whittaker functions and harmonic analysiskhlee/conferences/seoul-nov-2015/se… · whittaker...

Whittaker functions and Harmonic analysis

Sergey OBLEZIN , Nottingham

Eisenstein series on Kac-Moody groups & Applications

19 November 2015 , KIAS, Seoul

Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 1

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1 A. Gerasimov, D. Lebedev, S. Oblezin Baxter operator formalism forMacdonald polynomials, Lett. Math.Phys. 104 (2014);

2 S. Oblezin, On parabolic Whittaker functions I & II, Lett. Math.Phys. 101 & Cent. Eur. J. Math. 10 (2012);

3 A. Gerasimov, D. Lebedev, S. Oblezin On a classical limit ofq-deformed Whittaker functions, Lett. Math. Phys., 100 (2012);

4 A. Gerasimov, D. Lebedev, S. Oblezin Parabolic Whittaker functionsand Topological field theories I, Comm. Number Th. Phys. 5 (2011);

5 A. Gerasimov, D. Lebedev, S. Oblezin On q-deformed Whittakerfunction I, II & III, Comm. Math. Phys 294 (2010) & Lett. Math.Phys 97 (2011);

6 A. Gerasimov, D. Lebedev, S. Oblezin On Baxter Q-operators andtheir arithmetic implications, Lett. Math. Phys. 88 (2009) .

7 A. Gerasimov, D. Lebedev, S. Oblezin Baxter operator andArchimedean Hecke algebra, Comm. Math. Phys. 284 (2008) .

8 A. Gerasimov, S Kharchev, D. Lebedev, S. Oblezin On aGauss-Givental representation of quantum Toda chain wave function,Int. Math. Res. Notices, 2006


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TopologicalQFT

"*

ss ++

SS

""

QuantumIntegrability

t|

KK

||

Representation Theory&

HarmonicAnalysis

Automorphic Forms&

Arithmetic Geometry


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Jacquet’s local Whittaker functionsThe Gauss (Bruhat) decomposition of G = G (F ):

G 0 = U− · A · U+ .

For λ = (λ1, . . . , λN) ∈ CN ,

χλ : B− = U−A −→ C∗ , χλ(ua) =N∏i=1

|ai |λi+ρi .

The principal series representation (πλ, Vλ) :

IndGB− χλ =

f ∈ Fun(G )∣∣∣ f (bg) = χλ(b) f (g) , b ∈ B−

The Whittaker function Ψλ(g) is a smooth function on G (F ) given by

Ψλ(g) =⟨ψL , πλ(g)ψR

⟩, ψL,R : U± −→ C∗ , (1)

attached to local character ψ : F → C∗ and U− = w−10 U+w0 :

ψR(u) =∏

simple roots

ψ(uαi

). ψL(u) = ψR

(uw−1

0

)−1


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Archimedean case: Spherical Whittaker functions

The Iwasawa decomposition of G = G (R):

G = K · A · U+ , H = K\G .

The spherical Whittaker function Ψsphλ (z) is a smooth function on

z ∈ H, analytic in λ given by

Ψsphλ (g) = eρ(g)

⟨ψK , πλ(g)ψR

⟩, (2)

with the K -invariant (spherical) vector ψK ∈ Vλ.

1 Ψsphλ (k · g · u) = ψ(u)Ψsph

λ (g) , for all k ∈ K and u ∈ U+ ;

2 D ·Ψsphλ (z) = cD(λ)Ψsph

λ (z) ,for any G -invariant differential operator D on H.


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Archimedean case: the quantum Toda D-module(Kazhdan, Kostant)

For G = G (R), generators Cr , r = 1, . . . ,N of the center ZU(g) definequantum Toda Hamiltonians:

Hr ·ΨRλ (ex) := e−ρ(x)

⟨ψK , πλ(Cr e−H(x))ψR

⟩. (3)

The G (R)-Whittaker function is an eigenfunction:

Hr ·ΨRλ (ex) = er (λ) ΨR

λ (ex) , (4)

er (λ) are r -symmetric functions in λ = (λ1, . . . , λN).

Example: G = GL(2; R)

H1 = −~( ∂

∂x1+

∂

∂x2

), H2 = −~2

( ∂2

∂x21

+∂2

∂x22

)+ ex1−x2 ,


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Archimedean case: the GL(2; R)-Whittaker functions

The Bessel function “of the third kind”:

ΨRλ1, λ2

(ex1 , ex2) =

∫RdT e

ı~λ2(x1+x2−T )+ ı

~λ1T − 1~

(ex1−T +eT−x2

)(5)

= eλ1+λ2

2 ex1+x2

2 Kλ1−λ2~

(2

~e

x1−x22).

The Mellin-Barnes integral representation:

ΨRλ1, λ2

(ex1 , ex2) =

∫R−ıε

dγ eı~ x2(λ1+λ2−γ)+ ı

~ x1γ2∏

i=1

~λi−γ

~ Γ(λi − γ

~

)(6)

Both integral representations can be generalized to GL(N; R) by inductionover the rank N, using the Baxter Q-operator formalism, [GLO:08,09,14].


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Archimedean case: Baxter operator [GLO’08]The Gelfand pair G = GLN(R), K = ON(R)

=⇒ the local Hecke algebra H∞(G , K )

The dual group: G∨(C) = GLN(C)

Baxter Q-operator and Hecke algebra

The one-parameter family of K -bi-invariant functions on G (R),

Qs(g) = 2N | det g |ıs+N−12 e−πTr(g

T g) , (7)

acting on a spherical Whittaker function produces the L-function:(Qs ∗Ψsph

λ

)(g) =

∫G

dh Qs(gh−1) Ψsphλ (h) = L∞(s; Vλ) Ψsph

λ (g) . (8)

The local L-functions for G = GL(1; F )

Lp(s; V ) =1

1− pλ−s, L∞(s; V ) = h

λ−s~ Γ

(λ− s

~

).


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Non-Archimedean case: Spherical Whittaker functions

The Gelfand pair G = GLN(Qp), K = GLN(Zp) =⇒ Hp(G , K )

The dual group: G∨(C) = GLN(C)

ξλ : H(G , K )→ C is a Hecke character;

σλ ⊂ G∨(C) is the (semisimple) conjugacy class, Satake-dual to ξλ;

ψ : U+ −→ C× is a unipotent character.

The class-one GL(N; Qp)-Whittaker function:

1 ΨQp

λ (kgu) = ψ(u) ΨQp

λ (g) ;

2∫G

dh ΨQp

λ (gh)φ(h−1) = ξλ(φ) ΨQp

λ (g) for any φ ∈ H(G , K ) ;

3 ΨQp

λ (1) = 1 .


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Non-Archimedean case: the Baxter operator

Tωn = characteristic function of K ·(

p·Idn 00 IdN−n

)· K

ωnNn=1 = fundamental weights of G∨(C) = GLN(C)

Given finite-dimensional ρV : GLN(C) −→ GL(V ),

TV ·ΨQp

λ (g) :=

∫Gp

dh TV (h)ΨQp

λ (gh) = chV (σλ) ΨQp

λ (g)

p-adic substitute of the Baxter operator [Piatetski-Shapiro], [GLO’08]

Let QQps =

∑n≥0

p−n s TSymnCN , then

QQps ·ΨQp

λ = Lp(s, CN) ΨQp

λ , (9)

where Lp(s, CN) = 1

detCN(

1− p−sρCN (σλ)) .


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Non-Archimedean case: The Langlands-Shintani formulaL− functions

\\

Baxteroperators

Whittaker functions

19

Characters of G∨

19

oo Local Langlands

Reciprocity// Matrix elements

%-

Class-one GL(N ; Qp)-Whittaker function == GL(N ; C)-character

ΨQp

λ (pn) =

p−%(n) chVn

(pλ1

. . .pλN

), n = (n1 ≥ . . . ≥ nN)

0 , n non-dominant


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Archimedean case: Explicit formulas [Givental] [GLO’05]

ΨRλ

(exN)

=

∫C

∏k≤n<N

dxnk eFλ(xN , xnk ) , C ∼ RN(N−1)

2 ⊂ CN(N−1)

2 , (10)

Fλ(xN , xnk) =N∑

n=1

ıλn

( n∑k=1

xn,k−n−1∑i=1

xn−1, i

)−∑

arrowsetarget(a)− source(a)

summed over the arrows from Gelfand-Zetlin (GZ) graph:

xN,1

$$

xN,2 . . .

$$

xNN

xN−1, 1

::

##

. . .

xN−1,N−1

::

. . .

==

!!

...

;;

x11

>>


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Archimedean case: Explicit formulas [Kharchev-Lebedev]

ΨλN, 1,...,λNN (ex) =∨QglNglN−1

(xN) ∗ΨλN−1, 1,...,λN−1,N−1(ex1 , . . . , exN−1) (11)

=

∫S

∏1≤k≤n<N

dλnk e− ı

~

N−1∑n=1

xn+1

(n+1∑j=1

λn+1, j −n∑

j=1λn,j

)×

×N−1∏n=1

n+1∏m=1

n∏k=1

Γ(ıλn,k−ıλn+1,m

~

)∏s 6=p

Γ(ıλn, s−ıλn, p

~

) . (12)

Contour S : maxjIm(λkj) < min

mIm(λk+1,m) , k = 1, . . . ,N − 1

The dual recursion operator

∨Q

glNglN−1

(λ, γ|x) = e− ı

~ x( N∑i=1

λi −N−1∑j=1

γj

)×

N∏i=1

N−1∏j=1

Γ( ıλi − ıγj

~

).


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Archimedean case: The dual Baxter operator [GLO’08]One-parameter family of integral operators:( ∨Qz ∗Ψ(ex)

)λ

=

∫RN

dγ∨

Qz(λ; γ) Ψγ(ex) = L∨(z ; x) Ψλ(ex) ; (13)

with the integral kernel:

∨Qz(λ, γ) = e

− ı~

N∑i=1

(γi −λi )z N∏i ,j=1

Γ( ıγi − ıλj

~

).

The eigenvalue, the “dual L-function”:

L∨(z ; x1, . . . , xN) = e−1~ ez−xN .

Recursion == composition of the dual pair of Baxter operators

QglNglN

= eıλNxN ×QglN−1

λN ∨QglN−1

xN . (14)


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Unification: Archimedean VS non-Archimedean [GLO’09’12]

Ψqz (Λ) & Lq

y (z)

q→1

))

q→0

z=pλ

uu

ΨQp

λ (a) & Lp(s) ΨRλ (a) & L∞(s)

The q-deformed Whittaker function === q-Toda eigenfunction:

Hr ·Ψqz (Λ) = er (z) Ψq

z (Λ) . (15)

The q-deformed Baxter operator:

Qy ∗Ψqz = Lq

y (z) Ψqz , Lq

y (z) =N∏i=1

∏n≥0

1

1− y−1ziqn(16)

Example: G = GL(2)

H1 =(1− qΛ1−Λ2+1

)TΛ1 + TΛ2 , H2 = TΛ1TΛ2


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Further unification: The Macdonald polynomials

Λq,t =←−limN

Q(q, t)[x1, . . . , xN ]SN ,

Macdonald’s polynomials Pq,tΛ (x) = basis in Λq,t , labeled by partitions Λ

(i) 〈PΛ, Pµ〉q,t = 0 , iff Λ 6= µ ,

(ii) PΛ =∑µ≥Λ

uΛµmµ , with uΛΛ = 1 , .

The main diagramme

ΨRλ (a)

P q,tΛ (z)

t→0

&&

t→+∞

88Ψq

z (Λ)

q→166

q→0

z=pλ ((Ψ

Qp

λ (a)


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The Ruijsenaars-Macdonald quantum system

Mr = tr(r−1)

2

∑Ir

∏i∈Irj /∈Ir

txi − xjxi − xj

Tq, xi , Tq, x · f (x) = f (qx) .

Eigenvalue problem:

Mr · PΛ = χr (t%qΛ) PΛ , χr (y) =∑Ir

yi1 · · · yir . (17)

The new scalar product:

〈a, b〉′q,t =1

N!

∮T

d×z ∆(z) a(z) b(z−1) , q ∈ C , |q| < 1 ,

T =

z ∈ CN : |zi | = 1, ∆(z) =

N∏i,j=1i 6=j

Γq,t

(z−1i zj

)−1,

where

Γq,t(y) =∏n≥0

1− tyqn

1− yqn, Γq, t(z)Γq, t−1(qz−1) = t1/2 θ1

((tz)1/2; q

)θ1

(z1/2; q

) .


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The dual Ruijsenaars-Macdonald quantum systemNormalized Macdonald polynomials

ΦΛ(x ; q, t) := tρ(Λ)N∏

a,b=1a<b

Γq,t(tb−aqλa−λb)× Pq,tΛ (x) ,

Remarkable bispectral symmetry [Koornwinder]

ΦΛ

(qµ−kρ; q, q−k

)= Φµ

(qΛ−kρ; q, q−k

). (18)

The dual Hamiltonians:

M∨r = tr(N−1)

2

∑Ir

∏i∈Irj /∈Irj<i

1− t i−j+1qλj−λi−1

1− t i−jqλj−λi−1

1− t i−j−1qλj−λi

1− t i−jqλj−λiTq, qλi ,

the dual eigenvalue problem:

M∨r · PΛ = χr (x) PΛ . (19)


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Baxter operator for Macdonald polynomials [GLO’14]

Qγ · P(x) =

∫T

d×y Qγ

(x , y)

∆(y) P(y−1) , γ ∈ Z , (20)

Qγ(x , y) =N∏i=1

(xiyi)γ N∏

i ,j=1

Γq,t(xiyj) .

Theorem

Macdonald polynomials are eigenfunctions under the action of (14):

Qγ · PΛ(x) = Lγ(λ) PΛ(x) , Lγ(Λ) =N∏i=1

Γq, tq−1(q)

Γq, tq−1(tN−iqΛi−γ+1),

when ΛN ≥ γ, and Qγ · PΛ(x) = 0 otherwise.

Proof uses (q, t)-analog of the Cauchy-Littlewood identity:n∏

i=1

m∏j=1

Γq,t(xiyj) =∑

Λ

PΛ(x) PΛ(y)

〈PΛ, PΛ〉q,t.


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Dual Baxter operator for Macdonald polynomials [GLO’14]

Theorem

For the Baxter operator with the kernel Qq, tx (µ,Λ) = x |µ−Λ|ϕµ/Λ ,

Qq,tx · PΛ(z) = L∨x (z) PΛ(z) , L∨x (z) =

N∏i=1

Γq,t(xzi ) , (21)

Proof uses the Pieri formula:

PΛ × P(n) =1

〈P(n), P(n)〉q,t

∑µi≥Λi≥µi+1|µ−Λ|=n

ϕµ/Λ Pµ ,

where

ϕµ/Λ =N∏

i,j=1i≤j

Γq,tq−1

(t j−iqµi−µj+1

)Γq,tq−1

(t j−iqµi−Λj+1

) Γq,tq−1

(t j−iqΛi−Λj+1+1

)Γq,tq−1

(t j−iqΛi−µj+1+1

) ,Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysis

Eisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 20/ 26

The t → +∞ limit: q-analog of the LS formula [GLO’10]

Ψqz (p

N) =

∑GZ

N∏n=1

z|p

n|−|p

n−1|

n−1∏i=1

(pn,i − pn, i+1)q!

n∏i=1

(pn,i − pn−1, i )q! (pn−1,i − pn, i+1)q!

(22)

(m)q! := (1− q) · . . . · (1− qm) ,summed over the Gelfand-Zetlin (GZ) patterns:

pN,1 pN,2 . . . pNN

pN−1, 1 . . . pN−1,N−1

. . .. . .

p11

pnk ≥ pn−1, k ≥ pn, k+1 ,1 ≤ k ≤ n < N

Uq(glN)-Whittaker function === character of glN-Demazure module

Ψqλ(p) =

∆q(λ)−1 chVw (p′) , p = (p1 ≥ . . . ≥ pN)

0 , p non-dominant


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Archimedean limit q → 1 [GLO’12]q = e−ε , mε = −

[ε−1 log ε

]Lemma

Let fα(y ; ε) :=(ε−1y + αmε

)q! , then as ε→ +0

fα(y ; ε) ∼

eA(ε) + e−y +O(ε) , α = 1

eA(ε) +O(εα−1) , α > 1, A(ε) = −π

2

6− 1

2ln

ε

2π.

Theorem

Set

pn,k = (n + 1− 2k)mε +xn,kε, zn = eı εΛn , 1 ≤ n ≤ k ≤ N ,

then for the general partition pN

=(pN,1 > pN,2 > . . . > pNN

):

ΨRλ

(exN)

= limε→+0

[ε

N(N−1)2 e

(N−1)(N+2)2

A(ε) Ψqz (p

N)]. (23)


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Example: minimal parabolic, pN

= (n, . . . , n︸︷︷︸m

, 0, . . . , 0), [O]

limε→+0

[εm(N−m) e [m(N−m)+1]A(ε) Ψq

z (nm, 0N−m)]

=

∫Cm

∏k,i

dxnk eFλ(xnk ) ,

Fλ(xk,i ) = Fm(λ) −∑

arrowsetarget(a)− source(a)

is the superpotential in type B sigma-model.

x

xN−m, 1

// . . .

// xN−1,m

...

// . . .

// ...

x11

// . . . // xm,m // 0


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Archimedean analog of the LS formula [GLO’11,O]

Theorem (in progress)

The J-parabolic GL(N; R)-Whittaker function possesses the int rep

ΨJλ(ex) = e−ρ(x)

⟨ψL , πλ

(e−H

J(x))ψJR

⟩∣∣∣ xi=0

i /∈J

=

∫S

∏n,k

dγnk

r∏n=1

e

xn~

( Jn∑i=1

γJn, i−Jn+1∑j=1

γJn+1, j

) Jn∏i=1

Jn+1∏j=1

Γ(γJn, i−γJn+1, j

~)

Jn∏i,k=1i 6=k

Γ(γJn, i−γJn, k

~) , (24)

and can be identified with the S1 × U(N)-equivariant volume of the space

M = Maphol(D, Fl∨J (C))

of holomorphic maps of the disc D = z ∈ C : |z | ≤ 1 into the complexflag variety FlJ = GL(N,C)/PJ .


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Archimedean Langlandscorrespondence == Mirror Symmetry [GLO’11]

• Classical (Type B) integral representation:

a−s Γ(s) =

+∞∫−∞

dt eWs(t;a) , Ws(t; a) = st − aet .

MB = Map(

D → (C2, σ, Ws)).

• Type A integral representation as an S1 × U(1)-equivariant volume:

a−(s−λ) Γ(s − λ) =

∫MA

eΩ + s HS1 + λHU(1) = volS1×U(1)(MA) ,

Key Observation

MA = Maphol(D → C) is Mirror Symmetric to MB .


/ 26

Further directions:

1 Qq,t-operators VS Hecke type algebras;

2 Extension to other types of G ;

3 Applications to the Langlands-Shahidi method;

4 TQFT VS Macdonald polynomials;

5 Automorphic and arithmetic interpretations of the Lq,tx (z)-functions.


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whittaker functions and harmonic analysiskhlee/conferences/seoul-nov-2015/se… · whittaker...

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