Math. J. Okayama Univ. 61 (2019), 129–139

A LIMIT TRANSITION FROM THE HECKMAN-OPDAM
HYPERGEOMETRIC FUNCTIONS TO THE WHITTAKERFUNCTIONS ASSOCIATED WITH ROOT SYSTEMS
Nobukazu Shimeno
Abstract. We prove that the radial part of the class one Whittaker
function on a split semisimple Lie group can be obtained as an appropriate limit of the Heckman-Opdam hypergeometric function.

Introduction
Among quantum integrable systems associated with root systems, there
are three classes containing well-behaved joint eigenfunctions closely relatedwith Lie theory. They are the trigonometric Calogero-Moser model, therational CM model, and the Toda model. For the CM model, the eigenfunction is the Heckman-Opdam hypergeometric function and the Bessel
function corresponding to trigonometric and rational cases respectively (cf.
[12, 19, 20]). Among other eigenfunctions, they are up to constant multiples
unique globally defined analytic functions. For special parameter they are
the radial part of the spherical functions on a Riemannian symmetric spaceof the non-compact type and the Euclidean type respectively (cf. [14]). For
the non-periodic Toda model, the eigenfunction is the class one Whittaker
function defined by the Jacquet integral on a Riemannian symmetric spaceof the non-compact type (cf. [17, 11]). Among other eigenfunctions, it is up
to a constant multiple unique eigenfunction of moderate growth.
On the other hand, there are two limit transitions between the Hamiltonians. One is from the trigonometric CM model to the rational CM model,
and the other is from the trigonometric CM model to the Toda model (cf.
[4, 16]). In the rank one case, corresponding limit transitions are one fromthe Gauss hypergeometric function to the Bessel function, and the other isfrom the Gauss hypergeometric function to the Macdonald function. In general, a limit transition for eigenfunctions in the former case was establishedby Ben Saı̈d-Ørsted [1] and de Jeu [4]. In this paper we establish a limit
transition in the latter case. Namely we prove that a limit of the HeckmanOpdam hypergeometric function is the radial part of the Whittaker function
on a split semisimple Lie group (Theorem 5). Similar result for functions on
Sp(2, R) was proved by Hirano-Ishii-Oda [15], which motivated the study ofthis paper.
Mathematics Subject Classification. Primary 22E30, 33C67; Secondary 43A90.
Key words and phrases. root system, hypergeometric function, Whittaker function.
129

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1. Preliminaries
1.1. The Heckman and Opdam hypergeometric function. In thissubsection, we review on the Heckman-Opdam hypergeometric function associated with a root system. See [12] and [20] for details.
Let a be a Euclidean space of dimension n equipped with an inner product( , ). We identify a∗ with a as usual. For α ∈ a∗ \ {0} define
α∨ =

2α
.
(α, α)Let R denote a reduced root system in a∗ . Choose a positive system R+ ⊂ R
and let B denote the set of the simple roots. Let W denote the Weyl groupfor R. For α ∈ R let kα be a non-negative number such that kwα = kα for
all w ∈ W . We call k : α 7→ kα a multiplicity function. We put
1 ∑ρ(k) =
kα α.
2
α∈R+

Let A = exp a and
A+ = {a ∈ A : α(log a) > 0 for all α ∈ R+ }.
Let {ξ1 , . . . , ξn } be an orthonormal basis of a. Define(1)

L(k) =n∑i=1∂ξ2i +∑α∈R+kα

1 + e−α∂α .
1 − e−αThere exist a commutative algebra D(k) of diﬀerential operators containing
L(k) and we have an isomorphism γ : D(k) → S(aC )W , the set of W invariant elements of the symmetric algebra of aC = a ⊗R C.
Let Q be the Z-span of R and let Q+ be the Z+ -span of R+ . There exists
a solution ϕ(a) = Φ(λ, k; a) for(2)

Dφ = γ(D)(λ)φ

of the form
(3)

Φ(λ, k; a) =∑Γµ (λ, k)e(λ−ρ(k)−µ)(log a) ,

Γ0 (λ, k) = 0.

µ∈Q+

The coeﬃcients Γµ (λ, k) are determined by recurrence relations coming fromL(k).
If λ ∈ a∗C = a∗ ⊗R C satisfies the condition(4)(2λ + µ, µ) ̸= 0 for all µ ∈ Q \ {0},then {Φ(wλ, k; a) : w ∈ W } forms a basis of the solution space of (2) on
A+ .

A LIMIT TRANSITION FROM HECKMAN-OPDAM TO WHITTAKER

Definec̃(λ, k) =∏131

Γ((λ, α∨ )),Γ((λ, α∨ ) + kα )α∈R+where Γ( ) is the gamma function andc(λ, k) =Define(5)

F (λ, k; a) =∑c̃(λ, k).
c̃(ρ(k), k)c(wλ, k)Φ(wλ, k; a).

w∈W

The function F is called the Heckman-Opdam hypergeometric function for
the root system R. It is well behaved compared to Φ.
Theorem 1 (Heckman-Opdam). F (λ, k; a) is a unique W -invariant solution for (2) that is analytic in a ∈ A, is holomorphic in λ ∈ a∗C , and satisfies
F (wλ, k; a) = F (λ, k; a)(w ∈ W ),F (λ, k; wa) = F (λ, k; a)(w ∈ W ).

1.2. Notation on Lie groups. Let G be a normal real form of a connectedcomplex semisimple Lie group and K a maximal compact subgroup. Let
g and k be the Lie algebras of G and K respectively. Let θ denote thecorresponding Cartan involution of g and g = k + p be the decomposition
into ±1 eigenspaces of θ. Equip an inner product ( , ) on g induced by the
Killing form on g.
Fix a maximal abelian subspace a of p. Let Σ = Σ(g, a) denote the set ofthe restricted roots. Fix a positive system Σ+ and let Π denote the set ofthe simple roots. Notice that each∑root space has dimension 1, because we
assume that G is split. Put ρ = 12 α∈Σ+ α. Let W denote the Weyl groupof Σ. It is isomorphic to NK (a)/ZK (a), where NK (a) (resp. ZK (a)) is thenormalizer (resp. centralizer) of a in K.
Let n be the sum of root spaces for the positive roots. Put N = exp n and
A = exp a. Then we have the Iwasawa decomposition G = N AK = KAN .
Remark. In the previous section, we adopt notation of Heckman and Opdam.
Relations between notation in the previous section and this section are given
by
1
1
(6)
R = 2Σ, R+ = 2Σ+ , B = 2Π, k2β = mβ =(β ∈ Σ).
2
2
The operator L(k) in (1) is the radial part of the Laplace-Beltrami operator
LG/K on G/K, the algebra D(k) consists of the radial parts of invariant
diﬀerential operators on G/K with respect to the Cartan decomposition

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N. SHIMENO

G = KAK, c(λ, k) is Harish-Chandra’s c-function, and the hypergeometricfunction F (λ, k; a) is the radial part of the zonal spherical function on G/K.
1.3. The Whittaker function on a semisimple Lie group. In thissubsection, we review the class one Whittaker functions on a split semisimple
Lie group following Hashizume [11].
Let ψ√be a unitary character of N . We denote the diﬀerential characterof n to −1R by the same letter ψ. Let Cψ∞ (G/K) denote the space ofC ∞ -functions on G satisfying u(ngk) = ψ(n)u(g) for all n ∈ N , g ∈ G,
and k ∈ K. By the Iwasawa decomposition G = N AK, the values ofu ∈ Cψ∞ (G/K) are completely determined by u|A . Let D(G/K) denote thecommutative algebra of left G-invariant diﬀerential operators on G/K andχλ : D(G/K) → C the Harish-Chandra homomorphism. Let Aψ (G/K, Mλ )be the subspace of Cψ∞ (G/K) defined byAψ (G/K, Mλ ) = {u ∈ Cψ∞ (G/K) : Du = χλ (D)u for all D ∈ D(G/K)}.
Notice that Cψ∞ (G/K, Mλ ) consists of real analytic functions, because LG/K
is an elliptic diﬀerential operator.
For √β ∈ Π let Xβ ∈ gβ be a unit root vector. For α ∈ B = 2Π putlα = − −1ψ(Xα/2 ). For u ∈ Aψ (G/K, Mλ ), φ = e−ρ u|A satisfies
(7)

n
(∑

∂ξ2i − 2i=1∑)lα2 eα φ = (λ, λ)φ.

α∈B

There exists a solution ΨT (λ, ψ; a) of the equation (7) on A+ of the form∑bµ (λ)aµ , b0 (λ) = 0.
(8)
ΨT (λ, ψ; a) = aλµ∈Q+Moreover, extending function u(a) = eρ ΨT (λ, ψ; a) on A to G so thatu ∈ Cψ∞ (G/K) is also a joint eigenfunction of D(G/K) and belongs toAψ (G/K, Mλ ). If λ ∈ a∗C satisfies the condition (4), then {eρ ΨT (wλ, ψ; a) :w ∈ W } forms a basis of Aψ (G/K, Mλ )|A (cf. [11, Corollary 5.3, Theorem5.4]).
For λ ∈ a∗C define function 1λ on G by
1λ (nak) = aλ+ρ(n ∈ N, a ∈ A, k ∈ K).

For g ∈ G let H(g) denote the element of a defined by g ∈ K exp H(g)N .
We normalize the Haar measure dn and dn̄ on N and N̄ = θN by∫e−2ρ(H(n̄)) dn̄ = 1
θ(dn) = dn̄,N̄A LIMIT TRANSITION FROM HECKMAN-OPDAM TO WHITTAKER

133(cf. [14, Ch. IV, §6]). Define the class one Jacquet integral∫(9)
W (λ, ψ; g) =1λ (w̄0−1 ng)ψ(n)−1 dn.
N

Here w̄0 is a representative in NK (a) of the longest element w0 ∈ W . For
λ ∈ a∗C with Re (λ, α) > 0 for all α ∈ Σ+ , the class one Jacquet integral
W (λ, ψ; g) converges absolutely and uniformly and belongs to Aψ (G/K, Mλ ).
Moreover W (λ, ψ; g) is continued to a meromorphic function of λ ∈ a∗C as anelement of Aψ (G/K, Mλ ) (cf. [11, Theorem 6.6]). We call this meromorphic
continuation of W (λ, ψ; a) the Whittaker function.
The Whittaker function W (λ, ψ; g) is up to a constant multiple a uniqueelement of Aψ (G/K, Mλ ) that is of moderate growth (cf. [3, Theorem 9.1]).
The analytic properties of the integral (9) were studied by Jacquet [17],Schiﬀmann [21], Goodman-Wallach [7], and Hashizume [10, 11], etc. The
function W (λ, ψ; g) satisfies the following functional equation.
(10)

W (λ, ψ; g) = M (w, λ, ψ)W (wλ, ψ; g) (w ∈ W ).

Here the factor M (w, λ, ψ) is independent of g and is given by the productformula(11)(12)

M (ww′ , λ, ψ) = M (w′ , λ, ψ)M (w, w′ λ, ψ) (w, w′ ∈ W ),()(λ,α∨ )2lα2
Γ(−(λ, α∨ ) + 1/2)
M (sα , λ, ψ) =(α ∈ B),(α, α)
Γ((λ, α∨ ) + 1/2)

where sα denote the simple reflection corresponding to α ∈ B. (cf. [11,(7.5)–(7.7)]. Notice again that R = 2Σ and G is split.)
Let c(λ) denote the Harish-Chandra c-function for the split Lie group G,
which is given by c(λ) = c(λ, k) with kα = 1/2 for all α ∈ R. That is
c̃(λ)
,
c̃(ρ)
∏
Γ((λ, α∨ ))c̃(λ) =.
Γ((λ, α∨ ) + 12 )α∈Rc(λ) =+

Hashizume [11, Theorem 7.8] expressed the Whittaker function W (λ, ψ; a)
as a linear combination of ΨT (wλ, ψ; a) explicitly.
Theorem 2 (Hashizume). Let ψ be a non-degenerate character of N andassume that λ ∈ a∗C satisfies (4). Then∑(13)
W (λ, ψ; a) = aρM (w0 w, λ, ψ)c(w0 wλ)ΨT (wλ, ψ; a).
w∈W

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N. SHIMENO

2. Limit transition from the Heckman-Opdam hypergeometricfunction to the Whittaker function
2.1. Limit transition from the Calogero-Moser Hamiltonian to the
Toda Hamiltonian. In this subsection, we review the limit transition fromthe quantum trigonometric Calogero-Moser model to the Toda model.
Define a function δ(k) = δ(k; a) by∏ 1
1
δ(k)1/2 =(e 2 α − e− 2 α )kα .
α∈R+We have(14)

δ(k)1/2 ◦ {L(k) + (ρ(k), ρ(k))} ◦ δ(k)−1/2=n∑∂ξ2i +

i=1∑ kα (1 − kα )(α, α)
4 sinh2 21 αα∈R+

We denote the right hand side of (14) by HCM (k). It is the Hamiltonian for
the trigonometric Calogero-Moser model.
Recall that R = 2Σ (Remark 1.2) and let B = 2Π be the simple system
of R+ = 2Σ+ . We assume that lα = 1 for all α ∈ B. This means we assume
that ψ is a special non-degenerate unitary character of N . The left handside of (7) gives the Hamiltonian for the quantum Toda model
(15)

HT =

n
∑

∂ξ2i − 2

i=1

∑

eα .

α∈B

Let M be a positive real number. Define a positive multiplicity function
kM bykM (α)(kM (α) − 1)(α, α) = 2e2M
and define aM ∈ A bylog aM = w0 log a + M ρ∨ ,where w0 is the longest element of W . Notice that
∑
1 ∑ ∨
ρ∨ =
β =
α∨
2
β∈Σ+

α∈R+

is the Weyl vector of Σ∨ = 2R∨ and (α, ρ∨ ) = 1 for all α ∈ Π = 21 B (cf.
Bourbaki [2, Ch VI Proposition 29]).
We shall consider a limit of the hypergeometric function when M → ∞.
Taking a limit of HCM (k), we have the following lemma.
Lemma 3 (Inozemtsev). For any φ ∈ C ∞ (A),(16)

lim HCM (kM ) φ(aM ) = HT φ(a).

M →∞A LIMIT TRANSITION FROM HECKMAN-OPDAM TO WHITTAKER

135

This limit procedure was proved by Inozemtsev [16] (see also [5, Section
7] and [18]).
2.2. Limit transition of eigenfunctions. DefineΨCM (λ, k; a) = δ(k; a)1/2 Φ(λ, k; a).
By (3), φ(a) = ΨCM (λ, k; a) is of the form∑bµ (λ, k)e(λ−µ)(log a) ,(17)ΨCM (λ, k; a) =b0 (λ, k) = 1.

µ∈ΛBy (14), it is also a solution of(18)

HCM (k) φ = (λ, λ) φ.

On the other hand, as we have seen in Subsection 1.3, there is a series
solution φ(a) = ΨT (λ; a) of
HT φ = (λ, λ) φ.
Proposition 4. If λ ∈ a∗C satisfies the condition (4), then(19)∨

lim e−(λ,ρ )M ΨCM (λ, kM ; aM ) = ΨT (w0 λ; a)M →∞(a ∈ A+ ).

The convergence is uniform on each subchamber{a ∈ A+ : α(log a) > c > 0 (α ∈ B)},
where c > 0 is arbitrary.
Proof. The proof is an easy modification of the estimate of the HarishChandra series due to Gangolli [6] (see also Helgason [14, Ch IV §5]).
Let M be a positive number. Putting k = kM , a = aM , and writingbµ (λ, kM ) = b̃µ (λ, M ) in (17), we have(20)∑∨
ΨCM (λ, kM ; aM ) = e(λ,ρ )Mb̃µ (λ, M )e(w0 λ+µ)(log a) , b̃0 (λ, M ) = 1.
µ∈Q+

Then (18) becomes
(21)

n
(∑
i=1

∂ξ2i − 2

∑
α∈R+

e2M

∞
∑

)
∨
j e−j((α,ρ )M +w0 α(log a)) ΨCM (λ, kM ; aM )j=1

= (λ,λ)ΨCM (λ, kM ; aM )

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N. SHIMENO

by expanding the HCM (the right hand side of (14)) into power series. Equations (20) and (21) give the recurrence relation for b̃µ (λ, M ) such as(22)(2w0 λ+µ, µ)b̃µ (λ, M )
∑
∑
=2

∨

e(2−j(α,ρ ))M j b̃µ+jw0 α (λ, M ).

α∈R+ j≥1, µ+jw0 α∈Q+

Since (α, ρ∨ ) = 2 for α ∈ B = 2Π and (α, ρ∨ ) ≥ 4 for α ∈ R+ \ B, therecurrence relation (22) converges to∑(2w0 λ + µ, µ)b̃µ (λ, ∞) = 2(23)b̃µ−α (λ, ∞)α∈B

as M → ∞. (23) is nothing but the recurrence relation for the coeﬃcientsin expansion (8) for ΨT (w0 λ;∑a) (cf. [11, §4]).
∑
For µ ∈ Q+ we write µ = α∈B nα α and put n(µ) = α∈B nα . Choose
a constant c such that|2(w0 λ + µ, µ)| ≥ c n(µ)
for all µ ∈ Q+ . By (22) we have∑∑|b̃µ (λ, M )| ≤ 2c−1
α∈R+

≤ 2c−1∑∨e(2−(jα,ρ ))M j |b̃µ+jw0 α (λ, M )|j≥1, µ+jw0 α∈Q+



∑



α∈R+




j |b̃µ+jw0 α (λ, M )|j≥1, µ+jw0 α∈Q+

for M > 0. We can prove in the same way as the proof of [14, Ch IV Lemma
5.3, Lemma 5.6] that there exists a constant Ka,l such that|b̃µ (λ, M )| ≤ Kλ,a aµ
for all µ ∈ Λ. This estimate shows the convergence of the series (20) andalso guarantees the limit transition (19).
□Now we state and prove our main result:
Theorem 5. Assume that λ ∈ a∗C satisfies (4). Then∏(24)
lim δ(kM ; aM )1/2 c̃(ρ(kM ), kM )Γ(kM (α)) F (λ, kM ; aM )
M →∞

α∈R+

= c̃(ρ)f (λ) a−ρ W (λ, ψ; a),
where
(25)

f (λ) =∏ ( (α, α) )(λ,α∨ )/2α∈R+

2Γ((λ, α∨ ) + 12 )

A LIMIT TRANSITION FROM HECKMAN-OPDAM TO WHITTAKER

137

and ψ is a unitary character of N defined by lα = 1 (α ∈ B).
Proof. By the following formula for the Gamma function
lim

x→∞

Γ(µ + x)= 1,
Γ(x) xµ

we have∏ e−(λ,α∨ )Mc̃(λ, kM ) ∼ f (λ)c̃(λ)Γ(kM (α))

(26)

α∈R+

as M → ∞. By (5), Proposition 4, and (26), we have∏(27)
lim δ(kM ; aM )1/2 c̃(ρ(kM ), kM )Γ(kM (α)) F (λ, kM ; aM )
M →∞

α∈R+

=

∑

f (wλ)c̃(wλ)ΨT (w0 wλ; a).

w∈W

On the other hand, by Theorem 2, the right hand side of (24) is a linear
combination of ΨT (wλ; a) (w ∈ W ), where the coeﬃcient of ΨT (wλ; a) is
given by(28)d(w, λ) := f (λ)M (w0 w, λ, ψ)c̃(w0 wλ).

For β ∈ R, it follows from (11) and (12) thatd(w, sβ λ) = f (sβ λ)M (w0 w, sβ λ, ψ)c̃(w0 wsβ λ)= f (sβ λ)M (sβ , λ, ψ)−1 M (w0 wsβ , λ, ψ)c̃(w0 wsβ λ)= d(wsβ , λ).
The last equality follows from sβ (B \ {β}) = B \ {β}. Thus the right handside of (24) is W -invariant with respect to λ. We have d(w0 , λ) = f (λ)c̃(λ)
and it coincides with the coeﬃcient of ΨT (λ, l; a) in the right hand side of(27), hence the result follows.
□
Example. For R of type A1
F (λ, k; at ) = 2 F1 ( 12 (k − λ), 12 (k + λ) ; k + 12 ; − sinh2 t),
where 2 F1 (a, b, c; z) is the Gauss hypergeometric function. Then Theorem 5
reads
1
lim k −1/2 2−k sinhk (−t + M )F (λ, k; a−t+M ) = √ Kλ (et /2),k→∞πwhere Kλ (z) is the Macdonald function.

138

N. SHIMENO

Remark. We restrict ourselves to split semisimple Lie groups, because the
Hamiltonian (15) of the Toda lattice depends only on the reduced rootsystem. The class one Whittaker function given by the Jacquet integral
for a non-split semisimple Lie group is a constant multiple of the Whittaker
function for the split Lie group of the same indivisible restricted roots.
We can change parameters lα2 (α ∈ B) in the left hand side of (7) bymaking a shift of variables, as it was pointed out by [18, §2.1]. Let ϖα (α ∈B) denote the fundamental weights corresponding to B. If we put∑
2
log a = log a′ −ϖα log lα2 ,(α, α)
α∈B, lα ̸=0

then{α(log a′ ) − log lα2 (α ∈ B, lα ̸= 0)α(log a) =α(log a′ ) (α ∈ B, lα = 0).
∑∑∑
Thus in the new coordinates ni=1 ∂ξ2i − 2 α∈B lα2 eα becomes ni=1 ∂ξ2i −∑2 α∈B, lα ̸=0 eα . Moreover, if lα = 0 for some α, then the Whittaker function
can be reduced to lower rank cases. This is the reason why we assume lα = 1
for all α ∈ B.
Acknowledgement
This paper is based on a lecture given by the author in “Workshop on
representation theory” held at Tambara Institute of Mathematical Sciences
The University of Tokyo on October 8, 2006. The author thanks Professor
Toshio Oshima and Professor Hisayoshi Matumoto for helpful discussions
and comments. The author also thank the referee for careful reading ourmanuscript and for giving useful comments.
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Nobukazu Shimeno
School of Science and Technology
Kwansei Gakuin University
Sanda 669-1337 Japane-mail address: shimeno@kwansei.ac.jp(Received May 31, 2017 )(Accepted September 21, 2017 )