ( t ) := t n − s F ( − n + s − , − s − , n − s t ) . Then a [ s ; n ] ( t ) and b [ s ; n ] ( t ) form fundamental set of solutions to (4.45) for suitable parameters( s, n ) ∈ C × Z ≥ for which a [ s ; n ] ( t ) and b [ s ; n ] ( t ) are well-defined. Therefore we haveSol II ( s ; n ) ⊂ C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) . (4.46)The precise conditions of ( s, n ) such that a [ s ; n ] ( t ) , b [ s ; n ] ( t ) ∈ Sol II ( s ; n ) will be investigated in Section5.1.4.4. Recipe for the K -type decomposition of S ol ( s ; σ ) K . In Section 3.7, we gave a generalrecipe to determine the K -type formula (3.20) of S ol ( u ; λ ) ( σ ) K . We modify the recipe in such a waythat it will fit well for S ol ( s ; σ ) K with S ol ( s ; σ ) K ≃ M n ≥ Pol n [ t ] ⊗ Hom M (Sol J ( s ; n ) , σ ) . We first fix an identification Ω J : k ∼ −→ sl (2 , C ) for J ∈ { I , II } . Then the K -type formula of S ol ( s ; σ ) K may be determined in the following three steps. Step A:
Classify ( s, n ) ∈ C × Z ≥ such that Sol J ( s ; n ) = { } . It follows from (4.36) and (4.46) thatthis is indeed equivalent to classifying ( s, n ) ∈ C × Z ≥ such that • u [ s ; n ] ( t ) ∈ Pol n [ t ] or v [ s ; n ] ( t ) ∈ Pol n [ t ] for J = I; • a [ s ; n ] ( t ) ∈ Pol n [ t ] or b [ s ; n ] ( t ) ∈ Pol n [ t ] for J = II. Step B:
For ( s, n ) ∈ C × Z ≥ with Sol J ( s ; n ) = { } , classify the M -representations on Sol J ( s ; n ). Step C:
Given σ ∈ Irr( M ), classify ( s, n ) ∈ C × Z ≥ with Sol J ( s ; n ) = { } such thatHom M (Sol J ( s ; n ) , σ ) = { } . In Sections 5 and 6, we shall proceed Steps A, B, and C for J = II and J = I, respectively.5. Hypergeometric model dπ II n ( D ♭s ) f ( t ) = 0The aim of this section is to classify the K -type formulas for S ol ( s ; σ ) K by using the hypergeo-metric model dπ II n ( D ♭s ) f ( t ) = 0. The decomposition formulas are achieved in Theorem 5.36.5.1. The classification of
Sol II ( s ; n ) . As Step A of the recipe in Section 4.4, we first wish toclassify ( s, n ) ∈ C × Z ≥ such that Sol II ( s ; n ) = { } , whereSol II ( s ; n ) = { p ( t ) ∈ Pol n [ t ] : dπ II n ( D ♭s ) p ( t ) = 0 } . Recall from (4.46) that we have Sol II ( s ; n ) ⊂ C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t )with a [ s ; n ] ( t ) = F ( − n , − n + s − , − n + s t ) , (5.1) b [ s ; n ] ( t ) = t n − s F ( − n + s − , − s − , n − s t ) . (5.2)It thus suffices to classify ( s, n ) ∈ C × Z ≥ such that a [ s ; n ] ( t ) ∈ Pol n [ t ] or b [ s ; n ] ( t ) ∈ Pol n [ t ].Given n ∈ Z ≥ , we define I ± , J , I , I ± , J , I ⊂ Z as follows: I ± := {± (3 + 4 j ) : j = 0 , , . . . , n − } ( n ∈ Z ≥ ); J := { j : j = 0 , , . . . , n − } ( n ∈ Z ≥ ); I := {± j : j = 0 , , . . . , h n i } ; I ± := {± (1 + 4 j ) : j = 0 , , . . . , h n i } ; J := { j : j = 0 , , . . . , h n i − } ; I := {± (2 + 4 j ) : j = 0 , , . . . , h n i } . (5.3)We start by observing when a [ s ; n ] ( t ) ∈ Pol n [ t ]. A key observation is that, for ( a, b, c ) ∈ C forwhich the hypergeometric function F ( a, b, c ; z ) is well-defined, we have F ( a, b, c ; z ) ∈ Pol n [ z ] ⇐⇒ a ∈ { , − , . . . , − n } or b ∈ { , − , . . . , − n } . Proposition 5.4.
The following conditions on ( s, n ) ∈ C × Z ≥ are equivalent. (i) a [ s ; n ] ( t ) ∈ Pol n [ t ] . (ii) One of the following conditions hold: (a) n ≡ s ∈ C \ J ;(b) n ≡ s ∈ I ;(c) n ≡ s ∈ C \ J ;(d) n ≡ s ∈ I .Proof. The proposition follows from a careful observation for the parameters of a [ s ; n ] ( t ) in (5.1).Indeed, suppose that n is even. Then the parameters of a [ s ; n ] ( t ) imply that a [ s ; n ] ( t ) / ∈ Pol n [ t ] if andonly if the following conditions are satisfied3 − n + s ∈ − Z ≥ , − n < − n + s , and − n + s − < − n + s , which is equivalent to s ∈ J for n ≡ s ∈ J for n ≡ Next suppose that n is odd. In this case it follows from (5.1) that a [ s ; n ] ( t ) ∈ Pol n [ t ] if and only if0 ≤ n + s − ≤ h n i and n + s − ∈ Z ≥ , which is equivalent to s ∈ I for n ≡ s ∈ I for n ≡ s ∈ I ∪ I , we have − n − s / ∈ Z . This concludes the proposition. (cid:3) Suppose that a [ s ; n ] ( t ) ∈ Pol n [ t ]. Then generically we have deg a [ s ; n ] ( t ) = n . Lemma 5.5 belowclassifies the singular parameters of s ∈ C for a [ s ; n ] ( t ) in a sense that deg a [ s ; n ] ( t ) < n (see Section5.2.4). Lemma 5.5.
Suppose that n ≡ k (mod 4) and s ∈ C \ J k for k = 0 , . Then the following conditionson ( s, n ) ∈ C × Z ≥ are equivalent. (i) deg a [ s ; n ] ( t ) < n . (ii) deg a [ s ; n ] ( t ) = n + s − . (iii) One of the following conditions holds: (a) n ≡ s ∈ I − ;(b) n ≡ s ∈ I − .Proof. It follows from the parameters of a [ s ; n ] ( t ) that deg a [ s ; n ] ( t ) < n if and only if2 · n + s − < n and n + s − ∈ Z ≥ , (5.6)which shows the equivalence between (i) and (ii). Moreover, (5.6) is equivalent to s ∈ {− n + 1 + 4 j : j = 0 , , , . . . , n − } . (5.7)One can readily verifty that, under the condition s / ∈ J k , (5.7) is indeed equivalent to s ∈ I − k for k = 0 , (cid:3) We next consider b [ s ; n ] ( t ) in (5.2). Proposition 5.8.
The following conditions on ( s, n ) ∈ C × Z ≥ are equivalent. (i) b [ s ; n ] ( t ) ∈ Pol n [ t ] with b [ s ; n ] ( t ) = a [ s ; n ] ( t ) . (ii) One of the following conditions holds. (a) n ≡ s ∈ I +0 ∪ I − ∪ J ;(b) n ≡ s ∈ I ;(c) n ≡ s ∈ I +2 ∪ I − ∪ J ;(d) n ≡ s ∈ I ; Proof.
Observe that if b [ s ; n ] ( t ) ∈ Pol[ t ], then the exponent n − s for t n − s in (5.2) must satisfy n − s ∈ Z ≥ , which in particular forces n − s / ∈ − Z ≥ . Moreover, if n − s = 0, then b [ n +1; n ] ( t ) = F ( − n , − n , t ) = a [ n +1; n ] ( t ) . Consequently, we have b [ s ; n ] ( t ) ∈ Pol n [ t ] with b [ s ; n ] ( t ) = a [ s ; n ] ( t ) if and only if either1 + n − s ∈ Z ≥ , n + s − ∈ Z ≥ , and 1 + n −
22 + 2 · n + s − ≤ n, (5.9)or 1 + n − s ∈ Z ≥ , s − ∈ Z ≥ , and 1 + n −
22 + 2 · s − ≤ n. (5.10)A direct observation shows that the conditions (5.9) and (5.10) are equivalent to s ∈ ( − n + 1 + 4 Z ≥ ) ∩ ( −∞ , n + 1) (5.11)and n is even and s ∈ (1 + 2 Z ≥ ) ∩ ( −∞ , n + 1) , (5.12)respectively. One can directly verify that (5.11) and (5.12) are equivalent to the conditions on s ∈ C stated in Proposition 5.8 for n ≡ k (mod 4) for k = 0 , , ,
3. Indeed, if n ≡ s ∈ I − ∪ J and s ∈ I +0 ∪ J , respectively. Since the other three casescan be shown similarly, we omit the proof. (cid:3) It follows from Propositions 5.4 and 5.8 that if n ≡ k (mod 4) for k = 0 ,
2, thenSol II ( s ; n ) = C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) for s ∈ I + k ∪ I − k . (5.13)For a later purpose for determining the M -representations on Sol II ( s ; n ), for such n ≡ k (mod 4),we define c ± [ s ; n ] ( t ) := a [ s ; n ] ( t ) ± C ( s ; n ) b [ s ; n ] ( t ) for s ∈ I − k , where C ( s ; n ) := ( − n , [ n ])[ n ] ! if n ≡ s = − , ( − n , n + s − )( − n + s − , n + s − )( − n + s , n + s − ) · ( n + s − ) ! otherwise . (5.14)Here ( ℓ, m ) stands for the shifted factorial, namely, ( ℓ, m ) = Γ( ℓ + m )Γ( ℓ ) . Then, for n ≡ k (mod 4) for k = 0 ,
2, the space Sol II ( s ; n ) may be described asSol II ( s ; n ) = ( C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) if s ∈ I + k , C c +[ s ; n ] ( t ) ⊕ C c − [ s ; n ] ( t ) if s ∈ I − k . (5.15)We then summarize the parameters ( s, n ) such that Sol II ( s ; n ) = { } as follows. Theorem 5.16.
The following conditions on ( s, n ) ∈ C × Z ≥ are equivalent. (i) Sol II ( s ; n ) = { } . (ii) One of the following conditions is satisfied. • n ≡ s ∈ C . • n ≡ s ∈ I . • n ≡ s ∈ C . • n ≡ s ∈ I .Further, for such ( s, n ) , the space Sol II ( s ; n ) may be given as follows. (1) n ≡ II ( s ; n ) = C a [ s ; n ] ( t ) if s ∈ C \ ( I +0 ∪ I − ∪ J ) , C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) if s ∈ I +0 , C b [ s ; n ] ( t ) if s ∈ J , C c +[ s ; n ] ( t ) ⊕ C c − [ s ; n ] ( t ) if s ∈ I − . (2) n ≡ II ( s ; n ) = C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) for s ∈ I . (3) n ≡ II ( s ; n ) = C a [ s ; n ] ( t ) if s ∈ C \ ( I +2 ∪ I − ∪ J ) , C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) if s ∈ I +2 , C b [ s ; n ] ( t ) if s ∈ J , C c +[ s ; n ] ( t ) ⊕ C c − [ s ; n ] ( t ) if s ∈ I − . (4) n ≡ II ( s ; n ) = C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) for s ∈ I . Proof.
This is a summary of the results in Propositions 5.4 and 5.8 and (5.15). (cid:3)
Remark . Theorem 5.16 shows that the structure of Sol II ( s ; n ) (hypergeometric model) is some-what complicated. It will be shown in Theorem 6.7 that that of Sol I ( s ; n ) (Heun model) is morestraightforward. Remark . Theorem 5.16 also completely classifies the dimension dim C Sol II ( s ; n ). When n is even, the dimension dim C Sol II ( s ; n ) (= dim C Sol I ( s ; n )) was determined in [16, Thm. 5.13] byfactorization formulas of certain tridiagonal determinants (see Theorem 7.18 and Remark 7.21).We shall study such determinants in Section 7 from a different perspective from [16].5.2. The M -representations on Sol II ( s ; n ) . As Step B of the recipe in Section 4.4, we nextclassify the M -representations on Sol II ( s ; n ). Here is the classification. Theorem 5.19.
For each ( s ; n ) ∈ C × Z ≥ determined in Theorem 5.16, the M -representations on Sol II ( s ; n ) are classified as follows. (1) n ≡ II ( s ; n ) ≃ (+,+) if s ∈ C \ ( I +0 ∪ I − ∪ J ) , (+,+) ⊕ (+, − ) if s ∈ I +0 , (+,+) if s ∈ J , (+,+) ⊕ ( − ,+) if s ∈ I − . (2) n ≡ II ( s ; n ) ≃ H for s ∈ I . (3) n ≡ II ( s ; n ) ≃ ( − , − ) if s ∈ C \ ( I +2 ∪ I − ∪ J ) , ( − , − ) ⊕ ( − ,+) if s ∈ I +2 , ( − , − ) if s ∈ J , ( − , − ) ⊕ (+, − ) if s ∈ I − . (4) n ≡ II ( s ; n ) ≃ H for s ∈ I . Here the characters ( ε, ε ′ ) stand for the ones on C a [ s ; n ] ( t ) , C b [ s ; n ] ( t ) , and C c ± [ s ; n ] ( t ) at the sameplaces in Theorem 5.16. We prove Theorem 5.19 by considering the following cases separately. • Case 1: n is odd. • Case 2: n is even. Let k ∈ { , } . – Case 2a: n ≡ k (mod 4) and s ∈ C \ ( I − k ∪ J k ) – Case 2b: n ≡ k (mod 4) and s ∈ I + k ∪ J k – Case 2c: n ≡ k (mod 4) and s ∈ I − k Case 1.
We start with the case that Sol II ( s ; n ) is a two-dimensional representation of M . Lemma 5.20.
Suppose that n ≡ k (mod 4) and s ∈ I k for k = 1 , . Then, as an M -representation,we have C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) ≃ H . Proof.
In this case a [ s ; n ] ( t ) and b [ s ; n ] ( t ) are even and odd functions, respectively. Then the assertioncan be shown by essentially the same argument as the one for [27, Prop. 6.11] by replacing u n ( t )and v n ( t ) with a [ s ; n ] ( t ) and b [ s ; n ] ( t ) , respectively. Hence we omit the proof. (cid:3) Cases 2a.
We next consider the characters on C a [ s ; n ] ( t ). Lemma 5.21.
Suppose that n ≡ k (mod 4) and s ∈ C \ ( I − k ∪ J k ) for k = 0 , , Then M acts on C a [ s ; n ] ( t ) as a character.Proof. We only give a proof for the case k = 0, namely, n ≡ s ∈ C \ ( I − ∪ J ); theother case can be shown similarly. As C \ ( I − ∪ J ) = ( C \ ( I +0 ∪ I − ∪ J )) ∪ I +0 , we consider the cases s ∈ C \ ( I +0 ∪ I − ∪ J ) and s ∈ I +0 , separately.First suppose that s ∈ C \ ( I +0 ∪ I − ∪ J ). By Theorem 5.16, we have Sol II ( s ; n ) = C a [ s ; n ] ( t ).Since Sol II ( s ; n ) is an M -representation by Lemma 2.9, the assertion clearly holds for this case.We next suppose that s ∈ I +0 . In this case we have Sol II ( s ; n ) = C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) by Theorem5.16. As the exponent n − s for t n − s of b [ s ; n ] ( t ) is odd, a [ s ; n ] ( t ) and b [ s ; n ] ( t ) are an even andodd function, respectively. The transformation laws (4.27) imply that the action of M on Pol n [ t ]preserves the parities of the polynomials for n even. Hence M acts both on C a [ s ; n ] ( t ) and C b [ s ; n ] ( t )as a character. (cid:3) We next determine the characters on a [ s ; n ] ( t ) explicitly. It follows from Lemma 5.5 that a [ s ; n ] ( t )has degree deg a [ s ; n ] ( t ) = n for ( s, n ) with n ≡ k (mod 4) and s ∈ C \ ( I − k ∪ J k ) for k = 0 ,
2. Thus,in this case, the hypergeometric polynomial a [ s ; n ] ( t ) is given as a [ s ; n ] ( t ) = n/ X j =0 A j ( s ; n ) t j (5.22)with A j ( s ; n ) = (cid:0) − n , j (cid:1) (cid:0) − n + s − , j (cid:1)(cid:0) − n + s , j (cid:1) · j ! , where ( ℓ, m ) stands for the shifted factorial. Recall from (4.25) that M = {± m II j : j = 0 , , , } with m II0 = (cid:18) (cid:19) , m II1 = (cid:18) − (cid:19) , m II2 = (cid:18) √− −√− (cid:19) , m II3 = − (cid:18) √− √− (cid:19) . As m II3 = m II1 m II2 , it suffices to check the actions of m II1 and m II2 on a [ s ; n ] ( t ) = P n/ j =0 A j ( s ; n ) t j . Proposition 5.23.
Under the same hypothesis in Lemma 5.21, the character ( ε, ε ′ ) on C a [ s ; n ] ( t ) is given as follows. (1) n ≡ and s ∈ C \ ( I − ∪ J ) : C a [ s ; n ] ( t ) ≃ (+,+) . (2) n ≡ and s ∈ C \ ( I − ∪ J ) : C a [ s ; n ] ( t ) ≃ ( − , − ) . Proof.
Since the second assertion can be shown similarly, we only give a proof for the first assertion.Let n ≡ s ∈ C \ ( I − ∪ J ). We wish to show that both m II1 and m II2 act trivially.First, it is easy to see that the action of m II2 is trivial. Indeed, since n ≡ a [ s ; n ] ( t ) isan even function, the transformation law of (4.27) shows that m II2 : a [ s ; n ] ( t ) −→ ( √− n a [ s ; n ] ( − t ) = a [ s ; n ] ( t ) . In order to show that m II1 also acts trivially, observe that, by (4.27) and (5.22), we have m II1 : a [ s ; n ] ( t ) −→ t n a [ s ; n ] (cid:18) − t (cid:19) = n/ X j =0 A n − j ( s ; n ) t j . On the other hand, by Lemma 5.21 and Table 4, m II1 acts on a [ s ; n ] ( t ) by ±
1. Therefore, n/ X j =0 A n − j ( s ; n ) t j = ± n/ X j =0 A j ( s ; n ) t j . An easy computation shows that we have A n ( s ; n ) = 1 = A ( s ; n ), which forces t n a [ s ; n ] (cid:0) − t (cid:1) = a [ s ; n ] ( t ). Hence m II1 also acts on C a [ s ; n ] ( t ) trivially. (cid:3) Cases 2b.
Next we consider the characters on C b [ s ; n ] ( t ). Proposition 5.24.
Suppose that n ≡ k (mod 4) and s ∈ I + k ∪ J k for k = 0 , . Then M acts on C b [ s ; n ] ( t ) as a character as follows. (1) n ≡ and s ∈ I +0 ∪ J : C b [ s ; n ] ( t ) ≃ ( (+, − ) if s ∈ I +0 , (+,+) if s ∈ J . (2) n ≡ and s ∈ I +2 ∪ J : C b [ s ; n ] ( t ) ≃ ( ( − ,+) if s ∈ I +2 , ( − , − ) if s ∈ J . Proof.
Since the assertions can be shown similarly to Lemma 5.21 and Proposition 5.23, we omitthe proof. We remark that it is already shown in the proof of Lemma 5.21 that M acts on C b [ s ; n ] ( t )as a character for ( s, n ) with n ≡ s ∈ I +0 . (cid:3) Case 2c.
Now we consider C c ± [ s ; n ] ( t ) for n ≡ k (mod 4) and s ∈ I − k for k = 0 ,
2, where C c ± [ s ; n ] ( t ) is given as c ± [ s ; n ] ( t ) = a [ s ; n ] ( t ) ± C ( s ; n ) b [ s ; n ] ( t )with C ( s ; n ) in (5.14).Observe that, in this case, a [ s ; n ] ( t ) has degree deg a [ s ; n ] ( t ) = n + s − < n by Lemma 5.5. Therefore, a [ s ; n ] ( t ) is of the form a [ s ; n ] ( t ) = ( n + s − / X j =0 e A j ( s ; n ) t j . (5.25)To give an explicit formula of e A j ( s ; n ), first remark that when n ≡ s = −
1, wehave − n + s − = − n + s ∈ Z ≥ . Namely, for n = 4 ℓ + 2 and s = −
1, the hypergeometric polynomial a [ − ℓ +2] ( t ) is a [ − ℓ +2] ( t ) = F ( − ℓ − , − ℓ, − ℓ ; t ) . (5.26)In this paper we regard (5.26) as a [ − ℓ +2] ( t ) = k X j =0 ( − ℓ − , j ) j ! t j so that a [ − ℓ +2] ( t ) and b [ − ℓ +2] ( t ) still form a fundamental solutions to the hypergeometric equa-tion (4.45). Therefore e A j ( s ; n ) in (5.25) is given as e A j ( s ; n ) = ( − n ,j ) j ! if n ≡ s = − − n ,j )( − n + s − ,j )( − n + s ,j ) · j ! otherwise . In particular, by (5.14), we have C ( s ; n ) = e A n + s − ( s ; n ) . (5.27) It follows from (5.2) that b [ s ; n ] ( t ) is given as b [ s ; n ] ( t ) = t n − s ( n + s − / X j =0 B j ( s ; n ) t j (5.28)with B j ( s ; n ) = (cid:0) − n + s − , j (cid:1) (cid:0) − s − , j (cid:1)(cid:0) n − s , j (cid:1) · j ! . Lemma 5.29 below plays a key role to determine the M -representation on C c ± [ s ; n ] ( t ). Lemma 5.29.
Suppose that n ≡ k (mod 4) and s ∈ I − k for k = 0 , . Then, π n ( m II1 ) a [ s ; n ] ( t ) = C ( s ; n ) b [ s ; n ] ( t ) . Proof.
We only show the case of k = 0; the other case can be shown similarly, including theexceptional case for n ≡ s = −
1. Let n ≡ s ∈ I − . It follows from(5.13) that we have Sol II ( s ; n ) = C a [ s ; n ] ( t ) ⊕ C b [ s ; n ] ( t ) . Therefore there exist some constants c , c ∈ C such that π n ( m II1 ) a [ s ; n ] ( t ) = c a [ s ; n ] ( t ) + c b [ s ; n ] ( t ) . (5.30)On the other hand, by (4.27) and (5.25), we have π n ( m II1 ) a [ s ; n ] ( t ) = t n a [ s ; n ] (cid:18) − t (cid:19) = ( n + s − / X j =0 e A j ( s ; n ) t n − j . (5.31)Since n − n + s − > n + s −
12 = deg a [ s ; n ] ( t )for n ≡ s ∈ I − , all exponents n − j for t n − j in (5.31) is n − j > deg a [ s ; n ] ( t ).Therefore, π n ( m II1 ) a [ s ; n ] ( t ) = c b [ s ; n ] ( t ) . Further, by (5.31) and (5.28), we then have e A n + s − ( s ; n ) = c B ( s ; n ) = c . Now (5.27) concludes the assertion. (cid:3)
Proposition 5.32.
Suppose that n ≡ k (mod 4) and s ∈ I − k for k = 0 , . Then M acts on C c ± [ s ; n ] ( t ) as the following characters. (1) n ≡ and s ∈ I − . C c +[ s ; n ] ( t ) ≃ (+,+) and C c − [ s ; n ] ( t ) ≃ ( − ,+) . (2) n ≡ and s ∈ I − . C c +[ s ; n ] ( t ) ≃ ( − , − ) and C c − [ s ; n ] ( t ) ≃ (+, − ) . Proof.
As in Lemma 5.29, we only show the case of k = 0; the other case can be treadted similarly(including the exceptional case for n ≡ s = − n ≡ s ∈ I − . Toshow the assertion it suffices to consider only m II1 . Indeed, since the exponent n − s for t n − s in(5.28) is even, a [ s ; n ] ( t ) and b [ s ; n ] ( t ) are both even functions; thus, m II2 acts on C c ± [ s ; n ] ( t ) trivially by(4.27). To consider the action of m II1 on C c ± [ s ; n ] ( t ), observe that, by Lemma 5.29, we have π n ( m II1 ) b [ s ; n ] ( t ) = C ( s ; n ) − a [ s ; n ] ( t ) . Therefore, for ε ∈ { + , −} , m II1 transforms c ε [ s ; n ] ( t ) as m II1 : c ε [ s ; n ] ( t ) −→ εc ε [ s ; n ] ( t ) . Now Table 4 concludes the assertion. (cid:3)
The classification of
Hom M (Sol II ( s ; n ) , σ ) . As Step C in the recipe in Section 4.4, we nowclassify ( σ, s, n ) ∈ Irr( M ) × C × Z ≥ such that Hom M (Sol II ( s ; n ) , σ ) = { } . Given σ ∈ Irr( M ), wedefine I ± ( (+, − ) ), I ± ( ( − ,+) ), I ( H ) ⊂ Z × Z ≥ as follows. I + ( (+, − ) ) := { ( s, n ) ∈ (3 + 4 Z ≥ ) × (4 Z ≥ ) : n > s } ,I − ( (+, − ) ) := { ( s, n ) ∈ − (1 + 4 Z ≥ ) × (2 + 4 Z ≥ ) : n > | s |} ,I + ( ( − ,+) ) := { ( s, n ) ∈ (1 + 4 Z ≥ ) × (2 + 4 Z ≥ ) : n > s } ,I − ( ( − ,+) ) := { ( s, n ) ∈ − (3 + 4 Z ≥ ) × (4 Z ≥ ) : n > | s |} ,I ( H ) := { ( s, n ) ∈ (2 Z ) × (1 + 2 Z ≥ ) : n > | s |} . (5.33) Theorem 5.34.
The following conditions on ( σ, s, n ) ∈ Irr( M ) × C × Z ≥ are equivalent. (i) Hom M (Sol II ( s ; n ) , σ ) = { } . (ii) dim C Hom M (Sol II ( s ; n ) , σ ) = 1 . (iii) One of the following conditions holds. • σ = (+,+) : ( s, n ) ∈ C × Z ≥ . • σ = ( − , − ) : ( s, n ) ∈ C × (2 + 4 Z ≥ ) . • σ = (+, − ) : ( s, n ) ∈ I + ( (+, − ) ) ∪ I − ( (+, − ) ) . • σ = ( − ,+) : ( s, n ) ∈ I + ( ( − ,+) ) ∪ I − ( ( − ,+) ) . • σ = H : ( s, n ) ∈ I ( H ) .Further, for such ( σ, s, n ) , the space Hom M (Sol II ( s ; n ) , σ ) is given as follows. (1) σ = (+,+) : For n ∈ Z ≥ , we have Hom M (Sol II ( s ; n ) , (+,+) ) = C a [ s ; n ] ( t ) if s ∈ C \ ( I − ∪ J ) , C b [ s ; n ] ( t ) if s ∈ J , C c +[ s ; n ] ( t ) if s ∈ I − . (2) σ = ( − , − ) : For n ∈ Z ≥ , we have Hom M (Sol II ( s ; n ) , ( − , − ) ) = C a [ s ; n ] ( t ) if s ∈ C \ ( I − ∪ J ) , C b [ s ; n ] ( t ) if s ∈ J , C c +[ s ; n ] ( t ) if s ∈ I − . (3) σ = (+, − ) : We have
Hom M (Sol II ( s ; n ) , (+, − ) ) = ( C b [ s ; n ] ( t ) if ( s, n ) ∈ I + ( (+, − ) ) , C c − [ s ; n ] ( t ) if ( s, n ) ∈ I − ( (+, − ) ) . (4) σ = ( − ,+) : We have
Hom M (Sol II ( s ; n ) , ( − ,+) ) = ( C b [ s ; n ] ( t ) if ( s, n ) ∈ I + ( ( − ,+) ) , C c − [ s ; n ] ( t ) if ( s, n ) ∈ I − ( ( − ,+) ) . (5) σ = H : We have
Hom M (Sol II ( s ; n ) , H ) = C ϕ ( s ; n )II for ( s, n ) ∈ I ( H ) , where ϕ ( s ; n )II is a non-zero M -isomorphism ϕ ( s ; n )II : Sol II ( s ; n ) ∼ −→ H . Proof.
The theorem simply follows from Theorems 5.16 and 5.19. Indeed, for instance, supposethat σ = (+,+) . It then follows from Theorem 5.19 that Hom M (Sol II ( s ; n ) , (+,+) ) = { } if and onlyif n ≡ M (Sol II ( s ; n ) , (+,+) ) = C a [ s ; n ] ( t ) if s ∈ C \ ( I − ∪ J ) , C b [ s ; n ] ( t ) if s ∈ J , C c +[ s ; n ] ( t ) if s ∈ I − . This concludes the assertion for σ = (+,+) . Similarly, suppose that σ = H . Then, by Theorem5.19, we have Hom M (Sol II ( s ; n ) , H ) = { } if and only if ( s, n ) ∈ I ( H ). Furthermore, in thiscase, Sol II ( s ; n ) ≃ H . Therefore, Hom M (Sol II ( s ; n ) , H ) is spanned by a non-zero M -isomorphism ϕ ( s ; n )II : Sol II ( s ; n ) ∼ → H . Since the other cases can be handled similarly, we omit the proof. (cid:3) Remark . As for Theorem 5.16, it will be shown in Theorem 6.9 that Hom M (Sol I ( s ; n ) , σ ) issimpler than Hom M (Sol II ( s ; n ) , σ ).Now we give the K -type formulas for S ol ( s ; n ) K for each σ ∈ Irr( M ) in the polynomial realization(3.14) of Irr( K ). Theorem 5.36.
The following conditions on ( σ, s ) ∈ Irr( M ) × C are equivalent. (i) S ol ( s ; σ ) K = { } . (ii) One of the following conditions holds. • σ = (+,+) : s ∈ C . • σ = ( − , − ) : s ∈ C . • σ = (+, − ) : s ∈ Z . • σ = ( − ,+) : s ∈ Z . • σ = H : s ∈ Z Further, the K -type formulas for S ol ( s ; σ ) K may be given as follows. (1) σ = (+,+) : S ol ( s ; (+,+) ) K ≃ M n ≥ Pol n [ t ] for all s ∈ C . (2) σ = ( − , − ) : S ol ( s ; ( − , − ) ) K ≃ M n ≥ Pol n [ t ] for all s ∈ C . (3) σ = (+, − ) : S ol ( s ; (+, − ) ) K ≃ M n ≥ Pol | s | +1+4 n [ t ] for s ∈ Z . (4) σ = ( − ,+) : S ol ( s ; ( − ,+) ) K ≃ M n ≥ Pol | s | +1+4 n [ t ] for s ∈ Z . (5) σ = H : S ol ( s ; H ) K ≃ M n ≥ Pol | s | +1+4 n [ t ] for s ∈ Z . Proof.
We only give a proof for σ = (+, − ) ; the other cases may be handled similarly. Suppose that σ = (+, − ) . By (4.22) with J = II, we have S ol ( s ; (+, − ) ) K ≃ M n ≥ Pol n [ t ] ⊗ Hom M (Sol II ( s ; n ) , (+, − ) ) . Thus, S ol ( s ; (+, − ) ) K = { } if and only if Hom M (Sol II ( s ; n ) , (+, − ) ) = { } for some n ∈ Z ≥ , whichis further equivalent to s ∈ (3 + 4 Z ≥ ) ∪ ( − (1 + 4 Z ≥ )) = 3 + 4 Z by Theorem 5.34. This shows the assertion between (i) and (ii) for σ = (+, − ) .To determine the explicit branching law, observe that, by the Frobenius reciprocity, we haveHom K (Pol n [ t ] , S ol ( s ; (+, − ) ) K ) = { } if and only if Hom M (Sol II ( s ; n ) , (+, − ) ) = { } , which is equivalent to n ∈ Z ≥ with n > s for s ∈ Z ≥ or n ∈ Z ≥ with n > | s | for s ∈ − (1 + 4 Z ≥ )by Theorem 5.34. It also follows from Theorem 5.34 that S ol ( s ; (+, − ) ) K is multiplicity-free. There-fore, we have S ol ( s ; (+, − ) ) K ≃ M n ≡ n>s Pol n [ t ] if s ∈ Z ≥ , M n ≡ n> | s | Pol n [ t ] if s ∈ − (1 + 4 Z ≥ ) , which is equivalent to the proposed formula. (cid:3) Remark . The K -type formula for the case of σ = H is obtained also in [36, Prop. 5.4.6].We close this section by giving a proof of Theorem 1.7 in the introduction. Proof of Theorem 1.7.
The assertions simply follow from Theorem 5.36. Indeed, as S ol ( s ; n ) K isdense in S ol ( s ; n ), we have S ol ( s ; n ) K = { } if and only if S ol ( s ; n ) = { } . Then the equivalence( π n , Pol n [ t ]) ≃ ( n/
2) concludes the theorem. (cid:3) Heun model dπ I n ( D ♭s ) f ( t ) = 0The purpose of this short section is to interpret the results in Section 5 to the Heun model dπ I n ( D ♭s ) f ( t ) = 0. In particular we give a variant of Theorem 5.34 for Hom M (Sol I ( s ; n ) , σ ) inTheorem 6.9. We remark that the results in this section will play a key role to compute certaintridiagonal determinants in Section 7.6.1. Relationships between a [ s ; n ] ( t ) , b [ s ; n ] ( t ) , c ± [ s ; n ] ( t ) and u [ s ; n ] ( t ) , v [ s ; n ] ( t ) . As for the hyperge-ometric model dπ II n ( D ♭s ) f ( t ) = 0, we start with Step A of the recipe in Section 4.4, that is, theclassification of ( s, n ) ∈ C × Z ≥ such that Sol I ( s ; n ) = { } . Recall from (4.36) that we haveSol I ( s ; n ) ⊂ C u [ s ; n ] ( t ) ⊕ C v [ s ; n ] ( t ) (6.1)with u [ s ; n ] ( t ) = Hl ( − , − ns − n , − n − , , − n − s t ) ,v [ s ; n ] ( t ) = tHl ( − , − n − s ; − n − , − n − , , − n − s t ) . Thus, equivalently, we wish to classify ( s, n ) such that u [ s ; n ] ( t ) ∈ Pol n [ t ] or v [ s ; n ] ( t ) ∈ Pol n [ t ].To make full use of the results for Sol II ( s ; n ), we first consider a transfer of elements in Sol II ( s ; n )to Sol I ( s ; n ). Recall from (4.18) and (4.23) that k = 1 √ (cid:18) √− − −√− (cid:19) ∈ SU (2)gives an M -isomorphism π n ( k ) : Sol II ( s ; n ) ∼ −→ Sol I ( s ; n ) . (6.2)For p ( t ) , q ( t ) ∈ Pol n [ t ] with π n ( k ) p ( t ) ∈ C q ( t ), we write p ( t ) k ∼ q ( t ) . Then, for n ∈ Z ≥ , the polynomials a [ s ; n ] ( t ) , b [ s ; n ] ( t ) , c ± [ s ; n ] ( t ) ∈ Sol II ( s ; n ) are transferred to u [ s ; n ] ( t ) , v [ s ; n ] ( t ) ∈ Sol I ( s ; n ) via π n ( k ) as follows. Lemma 6.3.
Let n ∈ Z ≥ . We have the following. (1) n ≡ a [ s ; n ] ( t ) k ∼ u [ s ; n ] ( t ) for s ∈ C \ ( I − ∪ J ) . (b) b [ s ; n ] ( t ) k ∼ u [ s ; n ] ( t ) for s ∈ J . (c) c +[ s ; n ] ( t ) k ∼ u [ s ; n ] ( t ) for s ∈ I − . (d) b [ s ; n ] ( t ) k ∼ v [ s ; n ] ( t ) for s ∈ I +0 . (e) c − [ s ; n ] ( t ) k ∼ v [ s ; n ] ( t ) for s ∈ I − . (2) n ≡ a [ s ; n ] ( t ) k ∼ v [ s ; n ] ( t ) for s ∈ C \ ( I − ∪ J ) . (b) b [ s ; n ] ( t ) k ∼ v [ s ; n ] ( t ) for s ∈ J . (c) c +[ s ; n ] ( t ) k ∼ v [ s ; n ] ( t ) for s ∈ I − . (d) b [ s ; n ] ( t ) k ∼ u [ s ; n ] ( t ) for s ∈ I +2 . (e) c − [ s ; n ] ( t ) k ∼ u [ s ; n ] ( t ) for s ∈ I − .Proof. We only give a proof of (1)(a); the other cases can be shown similarly. Let n ≡ s ∈ C \ ( I − ∩ J ). It follows from Theorem 5.16 that in this case a [ s ; n ] ( t ) ∈ Sol II ( s ; n ) and thus π n ( k ) a [ s ; n ] ( t ) ∈ Sol I ( s ; n ). By (6.1), this implies that π n ( k ) a [ s ; n ] ( t ) = c u [ s ; n ] ( t ) + c v [ s ; n ] ( t ) forsome constants c , c ∈ C . We wish to show c = 0. To do so, it suffices to show that π n ( k ) a [ s ; n ] ( t )is an even function, as u [ s ; n ] ( t ) and v [ s ; n ] ( t ) are even and odd functions, respectively. It follows fromTheorem 5.19 that m II1 acts on a [ s ; n ] ( t ) trivially. Since the linear map π n ( k ) : Sol II ( s ; n ) → Sol I ( s ; n )is an M -isomorphism, by (4.24), this implies that m I1 acts on π n ( k ) a [ s ; n ] ( t ) trivially. Then, by (3.16)and the assumption n ≡ π n ( k ) a [ s ; n ] ( − t ) = ( √− n π n ( k ) a [ s ; n ] ( − t )= π n ( m I1 ) π n ( k ) a [ s ; n ] ( t )= π n ( k ) a [ s ; n ] ( t ) . Now the assertion follows. (cid:3)
Remark . Lemma 6.3 in particular gives Gauss-to-Heun transformations. Moreover, by Remark4.37, the transformations (1)(a) and (2)(a) reduce to Gauss-to-Gauss transformations for s = 0. Proposition 6.5.
Given n ∈ Z ≥ , the following conditions on s ∈ C are equivalent. (i) u [ s ; n ] ( t ) ∈ Pol n [ t ] . (ii) One of the following holds. (a) n ≡ s ∈ C . (b) n ≡ s ∈ I . (c) n ≡ s ∈ I +2 ∪ I − . (d) n ≡ s ∈ I .Proof. For the case of n even, the assertions simply follow from Theorem 5.16 and Lemma 6.3. Forthe case of n odd, suppose that n ≡ II ( s ; n ) = { } if andonly if s ∈ I . Moreover, for such s ∈ I , the dimension dim C Sol II ( s ; n ) is dim C Sol II ( s ; n ) = 2.Thus dim C Sol I ( s ; n ) = 2 for s ∈ I . Now the assertion follows from (6.1). Since the case of n ≡ (cid:3) Proposition 6.6.
Given n ∈ Z ≥ , the following conditions on s ∈ C are equivalent. (i) v [ s ; n ] ( t ) ∈ Pol n [ t ] . (ii) One of the following holds. (a) n ≡ s ∈ I +0 ∪ I − . (b) n ≡ s ∈ I . (c) n ≡ s ∈ C . (d) n ≡ s ∈ I .Proof. Since the proof is similar to the one for Proposition 6.5, we omit the proof. (cid:3)
Theorem 6.7.
The following conditions on ( s, n ) ∈ C × Z ≥ are equivalent. (i) Sol I ( s ; n ) = { } . (ii) One of the following conditions is satisfied. • n ≡ s ∈ C . • n ≡ s ∈ I . • n ≡ s ∈ C . • n ≡ s ∈ I .Moreover, for such ( s, n ) , the space Sol I ( s ; n ) may be described as follows. (1) n ≡ I ( s ; n ) = C u [ s ; n ] ( t ) if s ∈ C \ ( I +0 ∪ I − ) , C u [ s ; n ] ( t ) ⊕ C v [ s ; n ] ( t ) if s ∈ I +0 , C u [ s ; n ] ( t ) ⊕ C v [ s ; n ] ( t ) if s ∈ I − . (2) n ≡ I ( s ; n ) = C u [ s ; n ] ( t ) ⊕ C v [ s ; n ] ( t ) for s ∈ I . (3) n ≡ I ( s ; n ) = C v [ s ; n ] ( t ) if s ∈ C \ ( I +2 ∪ I − ) , C u [ s ; n ] ( t ) ⊕ C v [ s ; n ] ( t ) if s ∈ I +2 , C u [ s ; n ] ( t ) ⊕ C v [ s ; n ] ( t ) if s ∈ I − . (4) n ≡ I ( s ; n ) = C u [ s ; n ] ( t ) ⊕ C v [ s ; n ] ( t ) , for s ∈ I . Proof.
This is a summary of the results in Propositions 6.5 and 6.6. (cid:3)
Theorem 6.8.
For each ( s, n ) ∈ C × Z ≥ determined in Theorem 6.7, the M -representations on Sol II ( s ; n ) are classified as follows. (a) n ≡ I ( s ; n ) ≃ (+,+) if s ∈ C \ ( I +0 ∪ I − ) , (+,+) ⊕ (+, − ) if s ∈ I +0 , (+,+) ⊕ ( − ,+) if s ∈ I − . (b) n ≡ I ( s ; n ) ≃ H for s ∈ I . (c) n ≡ I ( s ; n ) ≃ ( − , − ) if s ∈ C \ ( I +2 ∪ I − ) , ( − ,+) ⊕ ( − , − ) if s ∈ I +2 , (+, − ) ⊕ ( − , − ) if s ∈ I − . (d) n ≡ I ( s ; n ) ≃ H for s ∈ I . Here the characters ( ε, ε ′ ) stand for the ones on C u [ s ; n ] ( t ) and C v [ s ; n ] ( t ) at the same places inTheorem 6.7.Proof. The case of n odd, the assertions simply follow from Theorem 5.19 via the M -isomorphism π n ( k ) : Sol II ( s ; n ) ∼ → Sol I ( s ; n ). Similarly, the assertions for n even are drawn by Lemma 6.3,Theorem 6.7, and Propositions 5.23, 5.24, and 5.32. (cid:3) Recall from (5.33) the subsets I ± ( (+, − ) ), I ± ( ( − ,+) ), I ( H ) ⊂ Z × Z ≥ . We now give the explicitdescription of Hom M (Sol I ( s ; n ) , σ ) = { } . Theorem 6.9.
The following conditions on ( σ, s, n ) ∈ Irr( M ) × C × Z ≥ are equivalent. (i) Hom M (Sol I ( s ; n ) , σ ) = { } . (ii) dim C Hom M (Sol I ( s ; n ) , σ ) = 1 . (iii) One of the following conditions holds. • σ = (+,+) : ( s, n ) ∈ C × Z ≥ . • σ = ( − , − ) : ( s, n ) ∈ C × (2 + 4 Z ≥ ) . • σ = (+, − ) : ( s, n ) ∈ I + ( (+, − ) ) ∪ I − ( (+, − ) ) . • σ = ( − ,+) : ( s, n ) ∈ I + ( ( − ,+) ) ∪ I − ( ( − ,+) ) . • σ = H : ( s, n ) ∈ I ( H ) .Moreover, for such ( σ, s, n ) , the space Hom M (Sol I ( s ; n ) , σ ) is given as follows. (1) σ = (+,+) : For n ∈ Z ≥ , we have Hom M (Sol I ( s ; n ) , (+,+) ) = C u [ s ; n ] ( t ) for all s ∈ C . (2) σ = ( − , − ) : For n ∈ Z ≥ , we have Hom M (Sol I ( s ; n ) , ( − , − ) ) = C v [ s ; n ] ( t ) for all s ∈ C . (3) σ = (+, − ) : We have
Hom M (Sol I ( s ; n ) , (+, − ) ) = ( C v [ s ; n ] ( t ) if ( s, n ) ∈ I + ( (+, − ) ) , C u [ s ; n ] ( t ) if ( s, n ) ∈ I − ( (+, − ) ) . (4) σ = ( − ,+) : We have
Hom M (Sol I ( s ; n ) , ( − ,+) ) = ( C u [ s ; n ] ( t ) if ( s, n ) ∈ I + ( ( − ,+) ) , C v [ s ; n ] ( t ) if ( s, n ) ∈ I − ( ( − ,+) ) . (5) σ = H : We have
Hom M (Sol I ( s ; n ) , H ) = C ϕ ( s ; n )I for ( s, n ) ∈ I ( H ) , where ϕ ( s ; n )I is a non-zero M -isomorphism ϕ ( s ; n )I : Sol I ( s ; n ) ∼ −→ H . Proof.
Since the theorem can be shown similarly to Theorem 5.34, we omit the proof. (cid:3) Sequences { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 of tridiagonal determinants The aim of this section is to discuss about two sequences { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 of k × k tridiagonal determinants associated to polynomial solutions to the Heun model dπ I n ( D ♭s ) f ( t ) = 0.We give factorization formulas for P [ n +22 ]( x ; n ) and Q [ n +12 ]( x ; n ) as well as palindromic property of { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 for n even. (See Definition 1.15 for the definition of palindromicproperty.) These are achieved in Theorems 7.18 and 7.22 (factorization formulas), and 7.29 and7.41 (palindromic property).7.1. Sequences { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 of tridiagonal determinants. We start withthe definitions of { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 . Let a ( x ) and b ( x ) be two polynomials such that a ( x ) = 2 x (2 x −
1) and b ( x ) = 2 x (2 x + 1) . (7.1)For instance, for x = 1 , , , . . . , we have a (1) = 1 · , a (2) = 3 · , a (3) = 5 · , . . . ,b (1) = 2 · , b (2) = 4 · , b (3) = 6 · , . . . . Similarly, for x = , , , . . . , we have a (cid:18) (cid:19) = 0 · , a (cid:18) (cid:19) = 2 · , a (cid:18) (cid:19) = 4 · , . . . ,b (cid:18) (cid:19) = 1 · , b (cid:18) (cid:19) = 3 · , b (cid:18) (cid:19) = 5 · , . . . . Clearly, the polynomials a ( x ) and b ( x ) satisfy a ( x ) = b (cid:18) x − (cid:19) and b ( x ) = a (cid:18) x + 12 (cid:19) . (7.2)We define k × k tridiagonal determinants P k ( x ; y ) and Q k ( x ; y ) in terms of a ( x ) and b ( x ) asfollows.(1) P k ( x ; y ) : • k = 0 : P ( x ; y ) = 1, • k = 1 : P ( x ; y ) = yx , • k ≥ P k ( x ; y ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) yx a (1) − a (cid:0) y (cid:1) ( y − x a (2) − a (cid:0) y − (cid:1) ( y − x a (3) . . . . . . . . . − a (cid:16) y − k +62 (cid:17) ( y − k + 8) x a ( k − − a (cid:16) y − k +42 (cid:17) ( y − k + 4) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .(2) Q k ( x ; y ) : • k = 0 : Q ( x ; y ) = 1, • k = 1 : Q ( x ; y ) = ( y − x , • k ≥ Q k ( x ; y ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y − x b (1) − b (cid:0) y − (cid:1) ( y − x b (2) − b (cid:0) y − (cid:1) ( y − x b (3) . . . . . . . . . − b (cid:16) y − k +42 (cid:17) ( y − k + 6) x b ( k − − b (cid:16) y − k +22 (cid:17) ( y − k + 2) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Example . Here are a few examples of P k ( x ; 4) and Q k ( x ; 6) for k = 2 , , P k ( x ; 4) : P ( x ; 4) = (cid:12)(cid:12)(cid:12)(cid:12) x · − · (cid:12)(cid:12)(cid:12)(cid:12) , P ( x ; 4) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · − · · − · − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , P ( x ; 4) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · − · · − · − x · − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2) Q k ( x ; 6) : Q ( x ; 6) = (cid:12)(cid:12)(cid:12)(cid:12) x · − · (cid:12)(cid:12)(cid:12)(cid:12) , Q ( x ; 6) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · − · · − · − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , Q ( x ; 6) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · − · · − · − x · − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Moreover, for instance, for y = 5 and k = 3, we have P ( x ; 5) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · − · x · − · − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and Q ( x ; 5) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · − · − x · − · − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Remark . In general P k ( x ; y ) and Q k ( x ; y ) satisfy the following properties for specific y and k .(1) If y = n ∈ Z ≥ , then P n +22 ( x ; n ) is anti-centrosymmetric: P n +22 ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nx a (1) − a (cid:0) n (cid:1) ( n − x a (2) − a (cid:0) n − (cid:1) ( n − x a (3) . . . . . . . . . − a (2) − ( n − x a (cid:0) n (cid:1) − a (1) − nx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7.5) (2) If y = n ∈ Z ≥ , then Q n ( x ; n ) is also anti-centrosymmetric: Q n ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( n − x b (1) − b (cid:0) n − (cid:1) ( n − x b (2) − b (cid:0) n − (cid:1) ( n − x b (3) . . . . . . . . . − b (2) − ( n − x b (cid:0) n − (cid:1) − b (1) − ( n − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7.6)(3) It follows from (7.2) that for y = n ∈ Z ≥ , we have P n +12 ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nx a (1) − b (cid:0) n − (cid:1) ( n − x a (2) − b (cid:0) n − (cid:1) ( n − x a (3) . . . . . . . . . − b (2) − ( n − x a (cid:0) n − (cid:1) − b (1) − ( n − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (7.7) Q n +12 ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( n − x b (1) − a (cid:0) n − (cid:1) ( n − x b (2) − a (cid:0) n − (cid:1) ( n − x b (3) . . . . . . . . . − a (2) − ( n − x b (cid:0) n − (cid:1) − a (1) − nx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7.8)We also have P ( x ; 1) = x and Q ( x ; 1) = − x . Therefore, Q n +12 ( x ; n ) = ( − n +12 P n +12 ( x ; n ) for n ∈ Z ≥ . (7.9)The tridiagonal determinants P k ( x ; y ) and Q k ( x ; y ) enjoy the following property. Lemma 7.10.
For k ∈ Z ≥ , we have P k ( − x ; y ) = ( − k P k ( x ; y ) and Q k ( − x ; y ) = ( − k Q k ( x ; y ) . Proof.
The identities for k = 0 , k ≥ k × k tridiagonal determinants: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − a b c − a b c − a b . . . . . . . . .c k − − a k − b k − c k − − a k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a b c a b c a b . . . . . . . . .c k − a k − b k − c k − a k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7.11) (cid:3) From the next subsections y is taken to be y = n ∈ Z ≥ and we shall discuss several propertiesof { P k ( x ; n ) } ∞ k =0 and { Q k ( x ; n ) } ∞ k =0 .7.2. Generating functions of { P k ( x ; n ) } ∞ k =0 and { Q k ( x ; n ) } ∞ k =0 . We first give the generatingfunctions of { P k ( x ; n ) } ∞ k =0 and { Q k ( x ; n ) } ∞ k =0 . Let u [ s ; n ] ( t ) be the local Heun function defined in(4.34). Write u [ s ; n ] ( t ) = P ∞ k =0 U k ( s ; n ) t k for the power series expansion at t = 0. It follows from (9.12) with (9.9) in Section 9 that each coefficient U k ( s ; n ) can be given as U k ( s ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E u − F u E u − F u E u − . . . . . . . . .F uk − E uk − − F uk − E uk − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (7.12)with E u = nsa (1) , E uk = ( n − k ) sa ( k + 1) , and F uk = a (cid:0) n − k +22 (cid:1) a ( k + 1) . (7.13)Similarly, equation (9.13) with (9.11) in Section 9 shows that the coefficients V k ( s ; n ) of the powerseries expansion v [ s ; n ] ( t ) = P ∞ k =0 V k ( s ; n ) t k +1 of the second solution v [ s ; n ] ( t ) to the Heun equation(4.33) at t = 0 are given as V k ( s ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E v − F v E v − F v E v − . . . . . . . . .F vk − E vk − − F vk − E vk − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (7.14)with E v = ( n − sb (1) , E vk = ( n − k − sb ( k + 1) , and F vk = b (cid:0) n − k (cid:1) b ( k + 1) . (7.15)Proposition 7.16 below then shows that u [ s ; n ] ( t ) and v [ s ; n ] ( t ) are in fact the “hyperbolic cosine”generating function of { P k ( s ; n ) } ∞ k =0 and “hyperbolic sine” generating function of { Q k ( s ; n ) } ∞ k =0 ,respectively. Proposition 7.16.
We have u [ s ; n ] ( t ) = ∞ X k =0 P k ( s ; n ) t k (2 k )! and v [ s ; n ] ( t ) = ∞ X k =0 Q k ( s ; n ) t k +1 (2 k + 1)! . Proof.
We only show the identity for u [ s ; n ] ( t ); the assertion for v [ s ; n ] ( t ) can be shown similarly.We wish to show that U k ( s ; n ) = P k ( s ; n ) / (2 k )! for all k ∈ Z ≥ . By definition, it is clear that U ( s ; n ) = 1 = P ( s ; n ) and U ( s ; n ) = ns/ P ( s ; n ) / k ≥
2. It followsfrom (7.13) that U k ( s ; n ) can be given as U k ( s ; n ) = ( − k Q kj =1 a ( j ) P k ( − s ; n ) = P k ( s ; n ) Q kj =1 a ( j ) . Here, Lemma 7.10 is applied from the second identity to the third. Now the assertion follows fromthe identity Q kj =1 a ( j ) = (2 k )!. (cid:3) For R k ( s ; n ) ∈ { P k ( s ; n ) , Q k ( s ; n ) } , we define S ol k ( R ; n ) := { s ∈ C : R k ( s ; n ) = 0 } . We recall from (5.3) the subsets I ± , I , I ± , I ⊂ Z . Corollary 7.17.
Let n ∈ Z ≥ . Then S ol n +22 ( P ; n ) and S ol n ( Q ; n ) are given as S ol n +22 ( P ; n ) = ( C if n ≡ ,I +2 ∪ I − if n ≡ and S ol n ( Q ; n ) = ( I +0 ∪ I − if n ≡ , C if n ≡ . Further, for n ∈ Z ≥ , the sets S ol n +12 ( P ; n ) and S ol n +12 ( Q ; n ) are given as S ol n +12 ( P ; n ) = S ol n +12 ( Q ; n ) = ( I if n ≡ ,I if n ≡ . Proof.
It follows from Propositions 6.5 and 7.16 that the following conditions on s ∈ C are equivalentfor n ∈ Z ≥ :(i) s ∈ S ol n +22 ( P ; n );(ii) u [ s ; n ] ( t ) ∈ Pol n [ t ];(iii) s ∈ ( C if n ≡ ,I +2 ∪ I − if n ≡ . This concludes the assertion for S ol n +22 ( P ; n ). The other cases can be shown similarly. (cid:3) Factorization formulas of P [ n +22 ]( x ; n ) and Q [ n +12 ]( x ; n ) . We next show the factorizationformulas for P n +22 ( x ; n ) and Q n ( x ; n ) for n even, and for P n +12 ( x ; n ) and Q n +12 ( x ; n ) for n odd. Weremark that any product of the form Q j − ℓ = j c ℓ with c ℓ ∈ Pol[ t ] (in particular, c ℓ ∈ C ) is regarded as Q j − ℓ = j c ℓ = 1.7.3.1. Factorization formulas of P n +22 ( x ; n ) and Q n ( x ; n ) . We start with P n +22 ( x ; n ) and Q n ( x ; n )for n even (see (7.5) and (7.6)). For n ∈ Z ≥ , let α n and β n be the products of the coefficients of x on the main diagonal of P n +22 ( x ; n ) and Q n ( x ; n ), respectively, Namely, we have α n = n Y ℓ =0 ( n − ℓ ) and β n = n − Y ℓ =0 ( n − − ℓ ) . It is remarked that α n and β n may be given as follows. α n = n ≡ , ( − n +24 n − Q ℓ =0 (1 + 2 ℓ ) if n ≡ β n = ( − n n − Q ℓ =0 (1 + 2 ℓ ) if n ≡ , n ≡ . Theorem 7.18 (Factorization formulas of P n +22 ( x ; n ) and Q n ( x ; n )) . For n ∈ Z ≥ , the polyno-mials P n +22 ( x ; n ) and Q n ( x ; n ) are either or factored as follows. P n +22 ( x ; n ) = n ≡ ,α n n − Q ℓ =0 ( x − (4 ℓ + 1) ) if n ≡ and Q n ( x ; n ) = β n n − Q ℓ =0 ( x − (4 ℓ + 3) ) if n ≡ , n ≡ . (7.20) Proof.
We only demonstrate the proof for P n +22 ( x ; n ); the assertion for Q n ( x ; n ) can be shownsimilarly. It follows from Corollary 7.17 that S ol n +22 ( P ; n ) = ( C if n ≡ ,I +2 ∪ I − if n ≡ . In particular, as α n is the product of the coefficients of x in P n +22 ( x ; n ), we have P n +22 ( x ; n ) = n ≡ ,α n Q s ∈ I +2 ∪ I − ( x − s ) if n ≡ . Now the assertion follows from the identity Q s ∈ I +2 ∪ I − ( x − s ) = Q n − ℓ =0 ( x − (4 ℓ + 1) ). (cid:3) Remark . The factorization formulas (7.19) and (7.20) can also be obtained from [18, Prop. 5.11].In fact, in [18], the dimension dim C Sol I ( s ; n )(= dim C Sol II ( s ; n )) was determined by using (7.19)and (7.20) (see Remark 5.18).7.3.2. Factorization formula of P n +12 ( x ; n ) . We next consider P n +12 ( x ; n ) and Q n +12 ( x ; n ) for n odd(see (7.7) and (7.8)). As shown in (7.9), the determinant Q n +12 ( x ; n ) is given as Q n +12 ( x ; n ) = ( − n +12 P n +12 ( x ; n ) for n ∈ Z ≥ . It thus suffices to only consider P n +12 ( x ; n ). For n odd, let γ n be the product of the coefficients of x on the main diagonal of P n +12 ( x ; n ), namely, γ n = n − Y ℓ =0 ( n − ℓ ) . We remark that γ n may be given as γ n = ( − n − n − Q ℓ =0 (3 + 2 ℓ ) if n ≡ , ( − n +14 n − Q ℓ =0 (3 + 2 ℓ ) if n ≡ . Theorem 7.22 (Factorization formula of P n +12 ( x ; n )) . For n ∈ Z ≥ , the polynomial P n +12 ( x ; n ) is factored as P n +12 ( x ; n ) = γ n n − Y ℓ =0 ( x − ( n −
1) + 4 ℓ ) . (7.23) Equivalently, we have P n +12 ( x ; n ) = γ n x n − Q ℓ =1 ( x − (4 ℓ ) ) if n ≡ ,γ n n − Q ℓ =0 ( x − (4 ℓ + 2) ) if n ≡ . Proof.
Since the argument is similar to that for Theorem 7.18, we omit the proof. (cid:3)
The factorization formula of P n +12 ( x ; n ) in (7.23) suggests that one may express P n +12 ( x ; n ) interms of the so-called Sylvester determinant
Sylv( x ; n ) ((1.21)). The factorization formula (7.25)below for Sylv( x ; n ) was first observed by Sylvester in 1854 ([35]). Later, a number of people suchas Askey ([1]), Edelman–Kostlan ([10]), Holtz ([13]), Kac ([21]), Mazza ([30, p. 442]), Taussky-Todd ([37]), among others, gave several proofs. For more details about the Sylvester determinantSylv( x ; n ) as well as some related topics, see, for instance, [7, 8, 31, 32] and references therein. Weremark that the factorization formula (7.25) also readily follows from a general theory of sl (2 , C )representations (see Proposition 8.3 and Section 10). Fact . Let n ∈ Z ≥ . The Sylvester determinant Sylv( x ; n )in (1.21) is factored as Sylv( x ; n ) = n Y ℓ =0 ( x − n + 2 ℓ ) . (7.25)Equivalently, we have Sylv( x ; n ) = n − Q ℓ =0 ( x − (2 ℓ + 1) ) if n is odd ,x n Q ℓ =1 ( x − (2 ℓ ) ) if n is even . (7.26) Corollary 7.27.
For n odd, we have P n +12 ( x ; n ) = 2 n +1 Sylv( 12 ; n −
12 )Sylv( x n −
12 ) . Proof.
By a direct computation one finds n − Y ℓ =0 ( n − ℓ ) = 2 n +12 Sylv( 12 ; n −
12 )and n − Y ℓ =0 ( x − ( n −
1) + 4 ℓ ) = 2 n +12 Sylv( x n −
12 ) . Now the proposed identity follows from Theorem 7.22 as γ n = Q n − ℓ =0 ( n − ℓ ). (cid:3) Palindromic properties of { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 . We finish this section byshowing the palindromic properties of { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 (see Definition 1.15). For s ∈ R \{ } , we set sgn( s ) := ( +1 if s > , − s < . Palindromic property of { P k ( x ; y ) } ∞ k =0 . We start with { P k ( x ; y ) } ∞ k =0 . Recall from Corollary7.17 that S ol n +22 ( P ; n ) for n ∈ Z ≥ is given as S ol n +22 ( P ; n ) = ( C if n ≡ ,I +2 ∪ I − if n ≡ . We then define a map θ ( P ; n ) : S ol n +22 ( P ; n ) → {± } as θ ( P ; n ) ( s ) = ( n ≡ , sgn( s ) if n ≡ . (7.28) Theorem 7.29 (Palindromic property of { P k ( x ; y ) } ∞ k =0 ) . The pair ( { P k ( x ; y ) } ∞ k =0 , { (2 k )! } ∞ k =0 ) isa palindromic pair with degree n . Namely, let n ∈ Z ≥ and s ∈ S ol n +22 ( P ; n ) . Then we have P k ( s ; n ) = 0 for k ≥ n +22 and P k ( s ; n )(2 k )! = θ ( P ; n ) ( s ) P n − k ( s ; n )( n − k )! for k ≤ n . (7.30) Equivalently, for k ≤ n , the palindromic identity (7.30) is given as follows. (1) n ≡ We have P k ( s ; n )(2 k )! = P n − k ( s ; n )( n − k )! for all s ∈ C . (7.31)(2) n ≡ We have P k ( s ; n )(2 k )! = sgn( s ) P n − k ( s ; n )( n − k )! for s ∈ I +2 ∪ I − . (7.32) Proof.
Take s ∈ S ol n +22 ( P ; n ). It follows from the equivalence given in the proof of Corollary 7.17that P k ( s ; n ) = 0 for k ≥ n +22 . Thus, to prove the theorem, it suffices to show (7.31) and (7.32) for k ≤ n . We first show that P k ( s ; n )(2 k )! = ± P n − k ( s ; n )( n − k )! . (7.33)It follows from Theorem 6.8 and Corollary 7.17 that M acts on u [ s ; n ] ( t ) as a character χ ( ε,ε ′ ) in(3.12); in particular, we have u [ s ; n ] ( t ) = χ ( ε,ε ′ ) ( m I2 ) π n ( m I2 ) u [ s ; n ] ( t ) . (7.34)By (3.16), the identity (7.34) is equivalent to u [ s ; n ] ( t ) = χ ( ε,ε ′ ) ( m I2 ) t n u [ s ; n ] (cid:18) − t (cid:19) . (7.35) As s ∈ S ol n +22 ( P ; n ), by Proposition 7.16, we have u [ s ; n ] ( t ) = n X k =0 P k ( s ; n ) t k (2 k )! . The identity (7.35) thus yields the identity n X k =0 P k ( s ; n ) t k (2 k )! = χ ( ε,ε ′ ) ( m I2 ) n X k =0 P n − k ( s ; n ) t k ( n − k )! . (7.36)Since χ ( ε,ε ′ ) ( m I2 ) ∈ {± } , the identity (7.33) follows from (7.36).In order to show (7.31) and (7.32), observe that Theorem 6.8 shows that the character χ ( ε,ε ′ ) isgiven as • χ ( ε,ε ′ ) = χ (+,+) for n ≡ s ∈ C , • χ ( ε,ε ′ ) = χ ( − ,+) for n ≡ s ∈ I +2 , • χ ( ε,ε ′ ) = χ (+, − ) for n ≡ s ∈ I − .By Table 3, we have χ (+,+) ( m I2 ) = χ ( − ,+) ( m I2 ) = 1 and χ (+, − ) ( m I2 ) = −
1. Now (7.31) and (7.32)follow from (7.36). (cid:3)
Theorem 7.29 in particular implies the factorial identity of P n ( s ; n ) (see (1.19)) as follows. Corollary 7.37 (Factorial identity of P n ( x ; n )) . Let n ∈ Z ≥ . Then the following hold. (1) n ≡ We have P n ( s ; n ) = n ! for all s ∈ C . (2) n ≡ We have P n ( s ; n ) = sgn( s ) n ! for s ∈ I +2 ∪ I − . (7.38) Proof.
This is simply the case of k = n in (7.31) and (7.32). (cid:3) We now give a proof for Corollary 1.24 in the introduction.
Proof of Corollary 1.24.
It follows from Corollary 7.37 that it suffices to show (1.25). Suppose n ≡ I +2 ∪ I − = (cid:26) ± (4 j + 1) : j = 0 , , , . . . , n − (cid:27) , we have I +2 ∪ I − = {± , ± , ± . . . , ± ( n − } . The identity (7.38) then concludes the assertion. (cid:3)
Theorem 7.29 shows that ( { P k ( x ; y ) } ∞ k =0 , { (2 k )! } ∞ k =0 ) is a palindromic pair for n even. Proposition7.39 below shows that it is not the case for n odd. Proposition 7.39.
The pair ( { P k ( x ; y ) } ∞ k =0 , { (2 k )! } ∞ k =0 ) is not a palindromic pair for n odd. Proof.
Proof by contradiction. Suppose the contrary, that is, there exist n ∈ Z ≥ and a map d : Z ≥ → R with d ( n ) ∈ Z ≥ such that, for all s ∈ S ol d ( n )+1 ( P ; n ), we have P k ( s ; n ) = 0 for k ≥ d ( n ) + 1 and P k ( s ; n )(2 k )! = ± P d ( n ) − k ( s ; n )(2 d ( n ) − k )! for k ≤ d ( n ) . Take s ∈ S ol d ( n )+1 ( P ; n ). It follows from Proposition 7.16 that the palindromity of the pair( { P k ( x ; y ) } ∞ k =0 , { (2 k )! } ∞ k =0 ) implies u [ s ; n ] ( t ) = d ( n ) X k =0 P k ( s ; n ) t k (2 k )! ∈ Pol d ( n ) [ t ]and t d ( n ) u [ s ; n ] (cid:18) t (cid:19) = ± u [ s ; n ] ( t ) , which is, by (3.16), equivalent to π d ( n ) ( m I2 ) u [ s ; n ] ( t ) = ± u [ s ; n ] ( t ) . Since u [ s ; n ] ( − t ) = u [ s ; n ] ( t ), this further implies that C u [ s ; n ] ( t ) is a one-dimensional representationof M for such ( s, n ). On the other hand, as n ∈ Z ≥ , it follows from Proposition 6.5 that,for n ≡ ℓ (mod 4) for ℓ = 1 ,
3, we have s ∈ I ℓ ∩ S ol d ( n )+1 ( P ; n ) . In particular, by Theorem 6.8, the space C u [ s ; n ] ( t ) ⊕ C v [ s ; n ] ( t ) forms the unique two-dimensionalirreducible representation of M , which is a contradiction. Now the proposed assertion follows. (cid:3) Palindromic property of { Q k ( x ; y ) } ∞ k =0 . We next consider { Q k ( x ; y ) } ∞ k =0 . Corollary 7.17shows that S ol n ( Q ; n ) = ( I +0 ∪ I − if n ≡ , C if n ≡ . We then define a map θ ( Q ; n ) : S ol n ( Q ; n ) → {± } as θ ( Q ; n ) ( s ) = ( sgn( s ) if n ≡ , n ≡ . (7.40) Theorem 7.41 (Palindromic property of { Q k ( x ; y ) } ∞ k =0 ) . The pair ( { Q k ( x ; y ) } ∞ k =0 , { (2 k + 1)! } ∞ k =0 ) is a palindromic pair with degree n − . Namely, let n ∈ Z ≥ ) and s ∈ S ol n ( Q ; n ) . Then wehave Q k ( s ; n ) = 0 for k ≥ n and Q k ( s ; n )(2 k + 1)! = θ ( Q ; n ) ( s ) Q n − − k ( s ; n )( n − k − k ≤ n − . (7.42) Equivalently, for k ≤ n − , the palindromic identity (7.42) is given as follows. (1) n ≡ We have Q k ( s ; n )(2 k + 1)! = sgn( s ) Q n − − k ( s ; n )( n − k − s ∈ I +0 ∪ I − . (7.43) (2) n ≡ We have Q k ( s ; n )(2 k + 1)! = Q n − − k ( s ; n )( n − k − s ∈ C . (7.44) Proof.
Since the argument goes similarly to the one for Theorem 7.29, we omit the proof. (cid:3)
Corollary 7.45 (Factorial identity of Q n − ( x ; n )) . Let n ∈ Z ≥ ) . Then the following holdfor Q n − ( s ; n ) (see (1.20) ). (1) n ≡ We have Q n − ( s ; n ) = sgn( s )( n − s ∈ I +0 ∪ I − . (7.46)(2) n ≡ We have Q n − ( s ; n ) = ( n − s ∈ C . Proof.
This is the case of k = n − in (7.43) and (7.44). (cid:3) Here is a proof of Corollary 1.26 in the introduction.
Proof of Corollary 1.26.
By Corollary 7.45, it suffices to show (1.27). Suppose n ≡ I +0 ∪ I − = (cid:26) ± (4 j + 3) : j = 0 , , , . . . , n − (cid:27) , we have I +0 ∪ I − = {± , ± , ± , . . . , ± ( n − } . Then (7.46) concludes the proposed identity. (cid:3)
We close this section by stating about the palindromity of the pair ( { Q k ( x ; y ) } ∞ k =0 , { (2 k +1)! } ∞ k =0 )for n odd. Proposition 7.47.
The pair ( { Q k ( x ; y ) } ∞ k =0 , { (2 k + 1)! } ∞ k =0 ) is not a palindromic pair for n odd.Proof. As the proof is similar to the one for Proposition 7.39, we omit the proof. (cid:3) Cayley continuants { Cay k ( x ; y ) } ∞ k =0 and Krawtchouk polynomials {K k ( x ; y ) } ∞ k =0 In Section 7, it was shown that { P k ( x ; y ) } ∞ k =0 and { Q k ( x ; y ) } ∞ k =0 admit palindromic proper-ties. In this short section we show that the sequences of Cayley continuants { Cay k ( x ; y ) } ∞ k =0 andKrawtchouk polynomials {K k ( x ; y ) } ∞ k =0 also admit palindromic properties. These are achieved inTheorem 8.8 for Cayley continuants and in Theorem 8.19 for Krawtchouk polynomials.8.1. Cayley continuants { Cay k ( x ; y ) } ∞ k =0 . We start with the definition of { Cay k ( x ; y ) } ∞ k =0 . Foreach k ∈ Z ≥ , the k × k tridiagonal determinants Cay k ( x ; y ) are defined as follows ([4, 31] and [30,p. 429]). • k = 0 : Cay ( x : y ) = 1, • k = 1 : Cay ( x ; y ) = x , • k ≥ k ( x ; y ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x y x y − x . . . . . . . . .y − k + 3 x k − y − k + 2 x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .Following [31], we refer to each Cay k ( x ; y ) as a Cayley continuant . There are some combinatorialinterpretations of them. For instance, they can be thought of as a generalization of the raisingfactorial (shifted factorial) x k := x ( x + 1) · · · ( x + k − x k := x ( x − · · · ( x − k + 1), and the factorial k !. Indeed, we haveCay k ( x ; x ) = x k , Cay k ( x ; − x ) = x k , and Cay k (1; −
1) = k ! . For more details on a relationship with combinatorics, see [31].
Remark . When y = n ∈ Z ≥ , the Cayley continuant Cay n +1 ( x ; n ) is the Sylvester determi-nant Sylv( x ; n ) (see (1.21)). Thus, by (7.25), we haveCay n +1 ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x n x n − x . . . . . . . . . x n − x n x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Sylv( x ; n ) = n Y ℓ =0 ( x − n + 2 ℓ ) . (8.2)The generating function of the Cayley continuants { Cay k ( x ; y ) } ∞ k =0 is known and several proofsare in the literature (see, for instance, [21, 31, 37]). In Proposition 8.3 below, by utilizing the ideafor the proof of Proposition 7.16, we shall provide another proof for the generating function aswell as Sylvester’s formula (8.2). We remark that our argument for the formula (8.2) can be moreabstract. (See Section 10.) Proposition 8.3.
We have (1 + t ) y + x (1 − t ) y − x = ∞ X k =0 Cay k ( x ; y ) t k k ! . (8.4) In particular,
Cay n +1 ( x ; n ) = n Y ℓ =0 ( x − n + 2 ℓ ) . Proof.
Let E + and E − be the elements of sl (2 , C ) in (3.2). It then follows from (3.17) that thematrix dπ n ( E + + E − ) B with respect to the ordered basis B := { , t, t , . . . , t n } is given as dπ n ( E + + E − ) B = − − n − − ( n −
1) 0 − . . . . . . . . . − − ( n − − − n − . Therefore the Sylvester determinant Cay n +1 ( x ; n )(= Sylv( x ; n ))) is the characteristic polynomialof dπ n ( E + + E − ); thus, to show (8.2), it suffices to find the eigenvalues of dπ n ( E + + E − ). We set D [ x ; n ] := (1 − t ) ddt + nt − x. By (3.17), for p ( t ) ∈ Pol n [ t ], we have( dπ n ( E + ) + dπ n ( E − ) + x ) p ( t ) = 0 ⇐⇒ D [ x ; n ] p ( t ) = 0 . Thus it further suffices to determine s ∈ C for which D [ s ; n ] p ( t ) = 0 has a non-zero solution.By separation of variables, any solution to D [ x ; n ] f ( t ) = 0, where f ( t ) is not necessarily a poly-nomial, is of the from f ( t ) = c · (1 + t ) n + x (1 − t ) n − x for some constant c . It is clear that (1 + t ) n + x (1 − t ) n − x becomes a polynomial if and only if x = n − ℓ for ℓ = 0 , , . . . , n . Now Sylvester’s formula (8.2) follows.In order to show (8.4), suppose that P ∞ k =0 h k ( x ; n ) t k is the power series solution of D [ x ; n ] f ( t ) = 0with h ( x ; n ) = 1. Then, by a direct observation, the coefficients h k ( x ; n ) for k ≥ h ( x ; n ) = x,h k +1 ( x ; n ) = xk + 1 h k ( x ; n ) + k − n − k + 1 h k − ( x ; n ) . It then follows from Lemma 9.5 in Section 9 with (7.11) that each h k ( x ; n ) is given as h k ( x ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − − n x − − n − x −
1. . . . . . . . . − n − k +3 k − xk − − − n − k +2 k xk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Cay k ( x ; n ) k ! . Thus P ∞ k =0 Cay k ( x ; n ) k ! t k is a solution of D [ x ; n ] f ( t ) = 0; consequently, there exists a constant c suchthat c · (1 + t ) n + x (1 − t ) n − x = ∞ X k =0 Cay k ( x ; n ) k ! t k . As both sides are equal to 1 at t = 0, we have c = 1; thus, the identity (8.4) holds for y = n ∈ Z ≥ .Since both sides of (8.4) are well-defined for any y ∈ C for suitable values of t , the desired identitynow holds. (cid:3) Remark . The differential operator dπ n ( E + + E − ) is related to the study of K -type decompositionof the space of K -finite solutions to intertwining differential operators. In fact, let X be an nilpotentelement of sl (2 , C ) defined in (3.1). Then dπ n ( E + + E − ) is the realization of the differential operator R ( X ) on the K -type Pol n [ t ] via the identification Ω I : k ∼ → sl (2 , C ) in (3.7) (see [27, Sect. 5]).We write g C [ x ; n ] ( t ) := (1 + t ) n + x (1 − t ) n − x and set S ol k (Cay; n ) := { s ∈ C : Cay k ( s ; n ) = 0 } . It follows from (8.2) (or, equivalently, from (8.4)) that S ol n +1 (Cay; n ) = { n − ℓ : ℓ = 0 . , . . . , n } . (8.6) Corollary 8.7.
The following conditions of s ∈ C are equivalent. (i) g C [ s ; n ] ( t ) ∈ Pol n [ t ] . (ii) s ∈ S ol n +1 (Cay; n ) .Proof. A direct consequence of Proposition 8.3. (cid:3)
Palindromic property for { Cay k ( x ; y ) } ∞ k =0 . Now we show the palindromic property andfactorial identity for the Cayley continuants { Cay k ( x ; y ) } ∞ k =0 . Theorem 8.8 (Palindromic property of { Cay k ( x ; y ) } ∞ k =0 and factorial identity of Cay n ( x ; n )) . Thepair ( { Cay k ( x ; y ) } ∞ k =0 , { k ! } ∞ k =0 ) is a palindromic pair with degree n . Namely, let n ∈ Z ≥ and s ∈ S ol n +1 (Cay; n ) . Then we have Cay k ( s ; n ) = 0 for k ≥ n + 1 and Cay k ( s ; n ) k ! = ( − n − s Cay n − k ( s ; n )( n − k )! for k ≤ n. In particular we have
Cay n ( s ; n ) = ( − n − s n ! for s ∈ S ol n +1 (Cay; n ) . (8.9) Proof.
Take s ∈ S ol n +1 (Cay; n ). It follows from Corollary 8.7 that, for such s , we have g C [ s ; n ] ( t ) = n X k =0 Cay k ( s ; n ) k ! t k , (8.10)namely, Cay k ( s ; n ) = 0 for k ≥ n + 1. Further, (8.10) shows that t n g C [ s ; n ] (cid:0) t (cid:1) is given as t n g C [ s ; n ] (cid:18) t (cid:19) = n X k =0 Cay n − k ( s ; n )( n − k )! t k . On the other hand, as g C [ x ; n ] ( t ) = (1 + t ) n + x (1 − t ) n − x , we also have t n g C [ s ; n ] (cid:18) t (cid:19) = ( − n − s g C [ s ; n ] ( t ) . Therefore, n X k =0 Cay n − k ( s ; n )( n − k )! t k = ( − n − s n X k =0 Cay k ( s ; n ) k ! t k , which yields Cay k ( x ; n ) k ! = ( − n − s Cay n − k ( x ; n )( n − k )! for all k ≤ n. The second identity is the case k = n . This concludes the theorem. (cid:3) We close this section by showing a proof of Corollary 1.30 in the introduction.
Proof of Corollary 1.30.
As Cay n ( x ; n ) = Sylv( x ; n ), it follows from Sylvester’s factorization for-mula (1.22) or (7.26) that S ol n +1 (Cay; n ) = { s ∈ C : Sylv( s ; n ) = 0 } = ( { , ± , ± , . . . , ± n } if n is even , {± , ± , ± , . . . , ± n } if n is odd . Now the proposed identities are direct consequences of (8.9). (cid:3)
Krawtchouk polynomials {K k ( x ; y ) } ∞ k =0 . From the palindromic property of the Cayley con-tinuants { Cay k ( x ; y ) } ∞ k =0 , one may deduce that of a certain sequence {K k ( x ; y ) } ∞ k =0 of polynomials,where the polynomials K k ( x ; n ) for y = n ∈ Z ≥ are Krawtchouk polynomials in the sense of [28,p. 130] (equivalently, the case of p = 2 for [28, p. 151]). We close this section by discussing thepalindromic property of {K k ( x ; y ) } ∞ k =0 .We start with the definition of K k ( x ; y ). Definition . For k ∈ Z ≥ , we define a polynomial K k ( x ; y ) of x and y as K k ( x ; y ) := k X j =0 ( − j (cid:18) xj (cid:19)(cid:18) y − xk − j (cid:19) , where the binomial coefficient (cid:0) am (cid:1) is defined as (cid:18) am (cid:19) = a ( a − ··· ( a − m +1) m ! if m ∈ Z ≥ , m = 0 , . For y = n ∈ Z ≥ , the polynomials K k ( x ; n ) are called Krawtchouk polynomials ([28, p. 130]). Inthis paper, we also call the polynomials K k ( x ; y ) of two variables Krawtchouk polynomials. Remark . Symmetric Krawtchouk polynomials (see, for instance, [24, p. 237] ) are used to showSylvester’s factorization formula (8.2) in [1]. We remark that the definition of the Krawtchoukpolynomial K k ( x ; n ) in this paper is different from one in the cited paper; in particular, K k ( x ; n ) isnon-symmetric.It readily follows from Definition 8.11 that the Krawtchouk polynomials {K k ( x ; y ) } ∞ k =0 have thefollowing generating function: (1 + t ) y − x (1 − t ) x = ∞ X k =0 K k ( x ; y ) t k . (8.13)By comparing (8.13) with the generating function (8.4) of the Cayley continuants { Cay k ( x ; y ) } ∞ k =0 ,we have K k ( x ; y ) = Cay k ( y − x ; y ) k ! . (8.14)Then Cay k ( x ; y ) may be expressed explicitly as follows. Here we remark that in Proposition 8.15below we use the notation ( m ) j := m ( m + 1) · · · ( m + j −
1) for the raising factorial (shifted factorial) instead of ( m, j ) in (5.14) to make the comparison with the falling factorial ( m ) j := m ( m − · · · ( m − j + 1) clearer. Proposition 8.15.
We have
Cay k ( x ; y ) = k X j =0 (cid:18) kj (cid:19) (cid:18) x + y (cid:19) j (cid:18) x − y (cid:19) k − j . (8.16) Proof.
By (8.4) and (8.13), we haveCay k ( x ; y ) = ( k !) · K k ( y − x y ) = ( k !) · k X j =0 ( − j (cid:18) y − x j (cid:19)(cid:18) y + x k − j (cid:19) . As ( − j ( m ) j = ( − m ) j , this concludes the proposed identity. (cid:3) Remark . The identity (8.16) is also shown in [31, p. 356] slightly differently.Via the identity (8.13), we are now going to give the factorization formula, the palindromicproperty, and the factorial identity for the Krawtchouk polynomials {K k ( x ; y ) } ∞ k =0 . Proposition 8.18 (Factorization formula of K n +1 ( x ; n )) . For given n ∈ Z ≥ , we have K n +1 ( x ; n ) = ( − n +1 ( n + 1)! n Y ℓ =0 ( x − ℓ ) = ( − n +1 (cid:18) xn + 1 (cid:19) . Proof.
The proposed identity simply follows from (8.14) and Sylvester’s factorization formula (8.2). (cid:3)
We next show the palindromic property and factorial identity for Krawtchouk polynomials {K k ( x ; y ) } . We denote by { } ∞ k =0 the sequence { a k } ∞ k =0 such that a k = 1 for all k . Theorem 8.19 (Palindromic property of {K k ( x ; y ) } ∞ k =0 and factorial identity of K n ( x ; n )) . Thepair ( {K k ( x ; y ) } ∞ k =0 , { } ∞ k =0 ) is a palindromic pair with degree n . Namely, let n ∈ Z ≥ and s ∈ S ol n +1 ( K ; n ) . Then we have K k ( s ; n ) = 0 for k ≥ n + 1 and K k ( s ; n ) = ( − s K n − k ( s ; n ) for k ≤ n. In particular, we have K n ( s ; n ) = ( − s for s ∈ S ol n +1 ( K ; n ) . (8.20) Proof.
The proof is the same as that of Theorem 8.8, we omit the proof. (cid:3)
We close this section by giving a proof of Corollary 1.34 in the introduction.
Proof of Corollary 1.34.
It follows from Proposition 8.18 that S ol n +1 ( K ; n ) = { s ∈ C : K n +1 ( s ; n ) = 0 } = { , , , . . . , n } . Then (8.20) concludes the proposed identity. (cid:3) Appendix A: local Heun functions
In this appendix we collect several facts and lemmas for the local Heun function Hl ( a, q ; α, β, γ, δ ; z )at z = 0 that are used in the main part of this paper.9.1. General facts.
As in (4.30), we set D H ( a, q ; α, β, γ, δ ; z ) := d dz + (cid:18) γz + δz − εz − a (cid:19) ddz + αβz − qz ( z − z − a )with γ + δ + ε = α + β + 1. Then the equation D H ( a, q ; α, β, γ, δ ; z ) f ( z ) = 0is called Heun’s differential equation . The P -symbol is P a ∞ α z q − γ − δ − ε β . Let Hl ( a, q ; α, β, γ, δ ; z ) stands for the local Heun function at z = 0 ([38]). As in [38], wenormalize Hl ( a, q ; α, β, γ, δ ; z ) so that Hl ( a, q ; α, β, γ, δ ; 0) = 1. It is known that, for γ / ∈ Z , thefunctions Hl ( a, q ; α, β, γ, δ ; z ) and z − γ Hl ( a, q ′ ; α − γ + 1 , β − γ + 1 , − γ, δ ; z ) (9.1)with q ′ := q + (1 − γ )( ε + aδ ) are two linearly independent solutions at z = 0 to the Heun equation D H ( a, q ; α, β, γ, δ ; z ) f ( z ) = 0. (See, for instance, [29] and [34, p. 99].) Remark . In [34, p. 99], there seems to be a typographical error on the formula λ ′ = λ + ( a + b + 1 − c )(1 − c ) . This should read as λ ′ = λ + ( a + b + 1 − c + ( t − d )(1 − c ) . Let Hl ( a, q ; α, β, γ, δ ; z ) = P ∞ k =0 c k z k be the power series expansion at z = 0. In our normaliza-tion we have c = 1. Then c k for k ≥ − q + aγc = 0 , (9.3) P k c k − − ( Q k + q ) c k + R k c k +1 = 0 , (9.4)where P k = ( k − α )( k − β ) ,Q k = k [( k − γ )(1 + a ) + aδ + ε ] ,R k = ( k + 1)( k + γ ) a. Lemma 9.5 below shows that each coefficient c k for Hl ( a, q ; α, β, γ, δ ; z ) = P ∞ k =0 c k z k has adeterminant representation. Lemma 9.5 (see, for instance, [26, Lem. B.1]) . Let { a k } k ∈ Z ≥ be a sequence with a = 1 generatedby the following relations: • a = A ; • a k +1 = A k a k + B k a k − ( k ≥ for some A , A k , B k ∈ C . Then a k can be expressed as a k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A − B A − B A − . . . . . . . . .B k − A k − − B k − A k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The local solutions u [ s ; n ] ( t ) and v [ s ; n ] ( t ) . Now we consider the following local solutions to(4.33). u [ s ; n ] ( t ) = Hl ( − , − ns − n , − n − , , − n − s t ) , (9.6) v [ s ; n ] ( t ) = tHl ( − , − n − s ; − n − , − n − , , − n − s t ) . (9.7) Lemma 9.8.
Let u [ s ; n ] ( t ) = P ∞ k =0 U k ( s ; n ) t k be the power series expansion of u [ s ; n ] ( t ) at t = 0 .Then the coefficients U k ( s ; n ) for k ≥ satisfy the following recurrence relations. U ( s ; n ) = E u ,U k +1 ( s ; n ) = E uk U k ( s ; n ) + F uk U k − ( s ; n ) , where E u = ns , E uk = ( n − k ) s (2 k + 1)(2 k + 2) and F uk = ( n − k + 1)( n − k + 2)(2 k + 1)(2 k + 2) . (9.9) Proof.
This follows from (9.3) and (9.4) for the specific parameters in (9.6). (cid:3)
Lemma 9.10.
Let v [ s ; n ] ( t ) = P ∞ k =0 V k ( s ; n ) t k +1 be the power series expansion of v [ s ; n ] ( t ) at t = 0 .Then the coefficients V k ( s ; n ) ( k ≥ satisfy the following recurrence relations. V ( s ; n ) = E v ,V k +1 ( s ; n ) = E vk V k ( s ; n ) + F vk V k − ( s ; n ) , where E v = ( n − s , E vk = ( n − k − s (2 k + 2)(2 k + 3) and F vk = ( n − k )( n − k + 1)(2 k + 2)(2 k + 3) . (9.11) Proof.
This follows from (9.3) and (9.4) for the specific parameters in (9.7). (cid:3)
It follows from Lemmas 9.8, 9.10, and 9.5 that U k ( s ; n ) and V k ( s ; n ) have determinant represen-tations U k ( s ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E u − F u E u − F u E u − . . . . . . . . .F uk − E uk − − F uk − E uk − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (9.12) and V k ( s ; n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E v − F v E v − F v E v − . . . . . . . . .F vk − E vk − − F vk − E vk − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (9.13)10. Appendix B: A proof of Sylvester’s formula
In this short appendix we provide a proof of Sylvester’s factorization formulaSylv( x ; n ) = n Y ℓ =0 ( x − n + 2 ℓ ) , (10.1)based on a general theory of sl (2 , C ) representations. For other proofs and related topics, see theremark after Theorem 7.22 and Proposition 8.3. For possible wide audience of this appendix, weshall discuss a proof in two ways; one of which is abstract in nature (Section 10.1) and the other usesa concrete realization of irreducible represenations (Section 10.2). We remark that the argumentsin this appendix can be thought of as a baby case of the one discussed in Section 4.2.10.1. Sylvester determinant
Sylv( x ; n ) and sl (2 , C ) representations I. Let E + , E − , and E be the elements of sl (2 , C ) defined in (3.2), and let ( dρ n , V n ) be an irreducible representation of sl (2 , C ) with dim C V n = n + 1. Then there exists an ordered basis B of V n such that the matrix dρ n ( E + + E − ) B with respect to B is given as dρ n ( E + + E − ) B = − − n − − ( n −
1) 0 − . . . . . . . . . − − ( n − − − n − (10.2)(cf. [14, Sect. 7.2]). Then the Sylvester determinant Sylv( x ; n ) ((7.25)) is the characteristic poly-nomial of dρ n ( E + + E − ) B . As the leading coefficient of Sylv( x ; n ) is one, this implies thatSylv( x ; n ) = Y λ ∈ Spec( dρ n ( E + + E − )) ( x − λ ) , (10.3)where Spec( T ) denotes the set of eigenvalues of a linear map T . Thus, to show (10.1), it sufficesto find the eigenvalues of dρ n ( E + + E − ). Since E + + E − is conjugate E via an element of SU (2)(cf. [22, Thm. 4.34]), it is further sufficient to determine Spec( dρ n ( E )). It is well-known that theeigenvalues of dρ n ( E ) are n − j for j = 0 , , , . . . , n − , n (cf. [14, Sect. 7.2]). Therefore we haveSpec( dρ n ( E + + E − )) = { n − j : j = 0 , , , . . . , n − , n } . Now the desired factorization formula (10.1) follows from (10.3). Sylvester determinant
Sylv( x ; n ) and sl (2 , C ) representations II. We now give somedetails of the arguments given in Section 10.1 with a concrete realization of irreducible sl (2 , C )representations.Let ( π n , Pol n [ t ]) be the irreducible representation of SU (2) defined in (3.15) and ( dπ n , Pol n [ t ])the corresponding irreducible representation of sl (2 , C ), so that dπ n ( E j ) for j = + , − , dπ n ( E + + E − ) B withrespect to the ordered basis B := { , t, t , . . . , t n } is given as (10.2); therefore, we haveSylv( x ; n ) = Y λ ∈ Spec( dπ n ( E + + E − )) ( x − λ ) . (10.4)On the other hand, the matrix dπ n ( E ) B is given as dπ n ( E ) B = diag( n, n − , n − , . . . , − n + 2 , − n ) , where diag( m , m , . . . , m n ) denotes a diagonal matrix. Thus the eigenvalues of dπ n ( E ) on Pol n [ t ]are n − j for j = 0 , , , . . . , n − , n .For g := 1 √ (cid:18) − (cid:19) ∈ SU (2) , we have Ad( g )( E + + E − ) = E . Thus, dπ n ( E ) = π n ( g ) dπ n ( E + + E − ) π n ( g ) − . Therefore, Spec( dπ n ( E + + E − )) = { n − j : j = 0 , , , . . . , n − , n } . (10.5)Now (10.4) and (10.5) conclude the desired formula. Acknowledgements.
Part of this research was conducted during a visit of the first authorat the Department of Mathematics of Aarhus University. He is appreciative of their support andwarm hospitality during his stay.The authors are grateful to Anthony Kable for his suggestion to study the K -type formulasfor the Heisenberg ultrahyperbolic operator. They would also like to express their gratitude toHiroyuki Ochiai for the helpful discussions on the hypergeometric equation and Heun equation,especially for Lemma 4.32. Their deep appreciation also goes to Hiroyoshi Tamori for sending hisPh.D. dissertation, from which they thought of an idea for the hypergeometric model.The first author was partially supported by JSPS Grant-in-Aid for Young Scientists (JP18K13432). References
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