An explicit integral representation of Siegel-Whittaker functions on Sp(2,R) for the large discrete series representations
aa r X i v : . [ m a t h . N T ] D ec AN EXPLICIT INTEGRAL REPRESENTATION OFSIEGEL-WHITTAKER FUNCTIONS ON
Sp(2 , R ) FOR THE LARGEDISCRETE SERIES REPRESENTATIONS
YASURO GON AND TAKAYUKI ODA
In Memoriam: Fumiyuki Momose
Abstract.
We obtain an explicit integral representation of Siegel-Whittaker functions onSp(2 , R ) for the large discrete series representations. We have another integral expressiondifferent from that of Miyazaki [7]. Introduction
In this article, we study Siegel-Whittaker functions on G = Sp(2 , R ), the real symplecticgroup of degree two for the large discrete series representations. Let P S be the Siegelparabolic subgroup of G , which is a maximal parabolic subgroup with abelian unipotentradical N S . Let π be an admissible representation of G and ξ be a definite unitary characterof N S . Siegel-Whittaker model for an admissible π is a realization of π in the inducedmodule from a certain closed subgroup R which contains N S . (See (2.2) for definition of R .) We consider the intertwining space SW ( π ; η ) = Hom ( g C ,K ) ( π, C ∞ η ( R \ G )) , where η is an irreducible R -module such that η | N S contains ξ , K is an maximal compactsubgroup of G and g C is the complexification of the Lie algebra of G . A function in thisrealization is called a Siegel-Whittaker function for π .Takuya Miyazaki obtained a system of partial differential equations satisfied by Siegel-Whittaker functions for the large discrete representations in [7]. He also obtained multi-plicity one property and its formal power series solutions. In this article, we investigatefurther the system obtained by Miyazaki (Proposition 3.1) and give an explicit integral rep-resentation of the Siegel-Whittaker functions, which is of rapid decay, for a large discreteseries representation π . (Theorem 5.1.) In other words, we show that the rapidly de-creasing Siegel-Whittaker function for the large discrete series representation π is uniquelydetermined and (up to polynomials) described by the partially confluent hypergeometric Date : October 19, 2018.
Key words and phrases. large discrete series; Siegel-Whittaker functions.2000 Mathematics Subject Classification. 11F70; 22E45T. O. was partially supported by JSPS Grant-in-Aid for Scientific Research (A) no. 23244003. Y. G. waspartially supported by JSPS Grant-in-Aid for Scientific Research (C) no. 26400017. functions in the A -radial part ( a , a ) ∈ ( R > ) :(1.1) Ψ α,β ( a , a ) = e − π ( h a + h a ) Z F SW (cid:0) πh a t + 2 πh a (1 − t ) (cid:1) t α − (1 − t ) β − dt, with F SW ( x ) = F ( x ) = e x x − γ W κ,µ (2 x ) . Here, W κ,µ ( x ) is the Whittaker’s classical confluent hypergeometric function, α, β, γ > κ, µ are half integers depending on parameters of representations π and η . This kindof integral already appeared in Gon [1], treated the cases for the large discrete seriesrepresentations and P J -principal series representations of SU(2 , , R ), is the paperHirano-Ishii-Oda [2]. By this new integral expression (1.1), it seems possible to extendthe former argument in [2] of the confluence from Siegel-Whittaker functions to Whit-taker functions for P J -principal series, to the confluence from Siegel-Whittaker functionsto Whittaker functions [8] for the large discrete series of Sp(2 , R ). Here we recall thetemplate of the integral formulas of Whittaker functions in ( a , a ) ∈ ( R > ) :(1.2) µ W ( a , a ) Z ∞ F W ( t ) t − C e − A t /a − B a /t dt, where F W ( t ) = W ,γ ( t ) with a positive integral parameter γ , µ W ( a , a ) is a monomial in a , a times e δa ( δ >
0) and
A, B, C are positive real parameters.Probably we can proceed a bit more. Since Iida [3], Theorems 8.7 and 8.9 (Formulas(8.8) and (8.10), resp.) give analogous integral expressions for the matrix coefficient of P J -principal series, and also Oda [9] (p.247), Theorem 6.2 similarly gives analogous integralexpression for matrix coefficients of the large discrete series. We notice that the templateof formulas in Theorem 6.2 of [9] is given by(1.3) µ MC ( a , a ) Z F MC (cid:0) tx + (1 − t ) x (cid:1) t α − (1 − t ) β − dt, with x i = − ( a i − a − i ) / i = 1 , F MC ( x ) = F ( A, B ; C : x ) with A, B, C, α, β determined by Harish-Chandra parameters and the components of the minimal K -type.The elementary factor µ MC ( a , a ) is a monomial of (cid:0) ( a i ± a − i ) / (cid:1) ± / ( i = 1 , (1.1) fromthe matrix coefficients realization to Siegel-Whittaker realization in a very simple naturalway. We believe the similar argument is possible for the principal series: but in this casethe natures of integrands of Whittaker case Ishii [5], Theorem 3.2 and Siegel-Whittakercase [4], Theorem 10.1 are still difficult to deform.For the P J -principal series of the other group SU(2 , IEGEL-WHITTAKER FUNCTIONS ON Sp(2 , R ) 3 conjugations of a compact subgroup K of SU(2 , t → h t Kh − t = R. The deformation of the Siegel-Whittaker function of the P J -principal series to the Whit-taker function could handled in a similar way as in [2].In view of this observation, we may hope that similar phenomenon occurs for moregeneral groups SO(2 , q ) ( q > Preliminaries
Basic notations.
Let G be the real symplectic group of degree two:Sp(2 , R ) = (cid:26) g ∈ SL(4 , R ) (cid:12)(cid:12)(cid:12)(cid:12) t gJ g = J = (cid:18) − (cid:19)(cid:27) , with 1 the unit matrix of degree two and 0 the zero matrix of degree two.Fix a maximal compact subgroup K of G by K = (cid:26) k ( A, B ) = (cid:18)
A B − B A (cid:19) ∈ G (cid:12)(cid:12)(cid:12)(cid:12) A, B ∈ M(2 , R ) (cid:27) . It is isomorphic to the unitary group U(2) via the homomorphism K ∋ k ( A, B ) A + √− B ∈ U(2) . We define a certain spherical subgroup R of G as follows. Let P S = L S ⋉ N S be the Siegelparabolic subgroup with the Levi part L S and the abelian unipotent radical N S given by L S = (cid:26)(cid:18) A t A − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) A ∈ GL(2 , R ) (cid:27) ,N S = (cid:26) n ( T ) = (cid:18) T (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t T = T ∈ M(2 , R ) (cid:27) . Fix a non-degenerate unitary character ξ of N S by ξ ( n ( T )) = exp (cid:0) π √− H ξ T ) (cid:1) with H ξ = (cid:0) h h / h / h (cid:1) ∈ M(2 , R ) and det H ξ = 0. Consider the action of L S on N S byconjugation and the induced action on the character group c N S . Define SO( ξ ) to the identitycomponent of the subgroup of L S which stabilize ξ :SO( ξ ) := Stab L S ( ξ ) ◦ = (cid:26)(cid:18) A t A − (cid:19) ∈ L S (cid:12)(cid:12)(cid:12)(cid:12) t AH ξ A = H ξ (cid:27) . Then SO( ξ ) is isomorphic to SO(2) if det H ξ > ◦ (1 ,
1) if det H ξ <
0. In thisarticle we treat the case that ξ is a ‘definite’ character, that is det H ξ >
0. So we mayassume h , h > h = 0 without loss of generality. We sometimes identify the element Y. GON AND T. ODA of SO( ξ ) with its upper left 2 × χ m ( m ∈ Z ) ofSO( ξ ) ∼ = SO(2) by(2.1) χ m (cid:18)(cid:18) √ h √ h (cid:19) − (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) √ h √ h (cid:19)(cid:19) = exp( √− m θ ) . We define(2.2) R = SO( ξ ) ⋉ N S and η = χ m ⊠ ξ. Taking a maximal split torus A of G by A = { a = ( a , a ) = diag( a , a , a − , a − ) | a , a > } , we have the decomposition G = RAK .2.2.
Siegel-Whittaker functions.
We consider the space C ∞ η ( R \ G ) of complex valued C ∞ functions f on G satisfying f ( rg ) = η ( r ) f ( g ) ∀ ( r, g ) ∈ R × G. By the right translation, C ∞ η ( R \ G ) is a smooth G -module and we denote the same symbolits underlying ( g C , K )-module. For an irreducible admissible representation ( π, H π ) of G and the subspace H π,K of K -finite vectors, the intertwining space I η,π = Hom ( g C ,K ) ( H π,K , C ∞ η ( R \ G ))between the ( g C , K )-modules is called the space of algebraic Siegel-Whittaker functionals .For a finite-dimensional K -module ( τ, V τ ), denote by C ∞ η,τ ( R \ G/K ) the space (cid:8) φ : G → V τ , C ∞ | φ ( rgk ) = η ( r ) τ ( k − ) φ ( g ) ∀ ( r, g, k ) ∈ R × G × K (cid:9) . Let ( τ ∗ , V τ ∗ ) be a K -type of π and ι : V τ ∗ → H π be an injection. Here, τ ∗ means thecontragredient representation of τ . Then for Φ ∈ I η,π , we can find an element φ ι in C ∞ η,τ ( R \ G/K ) = C ∞ η ( R \ G ) ⊗ V τ ∗ ∼ = Hom K ( V τ ∗ , C ∞ η ( R \ G ))via Φ( ι ( v ∗ ))( g ) = h v ∗ , φ ι ( g ) i with h , i the canonical pairing on V τ ∗ × V τ .Since there is the decomposition G = RAK , our generalized spherical function φ ι isdetermined by its restriction φ ι | A , which we call the radial part of φ ι . For a subspace X of C ∞ η,τ ( R \ G/K ), we denote X | A = { φ | A ∈ C ∞ ( A ) | φ ∈ X } .Let us define the space SW( π, η, τ ) of Siegel-Whittaker functions and its subspaceSW( π, η, τ ) rap as follows:SW( π, η, τ ) = [ ι ∈ Hom K ( τ ∗ ,π ) { φ ι | Φ ∈ I η,π } and SW( π, η, τ ) rap = (cid:8) φ ι ∈ SW( π, η, τ ) | φ ι | A decays rapidly as a , a → ∞ (cid:9) . We call an element in SW( π, η, τ ) a
Siegel-Whittaker function for ( π, η, τ ). IEGEL-WHITTAKER FUNCTIONS ON Sp(2 , R ) 5 Parametrization of the discrete series representations.
Let E ij ∈ M ( R ) bethe matrix unit with 1 as its ( i, j )-component and 0 at the other entries. The root systemof G with respect to a compact Cartan subgroup T = exp (cid:0) R ( E − E ) + R ( E − E ) (cid:1) is given by a set of vectors in the Euclidean plane: {± ε , ± ε , ± ε ± ε } . Here, ε (cid:0) r ( E − E ) + r ( E − E ) (cid:1) = √− r ,ε (cid:0) r ( E − E ) + r ( E − E ) (cid:1) = √− r . We fix a subset of simple roots and the associated positive roots by { ε − ε , ε } , { ε , ε + ε , ε − ε , ε } respectively.Then the set of the unitary characters of T (or their derivatives) is identified naturallywith Z ⊕ Z , and the subset consisting of dominant integral weight isΞ = { ( n , n ) ∈ Z ⊕ Z | n ≥ n } . There is a bijection between b K and Ξ by highest weight theory. Because the half-sumof the positive root is integral, the discrete series representations of G = Sp(2 , R ) areparametrized by the subset of regular elements in Ξ:Ξ ′ = { ( n , n ) ∈ Z ⊕ Z | n > n , n = 0 , n = 0 , n + n = 0 } . Here the condition n > n means the positivity of weight ( n , n ) with respect to thecompact root ε − ε = (1 , − I = { ( n , n ) | n > n > } and Ξ IV = { ( n , n ) | > n > n } parametrize the holomorphic discrete series and the anti-holomorphic discrete series rep-resentations, respectively. SetΞ II = { ( n , n ) | n > > n , n + n > } , and Ξ III = { ( n , n ) | n > > n , > n + n } . Then the union Ξ II ∪ Ξ III parametrize the large discrete series representations of G .3. Miyazaki’s results
Miyazaki derived a system of partial differential equations satisfied by Siegel-Whittakerfunctions for the large discrete representations in [7]. He also obtained multiplicity oneproperty. We recall his results in this section.Let τ = τ ( λ ,λ ) = Sym λ − λ ⊗ det λ be the irreducible K -module with the highest weight( λ , λ ), then the dimension of τ is d + 1 with d = λ − λ . We take the basis { v j } dj =0 of V τ ∗ with τ ∗ = τ ( − λ , − λ ) as in [7, Lemma 3.1]. Y. GON AND T. ODA
We remark on a compatibility condition. For a non-zero function φ in C ∞ η,τ ∗ ( R \ G/K ),we have φ ( a ) = φ ( mam − ) = ( χ m ⊠ ξ )( m ) τ ( − λ , − λ ) ( m ) φ ( a ) , where, a ∈ A and m ∈ SO( ξ ) ∩ Z K ( A ) = {± } . If we take m = − , ( χ m ⊠ ξ )( m ) = χ m ( m ) = ( − m and τ ( − λ , − λ ) ( m ) = ( − d imply that ( m + d ) / Proposition 3.1 (Miyazaki [7]) . Let π = π Λ be a large discrete series representation of G with the Harish-Chandra parameter Λ = ( λ − , λ ) ∈ Ξ II and its minimal K -type τ = τ ( λ ,λ ) . Let ξ be a unitary character of N S associated with a positive definite matrix H ξ = (cid:0) h h (cid:1) . Put η = χ m ⊠ ξ , as in ( ) . Then we have the following: (i) we have dim C SW( π, η, τ ) ≤ and a function φ SW ( a ) = d X j =0 n ( p h a ) λ − j ( p h a ) λ + j e − π ( h a + h a ) c j ( a ) o v j is in the space SW( π Λ , η, τ ) | A if and only if { c j ( a ) } dj =0 is a smooth solution of thefollowing system: (3.1) h ∂ + j h a △ i c j − ( a ) + √− m h a △ c j ( a ) − ( d − j ) h a △ c j +1 ( a ) = 0 (1 ≤ j ≤ d ) , (3.2) j h a △ c j − ( a ) + √− m h a △ c j ( a ) + h ∂ − ( d − j ) h a △ i c j +1 ( a ) = 0 (0 ≤ j ≤ d − ,h a h ∂ − πh a − j h a △ + 2 λ − i c j − ( a ) − √− m h a h a △ c j ( a )(3.3) + h a h ∂ − πh a + 2( d − j ) h a △ + 2 λ − i c j +1 ( a ) = 0 (1 ≤ j ≤ d − , with ∂ i = a i ( ∂/∂a i ) ( i = 1 , and △ = h a − h a . (ii) dim C SW( π, η, τ ) rap ≤ . This is a paraphrase of Propositions 10.2, 10.7 and Theorem 11.5 of [7]. Here (3.1), (3.2),(3.3) are essentially identical equations to (10.4), (10.5), (10.6) of [7] deduced from Proposi-tion 10.2. However we replace the symbol χ ( Y η ) of [7] by its explicit value √− m / √ h h ,and the symbol D of [7] by △ .4. Partially Confluent hypergeometric functions in two variables
We introduce and study certain partially confluent hypergeometric functions in twovariables on A ≃ ( R > ) . These functions play a key role in describing Siegel-Whittakerfunctions φ SW ( a ). We remark that these types of confluent hypergeometric functions havealso appeared in [1], for the large discrete series representations and P J -principal seriesrepresentations of SU(2 , P J -principal series representations of Sp(2 , R ). IEGEL-WHITTAKER FUNCTIONS ON Sp(2 , R ) 7 Definition 4.1 (Partially confluent hypergeometric functions) . Let π = π Λ be a largediscrete series representation of G with the Harish-Chandra parameter Λ = ( λ − , λ ) ∈ Ξ II and its minimal K -type τ = τ ( λ ,λ ) . Let ξ be a unitary character of N S associated witha positive definite matrix H ξ = (cid:0) h h (cid:1) . Put η = χ m ⊠ ξ , as in (2.1).We assume that | m | ≥ d and | m | ≡ d (mod 2) . For 0 ≤ k ≤ d , define f k ( a ) = f k ( a ; [ π, η, τ ])= ( p h a ) k +1 ( p h a ) d +1 − k ( h a − h a ) | m |− d (4.1) × Z F (cid:0) πh a t + 2 πh a (1 − t ) (cid:1) t | m |− d − + k (1 − t ) | m | + d − − k dt, with F ( x ) = e x x − | m | + λ W λ −| m |− , λ (2 x ) . Here, W κ,µ ( z ) is Whittaker’s confluent hypergeometric function. (See [10] Chapter 16 fordefinition.)Since the indices α k − | m |− d − + k , β k − | m | + d − − k in the integrand of f k satisfy α k , β k > ≤ k ≤ d ), we see that f k ( a ) is a smooth function on A and of moderategrowth when each a , a tends to infinity. We have further more, Proposition 4.2.
Partially confluent hypergeometric functions { f k ( a ) } dk =0 satisfy the fol-lowing system of the difference-differential equations. (4.2) ∂ f k = − (2 k + 1) h a △ f k + ( | m | + d − − k ) h a △ f k +1 (0 ≤ k ≤ d − , (4.3) ∂ f k = (2 d − k + 1) h a △ f k − ( | m | − d − k ) h a △ f k − (1 ≤ k ≤ d ) , (cid:2) ( ∂ + ∂ ) + 2( λ − ∂ + ∂ ) − π ( h a ∂ + h a ∂ ) − λ − (cid:3) f k ( a ) = 0(4.4) (0 ≤ k ≤ d ) . Here, △ = h a − h a .Proof. For 0 ≤ k ≤ d , putˇ f k ( a ) = Z F (cid:0) πh a t + 2 πh a (1 − t ) (cid:1) t | m |− d − + k (1 − t ) | m | + d − − k dt. Then we can verify that ∂ f k = ( p h a ) k +1 ( p h a ) d +1 − k △ | m |− d h ∂ + (2 k + 1) + ( | m | − d ) h a △ i ˇ f k Y. GON AND T. ODA and ∂ ˇ f k = − ( | m | − d + 1 + 2 k ) h a △ ˇ f k + ( | m | + d − − k ) h a △ ˇ f k +1 for 0 ≤ k ≤ d −
1. Therefore, we obtain the formula (4.2). Similarly, we have ∂ f k = ( p h a ) k +1 ( p h a ) d +1 − k △ | m |− d h ∂ + (2 d + 1 − k ) − ( | m | − d ) h a △ i ˇ f k and ∂ ˇ f k = − ( | m | − d − k ) h a △ ˇ f k − + ( | m | + d + 1 − k ) h a △ ˇ f k for 1 ≤ k ≤ d . Thus, we have the formula (4.3).Let us prove (4.4). Put Ω and ˇΩ be the partial differential operators defined byΩ = ( ∂ + ∂ ) + 2( λ − ∂ + ∂ ) − π ( h a ∂ + h a ∂ ) − λ − ∂ + ∂ ) + 2( | m | + d + λ )( ∂ + ∂ ) − π ( h a ∂ + h a ∂ ) − π ( h a + h a )( | m | + 1) − π ( h a − h a )(2 k − d ) + ( | m | + d )( | m | + d + 2 λ ) . Then we can check thatΩ f k = ( p h a ) k +1 ( p h a ) d +1 − k △ | m |− d ˇΩ ˇ f k . By interchanging differentiation and integration, we haveˇΩ ˇ f k ( a ) = Z G (cid:0) πh a t + 2 πh a (1 − t ) (cid:1) t | m |− d − + k (1 − t ) | m | + d − − k dt with G ( x ) = 4 (cid:20) x d dx − (cid:8) x − ( | m | + λ + 1) (cid:9) x ddx − | m | + 1) x + ( | m | + λ ) − λ (cid:21) F ( x ) . Put F ( x ) = e x x − ( | m | + λ +1) / H ( x ). Then we have G ( x ) = 4 e x x − ( | m | + λ +1) / · x (cid:20) d dx + n − λ − | m | − x + 1 − λ x o(cid:21) H ( x ) . It is known that the differential equation (cid:20) d dx + n − λ − | m | − x + 1 − λ x o(cid:21) H ( x ) = 0has two linearly independent solutions: W λ −| m |− , λ (2 x ) , M λ −| m |− , λ (2 x ) . Here, W κ,µ ( z ) and M κ,µ ( z ) are Whittaker’s confluent hypergeometric functions. (See [10]Chapter 16 for definition.) Therefore, we have (4.4). It completes the proof. (cid:3) IEGEL-WHITTAKER FUNCTIONS ON Sp(2 , R ) 9 Main results
We state our main results on an explicit integral formula of Siegel-Whittaker functionswhich are of rapid decay for the large discrete series representations of Sp(2 , R ). Theorem 5.1.
Let π = π Λ be a large discrete series representation of G with the Harish-Chandra parameter Λ = ( λ − , λ ) ∈ Ξ II and its minimal K -type τ = τ ( λ ,λ ) . Let ξ bea unitary character of N S associated with a positive definite matrix H ξ = (cid:0) h h (cid:1) . Put η = χ m ⊠ ξ , as in ( ) .We assume that | m | ≥ d and | m | ≡ d (mod 2) . Then we have the following: (i) dim C SW( π, η, τ ) rap = 1 . (ii) Let { f k ( a ) } dk =0 be the partially confluent hypergeometric functions, defined in Defi-nition 4.1. We consider the following C -linear combinations { g j ( a ) } dj =0 of elementsof { f k ( a ) } dk =0 , given by (5.1) g j ( a ) = d X k =0 x jk f k ( a ) , where the complex numbers { x jk } ≤ j,k ≤ d are given by x jk =( − j + k (cid:0) sgn( m ) √− (cid:1) j (cid:18) d k − j (cid:19) | m | δ ( j ) k Y l =1 l − | m | − d + 2 l − × [ j/ X r =0 (cid:18) [ j/ r (cid:19) (cid:0) d − r − δ ( j ) (cid:1) ! d ! r Y l =1 (cid:26) k − j − l + 2 − δ ( j )2 k − l + 1 (cid:0) | m | − ( d − l + 2) (cid:1)(cid:27) ( δ ( j ) = 1 if j is odd, otherwise . Then the function (5.3) φ SW ( a ) = d X j =0 n ( p h a ) λ − j ( p h a ) λ + j e − π ( h a + h a ) g j ( a ) o v j gives a non-zero element in SW( π, η, τ ) rap | A which is unique up to constant multi-ple. Proof of main results
We claim that the coefficient functions c ( a ) , c ( a ) , . . . , c d ( a ) appearing in the Siegel-Whittaker function φ SW ( a ), in Proposition 3.1, are C -linear combination of the confluenthypergeometric functions f ( a ) , f ( a ) , . . . , f d ( a ) defined in Definition 4.1. Proposition 6.1.
We assume that | m | ≥ d and | m | ≡ d (mod 2) . Let { x jk } ≤ j,k ≤ d be a sequence of complex numbers, which satisfy (6.1) j x j − ,k + √− m x j,k + ( d − k + j + 1) x j +1 ,k − ( | m | − d + 2 k + 1) x j +1 ,k +1 = 0(0 ≤ j ≤ d − , ≤ k ≤ d ) , (6.2) ( j − k − x j − ,k + ( | m | + d − k + 1) x j − ,k − + √− m x j,k − ( d − j ) x j +1 ,k = 0(1 ≤ j ≤ d, ≤ k ≤ d ) , (6.3) x , = x , = · · · = x d, = 0 , and (6.4) x ,d = x ,d = · · · = x d − ,d = 0 . For ≤ j ≤ d , define g j ( a ) = d X k =0 x jk f k ( a ) , then { g j ( a ) } dj =0 is a smooth solution of the system of the difference-differential equations(3.1), (3.2) and (3.3) in Proposition 3.1.Proof. By using Proposition 4.2, we see that △ h a h ∂ + j h a △ i g j − ( a )= d X k =0 ( j − k − x j − ,k f k + d X k =0 ( | m | + d − k − x j − ,k f k +1 . Therefore, we have △ h a h ∂ + j h a △ i g j − ( a ) + √− m g j ( a ) − ( d − j ) g j +1 ( a )= d X k =0 ( j − k − x j − ,k f k + d X k =1 ( | m | + d − k + 1) x j − ,k − f k + √− m d X k =0 x j,k f k − ( d − j ) d X k =0 x j +1 ,k f k = 0 (1 ≤ j ≤ d ) . This is the desired (3.1) for { g j ( a ) } dj =0 . In the above calculation, we used the relations(6.2) and (6.4) on { x jk } ≤ j,k ≤ d . Again, by using Proposition 4.2, we see that △ h a h ∂ − ( d − j ) h a △ i g j +1 ( a )= d X k =0 ( d − k + j + 1) x j +1 ,k f k − d X k =0 ( | m | − d + 2 k − x j +1 ,k f k − . IEGEL-WHITTAKER FUNCTIONS ON Sp(2 , R ) 11 Therefore, we have jg j − ( a ) + √− m g j ( a ) + △ h a h ∂ − ( d − j ) h a △ i g j +1 ( a )= j d X k =0 x j − ,k f k + √− m d X k =0 x j,k f k + d X k =0 ( d − k + j + 1) x j +1 ,k f k − d − X k =0 ( | m | − d + 2 k + 1) x j +1 ,k +1 f k = 0 (0 ≤ j ≤ d − . This is the desired (3.2) for { g j ( a ) } dj =0 . In the above calculation, we used the relations(6.1) and (6.3) on { x jk } ≤ j,k ≤ d .By considering (3 . × h a + (3 . × h a + (3 . h a [( ∂ + ∂ ) − πh a + 2 λ − c j − ( a )+ h a [( ∂ + ∂ ) − πh a + 2 λ − c j +1 ( a ) = 0 (1 ≤ j ≤ d − . (6.5)By considering (3 . × h a − (3 . × h a , we have(6.6) h a ∂ c j − ( a ) − h a ∂ c j +1 ( a ) = 0 (1 ≤ j ≤ d − . Operating ∂ ( h a ) − on both hand sides of (6 . ∂ h a h a [( ∂ + ∂ ) − πh a + 2 λ − c j − ( a )+ [( ∂ + ∂ ) − πh a + 2 λ − ∂ c j +1 ( a ) = 0 (1 ≤ j ≤ d − . (6.7)Combining (6.7) and (6.6), we have, for 0 ≤ j ≤ d − (cid:2) ( ∂ + ∂ ) + 2( λ − ∂ + ∂ ) ∂ − π ( h a ∂ + h a ∂ ) − λ − (cid:3) c j ( a ) = 0 . Operating ∂ ( h a ) − on both hand sides of (6 .
5) and combining with (6.6), we have (6.8)for 2 ≤ j ≤ d . As a result, we have (6.8) for 0 ≤ j ≤ d .Lastly we prove that { g j ( a ) } dj =0 satisfy (3 . f ( a ) , f ( a ) , . . . , f d ( a )satisfy the same differential equation (6.8), therefore their C -linear combinations { g j ( a ) } dj =0 also satisfy (6.8). Since { g j ( a ) } dj =0 satisfy (6.6) and (6.8), we see that(6.9) ∂ F j ( a ) h a = ∂ F j ( a ) h a = 0 (1 ≤ j ≤ d − , where we put F j ( a ) = h a [( ∂ + ∂ ) − πh a + 2 λ − g j − ( a )+ h a [( ∂ + ∂ ) − πh a + 2 λ − g j +1 ( a ) . By (6.9), there exist constants β j (1 ≤ j ≤ d −
1) such that F j ( a ) = β j h a h a . Let us determine β j . For y >
0, define L = { ( a , a ) ∈ A | h a = h a = y } . We showthat ( F j | L )( y ) = 0 to deduce β j = 0. We can verify that( ∂ + ∂ ) f k = ( p h a ) k +1 ( p h a ) d +1 − k △ | m |− d (cid:2) ( ∂ + ∂ ) + ( d + | m | + 2) (cid:3) × Z F (cid:0) πh a t + 2 πh a (1 − t ) (cid:1) t | m |− d − + k (1 − t ) | m | + d − − k dt = ( p h a ) k +1 ( p h a ) d +1 − k △ | m |− d × Z F ∗ (cid:0) πh a t + 2 πh a (1 − t ) (cid:1) t | m |− d − + k (1 − t ) | m | + d − − k dt. Here, F ∗ ( x ) = (cid:18)(cid:16) x ddx + ( d + | m | + 2) (cid:17) F (cid:19) ( x ) . There are two cases:(i) If | m |− d ≥
2, then both ( ∂ + ∂ ) f k and f k have zeros at { h a − h a = 0 } . Hence,both ( ∂ + ∂ ) g j and g j have zeros at there. Therefore we have ( F j | L )( y ) = 0.(ii) Suppose that | m | = d , then we have (cid:0) ( ∂ + ∂ ) f k (cid:1)(cid:12)(cid:12) L ( y ) = y d +1 F ∗ (2 πy ) B (cid:16) k + 12 , d − k + 12 (cid:17) . Here, B ( α, β ) = Z t α − (1 − t ) β − dt = Γ( α ) Γ( β )Γ( α + β )is the Beta function. Then we can verify that (cid:0) ( ∂ + ∂ ) g j (cid:1)(cid:12)(cid:12) L ( y ) = (cid:18) d X k =0 x jk B (cid:16) k + 12 , d − k + 12 (cid:17)(cid:19) y d +1 F ∗ (2 πy )= (cid:0) sgn( m ) √− (cid:1) j − d π y d +1 F ∗ (2 πy ) . (6.10) To derive (6.10), we used (6.16) in Proposition 6.2: (We will prove later.) x jk = ( − j + k (cid:0) sgn( m ) √− (cid:1) j (cid:18) d k − j (cid:19) (when | m | = d ) , under the condition x , = 1, and the equality:(6.11) d X k =0 ( − k (cid:18) kj (cid:19)(cid:18) d − kd − j (cid:19)(cid:18) dk (cid:19) = ( − j d (cid:18) dj (cid:19) . Let S d,j be the left hand side of (6.11). We remark that the above equality (6.11)is proved by showing that S d,j = − dj S d − ,j − ( j ≥
1) and S d, = 2 d . See p.620 no. 63 in [11] for the equality S d, = 2 d . IEGEL-WHITTAKER FUNCTIONS ON Sp(2 , R ) 13 Similarly, we also obtain g j (cid:12)(cid:12) L ( y ) = (cid:0) sgn( m ) √− (cid:1) j − d π y d +1 F (2 πy ) . Therefore, we have (cid:0) ( ∂ + ∂ ) ( g j − + g j +1 ) (cid:1)(cid:12)(cid:12) L ( y ) = ( g j − + g j +1 ) (cid:12)(cid:12) L ( y ) = 0 , for 1 ≤ j ≤ d −
1. Therefore we have ( F j | L )( y ) = 0.In any cases, { g j ( a ) } dj =0 satisfy (6.5). Hence, { g j ( a ) } dj =0 satisfy (3.3) and it completes theproof. (cid:3) Proposition 6.2.
We assume that | m | ≥ d and | m | ≡ d (mod 2) . Let { x jk } ≤ j,k ≤ d be a sequence of complex numbers, which satisfy (6.12) j x j − ,k + √− m x j,k + ( d − k + j + 1) x j +1 ,k − ( | m | − d + 2 k + 1) x j +1 ,k +1 = 0(0 ≤ j ≤ d − , ≤ k ≤ d ) , (6.13) ( j − k − x j − ,k + ( | m | + d − k + 1) x j − ,k − + √− m x j,k − ( d − j ) x j +1 ,k = 0(1 ≤ j ≤ d, ≤ k ≤ d ) , (6.14) x , = x , = · · · = x d, = 0 , and (6.15) x ,d = x ,d = · · · = x d − ,d = 0 . Then the sequence { x jk } ≤ j,k ≤ d is uniquely determined and given by, up to a constantmultiple, x jk =( − j + k (cid:0) sgn( m ) √− (cid:1) j (cid:18) d k − j (cid:19) | m | δ ( j ) k Y l =1 l − | m | − d + 2 l − × [ j/ X r =0 (cid:18) [ j/ r (cid:19) (cid:0) d − r − δ ( j ) (cid:1) ! d ! r Y l =1 (cid:26) k − j − l + 2 − δ ( j )2 k − l + 1 (cid:0) | m | − ( d − l + 2) (cid:1)(cid:27) , (6.16) where, δ ( j ) = 1 if j is odd, otherwise .Proof. We may assume that x , = 1. Let us write down (6.12) with j = 0 , k replacedby k − , k , and (6.13) with j = 1 and k replaced by k − , k, k + 1. Then we have the following system of seven difference equations: √− m x ,k − + ( d − k + 3) x ,k − − ( | m | − d + 2 k − x ,k = 0 , √− m x ,k + ( d − k + 1) x ,k − ( | m | − d + 2 k + 1) x ,k +1 = 0 ,x ,k − + √− m x ,k − + ( d − k + 4) x ,k − − ( | m | − d + 2 k − x ,k = 0 ,x ,k + √− m x ,k + ( d − k + 2) x ,k − ( | m | − d + 2 k + 1) x ,k +1 = 0 , − (2 k − x ,k − + ( | m | + d − k + 3) x ,k − + √− m x ,k − − ( d − x ,k − = 0 , − k x ,k + ( | m | + d − k + 1) x ,k − + √− m x ,k − ( d − x ,k = 0 , − (2 k + 2) x ,k +1 + ( | m | + d − k − x ,k + √− m x ,k +1 − ( d − x ,k +1 = 0 . (6.17)We eliminate x j,k , x j,k ± ( j = 1 ,
2) in the above system. Then we have − k + 1)( | m | − d + 2 k + 1)( | m | − d + 2 k − x ,k +1 + (cid:8) k (4 d − k + 3) − d ( d − (cid:9) ( | m | − d + 2 k − x ,k + h d − k + 2) (cid:8) d ( d − − (2 k − d − k + 3) (cid:9) + (2 k − | m | + d − k + 1)( | m | − d + 2 k − i x ,k − + ( d − k + 4)( d − k + 3)( | m | + d − k + 3) x ,k − = 0 . (6.18)By solving the difference equation (6.18) for { x ,k } with x , = 1, we obtain(6.19) x ,k = ( − k (cid:18) d k (cid:19) k Y l =1 l − | m | − d + 2 l − . From the second equation of (6.17), (6.19) and (6.14), we have the following differenceequation for { x ,k } :( | m | − d + 2 k + 1) x ,k +1 = ( d − k + 1) x ,k + √− m ( − k (cid:18) d k (cid:19) k Y l =1 l − | m | − d + 2 l − x , = 0. Thus, we obtain(6.20) x ,k = ( − k +1 (cid:0) sgn( m ) √− (cid:1)(cid:18) d k − (cid:19) | m | d k Y l =1 l − | m | − d + 2 l − . IEGEL-WHITTAKER FUNCTIONS ON Sp(2 , R ) 15 We prove the formula (6.16) for general x jk by induction on the index j . For 0 ≤ p, p + 1 , k ≤ d , let us prove x p,k =( − k + p (cid:18) d k − p (cid:19) k Y l =1 l − | m | − d + 2 l − × p X r =0 (cid:18) pr (cid:19) (cid:0) d − r (cid:1) ! d ! r Y l =1 (cid:26) k − p − l + 22 k − l + 1 (cid:0) | m | − ( d − l + 2) (cid:1)(cid:27) , (6.21) x p +1 ,k =( − k + p +1 (cid:0) sgn( m ) √− (cid:1)(cid:18) d k − p − (cid:19) | m | k Y l =1 l − | m | − d + 2 l − × p X r =0 (cid:18) pr (cid:19) (cid:0) d − r − (cid:1) ! d ! r Y l =1 (cid:26) k − p − l k − l + 1 (cid:0) | m | − ( d − l + 2) (cid:1)(cid:27) . (6.22)Suppose that 2 p + 1 ≤ d and the above formulas are true for x p − ,k , x p,k (0 ≤ k ≤ d ),then substitute them into (6.13):(6.23) x p +1 ,k = 2 p − k − d − p x p − ,k + | m | + d − k + 1 d − p x p − ,k − + √− m d − p x p,k . Put a ( k ) = k Y l =1 l − | m | − d + 2 l − , b ( r ; k, p ) = r Y l =1 (cid:26) k − p − l k − l + 1 (cid:0) | m | − ( d − l + 2) (cid:1)(cid:27) . Then, we have( − k + p +1 (2 p − k − √− m ( d − p ) x p − ,k (6.24) = − p − k − d − p (cid:18) d k − p + 1 (cid:19) a ( k ) p − X r =0 (cid:18) p − r (cid:19) (cid:0) d − r − (cid:1) ! d ! b ( r ; k, p − a ( k ) d − p (cid:18) d k − p − (cid:19) ( d − k + 2 p + 1)( d − k + 2 p ) × p − X r =0 (cid:18) p − r (cid:19) (cid:0) d − r − (cid:1) ! d ! b ( r ; k, p )2 k − p − r , and( − k + p +1 ( | m | + d − k + 1) √− m ( d − p ) x p − ,k − (6.25)= | m | + d − k + 1 d − p (cid:18) d k − p − (cid:19) a ( k − p − X r =0 (cid:18) p − r (cid:19) (cid:0) d − r − (cid:1) ! d ! b ( r ; k − , p − a ( k ) d − p (cid:18) d k − p − (cid:19)(cid:8) | m | − ( d − k + 1) (cid:9) p − X r =0 (cid:18) p − r (cid:19) (cid:0) d − r − (cid:1) ! d ! b ( r ; k, p )2 k − r − a ( k ) d − p (cid:18) d k − p − (cid:19) p − X r =0 (cid:18) p − r (cid:19) (cid:0) d − r − (cid:1) ! d ! b ( r + 1; k, p )2 k − p − r − a ( k ) d − p (cid:18) d k − p − (cid:19) p − X r =0 (cid:18) p − r (cid:19) (cid:0) d − r − (cid:1) ! d ! (2 d − r − k + 1) b ( r ; k, p ) . Here, we used the equality: (cid:8) | m | − ( d − k + 1) (cid:9) = (cid:8) | m | − ( d − r ) (cid:9) + (2 d − r − k + 1)(2 k − r − d − k + 2 p + 1)( d − k + 2 p )2 k − p − r + (2 d − r − k + 1) = ( d − r )( d − r + 1)2 k − p − r − p, we have(6.24)+(6.25)= a ( k ) d − p (cid:18) d k − p − (cid:19) p − X r =0 (cid:18) p − r (cid:19) (cid:0) d − r − (cid:1) ! d ! (cid:26) ( d − r )( d − r + 1)2 k − p − r − p (cid:27) b ( r ; k, p )+ a ( k ) d − p (cid:18) d k − p − (cid:19) p X r =1 (cid:18) p − r − (cid:19) (cid:0) d − r + 1 (cid:1) ! d ! b ( r ; k, p )2 k − p − r = a ( k ) d − p (cid:18) d k − p − (cid:19) p X r =0 (cid:18) pr (cid:19) (cid:0) d − r + 1 (cid:1) ! d ! b ( r ; k, p )2 k − p − r − p a ( k ) d − p (cid:18) d k − p − (cid:19) p − X r =0 (cid:18) p − r (cid:19) (cid:0) d − r − (cid:1) ! d ! b ( r ; k, p ) . IEGEL-WHITTAKER FUNCTIONS ON Sp(2 , R ) 17 On the other hand, we have( − k + p +1 √− m ( d − p ) x p,k (6.26) = − d − p (cid:18) d k − p (cid:19) a ( k ) p X r =0 (cid:18) pr (cid:19) (cid:0) d − r (cid:1) ! d ! b ( r ; k, p − − a ( k ) d − p (cid:18) d k − p − (cid:19) ( d − k + 2 p + 1) p X r =0 (cid:18) pr (cid:19) (cid:0) d − r (cid:1) ! d ! b ( r ; k, p )2 k − p − r = a ( k ) d − p (cid:18) d k − p − (cid:19) p X r =0 (cid:18) pr (cid:19) (cid:0) d − r (cid:1) ! d ! b ( r ; k, p ) − a ( k ) d − p (cid:18) d k − p − (cid:19) p X r =0 (cid:18) pr (cid:19) (cid:0) d − r + 1 (cid:1) ! d ! b ( r ; k, p )2 k − p − r . Therefore, we have(6.24)+(6.25)+(6.26)= a ( k ) d − p (cid:18) d k − p − (cid:19) p X r =0 (cid:18) pr (cid:19) (cid:0) d − r (cid:1) ! d ! b ( r ; k, p ) − p a ( k ) d − p (cid:18) d k − p − (cid:19) p − X r =0 (cid:18) p − r (cid:19) (cid:0) d − r − (cid:1) ! d ! b ( r ; k, p )= a ( k ) (cid:18) d k − p − (cid:19) p X r =0 (cid:18) pr (cid:19) (cid:0) d − r − (cid:1) ! d ! b ( r ; k, p ) . By (6.23) and the above formula, we obtain x p +1 ,k = ( − k + p +1 (cid:0) m √− (cid:1) a ( k ) (cid:18) d k − p − (cid:19) p X r =0 (cid:18) pr (cid:19) (cid:0) d − r − (cid:1) ! d ! b ( r ; k, p ) . Therefore, the formula is valid for x p +1 ,k (0 ≤ k ≤ d ). Similarly, if we suppose that 2 p ≤ d and the formulas (6.21) and (6.22) are true for x p − ,k , x p − ,k (0 ≤ k ≤ d ), then we canprove the formula is also valid for x p,k (0 ≤ k ≤ d ). It completes the proof. (cid:3) Let us complete the proof of Theorem 5.1.
Proof of 5.1 . By the condition | m | ≥ d , | m | ≡ d (mod 2), Propositions 6.1 and 6.2, { g j ( a ) } dj =0 is a non-zero smooth solution of the system (3.1), (3.2) and (3.3) in Proposition3.1. Furthermore, we can check that all of { e − π ( h a + h a ) g j ( a ) } dj =0 are rapidly decreasingwhen each a , a tends to infinity, by Definition 4.1. We completes the proof. (cid:3) Remarks on { x jk } We have some remarks on the sequence of complex numbers { x jk } ≤ j,k ≤ d appearing inTheorem 5.1 and Proposition 6.2. Let X d = ( x jk ) ≤ j,k ≤ d ∈ M d +1 ( C ) be the square matrixof size ( d + 1), defined by the sequence { x jk } ≤ j,k ≤ d .Put ε = sgn( m ) √− t = | m | . Then x jk is given by x jk =( − j + k ε j (cid:18) d k − j (cid:19) t δ ( j ) k Y l =1 l − t − d + 2 l − × [ j/ X r =0 (cid:18) [ j/ r (cid:19) (cid:0) d − r − δ ( j ) (cid:1) ! d ! r Y l =1 (cid:26) k − j − l + 2 − δ ( j )2 k − l + 1 (cid:0) t − ( d − l + 2) (cid:1)(cid:27) , (7.1)where, δ ( j ) = 1 if j is odd, otherwise 0.Furthermore, we set(7.2) Z d = ( z jk ) ≤ j,k ≤ d ∈ M d +1 ( C ) with z jk = ε − j x jk . Though the Harish-Chandra parameter Λ = ( λ − , λ ) ∈ Ξ II implies that d = λ − λ ≥
4, we formally write down matrices Z d for 1 ≤ d ≤ Example 7.1 ( Z d for 1 ≤ d ≤ . Z = (cid:20) (cid:21) , Z = − t − tt − − t − , Z = − t − tt − − t − − t − tt −
00 0 − t − ,Z = − t − t − t − tt − − t ( t − t − − t − t +2( t − t − − t −
00 0 − t ( t − t − tt −
00 0 t − t − − t − ,Z = − t − t − t − tt − − t ( t − t −
2) 3( t − t − − t − t +5( t − t − − t ( t − t − − t ( t − t − t +5( t − t − − t −
00 0 t − t − − t ( t − t − tt −
00 0 0 t − t − − t − , IEGEL-WHITTAKER FUNCTIONS ON Sp(2 , R ) 19 Z = − t − t − t − − t − t − t − tt − − t ( t − t −
3) 15 t ( t − t − t − − t − t +9( t − t − − t +3)( t − t − t −
1) 3( t − t − − t ( t − t − t ( t +14)( t − t − t − − t ( t − t − t − t − − t +3)( t − t − t − t +9( t − t − − t −
00 0 0 t ( t − t − t − − t ( t − t − tt −
00 0 0 − t − t − t −
1) 45( t − t − − t − ,Z = − t − t − t − − t − t − t − tt − − t ( t − t −
4) 45 t ( t − t − t − − t − t − t − − t − t +14( t − t − − t +7)( t − t − t −
2) 15 t ( t − t − t − − t ( t − t − t ( t +26)( t − t − t − − t +7)( t − t − t −
2) 3( t − t − t − t − − t +7)( t − t − t − t ( t +26)( t − t − t − − t ( t − t − t ( t − t − t − − t +7)( t − t − t − t +14( t − t − − t −
00 0 0 − t − t − t −
2) 45 t ( t − t − t − − t ( t − t − tt −
00 0 0 0 − t − t − t −
2) 105( t − t − − t − . By direct calculation, we have
Proposition 7.2 (det Z d for 1 ≤ d ≤ . det Z = 1 , det Z = tt − , det Z = t − t − , det Z = t ( t − t − ( t − , det Z = ( t − ( t − ( t − ( t − , det Z = t ( t − ( t − ( t − ( t − ( t − , det Z = ( t − ( t − ( t − ( t − ( t − ( t − . Since det X d = ε d ( d +1) / det Z d , we have the following conjecture on det X d . Conjecture 7.3.
For a natural number q , we conjecture thatdet X q − = (cid:0) sgn( m ) √− (cid:1) q (2 q − q − Q l =1 (cid:0) t − (2 l − (cid:1) q − lq − Q l =1 ( t − l ) l , det X q = (cid:0) sgn( m ) √− (cid:1) q (2 q +1) t q q − Q l =1 (cid:0) t − (2 l ) (cid:1) q − lq Q l =1 ( t − l + 1) l − . In this article, we have shown that the Siegel-Whittaker functions of rapid decay forthe large discrete series representations of Sp(2 , R ) are described by the partially confluenthypergeometric functions { f k ( a ) } dk =0 . Conversely, if the above conjecture is true, we have, Corollary 7.4.
We assume that | m | ≥ d and | m | ≡ d (mod 2) . Let { c j ( a ) } dj =0 be a smooth solution of the system (3.1), (3.2) and (3.3) in Proposition 3.1,and suppose that all of { e − π ( h a + h a ) c j ( a ) } dj =0 are rapidly decreasing.If Conjecture 7.3 is true, then all of { f k ( a ) } dk =0 are C -linear combinations of elementsof { c j ( a ) } dj =0 .Proof. Since | m | ≥ d and | m | ≡ d (mod 2), we can check thatdet X d = 0 , and X − d exists. It completes the proof. (cid:3) References [1] Y. Gon, Generalized Whittaker functions on SU(2 ,
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