DOI 10.1007/s11005-012-0545-xLett Math Phys (2012) 100:279–290
On a Classical Limit of q-Deformed WhittakerFunctions
ANTON GERASIMOV1,2,3, DIMITRI LEBEDEV1 and SERGEY OBLEZIN1
1Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia.e-mail: [email protected]; [email protected] of Mathematics, Trinity College Dublin, Dublin 2, Ireland3Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland
Abstract. Previously, we derive a representation of q-deformed gl�+1-Whittaker functionas a sum over Gelfand–Zetlin patterns. This representation provides an analog of theShintani–Casselman–Shalika formula for q-deformed gl�+1-Whittaker functions. In thisnote, we provide a derivation of the Givental integral representation of the classical gl�+1-Whittaker function as a limit q →1 of the sum over the Gelfand–Zetlin patterns represen-tation of the q-deformed gl�+1-Whittaker function. Thus, Givental representation providesan analog the Shintani–Casselman–Shalika formula for classical Whittaker functions.
Keywords. q-deformed Whittaker function, Toda chain, Givental integral representation,Gelfand–Zetlin patterns.
Introduction
The q-deformed gl�+1-Whittaker functions can be defined as eigenfunctions ofthe q-deformed gl�+1-Toda chains [2,13]. Among various eigenfunctions of theq-deformed gl�+1-Toda chain, there exists a special class of eigenfunctions withthe support in the positive Weyl chamber. By analogy with the classical case, wecall such functions class one q-deformed gl�+1-Whittaker functions. In [4–6], anexplicit representation of the class one q-deformed gl�+1-Whittaker function as asum over Gelfand–Zetlin patterns that has a remarkable integrality and positivityproperties was proposed. More precisely each term in the sum is a positive inte-ger multiplied by a weight factor qwt and a character of the torus U �+1
1 . Thisallows one to represent the q-deformed gl�+1-Whittaker function as a characterof a C
∗ ×U�+1-module (i.e. it allows a categorification). The interpretation of theq-deformed gl�+1-Whittaker function as a character shall be considered as aq-version of the Shintani–Casselman–Shalika formula [1,14]. Indeed in the limit
280 ANTON GERASIMOV ET AL.
q → 0, the q-deformed gl�+1-Whittaker function can be identified with the non-Archimedean Whittaker function and the representation of the q-deformed Whit-taker function as a character reduces to the standard Shintani–Casselman–Shalikaformula for non-Archimedean Whittaker function [1,14].
There is another interesting limit of the q-Whittaker functions reproducing clas-sical Whittaker functions. It was pointed out in [4] that in this limit explicit sumrepresentation of the q-deformed gl�+1-Whittaker function turns into the Given-tal integral representation for class one gl�+1-Whittaker function [12] (see also [3]).Thus, the Givental integral representation shall be considered as the Archime-dean counterpart of the Shintani–Casselman–Shalika formula (for more details see[7,8,11]). In this note, we provide a precise description of the q →1 limit reducingq-deformed gl�+1-Whittaker function to its classical analog and explicitly demon-strate that the Givental integral representation arises as the limit of the sum rep-resentation of the q-deformed gl�+1-Whittaker function. The main result is givenby Theorem 3.1. The established relation between a sum over the Gelfand–Zetlinpatters and the Givental integrals for gl�+1 is a special case of a general relationbetween the Gelfand–Zetlin patterns and the Givental type integrals for classicalseries of Lie algebras [9]. Thus, in particular this connection implies the identi-fication of the Givental and the Gelfand–Zetlin graphs noticed in [9]. The rela-tion between the Gelfand–Zetlin and Givental constructions described in this noteshould be compared with the duality type relation introduced in [10]. We are goingto discuss the general form of the relation between the Gelfand–Zetlin and theGivental constructions for classical groups elsewhere.
1. q -Deformed gl�+1-Whittaker Function
In this section, we recall the explicit construction of the class one q-deformedgl�+1-Whittaker functions derived in [4]. The quantum q-deformed gl�+1-Toda chain(see e.g. [2,13]) is defined by a set of �+1 mutually commuting functionally inde-pendent quantum Hamiltonians Hgl�+1
r , r =1, . . . , �+1:
Hgl�+1r (p
�+1)=∑
Ir
(X̃
1−δi2−i1,1
i1· . . . · X̃
1−δir −ir−1,1
ir−1· X̃
1−δir+1−ir ,1
ir
)Ti1 · . . . · Tir , (1.1)
where r = 1, . . . , �+ 1, ir+1 := �+ 2 and p�+1
= (p�+1,1, . . . , p�+1,�+1) ∈ Z�+1. The
summation in (1.1) goes over all ordered subsets Ir ={i1< i2< · · ·< ir } of {1,2, . . . ,�+1}. Here we use the notations
where we use the notation (n)q != (1−q) . . . (1−qn). When p�+1
is outside the dom-inant domain we set
�gl�+1z1,...,z�+1(p�+1,1, . . . , p�+1,�+1)=0.
EXAMPLE 1.1. Let g = gl2, p2,1 := p1 ∈ Z, p2,2 := p2 ∈ Z and p1,1 := p ∈ Z. Thefunction
�gl2z1,z2(p1, p2)=
∑
p2≤p≤p1
z p1 z p1+p2−p
2
(p1 − p)q !(p − p2)q ! , p1 ≥ p2 ,
�gl2z1,z2(p1, p2)=0, p1< p2,
(1.5)
is a common eigenfunction of mutually commuting Hamiltonians
Hgl21 = (1−q p1−p2+1)T1 + T2, Hgl2
2 = T1T2.
The formula (1.4) can be easily rewritten in the recursive form.
282 ANTON GERASIMOV ET AL.
COROLLARY 1.1. The following recursive relation holds
�gl�+1z1,...,z�+1(p�+1
)=∑
p�∈P�+1,�(p
�+1)
�(p�) z∑
i p�+1,i −∑
i p�,i�+1
×Q�+1,�(p�+1, p
�|q)�gl�
z1,...,z� (p�), (1.6)
where
Q�+1,�(p�+1, p
�|q)= 1
∏�i=1(p�+1,i − p�,i )q ! (p�,i − p�+1,i+1)q ! ,
�(p�)=
�−1∏
i=1
(p�,i − p�,i+1)q !.(1.7)
The following representations of the class one q-deformed gl�+1-Whittaker func-tion are a consequence of the positivity and integrality of the coefficients of theq-series expansions of each term in the sum (1.4) (see [4,6] for details).
PROPOSITION 1.1. (i) There exists a C∗ × GL�+1(C) module V such that the
common eigenfunction (1.4) of the q-deformed Toda chain allows the followingrepresentation for p�+1,1 ≥ p�+1,2 ≥· · ·≥ p�+1,�+1 :
�gl�+1λ (p
�+1)=Tr V q L0
�+1∏
i=1
qλi Hi , (1.8)
where Hi := Ei,i , i = 1, . . . , �+ 1 are Cartan generators of gl�+1 = Lie(GL�+1)
and L0 is a generator of Lie(C∗).(ii) There exists a finite-dimensional C
∗ × GL�+1(C) module V f such that the fol-lowing representation holds for p�+1,1 ≥ p�+1,2 ≥· · ·≥ p�+1,�+1:
�̃gl�+1λ (p
�+1)=�(p
�+1) �
gl�+1λ (p
�+1)=Tr V f q L0
�+1∏
i=1
qλi Hi . (1.9)
The module V entering (1.8) and the module V f entering (1.9) have a structureof modules under the action of (quantum) affine Lie algebras.
2. Classical Limit of q -Deformed Toda Chain
In this section, we define a limit q →1 of the q-deformed gl�+1-Toda chain repro-ducing the classical gl�+1-Toda chain. We provide an explicit check that the firsttwo generators of the ring of quantum Hamiltonians of gl�+1-Toda chain ariseas a limit of some combinations of the following quantum Hamiltonians of the
ON A CLASSICAL LIMIT OF q-DEFORMED WHITTAKER FUNCTIONS 283
q-deformed Toda chain
Hgl�+11 (p
�+1|q)=
�∑
i=1
(1 − 1
2�2q p�+1,i −p�+1,i+1+1
)Ti + T�+1, (2.1)
Hgl�+1�+1 (p
�+1|q)= T1T2 · · · T�+1. (2.2)
Let us introduce the following parametrization:
q = e−ε�, p�+1,i =i∑
m=1
n�+1,m(ε)+ x�+1,i (ε�)−1, (2.3)
where n�+1,i (ε)∈Z and satisfy the conditions
m�+1,i :=n�+1,i −n�+1,i+1 = −[2(ε�)−1 ln(ε �)].Here [x]∈Z is the integer part of x . Note that if we consider x�+1,i to be contin-uous variables the operators Ti can be written in the following form:
Ti = e−ε� ∂
∂x�+1,i . (2.4)
PROPOSITION 2.1. The following limiting relations hold:
Hgl�+11 (x�+1)= lim
ε→0
1ε
[Hgl�+1
1 (p�+1(x, ε)|q(ε))− (�+1)
],
Hgl�+12 (x�+1)=− lim
ε→0
1ε2
[2Hgl�+1
1 (p�+1(x, ε)|q(ε))
+ 12
(Hgl�+1�+1 (p
�+1(x, ε)|q(ε)))2
−3Hgl�+1�+1 (p
�+1(x, ε)|q(ε))+ 1
2−2�
],
where Hgl�+1i , i =1,2 are the Hamiltonians of the quantum gl�+1-Toda chain:
Hgl�+11 (x�+1)=−�
�+1∑
i=1
∂
∂x�+1,i,
Hgl�+12 (x�+1)=−1
2�
2�+1∑
i=1
∂2
∂x2�+1,i
+�∑
k=1
ex�+1,k+1−x�+1,k .
Proof. Let us consider operators (2.1), (2.2) as acting on the space of functionsof continues variables {x�+1,i } ∈R
�+1 using (2.3) and (2.4). We have the followingestimate:
Now the limiting formulas can be straightforwardly verified.
It is easy to see that the eigenfunction problem (1.2) is transformed into thestandard eigenfunction problem if we use the following parametrization of thespectral variables zi = eı ελi , i =1, . . . , �+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) ofthe q-deformed gl�+1-Toda chain should go to the class one solution of the clas-sical gl�+1-Toda chain. In the classical setting, an integral representation for classone gl�+1-Whittaker function was constructed by Givental [12], (see [3] for a choiceof the contour realizing class one condition)
ψgl�+1λ1,...,λ�+1
(x1, . . . , x�+1)=∫
C
�∏
k=1
dxk eFgl�+1 (x), (3.1)
and the function Fgl�+1(x) is given by
Fgl�+1(x)= ı�+1∑
n=1
λn
( n∑
i=1
xn,i −n−1∑
i=1
xn−1,i
)−
�∑
k=1
k∑
i=1
(exk,i −xk+1,i + exk+1, i+1−xk,i
). (3.2)
Here C ⊂ N+ is a small deformation of the subspace R(�+1)�
2 ⊂C(�+1)�
2 making theintegral (3.2) convergent. Besides, we use the following notation: λ= (λ1, . . . , λ�+1);xi := x�+1,i , i =1, . . . , �+1.
ON A CLASSICAL LIMIT OF q-DEFORMED WHITTAKER FUNCTIONS 285
The integral representation (3.1) allows a recursive presentation analogous to(1.6)
ψgl�+1λ1,...,λ�+1
(x1, . . . , x�+1) =∫
R�
dx� Qgl�+1gl�
(x�+1; x�;λ�+1)ψgl�λ1,...,λ�
(x�,1, . . . , x��), (3.3)
where
Qgl�+1gl�
(x�+1; x�;λ�+1)= exp{
ıλ�+1
( �+1∑
i=1
x�+1,i −�∑
i=1
x�, i)
−�∑
i=1
(ex�, i −x�+1, i + ex�+1, i+1−x�, i
)}, (3.4)
and we assume Qgl1gl0(x11; λ1)= eıλ1x11 .
In the following, we demonstrate that in the previously defined limit q → 1 theclass one q-deformed gl�+1-Whittaker function given by the sum (1.4) indeed turnsinto the classical class one gl�+1-Whittaker function given by the integral represen-tation (3.1). In particular, iterative formula (1.6) turns into (3.3). For this purpose,we need the following asymptotic of the q-factorials entering (1.4).
LEMMA 3.1. Let us introduce the following functions
fα(y, ε)= (y/ε+αm(ε))q !, α∈[1,2],
where m(ε)=−[ε−1 log ε],q = e−ε . Then for ε→+0 the following expansions hold:
f1(y, ε)= eA(ε)+e−y+O(ε); (3.5)
fα>1(y, ε)= eA(ε)+O(εα−1), (3.6)
where A(ε)=−π2
61ε− 1
2 ln ε2π .
Proof. Taking into account the identity
logN∏
n=1
(1−qn)=N∑
n=1
log(1−qn)=−N∑
n=1
+∞∑
m=1
1m
qnm =−+∞∑
m=1
qm
m
(1−q Nm
1−qm
),
and using the substitution q = e−ε, N = ε−1 y +αm(ε) we obtain
log fα(y, ε)=−+∞∑
m=1
e−mε
m
(1− εαme−my
1− e−mε
).
286 ANTON GERASIMOV ET AL.
Now expanding over small ε we have
log fα(y, ε)=−+∞∑
m=1
e−mε
m
(1− εαme−my
1− e−mε
)
=−+∞∑
m=1
e−mε
m2ε
(1− εαme−my
1− 12 mε+ 1
3!m2ε2 +· · ·
),
and for the derivative we obtain
∂y log fα(y, ε)=−+∞∑
m=1
1mε
(εαme−my−mε
1− 12 mε+ 1
3!m2ε2 +· · ·
)=
+∞∑
k=−1
ck Iα,k(y, ε),
where
Iα,k(y, ε)=−+∞∑
m=1
εk+αmmke−ym =−εk+∞∑
m=1
tmmk, t = e−yεα,
and c−1 = 1. Let us separately analyze the term Iα,−1 and the other terms Iα,k≥0.We have
Iα,k≥0(y, ε)=−εk(
t∂
∂t
)k 11− t
=−εk ∂k
∂yk
11− εαe−y
,
and thus
Iα,k≥0 =−εk+αe−y +· · · , α=1,2.
Now consider the case of k =−1
Iα,−1(y, ε)=−1ε
+∞∑
m=1
tm
m
=−1ε
log(1− t)=−1ε
log(1− εαe−y))= εα−1e−y +· · · , t = e−yεα.
This gives (3.5), (3.6) with an unknown A(ε). To calculate A(ε) we take e−y = 0and notice that the resulting function does not depend on α. Thus, we should cal-culate the asymptotic of the following function:
log fα(y, ε)|ey=0 =−+∞∑
m=1
e−mε
m
(1− εmαe−my
1− e−mε
)∣∣∣ey=0
=−+∞∑
m=1
1m
(e−mε
1− e−mε
)
=−+∞∑
n=1
+∞∑
m=1
1m
e−nmε = log+∞∏
n=1
(1− e−nε).
ON A CLASSICAL LIMIT OF q-DEFORMED WHITTAKER FUNCTIONS 287
It can be easily done using the modular properties
η(−τ−1)=√−ıτ η(τ),
of the Dedekind eta function
η(τ)= eıπτ12
∞∏
n=1
(1− e2π ınτ ).
Namely, taking τ =2π ıε−1 we have
fα(y; ε)
∣∣∣e−y=0
=√
2πε−1 e1
24 εη(2π ıε−1)=√
2πε−1 e1
24 ε− π26 ε
−1∞∏
n=1
(1− e−ε−1(2π)2n).
This allows to infer the following result for the leading coefficients in the asymp-totic expansion of ln fα(y, ε
)∣∣∣e−y=0
:
A(ε) =−12
lnε
2π− π2
6ε−1, ε−→+0. (3.7)
This completes the proof of the Lemma.
THEOREM 3.1. Let us use the following parametrization
and we take �=1. The integral representa-tion (3.1) of the classical gl�+1-Whittaker function is given by the following limit ofthe q-deformed class one gl�+1-Whittaker function represented as a sum (1.4)
ψgl�+1λ1,...,λ�+1
(x1, . . . , x�+1)= limε→+0
[ε�(�+1)
2 e�(�+3)
2 A(ε)�gl�+1z1,...,z�+1(p�+1
)], (3.9)
with xi = x�+1,i , i =1, . . . , �+1.
Proof. We prove (3.9) by relating the recursive relation (1.6)
�gl�+1z1,...,z�+1(p�+1
)=∑
p�∈P�+1,�(p
�+1)
�(p�) z∑�+1
i=1 p�+1,i −∑�
j=1 p�, j
�+1
×Q�+1,�(p�+1, p
�|q)�gl�
z1,...,z� (p�),
where
Q�+1,�(p�+1, p
�|q)= 1
∏�i=1(p�+1,i − p�,i )q ! (p�,i − p�+1,i+1)q ! ,
�(p�)=∏�−1
i=1(p�,i − p�,i+1)q !.
(3.10)
with the recursive relation (3.3) for the classical Whittaker function.
288 ANTON GERASIMOV ET AL.
Let us introduce the following parametrization of the elements of the Gelfand-Zetlin patterns p
�∈P�+1, �(p�+1
):
p�, k = ε−1x�, k +akm(ε), m(ε)=−[ε−1 ln ε], (3.11)
where ak are some constants. The Gelfand–Zetlin conditions on weights p�+1
readsas follows:
p�+1, k ≥ p�, k ≥ p�+1, k+1, k =1, . . . , �,
and they lead to
ε−1x�+1, k + (�+2−2k)m(ε)≥ ε−1x�, k +akm(ε)≥ ε−1x�+1, k+1 + (�−2k)m(ε).
(3.12)
The requirement that the limit ε→+0 preserves the conditions (3.12) implies thefollowing restrictions on the parameters ak :
�−2k +2>ak >�−2k, k =1, . . . , �. (3.13)
Since p�= (p�,1, . . . , p�,�) ∈ Z
� the only consistent choice in the limit ε → +0 isak = �+ 1 − 2k, k = 1, . . . , �. Although the variables p�, k are restricted to be in thepositive Weyl chamber i.e. p�,k ≥ p�,k+1, in the limit ε→+0 the variables x�, k haveno such restrictions. This follows from a simple observation that the limit ε→+0the a
ε− b[ε−1 log ε]→+∞/−∞ depends only on the sign of non-zero coefficient
b. Thus we have
p�, k = ε−1x�, k + (�+1−2k)m(ε), z�+1 = eı ελ�+1 . (3.14)
Now using Lemma 3.1, it is easy to obtain the following limiting formulas:
limε→+0
e2�A(ε)Q�+1,�(p�+1, p
�|q)= lim
ε→+0
e2�A(ε)
∏�i=1 f1(x�+1, i − x�, i , ε) f1(x�, i − x�+1, i+1, ε)
= Qgl�+1gl�
(x�+1; x�; λ�+1
)∣∣∣λ�+1=0
, (3.15)
limε→+0
e(1−�)A(ε)�(p�)= lim
ε→+0e(1−�)A(ε)
�−1∏
i=1
f2(x�,i − x�,i+1, ε)=1, (3.16)
ON A CLASSICAL LIMIT OF q-DEFORMED WHITTAKER FUNCTIONS 289
where Qgl�+1gl�
(x�+1; x�; λ�+1
)is given by (3.4). This implies the following identity:
limε→+0
{ε�
∑
p�∈P�+1,�(p
�+1)
z∑�+1
i=1 p�+1,i −∑�
j=1 p�, j
�+1
[e(�+1)A(ε)Q�+1, �
(p�+1
; p�
∣∣q)�(p
�)]
× ε �(�−1)2 e
(�−1)(�+2)2 A(ε)�
gl�z1,...,z� (p�)
}
=∫
R�
dx� exp
⎧⎨
⎩ıλ�+1
⎛
⎝�+1∑
i=1
x�+1,i −�∑
j=1
x�, j
⎞
⎠
⎫⎬
⎭
× limε→+0
[e(�+1)A(ε)Q�+1, �
(p�+1(x�+1, ε); p
�(x�, ε)
∣∣q(ε))�(x�(x�, ε)
)]
× limε→+0
[ε�(�−1)
2 e(�−1)(�+2)
2 A(ε)�gl�z1,...,z�
(p�(x, ε)
)]. (3.17)
Thus we recover the recursive relations (3.3) for the Givental integrals directlyleading to the integral representation (3.1) for the classical gl�+1-Whittaker func-tion. Using (3.17) iteratively over � we obtain (3.9).
EXAMPLE 3.1. For �=1 we have the sum representation (1.5) of q-deformed gl2-Whittaker function. Using the parametrization
q = e−ε, p21 =m(ε)+ x21ε−1 p22 =−m(ε)+ x21ε
−1 zi = eıελi , i =1,2,
with m(ε)=−[ε−1 ln ε] we obtain
�gl2z1,z2(p21, p22)
=∑
x22−εm(ε)≤x11≤x2,1+εm(ε)
eλ1x11+λ2(x21+x22−x11)
((x11 − x22)/ε+m(ε))q ! ((x21 − x11)/ε+m(ε))q ! ,
where we use the notations p11 = x11/ε. Taking into account
The authors are grateful to A. Borodin, G. Olshanski for their interest in thiswork. The research was supported by Grant RFBR-09-01-93108-NCNIL-a. AGwas also partly supported by Science Foundation Ireland grant. The research ofSO was partially supported by P. Deligne’s 2004 Balzan Prize in Mathematics.
290 ANTON GERASIMOV ET AL.
References
1. Casselman, W., Shalika, J.: The unramified principal series of p-adic groups II. TheWhittaker function. Compositio Math. 41, 207–231 (1980)
2. Etingof, P.: Whittaker functions on quantum groups and q-deformed Toda operators.Amer. Math. Soc. Transl. Ser. 2, vol. 194, pp. 9–25. American Mathematical Society,Providence (1999). arXiv:math.QA/9901053
3. Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Gauss–Givental represen-tation of quantum Toda chain wave function. Int. Math. Res. Notices 23 (2006), AricleID96489. arXiv:math.RT/0505310
4. Gerasimov, A., Lebedev, D., Oblezin, S.: On q-deformed gl�+1-Whittaker functions I.Commun. Math. Phys. 294, 97–119 (2010). arXiv:0803.0145
5. Gerasimov, A., Lebedev, D., Oblezin, S.: On q-deformed gl�+1-Whittaker functions II.Commun. Math. Phys. 294, 121–143 (2010). arXiv:0803.0970
6. Gerasimov, A., Lebedev, D., Oblezin, S.: On q-deformed gl�+1-Whittaker functions III.Lett. Math. Phys. 97, 1–24 (2011). arXiv:0805.3754
7. Gerasimov, A., Lebedev, D., Oblezin, S.: Parabolic Whittaker Functions and Topolog-ical Field Theories I. Commun. Number Theory Phys. 5(1), (2011). arXiv:1002.2622
8. Gerasimov, A., Lebedev, D., Oblezin, S.: From Archimedean L-factors to TopologicalField Theories. Lett. Math. Phys. 96, 285–297 (2011). doi:10.1007/s11005-010-0407-3;Mathematische Arbeitstagung, MPIM 2009-401
9. Gerasimov, A., Lebedev, D., Oblezin, S.: New integral representations of Whittak-er functions for classical groups. Uspehi Math. Nauk 67(1), 1–94 (2012, in Russian).arXiv:math.RT/0705.2886
10. Gerasimov, A., Lebedev, D., Oblezin, S.: Baxter operator and archimedean Hecke alge-bras. Commun. Math. Phys. 284(3), 867–896 (2008). arXiv:0706.3476
11. Gerasimov, A.: A Quantum Field Theory Model of Archimedean Geometry, talk atRencontres Itzykson 2010: New trends in quantum integrability, 21–23 June, 2010,IPhT Saclay, France (see link on the webpage of the conference)
12. Givental, A.: Stationary phase integrals, quantum Toda lattices, flag manifolds andthe mirror conjecture. Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser., 2,vol. 180, pp. 103–115. AMS, Providence, Rhode Island (1997). arXiv:alg-geom/9612001
