On a classical limit of q-deformed Whittaker functions

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by  Anton Gerasimov

On a classical limit of -deformed Whittaker functions
q
Anton Gerasimov, Dimitri Lebedev and Sergey Oblezin
1
1
0
2 Abstract. WeprovideaderivationoftheGiventalintegralrepresentationoftheclassicalglℓ+1-
n Whittaker function as a limit q →1 of the q-deformed glℓ+1-Whittaker function represented as
a a sum over the Gelfand-Zetlin patterns.
J

4
2

] Introduction
G
A
. Theq-deformedgl -Whittaker functionscanbedefinedaseigenfunctions oftheq-deformedgl -
h ℓ+1 ℓ+1
t Toda chains [Ru], [Et]. Among various eigenfunctions there exists a special class of eigenfunctions
a
m with the support in the positive Weyl chamber. By analogy with the classical case we call such
functions the class one q-deformed gl -Whittaker functions. In [GLO1] an explicit representation
[ ℓ+1
1 of the class one q-deformed glℓ+1-Whittaker function as a sum over the Gelfand-Zetlin patterns
was proposed. This representation has remarkable integrality and positivity properties. Precisely
v
7 each term in the sum is a positive integer multiplied by a weight factor qwt and a character of the
6 torus Uℓ+1. This allows to represent the q-deformed gl -Whittaker function as a character of a
5 1 ℓ+1
4 C∗ Uℓ+1-module (i.e. it allows a categorification). The interpretation of the q-deformed glℓ+1-
×
. Whittaker function as a character shall be considered as a q-version of the Shintani-Casselman-
1
0 Shalika formula [Sh], [CS]. Indeed in the limit q 0 the q-deformed gl -Whittaker function
ℓ+1
→
1 can be identified with the non-Archimedean Whittaker function and the representation of the q-
1
deformed Whittaker function as a character reduces to the standard Shintani-Casselman-Shalika
:
v
formula for non-Archimedean Whittaker function [Sh], [CS].
i
X
In the limit q 1 the q-Whittaker functions reproduces the classical Whittaker functions. It
r →
a was pointed out in [GLO1] that in this limit an explicit sum type representation of the class one
q-deformed gl -Whittaker function turns into the Givental integral representation for the class
ℓ+1
one gl -Whittaker function [Gi] (see also [GKLO]). Thus the Givental integral representation
ℓ+1
shall be considered as the Archimedean counterpart of the Shintani-Casselman-Shalika formula
(for more details on this interpretation see [GLO2], [GLO3], [G]). In this note we provide a
precise description of the q 1 limit reducing the q-deformed gl -Whittaker function to its
ℓ+1
→
classical analog and explicitly demonstrate that the Givental integral representation arises as a
limit of the sum representation of the q-deformed gl -Whittaker function. This result is given by
ℓ+1
Theorem 3.1. The established relation between a sum over the Gelfand-Zetlin patterns for gl
ℓ+1
andtheGivental integrals forgl isaspecialcaseof ageneral relation between theGelfand-Zetlin
ℓ+1
patterns and the Givental type integrals for classical series of Lie algebras [GLO4]. This relation
elucidates the identification of the Givental and the Gelfand-Zetlin graphs noticed in [GLO4]. The
relation between the Gelfand-Zetlin and Givental constructions described in this note should be
also compared with the duality type relation introduced in [GLO5]. We are going to discuss the
1
general form of the relation between the Gelfand-Zetlin and the Givental constructions for classical
Lie algebras elsewhere.
Acknowledgments: The authors are grateful to A. Borodin and G. Olshanski for their interest
in this work. The research was supported by Grant RFBR-09-01-93108-NCNIL-a. AG was also
partly supported by Science Foundation Ireland grant. The research of SO was partially supported
by P. Deligne’s 2004 Balzan Prize in Mathematics.
1 q-deformed gl -Whittaker function
ℓ+1
In this Section we recall the explicit construction of the class one q-deformed gl -Whittaker
ℓ+1
functions derived in [GLO1]. Quantum q-deformed gl -Toda chain (see e.g. [Ru], [Et]) is defined
ℓ+1
gl
by a set of ℓ + 1 mutually commuting functionally independent quantum Hamiltonians ℓ+1,
r
H
r = 1,...,ℓ+1:
glℓ+1(p ) = X1−δi2−i1,1 ... X1−δir−ir−1,1 X1−δir+1−ir,1 T ... T ,
Hr ℓ+1 i1 · · ir−1 · ir i1 · · ir (1.1)
XIr (cid:0) (cid:1)
e e e
where r = 1,...,ℓ +1 and i = ℓ + 2. The summation in (1.1) goes over all ordered subsets
r+1
I = i < i < < i of 1,2, ,ℓ+1 . Here we use the notations
r 1 2 r
{ ··· } { ··· }
T f(p ) = f(p ), p = p +δ ,
i ℓ+1 ℓ+1 ℓ+1,k ℓ+1,k k,i
Xi = 1 qpℓ+1,i−pℓ+1e,i+1+1, i= 1,e...,ℓ, Xℓ+1 = 1.
−
The corresponding eigenvalue problem can be written in the following form:
e e
gl gl gl
ℓ+1(p )Ψ ℓ+1 (p ) = ( z ) Ψ ℓ+1 (p ),
Hr ℓ+1 z1,...,zℓ+1 ℓ+1 i z1,...,zℓ+1 ℓ+1 (1.2)
XIr iY∈Ir
and the first nontrivial Hamiltonian is given by
ℓ
H1glℓ+1(pℓ+1) = (1−qpℓ+1,i−pℓ+1,i+1+1)Ti + Tℓ+1. (1.3)
i=1
X
Oneofthemainresultsof[GLO1]nowcanbeformulatedasfollows. Givenp = (p ,...,p )
ℓ+1 ℓ+1,1 ℓ+1,ℓ+1
let us denote by (ℓ+1)(p ) a set of collections of the integer parameters p , k = 1,...,ℓ,
P ℓ+1 k,i
i = 1,...,k satisfying the Gelfand-Zetlin conditions p p p . Let (p ) be a
k+1,i ≥ k,i ≥ k+1,i+1 Pℓ+1,ℓ ℓ+1
set of p = (p ,...,p ), p Z, satisfying the conditions p p p .
ℓ ℓ,1 ℓ,ℓ ℓ,i ∈ ℓ+1,i ≥ ℓ,i ≥ ℓ+1,i+1
Theorem 1.1 A common solution of the eigenvalue problem (1.2) can be written in the following
form. For p being in the dominant domain p ... p , the solution is given by
ℓ+1 ℓ+1,1 ≥ ≥ ℓ+1,ℓ+1
ℓ+1
Ψglℓ+1 (p ) = zPipk,i−Pipk−1,i
z1,...,zℓ+1 ℓ+1 k
pk,i∈P(Xℓ+1)(pℓ+1) kY=1
ℓ k−1 (1.4)
(p p ) !
k,i k,i+1 q
−
k=2i=1
,
× ℓ k Q Q
(p p ) ! (p p ) !
k+1,i k,i q k,i k+1,i+1 q
− −
k=1i=1
Q Q
2
where we use the notation (n) ! = (1 q)...(1 qn). When p is outside the dominant domain
q
− − ℓ+1
we set
gl
Ψ ℓ+1 (p ,...,p ) = 0.
z1,...,zℓ+1 ℓ+1,1 ℓ+1,ℓ+1
Example 1.1 Let g = gl , p := p Z, p := p Z and p := p Z. The function
2 2,1 1 ∈ 2,2 2 ∈ 1,1 ∈
zpzp1+p2−p
Ψgz1l2,z2(p1,p2)= (p 1p)2!(p p ) !, p1 ≥ p2,
1 q 2 q
p2≤Xp≤p1 − −
Ψgzl12,z2(p1,p2) = 0, p1 < p2,
is a common eigenfunction of mutually commuting Hamiltonians
gl2 = (1 qp1−p2+1)T +T , gl2 = T T .
H1 − 1 2 H2 1 2
The formula (1.4) can be easily rewritten in the recursive form.
Corollary 1.1 The following recursive relation holds
Ψzg1lℓ,+..1.,zℓ+1(pℓ+1) = ∆(pℓ) zℓP+i1pℓ+1,i−Pipℓ,i Qℓ+1,ℓ(pℓ+1,pℓ|q)Ψgz1lℓ,...,zℓ(pℓ), (1.5)
pℓ∈Pℓ+X1,ℓ(pℓ+1)
where
1
Q (p ,p q) = ,
ℓ+1,ℓ ℓ+1 ℓ| ℓ
(p p ) ! (p p ) !
ℓ+1,i ℓ,i q ℓ,i ℓ+1,i+1 q
− −
i=1 (1.6)
Q
ℓ−1
∆(p )= (p p ) ! .
ℓ ℓ,i− ℓ,i+1 q
i=1
Y
The following representations of the class one q-deformed gl -Whittaker function are a con-
ℓ+1
sequence of the positivity and integrality of the coefficients of the q-series expansions of each term
in the sum (1.4) (see [GLO1] for details).
Proposition 1.1 (i). There exists a C∗ GL (C) module V such that the common eigenfunction
ℓ+1
×
(1.4) of the q-deformed Toda chain allows the following representation for p p ...
ℓ+1,1 ℓ+1,2
≥ ≥ ≥
p :
ℓ+1,ℓ+1
Ψλglℓ+1(pℓ+1) = TrV qL0 ℓi=+11qλiHi, (1.7)
where H := E , i= 1,...,ℓ+1 are Cartan generatorsQof gl = Lie(GL ) and L is a generator
i i,i ℓ+1 ℓ+1 0
of Lie(C∗).
(ii). There exists a finite-dimensional C∗ GL (C) module V such that the following repre-
ℓ+1 f
×
sentation holds for p p ... p :
ℓ+1,1 ℓ+1,2 ℓ+1,ℓ+1
≥ ≥ ≥
ℓ+1
Ψglℓ+1(p ) = ∆(p ) Ψglℓ+1(p ) = Tr qL0 qλiHi. (1.8)
λ ℓ+1 ℓ+1 λ ℓ+1 Vf
i=1
Y
e
The module V entering (1.7) and the module V entering (1.8) have a structure of modules under
f
the action of (quantum) affine Lie algebras [GLO1].
3
2 Classical limit of q-deformed Toda chain
InthisSectionwedefinealimitq 1oftheq-deformedgl -Todachainreproducingthestandard
ℓ+1
→
gl -Toda chain. We provide an explicit check that the firsttwo generators of the ring of quantum
ℓ+1
Hamiltonians of gl -Toda chain arise as a limit of some combinations of the following quantum
ℓ+1
Hamiltonians of the q-deformed Toda chain
ℓ
H1glℓ+1(pℓ+1|q) = 1 − qpℓ+1,i−pℓ+1,i+1+1 Ti + Tℓ+1,
Xi=1(cid:16) (cid:17)
gl
ℓ+1(p q) = T T T .
Hℓ+1 ℓ+1| 1 2··· ℓ+1
Let us introduce the following parametrization:
q = e−ǫ, p = (ℓ+2 2k)m(ǫ)+x ǫ−1. (2.1)
ℓ+1,k ℓ+1,k
−
Here m(ǫ) Z is given by
∈
m(ǫ) = [ǫ−1ln ǫ],
−
and [x] Z is the integer part of x.
∈
Proposition 2.1 The following limiting relations hold:
gl 1 gl
H ℓ+1(x ) = lim ℓ+1(p (x,ǫ)q(ǫ)) (ℓ+1) ,
1 ℓ+1 ǫ→0 ǫ H1 ℓ+1 | −
h i
gl 1 gl gl
H ℓ+1(x ) = lim ℓ+1(p (x,ǫ)q(ǫ)) ℓ+1(p (x,ǫ)q(ǫ)) ℓ
2 ℓ+1 − ǫ→0 ǫ2 H1 ℓ+1 | −Hℓ+1 ℓ+1 | −
h
+1( glℓ+1(p (x,ǫ)q(ǫ)) 1)2
2 Hℓ+1 ℓ+1 | −
i
gl
where H ℓ+1, i = 1,2 are the standard quantum Hamiltonians of the gl -Toda chain:
i ℓ+1
ℓ+1
gl ∂
H ℓ+1(x ) = ,
1 ℓ+1 ∂x
ℓ+1,i
i=1
X
Hglℓ+1(x ) = 1 ℓ+1 ∂2 + ℓ exi+1−xi.
2 ℓ+1 −2 ∂x2
i=1 i i=1
X X
Proof. Using the fact that exp(ǫ[2(ǫ)−1ln(ǫ)] = ǫ2(1+O(ǫ2/lnǫ)) we have
glℓ+1(p q) = (ℓ+1) + ǫℓ+1 ∂ +ǫ2 1 ℓ+1 ∂2 ℓ exℓ+1,k+1−xℓ+1,k + O(ǫ3),
H1 ℓ+1| ∂x 2 ∂x2 −
Xi=1 ℓ+1,i (cid:16) Xi=1 ℓ+1,i Xk=1 (cid:17)
glℓ+1(p q) = 1 + ǫℓ+1 ∂ + 1ǫ2 ℓ+1 ∂2 + O(ǫ3).
Hℓ+1 ℓ+1| ∂x 2 ∂x ∂x
ℓ+1,i ℓ+1,i ℓ+1,j
i=1 i,j=1
X X
4
Now the limiting formulas can be straightforwardly verified. We have
glℓ+1(p q) glℓ+1(p q) ℓ = ǫ2 1 ℓ+1 ∂2 ℓ exℓ+1,k+1−xℓ+1,k + O(ǫ3),
H1 ℓ+1| −Hℓ+1 ℓ+1| − − 2 ∂x ∂x −
ℓ+1,i ℓ+1,j
(cid:16) Xi6=j Xk=1 (cid:17)
1( glℓ+1(p q) 1)2 = 1ǫ2 ℓ+1 ∂2 + O(ǫ3),
2 Hℓ+1 ℓ+1| − 2 ∂x ∂x
ℓ+1,i ℓ+1,j
(cid:16)iX,j=1 (cid:17)
and thus
glℓ+1(p q) glℓ+1(p q) ℓ+ 1( glℓ+1(p q) 1)2
H1 ℓ+1| −Hℓ+1 ℓ+1| − 2 Hℓ+1 ℓ+1| −
1 ℓ+1 ∂2 ℓ
= ǫ2 exℓ+1,k+1−xℓ+1,k + O(ǫ3).
2 ∂x2 −
(cid:16) Xi=1 ℓ+1,i Xk=1 (cid:17)
2
It is easy to see that the eigenfunction problem (1.2) is transformed into the standard eigen-
function problem if we use the following parametrization of the spectral variables zi = eıǫλi, i =
1,...,ℓ+1.
3 Classical limit of class one Whittaker function
In the limit q 1 defined in the previous Section the class one solution (1.4) of the q-deformed
→
gl -Toda chain should goes to the class one solution of the classical gl -Toda chain. In the
ℓ+1 ℓ+1
classical setting an integral representation for class one gl -Whittaker function was constructed
ℓ+1
by Givental [Gi], (see [GKLO] for a choice of the contour realizing class one condition)
ℓ
ψglℓ+1 (x ,...,x ) = dx eFglℓ+1(x), (3.1)
λ1,...,λℓ+1 1 ℓ+1 k
ZCk=1
Y
and the function glℓ+1(x) is given by
F
ℓ+1 n n−1 ℓ k
glℓ+1(x) = ı λn xn,i xn−1,i exk,i−xk+1,i +exk+1,i+1−xk,i . (3.2)
F − −
nX=1 (cid:16)Xi=1 Xi=1 (cid:17) Xk=1Xi=1(cid:16) (cid:17)
Here C N+ is a small deformation of the subspace R(ℓ+21)ℓ C(ℓ+21)ℓ making the integral (3.2)
⊂ ⊂
convergent. Besides,weusethefollowingnotation: λ = (λ ,...,λ );x := x , i = 1,...,ℓ+1.
1 ℓ+1 i ℓ+1,i
The integral representation (3.1) allows a recursive presentation analogous to (1.5)
ψglℓ+1 (x ,...,x ) = dx Qglℓ+1(x ;x ;λ )ψglℓ (x ,...,x ),
λ1,...,λℓ+1 1 ℓ+1 ℓ glℓ ℓ+1 ℓ ℓ+1 λ1,...,λℓ ℓ,1 ℓℓ (3.3)
Z
Rℓ
where
ℓ+1 ℓ
gl
Q ℓ+1(x ;x ;λ ) = exp ıλ x x
glℓ ℓ+1 ℓ ℓ+1 ℓ+1 ℓ+1,i− ℓ,i
n (cid:16)Xi=1 Xi=1 (cid:17) (3.4)
ℓ
exℓ,i−xℓ+1,i +exℓ+1,i+1−xℓ,i ,
−
Xi=1(cid:16) (cid:17)o
5
and we assume Qgl1(x ; λ ) = eıλ1x11.
gl 11 1
0
In the following we demonstrate that in the previously defined limit q 1 the class one q-
→
deformed gl -Whittaker function given by the sum (1.4) indeed turns into the classical class one
ℓ+1
gl -Whittaker function given by the integral representation (3.1). In particular iterative formula
ℓ+1
(1.5)turnsinto (3.3). For this purposeweneed thefollowing asymptotic of theq-factorials entering
(1.4).
Lemma 3.1 Let us introduce the following functions
f (y,ǫ) =(y/ǫ+αm(ǫ)) !, α = 1,2,
α q
where m(ǫ)= [ǫ−1lnǫ], q = e−ǫ. Then for ǫ +0 the following expansions hold:
− →
f (y,ǫ) = eA(ǫ)+e−y+O(ǫ); (3.5)
1
f (y,ǫ) = eA(ǫ)+O(ǫα−1), (3.6)
2
where A(ǫ) = π21 1 ln ǫ .
− 6 ǫ − 2 2π
Proof. Taking into account the identity
N N N +∞ 1 +∞ qr 1 qNr
ln (1 qn) = ln(1 qn) = qnr = − ,
− − − r − r 1 qr
n=1 n=1 n=1r=1 r=1 (cid:18) − (cid:19)
Y X XX X
and using the substitution q = e−ǫ, N = ǫ−1y+αm(ǫ) we obtain
+∞ e−rǫ 1 e−αrǫm(ǫ)e−ry
ln f (y,ǫ) = − .
α − r 1 e−rǫ
!
r=1 −
X
Now expanding the denominator over small ǫ we have
+∞ e−rǫ 1 ǫαre−ry +∞ e−rǫ 1 ǫαre−ry
ln f (y,ǫ) = − + = − + ,
α − r 1 e−rǫ ··· − r2ǫ 1 1rǫ+ 1r2ǫ2+ ! ···
r=1 (cid:18) − (cid:19) r=1 − 2 3! ···
X X
and for the derivative we obtain
+∞ 1 ǫαre−ry−rǫ +∞
∂ ln f (y,ǫ) = + = c I (y,ǫ),
y α − rǫ 1 1rǫ+ 1r2ǫ2+ ! ··· k α,k
r=1 − 2 3! ··· k=−1
X X
where
+∞ +∞
I (y,ǫ) = ǫk+αrrke−yr = ǫk trrk, t =e−yǫα,
α,k
r=1 r=1
X X
and c = 1. Let us separately analyze the term I and the other terms I . We have
−1 α,−1 α,k≥0
−
∂ k 1 ∂k 1
I (y,ǫ) = ǫk t = ǫk ,
α,k≥0 ∂t 1 t ∂yk 1 ǫαe−y
(cid:18) (cid:19) − −
and thus
I = ǫk+αe−y + , α = 1,2.
α,k≥0
···
6
Now consider the case of k = 1
−
1 +∞ tr 1 1
c I (y,ǫ) = = ln(1 t) = ln(1 ǫαe−y)) = ǫα−1e−y + , t = e−yǫα.
−1 α,−1
−ǫ r −ǫ − −ǫ − ···
r=1
X
Thisgives (3.5),(3.6)withanunknownA(ǫ). Tocalculate A(ǫ)wetakee−y = 0andnoticethatthe
resulting function does not depend on α. Thus we should calculate the asymptotic of the following
function:
+∞ e−rǫ 1 ǫrαe−ry +∞ 1 e−rǫ
ln fα(y,ǫ)|ey=0 = − r 1− e−rǫ ey=0 =− r 1 e−rǫ =
Xr=1 (cid:18) − (cid:19)(cid:12) Xr=1 (cid:18) − (cid:19)
(cid:12)
(cid:12)
+∞+∞ +∞
1
= e−nrǫ = ln (1 e−nǫ).
− r −
n=1r=1 n=1
XX Y
It can be easily done using the modular properties
η( τ−1) = √ ıτ η(τ),
− −
of the Dedekind eta function
∞
η(τ) = eı1π2τ (1 e2πınτ).
−
n=1
Y
Namely, taking τ = ıǫ we have
2π
∞
fα y; ǫ = √2πǫ−1e−π62ǫ−1 1 e−ǫ−1(2π)2n .
e−y=0 −
(cid:0) (cid:1)(cid:12)(cid:12) nY=1(cid:0) (cid:1)
(cid:12)
This allows to infer the following result for the leading coefficients in the asymptotic expansion of
lnf (y,ǫ :
α
e−y=0
(cid:12)
(cid:1)(cid:12)
(cid:12) 1 ǫ π2
A(ǫ) = ln ǫ−1, ǫ +0. (3.7)
−2 2π − 6 −→
This completes the proof of Lemma. 2
Theorem 3.1 Let us use the following parametrization
q = e−ǫ, p = (ℓ+2 2k)m(ǫ)+ǫ−1x , z = eıǫλk, (3.8)
ℓ+1,k ℓ+1,k k
−
where k = 1,...,ℓ + 1, m(ǫ) = ǫ−1lnǫ . The integral representation (3.1) of the classical
−
gl -Whittaker function is given by the following limit of the q-deformed class one gl -Whittaker
ℓ+1 (cid:2) (cid:3) ℓ+1
function represented as a sum (1.4)
ψλgl1ℓ,+..1.,λℓ+1(x1,...,xℓ+1) = ǫ→lim+0 ǫℓ(ℓ2+1) eℓ(ℓ2+3)A(ǫ) Ψzgl1ℓ,+..1.,zℓ+1(pℓ+1) , (3.9)
h i
where A(ǫ) = π21 1 ln ǫ and x = x , i = 1,...,ℓ+1.
− 6 ǫ − 2 2π i ℓ+1,i
7
Proof. We prove (3.9) by relating the recursive relation (1.5)
ℓ+1 ℓ
P pℓ+1,i−P pℓ,j
Ψzg1lℓ,+..1.,zℓ+1(pℓ+1) = ∆(pℓ) zℓi=+11 j=1 Qℓ+1,ℓ(pℓ+1,pℓ|q) Ψgzl1ℓ,...,zℓ(pℓ),
pℓ∈Pℓ+X1,ℓ(pℓ+1)
where
1
Q (p ,p q) = ,
ℓ+1,ℓ ℓ+1 ℓ| ℓ
(p p ) ! (p p ) !
ℓ+1,i ℓ,i q ℓ,i ℓ+1,i+1 q
− −
i=1 (3.10)
Q
ℓ−1
∆(p )= (p p ) !,
ℓ ℓ,i− ℓ,i+1 q
i=1
Y
with the recursive relation (3.3) for the classical Whittaker function.
Let us introduce the following parametrization of the elements of the Gelfand-Zetlin patterns
p (p ):
ℓ ∈ Pℓ+1,ℓ ℓ+1
p = ǫ−1x + a m(ǫ), m(ǫ) = ǫ−1lnǫ , (3.11)
ℓ,k ℓ,k k
−
where a are some constants. The Gelfand-Zetlin conditions on(cid:2)weights(cid:3)p reads as follows:
k
ℓ+1
p p p , k = 1,...,ℓ,
ℓ+1,k ℓ,k ℓ+1,k+1
≥ ≥
and they lead to
ǫ−1x +(ℓ+2 2k)m(ǫ) ǫ−1x +a m(ǫ) ǫ−1x +(ℓ 2k)m(ǫ). (3.12)
ℓ+1,k ℓ,k k ℓ+1,k+1
− ≥ ≥ −
The requirement that the limit ǫ +0 preserves the conditions (3.12) implies the following re-
→
strictions on the parameters a :
k
ℓ 2k+2 > a > ℓ 2k, k = 1,...,ℓ. (3.13)
k
− −
Since p = (p ,...,p ) Zℓ the only consistent choice in the limit ǫ +0 is a = ℓ+1 2k,
ℓ ℓ,1 ℓ,ℓ ∈ → k −
k = 1,...,ℓ. Although the variables p are restricted to be in positive Weyl chamber i.e. p
ℓ,k ℓ,k
≥
p , in the limit ǫ +0 the variables x have no such restrictions. This follows from a simple
ℓ,k+1 ℓ,k
→
observation that the limit ǫ +0 the a b[ǫ−1lnǫ] + / depends only on the sign of
→ ǫ − → ∞ −∞
non-zero coefficient b. Thus we have
p = ǫ−1x + (ℓ+1 2k)m(ǫ). (3.14)
ℓ,k ℓ,k
−
Now using Lemma 3.1 it is easy to obtain the following limiting formulas:
e2ℓA(ǫ)
lim e2ℓA(ǫ) (p ,p q) = lim
ǫ→+0 Qℓ+1,ℓ ℓ+1 ℓ| ǫ→+0 ℓ
f (x x ,ǫ) f (x x ,ǫ)
1 ℓ+1,i − ℓ,i 1 ℓ,i − ℓ+1,i+1 (3.15)
i=1
Q
gl
= Q ℓ+1 x ;x ; λ ,
gl ℓ+1 ℓ ℓ+1
ℓ λℓ+1=0
(cid:12)
(cid:0) (cid:1)(cid:12)
(cid:12)
ℓ−1
lim e(1−ℓ)A(ǫ)∆(p ) = lim e(1−ℓ)A(ǫ) f (x x ,ǫ) = 1, (3.16)
ǫ→+0 ℓ ǫ→+0 2 ℓ,i− ℓ,i+1
i=1
Y
8
gl
where Q ℓ+1 x ;x ; λ is given by (3.4). This implies the following identity:
gl ℓ+1 ℓ ℓ+1
ℓ
(cid:0) (cid:1)
ℓ+1 ℓ
P pℓ+1,i−P pℓ,j
lim ǫℓ zi=1 j=1 e(ℓ+1)A(ǫ)Q p ;p q ∆(p )
ǫ→+0 ℓ+1 ℓ+1,ℓ ℓ+1 ℓ ℓ
npℓ∈Pℓ+X1,ℓ(pℓ+1) h (cid:0) (cid:12) (cid:1) i
(cid:12)
× ǫℓ(ℓ2−1)e(ℓ−1)2(ℓ+2)A(ǫ)Ψgzl1ℓ,...,zℓ(pℓ)
o
ℓ+1 ℓ
= dx exp ıλ x x
ℓ ℓ+1 ℓ+1,i ℓ,j
−
RZℓ n (cid:0)Xi=1 Xj=1 (cid:1)o
(3.17)
lim e(ℓ+1)A(ǫ)Q p (x , ǫ); p (x , ǫ) q(ǫ) ∆ x (x , ǫ)
×ǫ→+0 ℓ+1,ℓ ℓ+1 ℓ+1 ℓ ℓ ℓ ℓ
h ×ǫ→lim+0 ǫℓ(ℓ2−1(cid:0))e(ℓ−1)2(ℓ+2)A(ǫ)Ψgz1lℓ,...,zℓ p(cid:12)(cid:12)ℓ(x, ǫ(cid:1)) (cid:0). (cid:1)i
h i
(cid:0) (cid:1)
Thuswerecovertherecursiverelations(3.3)fortheGiventalintegralsdirectlyleadingtotheintegral
representation (3.1) for the classical gl -Whittaker function. Using (3.17) iteratively over ℓ we
ℓ+1
obtain (3.9). 2
Example 3.1 For ℓ = 1 we have
zp1,1zp2,1+p2,2−p1,1
Ψgz1l2,z2(p2,1,p2,2) = (p 1 p 2) !(p p ) !, p2,2 ≤ p2,1,
1,1 2,2 q 2,1 1,1 q
p2,2≤Xp1,1≤p2,1 − −
Ψ (p ,p ) = 0, p > p .
z1,z2 2,1 2,2 2,2 2,1
Using the parametrization
q = e−ǫ, p = m(ǫ)+x ǫ−1 p = m(ǫ)+x ǫ−1 z = eıǫλi, i = 1,2,
21 21 22 21 i
−
with m(ǫ) = [ǫ−1ln ǫ] we obtain
−
eıλ1x11+ıλ2(x21+x22−x11)
Ψgzl12,z2(p21,p22)= ((x x )/ǫ+m(ǫ)) ! ((x x )/ǫ+m(ǫ)) !,
11 22 q 21 11 q
x22−ǫm(ǫ)≤xX11≤x2,1+ǫm(ǫ) − −
where we use the notations p = x /ǫ. Taking into account
11 11
1 = e+π621ǫ+21ln2ǫπ−e−y+O(ǫ),
(y/ǫ+m(ǫ)) !
q
we obtain
ψλgl12,λ2(x1,x2) = ǫ→lim+0 ǫe−π321ǫ−ln2ǫπ Ψgz1l2,z2(p21,p22)
= dx eıλ1x11eıλ2(x21+x22−x11)e−ex11−x21−ex22−x11.
11
R
Z
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A.G.InstituteforTheoreticalandExperimentalPhysics,117259,Moscow,Russia;
SchoolofMathematics,TrinityCollegeDublin,Dublin2,Ireland;
Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland;
D.L. Institute for Theoretical and Experimental Physics, 117259,Moscow, Russia;
E-mail address: [email protected]
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